Burstone's Biomechanical Foundation of Clinical Orthodontics, Second Edition
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Library of Congress Cataloging-in-Publication Data
Names: Choy, Kwangchul, author. | Burstone, Charles J., 1928- Biomechanical
foundation of clinical orthodontics.
Title: Burstone's biomechanical foundation of clinical orthodontics /
Kwangchul Choy.
Other titles: Biomechanical foundation of clinical orthodontics
Description: Second edition. | Batavia, IL : Quintessence Publishing,
[2022] | Preceded by The biomechanical foundation of clinical
orthodontics / Charles J. Burstone and Kwangchul Choy. 2015. | Includes
bibliographical references and index. | Summary: "Explains and
illustrates basic force systems and how they function and then applies
these principles to the practice of clinical orthodontics, demonstrating
how to achieve specific tooth movements based on indication"-- Provided
by publisher.
Identifiers: LCCN 2021059935 | ISBN 9780867159493 (hardcover)
Subjects: MESH: Orthodontic Appliances | Biomechanical Phenomena |
Orthodontic Appliance Design | Malocclusion--therapy
Classification: LCC RK521 | NLM WU 426 | DDC 617.6/43--dc23/eng/20220110
LC record available at https://lccn.loc.gov/2021059935
A CIP record for this book is available from the British Library.
ISBN: 9780867159493
© 2022 Quintessence Publishing Co, Inc
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Editor: Leah Huffman
Design: Sue Zubek
Production: Sue Robinson
Printed in Croatia
Burstone's
Biomechanical
Foundation of
Clinical Orthodontics
SECOND EDITION
Kwangchul Choy, DDS, MS, PhD
Clinical Professor
Department of Orthodontics
College of Dentistry
Yonsei University
Seoul, Korea
Contents
Preface to the Second Edition vi
Preface to the First Edition viii
Contributors x
A Color Code Convention for Forces
xi
Part I The Basics and Single-Force Appliances 1
1 Why We Need Biomechanics
3
2 Concurrent Force Systems
11
3 Nonconcurrent Force Systems and Forces on a Free Body
4 Headgear
39
5 The Creative Use of Maxillomandibular Elastics
63
6 Single Forces and Deep Bite Correction by Intrusion
7 Deep Bite Correction by Posterior Extrusion
8 Equilibrium
119
137
Part II The Biomechanics of Tooth Movement
9 The Biomechanics of Altering Tooth Position
10 3D Concepts in Tooth Movement
Rodrigo F. Viecilli
11 Orthodontic Anchorage
Rodrigo F. Viecilli
205
199
163
161
91
25
Part III Advanced Appliance Therapy 215
12 Lingual Arches
217
13 Extraction Therapies and Space Closure
14 Forces from Wires and Brackets
263
311
15 Principles of Statically Determinate Appliances and
Creative Mechanics
357
Giorgio Fiorelli and Paola Merlo
Part IV Advanced Mechanics of Materials 381
16 The Role of Friction in Orthodontic Appliances
383
17 Properties and Structures of Orthodontic Wire Materials
A. Jon Goldberg and Charles J. Burstone
18 How to Select an Archwire
421
Part V Appendices 433
Hints for Developing Useful Force Diagrams
Glossary 441
Solutions to Problems 445
Bender's Tool Kit 491
Index
493
435
407
Preface to the
Second Edition
W
elcome aboard! This book will lead you on an
adventurous journey never before experienced
in dental school. It is my hope that this book
provides an exciting and refreshing foray into the world
of biomechanics.
When a force is applied to the tooth, there is a law
that governs its resulting orthodontic tooth movement. Though you may question the need to discuss Dr
Burstone’s work, it is imperative for me to state that the
law of orthodontic force was founded by him. It remains a
great privilege and point of pride to have studied biomechanics from him through years of personal discourse.
Looking back, my academic journey was much like
listening to Sherlock Holmes profiling a person before he
explains the reasons for his deductions or like revisiting
a movie I had not quite grasped the first time around.
Only after tackling many more journals was I able to
put scattered pieces of jigsaw puzzles together to answer
some of my many questions. This process is precisely
why I suggested to Dr Burstone that we publish an organized set of materials that could follow the journey I went
through to provide a more straightforward conceptual
understanding of biomechanics.
Many years later, all of our lectures of 35mm slide films
and blackboards filled with white chalk at the University of Connecticut and Yonsei University were bound
together into the first edition of this book—his legacy.
While working on the first edition of the book, we wanted
to focus on the following objective: that learning and
thinking about biomechanics is fun at its core. For both
students and teachers, we wanted this book to be an
entertaining thinking journey. We also added clinical
cases so the reader can see the theoretical principles
applied in clinical practice. The chapters of the book are
vi
organized by stage. In other words, the book follows a
somewhat cumulative method of learning and teaching;
as such, the reader should follow along in page order. The
first few chapters may be easy for anyone with an engineering background, but they will still help familiarize
the reader with new terminologies that are scientific but
unique to orthodontics as well as less scientific orthodontic terminologies as well. (We tried to eliminate jargon
as much as possible.) So it is my personal suggestion
that the introductory chapters NOT be skipped. Many
orthodontists regard biomechanics as a tricky subject,
but if by following the sequence of this book readers are
able to break that stereotype, I would consider it a great
personal success of mine.
In this new edition, I’ve tried to restructure the book
but remain true to our original mission. After all, this is
a book of WHY (the concepts of orthodontics) rather
than a book of HOW (the techniques of orthodontics).
Therefore, the larger infrastructure of the textbook
remains intact from its previous edition. As you might
have noticed by the new title, I believe this is the best
Watson can do without Sherlock around. In addition,
many suggestions from readers are reflected in the new
edition. Specifically, all the images of the book were recreated so as to provide higher resolution and more detailed
depictions. Also, video files have been added to supplement the concepts where still images can’t quite cut it. You
can readily access them by scanning the accompanying
QR codes using your smartphone or tablet.
We’ve also formed a discussion group to encourage
questions or comments: https://www.facebook.com/
groups/BiomechanicalFoundation. In fact, readers of the
first edition actually spotted a couple errors that have
since been corrected, so please engage with me!
Acknowledgments
It is simply impossible for me to list everyone who has
been of help in publishing this book. To the staff at Quintessence—Bryn Grisham, who directed the publishing
of this book; Leah Huffman, who spent time combing
through my rough writing; Sue Zubek, who offered a new
and fresh design; and to Sue Robinson, who put all the
pieces together in layout—thank you. To my students,
who ask intriguing questions at every step of our biomechanical journey. Your passion fills me with happiness
and excitement every moment in the classroom. Without
your curiosity and passion, this book would not have been
completed. I would also like to recognize Drs Nazario
Rinaldi and Wislei de Oliveira for their significant contribution in the conceptual developments. Lastly, to my
better half, Annie, and my daughter, Christa, who have
always been there for me and encouraged me.
“Now, let’s start a journey of deduction, the game is on!”
–Sherlock Holmes
Preface to the
First Edition
H
istorically, the mainstay of orthodontic treatment has been the appliance. Orthodontists
have been trained to fabricate and use appliances
and sequences of appliance shapes called techniques.
However, appliances are only the instrument to produce
force systems, which are the basis of tooth position and
bone modification. And yet a thorough understanding
of scientific biomechanics has not been a central part of
orthodontic training and practice. Both undergraduate
and graduate courses in most dental schools lack sound
courses in mechanics and physics. What makes this problem worse is that there are few textbooks that describe
biomechanics in a way that is suitable for the clinician.
The authors hope this text will fill this void.
This book was motivated by the request of orthodontists at all levels—from graduate students to experienced
clinicians—to learn, understand, and apply scientific
orthodontics and, in particular, efficiently manipulate
forces in their everyday practices. This is particularly
relevant at this time, when orthodontics is undergoing
a wide expansion in scope. Twenty-first-century orthodontics has introduced substantial changes in the goals
and procedures: bone modification by orthognathic
surgery and distraction osteogenesis, airway considerations, temporary anchorage devices, plates and implants,
brackets with controlled ligation forces, new wire materials, and nonbracket systems such as aligners. No longer
can clinicians depend entirely on their technical skills
in the fabrication and selection of appliance hardware
to adequately treat their patients. The establishment of
treatment goals and the force systems to achieve them has
become the paramount characteristic of contemporary
orthodontics.
Different orthodontic audiences can benefit in special
ways from a force-driven approach to treatment. The
clinician is aided in the selection of appliances, creative
appliance design, and treatment simulation. Simulation is
the most valuable because it allows the clinician to plan
different strategies using force systems and then select
viii
the best. It enables more predictive appliance shapes
that approach optimal forces. Unlike an older approach
of trying out new procedures directly in the mouth, it is
also cost-effective. Particularly in orthodontics, clinical
evaluation requires long-term observation. With sound
theory, many appliances can be evaluated so that longterm studies or trials can be avoided.
While commercial orthodontic companies may not
initially welcome clinical orthodontists who are knowledgeable in biomechanics, it is to their advantage when
new important products are introduced to be able to
discuss the innovations with scientifically trained clinicians. Researchers in orthodontic physics and material
science also need this background. Biologic research at
all levels also needs to control force variables. Studies
on experimental animals where forces or stresses are
delivered must control the force system to have valid
results. Many times biologists do not understand the
forces in their research and, hence, erroneous or insignificant results are obtained.
Because most orthodontists do not have a strong
background in physics and mathematics, the goal of this
book is simplicity and accuracy in developing a scientific
foundation for orthodontic treatment. In an orderly, stepby-step approach, important concepts are developed
from chapter to chapter, with most chapters building
on the previous one. From the most elementary to the
most advanced concepts, examples from orthodontic
appliances are used to demonstrate the biomechanical
principles; thus, the book reads like an orthodontic text
and not a physics treatise. Yet the principles, solutions,
and terminology are scientifically rigorous and accurate.
The biomechanics described in the book are ideal for
teachers and students. The simplest way to teach clinical orthodontics is to describe the force systems that
are used. Clear force diagrams are far better than vague
descriptions. The teaching of the past, such as “I make a
tip-back bend here” or “I put a reverse curve of Spee in
the arch,” is obviously lacking.
What is the best way to learn biomechanics? The
simplest approach is to carefully read each chapter and
to understand the fundamental principles. Then solve
each of the problems at the end of the chapter. It will be
quickly apparent if one genuinely understands the material. Over time, introduce biomechanics into your
practice. When undesirable side effects are observed, use
what has been learned to explain the problem. How could
the side effect be avoided with an altered force system
and appliance? Critically listening to lectures and reading articles can also be good training for developing a
high level of biomechanical competence. One learns to
bond a bracket quickly, but development of creativethinking skills using biomechanics will take time.
It was the intent of the authors to write a basic book
on orthodontic biomechanics that would be simple and
readable. Clear diagrams and clinical cases throughout
ensure that it is neither dull nor pedantic. Our philosophy
is that the creative thinking involved in manipulating
forces and appliance design should be fun.
Note on the metric system
The authors have adopted the metric system as their unit
system of choice. However, the long shadow of American
orthodontics has influenced the terminology in this book.
Because the United States is the only major country not to
fully adopt the metric system and is a major contributor
to the literature, some units used throughout the book
are not metric. Tradition and familiarity require some
inconsistencies: inches are used for wire and bracket slot
Sadly for us, after finishing this book, a giant fell.
Most of the contents of this book are based on Dr
Burstone’s energetic and rigorous research for more
than 200 research articles. The format of this book was
adapted from the lectures on biomechanics that we gave
at the University of Connecticut and Yonsei University
for many years. Over the last 3 years, my work with Dr
Burstone to convert those lectures and ideas into this
book was one of the most challenging, most exciting,
and the happiest moments in my life. As one of his
students, an old friend, and a colleague, I have to confess
that all of the concepts in this book are his.
In the beginning, Force was created with the Big Bang.
Fifteen billion years later, Newton discovered the Law
of Force in the universe. However, the knowledge of
sizes, and a nonstandard unit—the “gram force”—is the
force unit. It is our hope that the specialty of orthodontics will adhere fully to the International System of Units
in the future; therefore, future editions of this book will
most likely use only metric units.
Acknowledgments
This book would not have been possible without the
input of many graduate students and colleagues. One of
the authors (CJB) has been teaching graduate students
for over 62 years. Long-term teaching has guided us both
in how to most effectively present material and where
most difficulties lie in acquiring biomechanical skills in
a group of biologically trained orthodontists. This book
could not have been developed in this manner without
their intriguing questions and interaction.
Special thanks are given to the staff at Quintessence
Publishing for their valuable contribution in the development of this book: Lisa Bywaters, Director of Publications;
Sue Robinson, Production Manager, Book Division; and
particularly Leah Huffman, our editor, who worked so
hard on a difficult book combining biology, physics, and
clinical practice complicated by specialized dental and
physics terminologies and equations.
Dr Choy wishes to acknowledge the help he received
from his wife Annie and his daughter Christa in the preparation of the manuscript. He is also grateful to his student
Dr Sung Jin Kim for taking the time out of his busy schedule to review the questions and answers.
how to control orthodontic force remained an occult
practice that was only revealed through years of orthodontic apprenticeship. It was Dr Burstone who
uncovered the magic and found the principles governing this treatment method that was once thought to be
mysterious. There is no doubt that the Law of Orthodontic Force was his discovery.
I would like to share Dr Burstone’s words from his last
lecture with me on February 11, 2015, in Seoul: “Don’t
believe blindly in experience, but believe in theory, and
think creatively.”
My father shaped my body; you shaped my thoughts.
Charles, our dearest friend, may you rest in peace.
Kwangchul Choy
ix
Contributors
Charles J. Burstone, DDS, MS*
A. Jon Goldberg, PhD
Kwangchul Choy, DDS, MS, PhD
Paola Merlo, DDS
Professor Emeritus
Division of Orthodontics
School of Dental Medicine
University of Connecticut
Farmington, Connecticut
Clinical Professor
Department of Orthodontics
College of Dentistry
Yonsei University
Seoul, Korea
Giorgio Fiorelli, MD, DDS
Professor
Department of Orthodontics
University of Siena
Siena, Italy
Private Practice
Arezzo, Italy
*Deceased.
Professor
Department of Reconstructive Sciences
Institute for Regenerative Engineering
University of Connecticut
Farmington, Connecticut
Private Practice
Lugano, Switzerland
Rodrigo F. Viecilli, DDS, PhD
Associate Professor
Center for Dental Research
Department of Orthodontics
Loma Linda University School of Dentistry
Loma Linda, California
A Color Code Convention for Forces
This book has several force illustrations that are used for different applications; there are activation and
deactivation forces, equivalent forces, and resultant and component forces. To make it easier for the reader to
understand the logical development of important concepts, a color code convention is utilized in this book.
In situations where multiple forces must be shown, other colors may be utilized.
Solid straight arrows and solid curved
arrows represent forces and moments,
respectively. Red arrows are forces that
act on the teeth. Newton’s Third Law
tells us that there are equal and opposite forces acting on the wire or an
appliance.
Forces acting on a wire are drawn in blue. In
special situations, forces can act both on a
wire and on the teeth; in this book, therefore,
depending on the point of view, the function
being considered determines the color of the
arrow.
The green wire represents the wire in
a deactivated state. The orange wire
represents the wire in its activated state
when it is elastically bent. The gray wire
represents a rigid stabilizing archwire. This wire is
regarded as a rigid body that
has infinite stiffness.
Equivalent forces such as a force
and a couple or components are
identified with yellow arrows.
Gray arrows denote unknown or
incorrect forces.
Body motion including tooth motion
is shown by a dotted straight or curved
arrow. Motion arrows that describe linear
and angular displacement are purposely
different so that they are not confused
with forces or moments. The blue dot
represents the center of rotation.
The diagrams for the “Problems” in each chapter and their solutions at the end of
the book are kept simple, so the standard code above is not used. Problem figures
for emphasis show known and unknown forces as green arrows. Solutions are
shown in red arrows. Equilibrium diagrams (forces acting on a wire, for example)
can show force arrows in blue in the solution section.
xi
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s
1
Why We Need
Biomechanics
"Don’t believe blindly in experience, but believe in theory,
and think creatively."
—Charles J. Burstone
Dentofacial changes are primarily achieved by the orthodontist applying forces to teeth, the periodontium, and bone. Hence, the scientific
basis of orthodontics is physics and Newtonian mechanics applied
to a biologic system. The modern clinician can no longer practice or
learn orthodontics as a trade or a technique. The orthodontist must
understand forces and how to manipulate them to optimize active
tooth movement and anchorage. Communication with fellow clinicians
and other colleagues in other fields requires a common scientific
terminology and not a narrow “jargon.” There is no such thing as a
unique “orthodontics physics” divorced from the rest of the scientific
community. New appliances and treatment modalities will need a
sound biomechanical foundation for their development and most
efficient use.
3
1
E
Why We Need Biomechanics
very profession has its trade tools. The carpenter
uses a hammer and a saw. The medical doctor may
prescribe medication and is therefore a student of
proper drug selection and dosage. Traditionally, the
orthodontist is identified with brackets, wires, and other
appliances. Such hardware is only a means to an end
point: tooth alignment, bone remodeling, and growth
modification. The orthodontist achieves these goals by
manipulating forces. This force control within dentofacial
orthopedics is analogous to the doctor’s dosages. An
“orthodontic dosage” includes such quantities as force
magnitude, force direction, point of force application
(moment-to-force ratios), and force continuity.
Historically, because the end point for treatment is the
proper force system, one might expect the development
and usage of orthodontic appliances to be based on
concepts and principles from physics and engineering.
On the contrary, however, most appliances have been
developed empirically and by trial and error. For that
reason, treatment may not be efficient. Many times undesirable side effects are produced. If appliances “work,” at
a basic minimum the forces must be correct, which is
independent of the appliance, wires, or brackets.
Conversely, when bad things happen, there is a good
possibility that the force system is incorrect.
These empirically developed appliances rarely discuss
or consider forces. Forces are not measured or included
in the treatment plan. How is it possible to use such mechanisms for individualized treatment? The answer is that
they are shape driven rather than force driven. Different
shapes and configurations are taught and used to produce
the desired tooth movement. This approach is not unreasonable because controlled shapes can lead to defined
wire deflections that relate to the produced forces. Unfortunately, there is so much anatomical variation among
different patients that using a standard shape for a bracket
or a wire or even modifying that shape will not always
produce the desired results predictably.
An example of a shape-driven orthodontic appliance
is what E. H. Angle called the ideal arch. In a typical
application of this ideal arch, an archwire is formed with
a shape so that if crooked teeth (brackets) are tied into
the arch, the deflected wire will return to its original shape
and will correctly align the teeth. Today, wires have been
improved to deflect greater distances without permanent
deformation, and brackets may have compensations to
correct anatomical variation in crown morphology. The
principle is the same as Angle’s ideal arch, but this
approach is now called straight wire. Straight-wire appliances can efficiently align teeth but can also lead to
adverse effects in other situations. The final tooth
4
alignment may be correct, but the occlusal plane may be
canted or the arch widths disturbed. Intermediate secondary malocclusions can also occur. An understanding of
biomechanical principles can improve orthodontic treatment even with shape-driven appliances by identifying
possible undesirable side effects before any hardware is
placed. Aligners also use the shape-driven principle of
an ideal shape.
All orthodontic treatment modalities, including different brackets, wires, and techniques, can be improved by
applying sound biomechanics, yet much of clinical orthodontics today is delivered without consideration of forces
or force systems. This suggests that many clinicians
believe that a fundamental knowledge and application of
biomechanics has little relevance for daily patient care.
Scientific Biomechanics
There are many principles and definitions used in physics that are universally accepted by the scientific community. At one extreme, there is classical physics—concepts
developed by giants like Newton, Galileo, and Hooke.
There are also other scientific disciplines, such as quantum mechanics. What the author finds disturbing is
the hubris of what is called pseudo-biomechanics—new
physical principles developed by orthodontists that are
separate and at odds with classical mechanics. Orthodontists’ journal articles and lecture presentations are
filled with figures and calculations that do not follow the
principles of classical mechanics. Orthodontists may be
intelligent, but we should not think we can compete with
the likes of Newton.
There is another major advantage in adopting scientific
or classical mechanics. The methodology, terminology,
and guiding principles allow us to communicate with our
scientific colleagues and set the stage for collaborative
research. Imprecise words can confuse. We speak of
“power arms,” but power has a different meaning to an
engineer than it does to a politician or a clinician. Force
diagrams in orthodontic journals are difficult to decipher
and may not be in equilibrium. The concepts, symbols,
and terminology presented in this book are not trade jargon
but will be widely recognized in all scientific disciplines.
Note that the theme of this book is orthodontic biomechanics. The “bio” implies the union of biologic concepts
with scientific mechanics principles. Let us now consider
some specific reasons why the modern orthodontist
needs a solid background in biomechanics and the practical ways in which this background will enhance treatment efficacy.
Selecting or Designing a New Appliance
Optimization of Tooth Movement
and Anchorage
anchorage devices (TADs) may eliminate side effects. A
good biomechanical understanding is required to successfully use TADs; otherwise, adverse effects can still occur.
The application of correct forces and moments is necessary for full control during tooth movement, influencing
the rates of movement, potential tissue damage, and pain
response. Furthermore, different axes of rotation are
required that are determined by moment-to-force ratios
applied at the bracket. For example, if an incisor is to be
tipped lingually around an axis of rotation near the center
of the root, a lingual force is applied at the bracket. If the
axis of rotation is at the incisor apex, the force system
must change. A lingual force and lingual root torque with
a proper ratio must then be applied. These biomechanical principles are relevant to all orthodontic therapy and
appliances—headgears, functional appliances, sliding
mechanics, loops, continuous arches, segments, and
maxillomandibular elastics (also sometimes referred to
as intermaxillary elastics). The hardware is only the
means to produce the desired force system.
Equally important as active tooth movement is the
control over other teeth so that they do not exhibit undesirable movements. This is usually referred to as anchorage and depends in part on optimally combining and
selecting forces. Some orthodontists might think that
anchorage is determined by factors independent of forces.
For instance, the idea that more teeth means greater
anchorage is very limiting. Working with forces can be
more effective in enhancing anchorage, such as in pitting
tipping movement against translation. All archwires
produce multiple effects. Many of these effects are undesirable, which should also be considered anchorage loss.
In a sense, a new malocclusion is created, resulting in an
increased treatment time. Let us assume that translation
of teeth could be accomplished at the rate of 1 mm per
month. In a typical orthodontic patient, rarely does tooth
movement exceed 5 mm. Not considering any waiting
for growth, total treatment time should be no longer than
5 months. So why is treatment longer? Usually, more time
is required to correct side effects. The use of temporary
Selecting or Designing a New
Appliance
New appliances and variations of older existing appliances
are continually presented in journal articles or at meetings. What is the best way to evaluate these appliances?
One approach is to try them in your clinical practice. This
evaluation will be quite limited because there is a lot of
variation in a small sample of malocclusions. Moreover,
it is time-consuming and unfair to the patient. Because
treatment is so long term, it may take many years to arrive
at a conclusion on the efficacy of a new appliance. A better
approach would be an evaluation based on sound and
fundamental biomechanical principles. Drawing some
force diagrams is much easier than protracted treatment.
This is particularly valid when considering that most new
appliances and techniques do not stand the test of time.
Orthodontists have always been very creative. Not all
great research has come from university research laboratories. Whether in their own offices or on typodonts
in the lab, clinicians have made significant achievements
in bracket design, various wire configurations, and treatment sequences (techniques). It is much more efficient
to work with a pencil and a sheet of paper (or a computer)
than it is to go through the demanding trial and error
approach. The best appliances of the future will require
rigorous engineering and sound biomechanical methodology.
Let us assume for now that we have selected the best
appliance for our individual patient. There are still many
variables that require a sound biomechanical decision.
For example, what size wire should we use? A 0.014-inch
nickel-titanium (Ni-Ti) superelastic wire is not the same
as a 0.014-inch Nitinol wire. The choice between a 0.016and a 0.018-inch stainless steel (SS) archwire is significant.
The larger wire gives almost twice the force.
5
1
Why We Need Biomechanics
Research and Evaluation of
Treatment Results
The clinician can be surprised at the progress of a patient.
When the patient arrives for an appointment, mysterious
changes are sometimes observed. Why is there now an
open bite or a new reverse articulation (also referred to
as crossbite), or why is the malocclusion not improving?
These unexpected events may be attributed to biologic
variation. Or it may be the wrong appliance (or manufacturer). In reality, most of the clinical problems that
develop can be explained by deviation from sound biomechanical principles. Thus, an understanding of applied
biomechanics allows the orthodontist to determine both
why a puzzling and problematic treatment change
occurred and also what to do to correct it. Sometimes
the force system is almost totally incorrect; other times,
a small alteration of the force system can produce a
dramatic improvement.
The prediction of treatment outcomes requires precise
control and understanding of the applied force system as
well as the usual cephalometric and statistical techniques.
Good clinical research must control all of the known
variables if the efficacy of one appliance is to be compared
to another. Let us consider a study that is designed to
compare the different outcomes between a functional
appliance and an occipital headgear. It is insufficient to
simply specify headgear or even occipital headgear. Headgears can significantly vary not only in force magnitude
but also in direction and point of force application. It is
little wonder that some research studies lead to ambiguous and confusing conclusions.
A biomechanical approach to clinical studies opens up
new avenues for research to help predict patient outcomes.
The relationship between forces and tooth movement
and orthopedics requires more thorough investigation.
Relationships to be studied include force magnitudes,
force constancy, moment-to-force ratios at the bracket,
and stress-strain in bone and the periodontal ligament.
Force systems and “dosage” determine not only tooth
or bone displacement with its accompanying remodeling;
unwanted pathologic changes involving tissue destruction
can also occur. Root resorption, alveolar bone loss, and
pain are common undesirable events during treatment.
Some histologic and molecular studies suggest a relationship between force or stress and tissue destruction.
Although other variables may be involved, a promising
direction for research is between stress-strain and the
mechanisms of unwanted tissue changes. To control pain
6
and deleterious tissue destruction, it is likely that future
research will validate that “dosage” does count.
How Scientific Terminology Helps
As previously discussed, orthodontic appliances work by
the delivery of force systems. In this book, the methods
and terminology of the field of physics are adopted. Tooth
movement is only part of a subset of a broader field of
physics. This allows orthodontic scientists and clinicians
to communicate with the full scientific community
outside of dentistry, setting the stage for collaborative
research. Many of the specialized orthodontic terms
produce a jargon that is imprecise and certainly unintelligible to individuals in other disciplines. The orthodontist speaks of “torque.” Sometimes it means a moment
(eg, the force system). At other times, however, it means
tooth inclination (eg, “the maxillary incisor needs more
torque”). Imprecision leads to faulty appliance use, which
will be discussed later.
A universal biomechanical and scientific language is
the simplest way to describe an appliance and how it
works. It not only allows for efficient communication
with other disciplines for joint research but also offers
the best way to teach clinical orthodontics to residents or
other students. The old approach was primarily to teach
appliance fabrication. Treating patients was just following
a technique. An adjustment was how you shaped an arch:
“Watch how I make a tip-back bend, and duplicate it.”
Emphasis was on shape, and therefore we can call it
shape-driven orthodontics. The biomechanical approach
emphasizes principles and force systems. This approach,
force-driven orthodontics, is the theme of this book.
With clear terminology and sound scientific principles,
the learner can better understand how to fabricate and
use any appliance or configuration. It shortens the time
and confusion in teaching students. It is said that a
number of years of experience is required to complete
the education of an orthodontist. Some say as many as
10 years. Why? It is the time needed to make and learn
from your mistakes. If the student understands the
biomechanical basis of an appliance, many common
mistakes will never be made.
It is not only the beginning student who benefits from
sound biomechanical teaching. As new appliances are
developed, the experienced orthodontist can better learn
the “hows” and “whys” so that the learning interval is shortened. More importantly, fewer errors will be made. Lectures
at meetings will be shorter and easier to understand.
Advantages of Biomechanical Knowledge
FIG 1-1 Jacques Carelman painting of a pitcher. Although
the pitcher looks reasonable, it will not actually pour coffee,
much like some orthodontic appliances seem reasonable
but do not actually work.
Knowledge Transfer Among
Appliances
The orthodontist may feel comfortable treating with a
given appliance because routine treatment has become
satisfactory and predictable. However, if he or she wants
to change appliances (eg, moving from facial to lingual
orthodontics), the mechanics may not be the same. When
lingual orthodontic appliances were introduced a few
years ago, some orthodontists were troubled that their
mechanics (wire configurations and elastics) did not do
the same on the lingual that they did on the buccal.
Biomechanical principles that determine the equivalent
force system on the lingual are simple to apply. Clinicians
could have saved much “learning time” spent doing trial
and error experimentation. A few simple calculations
covered in chapter 3 could have helped the clinician avoid
any aggravation.
Advantages of Biomechanical
Knowledge
Historically, there have been many exaggerated claims
made by clinicians and orthodontic companies about the
superiority of appliances or techniques. Hyperbole is used
with such terms as controlled, hyper, biologic, and frictionless. Journals and orthodontic associations are now
doing a better job of monitoring possible conflicts of
interest. The best defense against unwanted salesmanship
is to stay vigilant and always apply scientific biomechan-
FIG 1-2 A wine bottle in a curved wine rack. Although it
would seem that the bottle would fall over, it is in a state of
static equilibrium so that it does not move. Similarly, some
orthodontic principles that seem illogical are actually quite
effective because they are based on sound biomechanics.
ics. What may look possible becomes clearly impossible
when the underlying principles are understood. The
pitcher in Jacques Carelman’s painting looks reasonable,
but it will not pour coffee (Fig 1-1). On the other hand,
a sound biomechanical background can make possible
what appears impossible. A filled wine bottle is placed in
a curved wine rack. The rack is not glued to the table, so
one might think that the bottle will tip over, but it does
not (Fig 1-2). As will be discussed later, the bottle is in
static equilibrium, and hence the impossible becomes
possible. Figure 1-3a shows an auxiliary root spring on
an edgewise arch designed to move the maxillary incisor
roots to the lingual. Is this possible or impossible? A labial
force is required to insert the spring (Fig 1-3b). After
insertion, the spring is bent to push lingually on the cervix
of the crown to produce lingual root torque. What is
easily overlooked in this situation is that the rectangular
wire in the slot will produce an equal and opposite force
with labial root torque, making this appliance impossible
(Fig 1-3c). Placing the auxiliary root spring on a round
or undersized wire makes the mechanism possible (Fig
1-3d).
The many advantages of biomechanical knowledge for
the clinician, including better and more efficient treatment, have been mentioned here. But what about for the
patient? Obviously, one benefit is better and shorter treatment. Another significant advantage is the elimination
of undesirable side effects. Side effects might require
more patient cooperation. To correct problems, new
elastics, headgear, surgery, or TADs may be prescribed.
With better mechanics, such anchorage loss would not
have happened. It is not fair to ask our patients to cover
7
1
Why We Need Biomechanics
FIG 1-3 (a) An auxiliary root spring on an edgewise arch designed
to move the maxillary incisor roots to the lingual (before insertion).
(b) A labial force is required to insert the spring. (c) After insertion, the
spring is bent to push lingually on the cervix of the crown to produce
lingual root torque. However, the rectangular wire in the slot will
produce an equal and opposite force with labial root torque, making
this appliance impossible. (d) Placing the auxiliary root spring on a
round or undersized wire makes the mechanism possible.
a
b
c
d
8
up our mistakes with added treatment time or added
therapy requiring considerable appliance wear, such as
headgear.
The future of the profession will be determined by how
well we train our residents. Currently, not all graduate
students are being trained in scientific biomechanics in
any depth. Ideally, when a student graduates from a
program, an understanding of biomechanics should be
second nature. Otherwise, he or she will not be able to
apply it clinically. Lectures and problem-solving sessions
are very useful; however, biomechanical principles must
be applied during chairside treatment. Carefully supervised patients and knowledgeable faculty are the key
ingredients to teaching biomechanics.
Conventional wisdom in orthodontics has emphasized
the appliance. Graduate students and orthodontists were
taught to fabricate appliances or make bends or adjustments in these appliances. Perhaps some lip service was
given to biomechanics or biology, but basically the clinician was a fabricator and user of appliances. Treatment
procedures were organized into a technique sequence.
This empirical approach to clinical practice led to the
development of different schools of thought, sometimes
identified with the name of a leading clinician. Shapedriven orthodontics (where forces are not considered) is
usually a standard sequence or cookbook approach that
does not adequately consider the individual variation
among patients.
The new wisdom is not appliance oriented. It involves
a thinking process in which the clinician identifies treatment goals, establishes a sequence of treatment, and then
develops the force systems needed for reaching those
goals. Only after the force systems have been carefully
established are the appliances selected to obtain those
force systems. This is quite a contrast to the older process
in which the orthodontist considered only wire shape,
bracket formulas, tying mechanisms, friction, play, etc,
without any consideration whatsoever of the forces
produced.
Advantages of Biomechanical Knowledge
It is easy for the clinician to harbor negative feelings
about orthodontic biomechanics. Some may believe that
treatment mechanics are only common sense and that
intuition and everyday knowledge are sufficient. Others
may regard biomechanics as too sophisticated, demanding, and complicated for daily practice. Indeed, many of
us became dentists and orthodontists because, as
students, we disliked mathematics and physics and
preferred the biologic disciplines. Fortunately, the physics used in orthodontics is not complicated, and many
simple principles and concepts can be broadly and practically applied. Orthodontics is not nuclear physics. Scientific biomechanical thinking is actually easier than vague
and disorderly thought processes, and it simplifies our
overall treatment.
The genius of pioneers such as Newton is that their
principles are anything but common sense. Aristotle
reasoned that if a heavy weight and a light weight were
dropped from the same height, the heavy weight would
hit the ground first. This seems like common sense. Galileo, on the other hand, thought that both weights would
hit the ground at the same time. He supposedly dropped
two different weights from the Leaning Tower of Pisa to
prove his point. Many common-sense ideas are false.
Common sense would tell you that the earth is flat and
that the sun revolves around the earth, and yet the earth
is round, and it revolves around the sun. As will be shown
in this text, many of our conventional and accepted orthodontic ideas from the past are invalid.
There are many textbooks and articles that describe
techniques involving different types of brackets, sequences
of wire change, and slot formulas, much like a recipe in
a cookbook. Many malocclusions might be successfully
treated following such cookbook procedures. However,
surprises can occur as unpredicted problems develop
during treatment. One or more recipes will not always
work because malocclusions vary so much. Therefore,
the clinician must seek sound biomechanical principles
rather than a technique to correct the problem. Thus,
bioengineering is needed not only for the challenging
situation but also for the routine patient who may show
an unexpected response to an appliance. Even if we typically treat by a certain technique, we must have
biomechanical knowledge and skill in reserve, which will
be required when unfortunate surprises strike. If that
knowledge is not readily available because we do not
continually apply it, we limit our ability to get out of
trouble. By way of analogy: The author recently tried to
do some simple plumbing. When the house became
flooded, an experienced plumber was called, and his
backup knowledge and expertise solved the problem.
Unfortunately, when the orthodontist gets into trouble,
he or she traditionally does not seek the advice of others,
leading to either a poorer result or a lengthier treatment
time.
What about the “easy” case we may routinely treat
successfully? It could be argued that applying creative
biomechanics could also improve our treatment result
or allow us to treat more efficiently. We might treat a
Class II patient without extraction with some leveling
arches and Class II elastics. A certain technique might
work, negating any biomechanical thinking. However,
the end point might be different than our treatment goals.
Perhaps the mandibular incisors are undesirably flared
or the occlusal plane angle steepened too much. The goals
and quality of treatment can vary so much that it is difficult to define what a routine or “easy” case entails. It takes
a very knowledgeable orthodontist to identify what an
“easy” case really is.
Technical competence is developed by fabricating and
inserting appliances, but understanding principles
involves thinking. Admittedly, technical skill is important
in daily practice. But performing techniques without
understanding the fundamental principles behind them
is risky. At the same time, principles without technique
lack depth. This book therefore explains the “hows” and
“whys” of orthodontic treatment, which are inseparable.
Orthodontic biomechanics is not just a theoretical
subject for academics and graduate students. It is the
core of clinical practice; orthodontists are biophysicists
in that daily bread-and-butter orthodontics is the creative
application of forces. The 21st century will be characterized by a major shift from shape-driven orthodontic
techniques to a biomechanical approach to treatment,
and with this shift will come rapid advancements in treatment and concepts.
9
2
Concurrent
Force Systems
“Goodbye, old friend. May the Force be with you."
— Obi-Wan Kenobi
The branch of physics dealing with forces is called mechanics. The
most relevant Laws of Newton are the First and Third Laws. Many
orthodontic questions and their solutions can be considered equilibrium situations, so Newton’s Second Law, which relates forces to
bodies that accelerate, is less important. The division of mechanics
describing bodies in equilibrium is called statics and for bodies that
accelerate, kinetics. The simplest force system is force acting on a
point; it is fully defined by force magnitude, force direction, and sense.
Manipulating a force system includes adding a number of forces to
obtain a resultant or breaking up a resultant into separate components.
Forces are vectors that must be added geometrically and cannot be
added algebraically. Simple orthodontic appliances that act on a point
are maxillomandibular elastics (also known as intermaxillary elastics),
finger springs, and cantilevers.
11
2
Concurrent Force Systems
M
edical doctors may use thousands of medicines to
treat their patients, but orthodontists use only one
treatment modality: force. No matter what kinds
of wires, springs, and brackets or appliances used, the hardware serves only as an intermediate tool to deliver a force
or a series of forces. With proper force positioning and
dosage, all kinds of tooth movement can be achieved.
Therefore, knowledge of force is essential for understanding
tooth movement. Because the word force has different
meanings in common language and in physics, important
definitions and concepts required for the application of
force analysis to the field of orthodontics are developed in
this chapter.
a
b
FIG 2-1 Simple coil spring demonstrating Newton’s First Law. (a)
Deactivated. (b) Activated. The spring is in equilibrium in both a and b.
The Field of Mechanics
Mechanics is the field of physics dealing with the study of
forces. Mechanics can be subcategorized into statics, kinetics, and materials science. Statics deals with a force on a body
with constant velocity, including a state of rest. Kinetics deals
with a force on a body with acceleration. Finally, materials
science deals with the effect of forces on materials.
The classic laws explaining the relationship between
force and bodies were presented by Newton in 1686.
Newton’s First Law (law of inertia) describes bodies at
rest or bodies with uniform velocity (no acceleration):
An object at rest tends to stay at rest, and an object in
motion tends to stay in motion with the same velocity
and in the same direction unless acted upon by an unbalanced force. This is the most important law for orthodontics, because it is the basis of all equilibrium applications. Activated appliances and restrained teeth within
the bone and periodontium are examples of the first law.
A simple orthodontic appliance component, a coil spring,
is shown in Fig 2-1. The deactivated spring in Fig 2-1a is
at rest; there are no forces acting on it. (This free-body
diagram purposely ignores gravity and other nonrelevant
forces.) The application of two forces in Fig 2-1b extends
the spring, which can now be placed in the mouth
between the anterior and posterior teeth for space closure.
The forces are equal (100 g) and opposite, allowing the
spring to remain in equilibrium. The spring deforms but
does not accelerate, demonstrating the First Law.
Newton’s Second Law (law of acceleration) states that
when force is applied to an object, the object accelerates
proportionally to the amount of force applied. The
famous Newtonian formula is
F = ma
12
where m is mass, a is acceleration, and F is force.
a
b
FIG 2-2 (a and b) Newton’s Third Law. Equal and
opposite forces act at the canine. FA (blue arrow),
activation force on the coil spring; FD (red arrow),
deactivation force on the canine.
This formula defines the nature of force—an ability to
accelerate an object. One would think that Newton’s
Second Law would have important applications in orthodontics. Are teeth not moving? Although they move, they
are not accelerating. Teeth are restrained objects and
hence are bodies in equilibrium and at rest. Imagine a
simple model in which a tooth is suspended by coil
springs on all sides. Similar to the spring in Fig 2-1, the
tooth is still in equilibrium after a force is applied to the
crown. Therefore, this book does not cover applications
in the field of kinetics.
Newton’s Third Law (law of action and reaction) states
that for every action there is always an equal and opposite
reaction (ie, for every force there is an equal and opposite
force). The commonly used example of this law is a rifle
shot where the bullet feels the force and one’s shoulder
Characteristics of a Force
feels the equal and opposite force. In Fig 2-2a, a coil spring
will be activated by a mesial force to allow its placement
on the canine hook. Because this force (FA) produces an
elongation of the elastic, it is called the activation force
(Fig 2-2b). This force extends the elastic during the act
of placement by the orthodontist, and later the canine
hook maintains the mesial activation force, holding the
elastic in place. At the canine hook, one observes the two
equal and opposite forces of Newton’s Third Law (see Fig
2-2b). The blue force (FA) is the activation force (the force
on the appliance), and the red force (FD) is the equal and
opposite force on the tooth or the hook. This equal and
opposite force (red arrow) is called the deactivation force
and is in the direction of the tooth movement. In other
words, the hook pulls on the elastic, and the elastic pulls
on the hook. These action and reaction forces occur at
the hook between two objects. In this example, it is also
true that the canine and the molar feel equal and opposite
forces, but this is not an expression of Newton’s Third
Law. Why? The elastic is in equilibrium; hence, the forces
on the elastic are equal and opposite. The explanation
lies in Newton’s First Law, which covers equilibrium on
bodies at rest (the elastic is not accelerating). Newton's
Third Law is properly used when both activation and
deactivation forces are showing on the canine (see Fig
2-2b).
This chapter introduces how to manipulate or handle
orthodontic forces. First, concurrent forces (ie, forces
acting on a point) are considered. In the next chapter,
this will be developed further to consider forces in three
dimensions on a body.
Characteristics of a Force
A force has three attributes: magnitude, direction and
sense, and point of force application. Figure 2-3 shows
three forces acting on a point (red dot), a hook on the
maxillary arch. Because the hook defines the point of
force application, only force magnitude and direction
require further description. From where do the forces
originate? Their source could be maxillomandibular elastics* or intra-arch elastics. Forces are vector quantities
that cannot be added algebraically but are rather added
geometrically. Note that the elastics have different angles
to each other, representing different lines of force application and denoting their vector properties.
FIG 2-3 Elastic forces acting on a point. The hook (red dot) defines
the point of force application. Different force magnitudes are represented by the length of the red arrows. The direction of the forces can
be measured by the force angle to the occlusal plane.
Force magnitude
The force magnitude is given in grams (g). The force
magnitudes in Fig 2-3 are represented by arrows; the length
of the arrow is proportional to the magnitude of the force.
Note that the 150-g maxillomandibular elastic arrow is
three times as long as the 50-g vertical elastic arrow and
half the length of the 300-g intra-arch elastic arrow.
Why are grams the unit of force in this example? This
unit is technically incorrect, as shown below. Historically
in America, ounces were used, and spring measuring
scales were calibrated in ounces. More universal metric
force gauges then became available, and the units were
in grams. Generally, scales used by the layman for
measuring body weight can be calibrated in pounds or
kilograms. For the physicist, these are not force (weight)
units but rather units for measuring mass. So let us briefly
consider the relationship between mass and force. Again,
the classic Newtonian formula is force equals mass times
acceleration (F = m × a). The force is the product of mass
(kilograms) and acceleration (m/s2). The unit of this
magnitude of force is therefore kg•m/s2, and 1 kg•m/s2
equals 1 Newton (N). The terms gram weight and kilogram weight are therefore incorrect.
Traditionally, orthodontists use the gram as the unit of
force. In the strict sense as explained above, this is incorrect
because grams are a unit of mass and not force. For example, gravity (a force) at sea level attracts a 100-g mass (the
amount of material). The calculated acceleration of gravity
is 9.8 m/s2. Let us now calculate how much force is acting
on the 100-g mass at sea level using Newton’s Second Law.
F=m×a
F = 100 g × 9.8 m/s2
F = 0.98 kg•m/s2 = 0.98 N = 98 cN
*Traditionally, orthodontists have used the term intermaxillary elastics to denote elastics placed between the maxillary and mandibular arches, because
both jaws used to be referred to as maxillae. However, because the mandible is no longer considered a maxilla, the term intermaxillary makes no
sense, hence the new term: maxillomandibular elastics.
13
2
Concurrent Force Systems
a
b
FIG 2-4 (a) Weight force depends on gravitational acceleration, so people weigh less on the moon than on Earth. (b) However, the activation
of an orthodontic appliance would produce the same force on the moon as on Earth.
Scientifically, a centi-Newton (cN) is the correct unit
for force, but in this book, grams will be used as the unit
of force because of its tradition in orthodontics; perhaps
this unit will be easier to understand for the clinician.
However, the authors recommend that scientific publications and presentations use Newtons or centi-Newtons
as the unit of choice. For purposes of practical conversion,
1 g equals 1 cN.*
Perhaps in the not-so-distant future, an orthodontist
might have a satellite office on the moon. If we attach a
100-g mass to a force gauge, as shown in Fig 2-4a, the
measured gravitational force will be about 1 N on Earth
but only 0.17 N on the moon. This is why people can
jump with less effort on the moon, because they actually
weigh less there. Let us now use this same force gauge to
measure the force from an orthodontic appliance (Fig
2-4b). This type of force gauge uses a calibrated spring
and has nothing to do with gravity. A spring gauge is
based on Hooke’s law, where force is proportional to wire
deflection. If the same appliance is used on the moon as
on Earth, there would be no difference in the forces,
provided the activation is the same (see Fig 2-4b). Our
imaginary orthodontist could therefore use the same
appliances and activations used on Earth, provided that
there were no biologic differences required in outer space.
Force direction and sense
Force also has sense and direction. The direction of the
force is defined by its line of action. This direction is
referred to as the sense. The arrows shown in Fig 2-3
demonstrate direction, sense, and the line of action of
14
*More accurately, 1 g equals 0.98 cN, and 1 cN equals 1.02 g.
three elastics. The origin of each arrow is the point of
force application (hook; red dot), the line (of action) indicates direction, and the arrowhead indicates the sense.
The direction of the force in Fig 2-5 is demonstrated by
the dotted line, and the arrowhead shows the sense.
To specify the direction of a force vector, a proper coordinate system is required; the direction of the force can
be represented by the angle between a given axis of the
coordinate system and the line of action. There are several
coordinate systems, but rectilinear Cartesian coordinates
are most frequently used. Figure 2-6a shows the three
axes of a Cartesian coordinate system and a sign convention in three dimensions. In this book, two-dimensional
diagrams, such as those in Fig 2-6b, are used for simplicity’s sake. Any coordinate system and sign convention is
acceptable, provided that it is clearly specified.
The orientation of a coordinate system can be set arbitrarily, depending on the problem to be studied. In an
orthodontic analysis, frequently used axes include the
occlusal plane, the Frankfort horizontal, the midsagittal
plane, and the long axis of a tooth. The direction of an
orthodontic force is specified in accordance with a given
established coordinate axis. For example, in Fig 2-7, a
crisscross elastic (red arrow) is applied at 90 degrees to
the mandibular right first molar relative to the midsagittal
plane. What is the best coordinate system to evaluate the
molar movement? Of the three shown (intersecting sets
of dotted lines), the author would most likely select the
system based on the mesiodistal or buccolingual axes of
the tooth. Resolving the force into rectilinear components
tells us that there are both mesial and lingual forces
(yellow arrows). It has been argued at some orthodontic
Characteristics of a Force
FIG 2-5 Force sense and direction. The line of action (dotted line)
demonstrates the direction, and the arrowhead denotes the sense
of the force.
a
b
FIG 2-6 Cartesian coordinates in three dimensions. (a) Three mutually perpendicular axes with a sign convention specified on each axis. (b)
Two-dimensional diagrams with the same coordinates as those shown in a. For simplicity’s sake, most diagrams in this text show only two
dimensions.
FIG 2-7 A crisscross elastic is attached at the buccal of the mandibular right molar. A coordinate system is selected that gives the
information that is most useful. Here the mesiodistal crown axis
system was selected. The elastic force (red arrow) has both lingual
and mesial components of force (yellow arrows).
15
2
Concurrent Force Systems
FIG 2-8 Law of transmissibility of force. The effect is the same
whether the force is applied at the mesial or the distal of a canine
as long as the force is along the same line of action (dotted line).
FIG 2-9 A force from an occipital headgear (FR , red arrow) can be
resolved graphically into two rectilinear horizontal (Fx ) and vertical
(Fy ) components (yellow arrows).
Manipulating Forces
Components
FIG 2-10 The same force from an occipital headgear (FR ) shown in
Fig 2-9 is resolved into Fx and Fy components mathematically using
simple trigonometric functions.
meetings that there are advantages to canine retraction
achieved by either pushing from the mesial or pulling
from the distal. As observed in Fig 2-8, however, there is
no difference in the line of action if the force is applied
at the mesial or the distal. A force acting anywhere along
this line of action has the same effect. In other words, a
force can be moved along its line of action without changing its effect. This principle is called the law of transmissibility of force. The appliance may differ with either an
open or closed coil spring, but if the force is along the
same line of action, the response should be the same
(assuming no other variables). A locomotion engine can
either push or pull a train car with the same effect.
16
It is convenient to resolve a force into rectilinear components (ie, two forces at 90 degrees to each other). Another
clinical way to look at direction is to ask how much force
is parallel to the occlusal plane and how much is vertically
directed. If distances are accurately drawn to represent
the forces, the solution can be obtained graphically. A
force from an occipital headgear is shown in Fig 2-9.
The direction is clearly shown as 30 degrees to the occlusal plane. Note that the headgear force (F R, red arrow)
can be understood by mentally walking from the hook
(application point) at 30 degrees upward and backward.
However, we can resolve this force into rectilinear components graphically by drawing two perpendicular lines:
Force X (Fx ) and Force Y (Fy ), with Fx parallel and Fy
perpendicular to the occlusal plane. Now let us take our
imaginary walk using these lines. Starting at the hook,
we walk along the occlusal plane (Fx ) to the right and
then walk upward at 90 degrees to the occlusal plane (Fy ).
This route may take longer, but we still end up at the apex
of the original red arrow. Forces are vectors, so we can
establish components using geometric addition. If
measured, the two component force lengths (yellow
arrows) tell us the magnitude and sense of the vertical
and horizontal rectilinear components of the original
force. Although the Fy component is depicted at the
arrowhead of Fx for analysis, Fy acts at the hook.
During clinical visits, many times a diagram may be
good enough to evaluate the rectilinear components of
Manipulating Forces
FIG 2-11 The two forces (red arrows) applied at the canine bracket
can be added arithmetically to give a resultant (yellow arrow) because
they act along the same line of action.
FIG 2-12 Two elastics (red arrows) applied at the canine hook. The
resultant, FR (yellow arrow), is determined using the parallelogram
method. The arrow (FR ) connects the origin at the hook with the
opposite corner of the parallelogram.
a force (graphic method). However, we might prefer to
use an analytical method, employing some simple trigonometry. Figure 2-10 is the same diagram as Fig 2-9,
where the angle of the headgear force can be any angle
(θ). Fx and Fy can be determined using the following trigonometric relationships:
The magnitudes of each force are the same as those in
Fig 2-11, but they lie on different lines of action. What is
the magnitude of the resultant? If you said 400 g, the
arithmetic total, the answer is incorrect. Forces are
vectors and must be added geometrically. Forces F1 and
F2 do not lie along the same line of action. The addition
must be done graphically.
Lines parallel to F1 and F2 are constructed, forming a
parallelogram. A diagonal line (FR, yellow arrow) is drawn
from the force origin (the hook) to the opposite corner
of the constructed parallelogram. This line represents the
vector sum of F1 and F2 and is the resultant force. The
length of the diagonal line proportionally represents the
force magnitude, and the angle to any plane represents
the sense and direction of the resultant. Note that the
length of FR (resultant) is not the arithmetic sum of the
lengths of F1 and F2, and its direction is different than the
applied individual elastics. The clinician might be advised
to place a single elastic (the resultant) for simplicity rather
than two, because the action on the arch will be the same.
Perhaps a more useful and universal graphic method
for force addition is the enclosed polygon method. Figure
2-13 shows the same component forces acting on the
hook as shown in Fig 2-12. Sequential forces will be
geometrically added instead of forming a parallelogram.
Starting with F1, an arrow is drawn downward and to the
distal. At the arrowhead of F1, F2 is drawn, keeping its
angle and magnitude the same as the original F2 in Fig
2-12. Connecting the origin (the hook) with the
Fy = FR sin θ
Fx = FR cos θ
Resultants
Often clinical situations require that we add a number
of forces. The canine shown in Fig 2-11 has two forces
acting at the bracket from two chain elastics (red arrows).
Because both elastics act along the same line of action,
we can find the sum of forces by simple arithmetic (addition), remembering that a force vector has a sense (direction), and therefore sign (+ or –) must be considered.
(–100 g) + (+300 g) = +200 g
The principle that forces along the same line of action
can be simply added together is important for orthodontists. The two elastics on the canine in Fig 2-11 produce
a total of +200 g (yellow arrow). This sum of all the forces
is called the resultant.
In Fig 2-12, two forces from maxillomandibular elastics
(F1 = 300 g, F2 = 100 g) are applied to the canine hook.
17
2
Concurrent Force Systems
arrowhead of the new F2 gives the resultant. In other words,
we can walk the short way (yellow arrow resultant) or take
the long way following the F1 and F2 components (red
arrows), ending up in the same place.
The closed polygon method is particularly useful if
more than two components are to be added. Four noncollinear forces are to be added in Fig 2-14. Each force is laid
out in sequence, arrowhead to tail. The resultant force
(FR , yellow) is a line connecting the origin hook and the
final component (F4 ) arrowhead.
Graphic methods for finding a resultant are very practical for the clinician. Most of the time, they are accurate
enough for patient care; more importantly, they do not
require complicated calculations. During chairside treatment, we are able to visualize the forces and come to
correct conclusions in our “mind’s eye” visualization of
force vectors and overall geometry. Nevertheless, an actual
diagram is most helpful as a starting place for manipulating forces, either by graphic or analytical methods.
Analytical method for determining a resultant
Instead of the graphic method, resultants can be calculated by using trigonometric functions and the Pythagorean theorem. Figure 2-15a shows two forces (red
arrows) acting on a hook mesial to the canine. F1 is a long
Class II elastic, and F2 is a short and more vertical Class
II elastic.
Step 1: Resolve all forces into components using a
common coordinate system.
In order to add forces, common lines of action can be
obtained by resolving F1 and F2 into x and y components.
Figure 2-15a shows the forces F1 and F2 resolved into
rectilinear components relative to an occlusal plane coordinate system. Fx is the horizontal component of force F,
and Fy is the vertical component of force F.
Using trigonometry,
Fx = F cos θ
Fy = F sin θ
Step 2: Add all x forces and y forces.
All forces on the x-axis are added. All forces on the y-axis
are added (Fig 2-15b).
Fx1 + Fx2 = Fx
Fy1 + Fy2 = Fy
18
Step 3: Draw a new right triangle using the summed
Fx and Fy values.
A new right triangle is drawn based on Fx (the sum of
Fx1 and Fx2) and Fy (the sum of Fy1 and Fy2) (Fig 2-15c).
Step 4: Calculate the magnitude and direction of the
resultant.
The magnitude of the resultant is calculated using the
Pythagorean theorem.
FR = Fx2 + Fy2
And the tangent function is used to calculate the direction (angle).
tan θ =
Fy
Fx
Below are some actual calculations using this method. Let
us suppose that F1 = 300 g and F2 = 100 g, with the direction specified in Fig 2-15a.
Step 1: Find the components of each force.
Fx1 = F1 cos θ1 = 300 g × cos 30° = 300 g × 0.87 = 261 g
Fy1 = F1 sin θ1 = 300 g × sin 30° = 300 g × 0.5 = 150 g
Fx2 = F2 cos θ2 = 100 g × cos 60° = 100 g × 0.5 = 50 g
Fy2 = F2 sin θ2 = 100 g × sin 60° = 100 g × 0.87 = 87 g
Step 2: Add each component.
Fx = Fx1 + Fx2 = 261 g + 50 g = 311 g
Fy = Fy1 + Fy2 = 150 g + 87 g = 237 g
Step 3: Now we have the x and y coordinates of a
resultant, and we can draw a new right triangle.
Step 4: Find the magnitude and direction of the
resultant.
FR = Fx2 + Fy2 = 3112 + 2372 = 391 (g)
tan θ =
Fy 237
=
= 0.76
Fx 311
Therefore, θ = 37.3°.
Clinical Applications
FIG 2-13 Enclosed polygon method for adding forces graphically.
Starting at the hook, each force is laid out tail to arrowhead, maintaining the magnitude, direction, and sense (red arrows). Connecting the
origin at the hook and the end point gives the resultant (yellow arrow).
a
b
FIG 2-14 The enclosed polygon method is useful, especially when
there are more than two components of force. FR (yellow arrow) is
the vector sum of all four components (red arrows).
c
FIG 2-15 Analytical method for determining a resultant. (a) Resolve all forces into rectilinear components (yellow arrows). (b) Add all x and y
forces. (c) Construct a new right triangle with the summed Fx and Fy values (yellow arrows). The hypotenuse (red arrow) is the resultant (FR ).
The magnitude and angle are found using the Pythagorean theorem and the tangent of θ.
Clinical Applications
This chapter has discussed important concepts relating
to a force or a group of forces acting on a point. A force
on a point was selected in one plane because it offers a
simple introduction to force manipulation. The same
principles will operate with forces on a body in two or
three dimensions. The major difference is the location of
the point of force application, which will be considered
in the next chapter. However, the clinician will be
confronted with many challenges that will involve forces
on a single point only, so let us now consider some of
these clinical applications.
Forces resolved into their rectilinear components are
always useful in planning the force system for proper
treatment. For example, we may want to know how large
the distal force component is in comparison to the occlusal (vertical) component using the occlusal plane as our
coordinate system.
Another clinical application is the simplification of the
orthodontic appliance. In Fig 2-16a, two maxillomandibular elastics are used, a Class II elastic and a vertical elastic. These elastics could be replaced with a single elastic,
the resultant (yellow arrow) in Fig 2-16b. The replacement
makes it easier for the patient and is therefore more likely
to ensure patient compliance. Conversely, sometimes it
19
2
Concurrent Force Systems
a
b
FIG 2-16 (a) A Class II elastic is applied. A vertical elastic is also used to close an open bite. (b) By using the enclosed polygon method, the
two elastics can be replaced with one elastic (yellow arrow), which is simpler for both the orthodontist and the patient.
a
b
FIG 2-17 Intrusive force from a cantilever (vertical red arrow) and distal force from an elastic chain (horizontal red arrow) produce a resultant force (yellow arrow) acting parallel to the long axis of the incisors. (a) Deactivated spring. (b) Activated spring.
is better to use two or more elastics that will produce the
same effect as a single elastic, because sometimes the
direction of the force needs to be changed slightly. For
example, the objective in Fig 2-17 is to deliver an intrusive
force parallel to the long axis of the incisors. Two forces
are used: (1) an intrusive force from an intrusive cantilever
attached to the first molar auxiliary tube and (2) a chain
elastic producing a distal force. Note that the resultant
force (yellow arrow) is parallel to the mean of the root
long axis. Moreover, multiple forces can replace a single
force when a single force cannot be placed clinically
because of anatomical limitations (eg, during canine
retraction, three or more forces are applied at the bracket
instead of one on the root).
Figure 2-18 shows an elastic chain engaged between
brackets and a transpalatal arch. What would be the resultant force acting on the maxillary right second premolar
and canine? Suppose the tension of the elastic is uniform;
we could easily imagine a parallelogram and find the
resultant graphically. The resultants (yellow arrows) on
the premolar and canine are in the right directions to
correct the malocclusion.
20
Suppose we want to apply lingual force on the canine.
Figure 2-19a shows a single force directly acting on a
canine using an auxiliary spring soldered to a passive
lingual arch. What if there is no lingual arch present, and
yet we need to apply a lingual force? The single lingual
force can be resolved along the arch into two components
(Fig 2-19b). Two simple elastics (component forces are
red arrows) will produce the same effect on the canine
as the auxiliary spring in Fig 2-19a. Anchorage, of course,
will be different. Note that components are not always
rectilinear.
Figure 2-20a seems to show the force system acting on
the molar using a temporary anchorage device (TAD)
and an elastic chain (gray arrow). But this diagram is
incorrect. One might think that an intrusive force would
be present because the elastic is wrapped above the entire
crown. However, only a buccal force (red arrow) is
produced from the elastic connecting the molar-bonded
hook and the TAD (Fig 2-20b). Figure 2-20c demonstrates
that although the chain elastic between the two hooks
on the molar is stretched, this part of the elastic produces
no force on the molar.
Clinical Applications
a
b
FIG 2-18 (a and b) One can easily imagine a parallelogram or enclosed polygon and estimate
the magnitude and direction of the resultants (yellow arrows) graphically. The predicted direction
of tooth movement is correct.
a
b
FIG 2-19 (a) A single force from a cantilever attached to a lingual arch gives a lingual force to the canine. (b) If no lingual
arch is present, two components (red arrows) from an elastic chain could deliver a similar force.
a
b
c
FIG 2-20 An elastic chain is attached from a buccal TAD to the molar. (a) The gray force does not exist. (b) Only the intrusive buccal force
(red arrow) is produced. (c) The elastic stretched between the two buttons on the molar delivers no vertical force to the molar because both
forces (red arrows) cancel to zero.
21
2
Concurrent Force Systems
Summary
Fiorelli G, Melsen B. Biomechanics in Orthodontics 4. Arezzo, Italy:
Libra Ortodonzia, 2013.
This chapter developed the key principles and methods
for manipulating forces acting on a point. In most orthodontic treatment, the clinician must plan for multiple
point applications on three-dimensional bodies. The
next chapter considers forces acting on more than one
point in two and three dimensions—nonconcurrent
forces. The principles and methods will be the same as
for concurrent forces. Determining the point or points
of force application will require consideration of an additional physical quantity—the moment.
Gottlieb EL, Burstone CJ. JCO interviews Dr. Charles J. Burstone on
orthodontic force control. J Clin Orthod 1981;15:266–278.
Recommended Reading
Burstone CJ. Application of bioengineering to clinical orthodontics.
In: Graber LW, Vanarsdall RL Jr, Vig KWL (eds). Orthodontics: Current
Principles and Techniques, ed 5. Philadelphia: Elsevier Mosby,
2011:345–380.
Burstone CJ. Biomechanical rationale of orthodontic therapy. In:
Melsen B (ed). Current Controversies in Orthodontics. Berlin: Quintessence, 1991:131–146.
Burstone CJ. Malocclusion: New directions for research and therapy.
J Am Dent Assoc 1973;87:1044–1047.
Burstone CJ. The biomechanics of tooth movement. In: Kraus B, Riedel
R (eds). Vistas in Orthodontics. Philadelphia: Lea and Febiger,
1962:197–213.
22
Halliday D, Resnick R, Walker J. Vectors. In: Fundamentals of Physics,
ed 8. New York: Wiley, 2008:38–115.
Isaacson RJ, Burstone CJ. Malocclusions and Bioengineering: A Paper
for the Workshop on the Relevance of Bioengineering to Dentistry
[DHEW publication no. (NIH) 771198m, 2042]. Washington, DC:
Government Printing Office, 1977.
Koenig HA, Vanderby R, Solonche DJ, Burstone CJ. Force systems
from orthodontic appliances: An analytical and experimental comparison. J Biomech Engineering 1980;102:294–300.
Melsen B, Fotis V, Burstone CJ. Biomechanical principles in orthodontics. I [in Italian]. Mondo Ortod 1985;10(4):61–73.
Nanda R, Burstone CJ. JCO interviews Charles J. Burstone. II: Biomechanics. J Clin Orthod 2007;41:139–147.
Nanda R, Kuhlberg A. Principles of biomechanics. In: Nanda R (ed).
Biomechanics in Clinical Orthodontics. Philadelphia: WB Saunders,
1996:1–22.
Nikolai RJ. Introduction to analysis of orthodontic force. In: Bioengineering: Analysis of Orthodontic Mechanics. Philadelphia: Lea and
Febiger, 1985:24–70.
Smith RJ, Burstone CJ. Mechanics of tooth movement. Am J Orthod
1984;85:294–307.
Problems
1. Compare A, B, C, and D. Is there a difference? The
force is acting on a very rigid, nondeformable wire in
B and C, and the force is applied on a very flexible wire
in D.
2. A 300-g force of headgear and a 100-g force of
intra-arch elastic act on the first molar. Find the
resultant.
3. A headgear (300 g) and a Class II elastic (100 g) act
at a hook on the archwire. Find the resultant.
4. Find the resultant of the forces from the lingual arch
and the crisscross elastic.
5. Resolve the 100-g force into two components parallel and perpendicular to the long axis of the tooth graphically
and analytically when the angle is (a) 60 degrees, (b) 45 degrees, and (c) 110 degrees.
a
b
c
23
Problems
6. Resolve the 150-g force from a Class II elastic into two components parallel and perpendicular to the occlusal plane
when the angle is (a) 20 degrees and (b) 45 degrees.
a
7. Resolve the 100-g crisscross elastic force attached at
the buccal tube of the first molar into buccolingual
and mesiodistal components.
9. A Class II elastic and a headgear are simultaneously
applied. The direction and magnitude of the elastic
are kept constant. The resultant force must lie along
the archwire axis. Find the angle when the headgear
force is (a) 200 g, (b) 600 g, and (c) 1,000 g.
24
b
8. Find the resultant of a 400-g headgear force and a
200-g Class II elastic force.
3
Nonconcurrent
Force Systems
and Forces on a
Free Body
“Measure what is measurable,
and make measurable what is not so.”
—Galileo Galilei
Teeth, segments, and arches are three-dimensional, and all appliance
forces may not act on a single point. Nonconcurrent forces and their
manipulation are described in this chapter. The principle of vector
addition or resolution is the same as with forces on a point. One new
parameter must be found: the point of force application. The concepts
of moment and moment of force are introduced in this chapter. The
point of force application of a resultant can be found by summing all
separate moments around an arbitrary point; the distance from that
arbitrary point to the resultant gives an identical moment. The useful
concept of equivalence is also introduced. Components and resultants are equivalent because their action is the same. Any force can
be replaced with a force and a couple that is equivalent. Force and
couple equivalents at the center of resistance of a tooth or an arch
or at a bracket are a powerful tool for understanding and predicting
tooth movement.
25
3
Nonconcurrent Force Systems and Forces on a Free Body
FIG 3-1 If forces are parallel, they can be added algebraically to
establish a resultant.
FIG 3-2 Nonparallel forces are resolved into components. Parallel
components can then be added, and the magnitude and direction
of the resultant can be determined.
FIG 3-3 The point of force application of a resultant on a body must
be determined among many gray arrows. The moment must be considered in order to determine the correct point of force application.
FIG 3-4 Resultants for angled elastics. The correct point of force
application as in Fig 3-3 is determined by considering moments.
I
n chapter 2, we considered forces acting at one point
and learned how to resolve a force into components
and find a resultant. As mentioned in that chapter,
however, most orthodontic treatment involves forces that
act on anatomical structures in three dimensions. This
chapter considers such three-dimensional (3D) force
systems (eg, more than one force acting at different points
on a full dental arch).
Determining the Magnitude and
Direction of the Resultant
Figure 3-1 shows a lateral view of a maxillary arch with
two vertical elastics applied at different points. Let us
find a resultant—a single elastic that will do the same
thing. A clarification is required before we start. For
simplicity, in this text we will look at separate perpendic26
ular projections while analyzing 3D clinical situations.
This can be problematic if major asymmetries influence
the location of the center of resistance (CR) of teeth (ie,
if the CR varies from one plane to another).
A simpler approach allows us to study one plane at a
time. Thus, in Fig 3-1, both forces are projected on the
xy plane (also called the z plane). Our analysis for now is
limited to just one plane. Both forces have the same direction (angle) and sense but lie on different lines of action.
Because they are parallel to each other, they can be added
algebraically, just like multiple forces on a point with a
common line of action.
F1 + F2 = FR
100 g + 100 g = 200 g
FR = 200 g
The resultant is therefore 200 g.
Moments and Couples
FIG 3-5 Downward force on the wrench tightens the bolt. Force (F)
times perpendicular distance (D) equals the moment (M), which is
10,000 gmm in this case. The moment is more specifically a moment
of force.
FIG 3-6 Equal and opposite forces not along the same line of action
produce a pure moment or a couple. The forces cancel each other out
so that the force magnitude is 0 g and the moment is –1,000 gmm.
Determining the magnitude of angled component
forces uses the same method presented in chapter 2. If
the forces are not parallel, they can be resolved into two
rectilinear components (Fig 3-2). Graphically or analytically, the resultant and its direction (angle) can then be
calculated using the Pythagorean theorem and trigonometric functions. With multiple forces on a single point,
the point of force application is given. However, on a 3D
body, this point of force application must be determined,
requiring an additional calculation, as shown in Fig 3-3.
The gray lines are the resultant. The magnitude is 200 g,
and the direction is parallel to the y-axis. But where is
the point of force application? Any of the gray lines is a
possibility. The same is true if the resultant is angled as
in Fig 3-4. Any of the lines could be the point of force
application. Finding the correct point of force application
involves an understanding and calculation of moment.
This quantity M is more specifically called the moment
of force, because it originates from the force on the
wrench. The unit of measurement is the gram-millimeter
(gmm) or centi-Newton-millimeter (cNmm), and its
direction is represented by the curved arrow in Fig 3-5.
If the force is doubled or if the distance is increased to
200 mm, the moment increases to 20,000 gmm. These
moments are positive (+) when they are in a clockwise
direction and negative (–) when they are in a counterclockwise direction in this book. The 100-mm distance
is called the moment arm. We all know from experience
that it is not the force alone but the moment that determines the ease of tightening the bolt. This moment (ie,
bolt tightening) is useful and readily visualized, but many
of the moments used in this book are imaginary and only
used for calculation. Thus, a broader definition of moment
of force is the force times a perpendicular distance to any
arbitrary or imaginary point. This concept is used to
determine the location of the point of force application
for a resultant of nonconcurrent forces.
Before locating the resultant, let us consider a special
type of moment where there is no force. Note that the
screw in Fig 3-6 has two forces applied. The forces are
parallel but have opposite senses. If we add them, the
resultant force is 0 g. The screw feels no force. The
moment, however, is one force times the perpendicular
distance to the other force.
Moments and Couples
What is moment? Moment is a measure of the tendency
of a body to rotate around a point or axis perpendicular
to any given plane. For example, let us use a wrench to
tighten a bolt in Fig 3-5. The 100 g is applied perpendicular to and 100 mm away from the bolt (known as the
perpendicular distance). The tendency to rotate, or the
moment (M), is calculated by multiplying the force (F)
times the perpendicular distance (D) to the bolt:
M = F × D = 100 g × 100 mm = 10,000 gmm
M = 50 g × 20 mm = 1,000 gmm (counterclockwise)
This special moment has the tendency to produce rotation only and can be called a pure moment or a couple.
27
3
Nonconcurrent Force Systems and Forces on a Free Body
FIG 3-7 A couple is a free vector. The moment effect is therefore the
same no matter where the couple is placed. The moment magnitude
can be the same if both force magnitude and distance are varied.
FIG 3-8 Direction of couples. (a) A couple is also a vector, and its
direction is depicted as a double-headed arrow along the rotation
axis to distinguish it from the force (b) or simply as a curved arrow.
Imagine that the right hand has grasped any rotation axis with four
fingers pointing in the direction of the desired moment; then the
thumb is in the direction of the double-headed arrow.
The direction is depicted in the curved arrow, which is
counterclockwise looking from above. With a moment
of force, a body (eg, a tooth) will feel both a force and a
couple. With a couple, however, no force is felt. A unique
property of a couple is that it acts as a free vector. This
means that it does not make any difference where it is
applied; the effect will be the same. If we move the forces
off-center as in Fig 3-7, the effect on the screw is the same.
The magnitude of the forces has been changed, but the
effect is the same because the moment is identical to that
in Fig 3-6.
movement are produced by changing the position of the
line of action of the force(s). By contrast, however, because
couples are free vectors, they can be located anywhere
on a diagram. Figure 3-9 shows a couple acting on a
canine. Two equal and opposite forces not along the same
line of action deliver a pure moment of –1,200 gmm (300
g × 4 mm) (Fig 3-9a). The moment can be correctly represented as a curved arrow anywhere on or off of the tooth
(Fig 3-9b). The tooth movement will always be the same
because a couple rotates a tooth around its CR (purple
circle). In Fig 3-10, a bar with three brackets is bonded
to the labial of a canine. Does it make any difference
whether the couple is applied at A, B, or C? No. Because
the canine will spin around its CR (purple circle); a couple
is a free vector not dependent on location of application.
Erroneously, some orthodontists have believed that a
couple causes a tooth to rotate between the two wings
of a bracket.
Three different views are shown in Fig 3-11. Traditional
orthodontic jargon has described the actions as rotation
(Fig 3-11a), tipping (Fig 3-11b), and torque (Fig 3-11c).
This terminology is confusing and certainly not in line
with the rest of the scientific and engineering specialties.
First, all teeth rotate around their CRs. Rotation occurs
in all views, not just from the occlusal. Second, the force
systems are identical in their application of couples. It is
simpler to classify all the force systems as couples and
recognize that these couples will produce rotation of the
teeth in all three planes of space. “Tip-back bends” and
“torque” both involve moments. Incidentally, an archwire
M = 100 g × 10 mm = 1,000 gmm (counterclockwise)
Because couples are free vectors and the force magnitudes and the distances between them can be varied, it
may not be necessary to show the equal and opposite
forces. For that reason, a curved arrow is used, provided
the magnitude of the moment is given with the correct
direction. Sometimes a double-headed arrow along the
rotation axis is used to represent a couple to distinguish
it from the force (Fig 3-8). Most of the time curved arrows
will be used in this book; a double-headed arrow is used
only when a curved arrow cannot represent the direction
in some perspectives.
The point of force application is critical in understanding appliance design and also the biomechanics of
tooth movement. During all planning and diagram drawing, a force must be located exactly where it will act.
Chapter 9 emphasizes that different types of tooth
28
Moments and Couples
a
b
FIG 3-9 A couple will rotate a canine around its CR (purple circle).
(a) Forces acting at bracket wings. Moment arm refers to the distance
between the two bracket wings. (b) A couple of –1,200 gmm is shown
as a curved arrow.
FIG 3-11 Orthodontic jargon
for the shown force systems: rotation (a), tipping (b), and torque
(c). These are all more simply
described as a couple. All produce rotation around the CR.
a
into the brackets would deliver a more complicated force
system than that shown in Fig 3-11, usually including
both forces and moments. Note that in Fig 3-11a, a couple
to rotate the molar counterclockwise will have the same
effect on the buccal as that applied on the lingual with a
lingual appliance.
Figure 3-12 illustrates a patient with a severe midline
discrepancy (Fig 3-12a). However, in the posteroanterior
cephalometric radiograph, there is no apical base discrepancy (Fig 3-12b). The maxillary incisors are tipped to the
left side, and the mandibular incisors are tipped to the
right. Correct bracket placement will produce couples at
each tooth in a favorable direction to correct the midline.
The couples on each tooth produced rotation of each
tooth around its CR, not the bracket. After leveling, the
roots became parallel to each other by rotation around
each CR, with the crown and root apex moving in
FIG 3-10 An imaginary appliance with three brackets
on a rigid bar attached to the canine. It does not make
any difference where the couple is applied. The tooth
will rotate around its CR.
b
c
opposite directions (Fig 3-12c). The midline was corrected
using the couples acting at the brackets without requiring
any lateral forces (Fig 3-12d).
Figure 3-13 illustrates a patient with a posterior crossbite on the right side at the second molars (Fig 3-13a).
The crossbite was mostly due to linguoversion of the
mandibular second molar (Fig 3-13b). A piece of 0.016
× 0.022–inch nickel-titanium wire was twisted and
inserted at the tubes of the mandibular second molar and
second premolar brackets, bypassing the first molar tube,
so that it produced equal and opposite couples, as
depicted by red arrows (Fig 3-13c). Even though the
couple is acting at the tube or bracket, the rotation still
occurs around each CR. The second molar rotated counterclockwise, and the second premolar rotated clockwise.
After the second molar was uprighted, the second premolar was extracted.
29
3
Nonconcurrent Force Systems and Forces on a Free Body
a
b
c
d
FIG 3-12 (a) A patient with a severe midline discrepancy (dotted lines). (b) In the posteroanterior cephalometric radiograph, there is no apical
base discrepancy. Correct bracket placement will produce couples at each tooth. (c) The couples on each tooth produce rotation of each tooth
around its CR. (d) The midline was corrected using the couples acting at the brackets without requiring any lateral force.
a
b
c
FIG 3-13 (a) A patient with a posterior crossbite on the right side at the second molars. (b) The crossbite was mostly due to linguoversion
of the mandibular second molar. (c) A piece of 0.016 × 0.022–inch nickel-titanium wire was twisted and inserted at the tubes of the second
molar and second premolar brackets, bypassing the first molar tube, so that it produced equal and opposite couples, as depicted by red arrows.
Determining the Point of Force
Application of the Resultant
Let us now go back to the determination of the resultant
in Fig 3-3. The principle of vector addition was used to
find the force magnitude of 200 g. The only unknown is
the position of the line of the force (Fig 3-14). Here a
moment formula is used that states: The sum of the
30
moments around any arbitrary point of the component
forces equals the moment of the resultant force around
that point (Fig 3-14a). Therefore,
∑M* = Rd
d=
∑M*
R
Determining the Point of Force Application of the Resultant
a
b
c
d
FIG 3-14 (a) To find the location of a resultant, an arbitrary point is selected (red dot). The sum of the component moments around that
point equals the resultant magnitude times the distance to the same point. (b) The point of force application of the resultant is found
at 15 mm from the arbitrary point (red dot). The moments from the red and yellow forces are equal (–3,000 gmm around the red dot).
(c) Selection of a point at which to sum the moments is based on convenience. Any point is valid. Even a point on the Taj Mahal or the Eiffel
Tower will give the correct answer as long as the distances are known, but they are not convenient. (d) The resultant can only be placed 15
mm anterior to the arbitrary point for the sense of the moment (–) to be correct. The red arrow is correct, while the gray arrow is incorrect.
where M is the moment measured in respect to any point
(*), R is the resultant magnitude, and d is the location of
the point of application of the resultant measured to the
same point.
First an arbitrary point is chosen. Because any point
will do, a convenient one is selected that is located at the
100-g force in the molar region (red dot). The sum of all
of the moments around that point is calculated (Fig
3-14b).
100 g × 30 mm = 3,000 gmm (counterclockwise)
Note that no moment is produced by the molar vertical elastic because the perpendicular distance to the
selected point is 0. Finally, dividing the sum of the
moments by the resultant magnitude gives the distance
to the selected point:
The resultant is located 15 mm from the molar elastic.
The two individual component elastics (red arrows) or
one resultant elastic (yellow arrow) delivers the same
moment to the selected point. Any point can be selected
on or off the body under consideration to sum the
moments, and the answer will be the same (Fig 3-14c).
Even a point at the Taj Mahal or the Eiffel Tower will work
if the distances are known. We select a point for convenience—here at the molar force to simplify the calculation. How do we know if the 15-mm distance is anterior
to or posterior to the selected molar point? This distance
must be anterior to the point for the direction of the
moment (counterclockwise) to be correct (Fig 3-14d).
The gray arrow is therefore incorrect.
3,000 gmm
= 15 mm
200 g
31
3
Nonconcurrent Force Systems and Forces on a Free Body
FIG 3-15 Equivalence. Force systems that are equivalent produce
the same effects. Examples include resultants and components.
FIG 3-16 Any force can be replaced with a force and a couple. A
single force applied at the root (red arrow) could produce root movement (rotation around the incisal edge, blue dot). A force and couple
(yellow arrows) at the bracket will do the same thing.
Equivalence of Forces
nose. Let us replace the force at the bracket. The equivalent force is still 200 g (yellow arrow) based on the first
formula. The bracket is picked as a convenient point to
sum the moments. The original 200-g force times the
12-mm perpendicular distance gives a +2,400-gmm
moment. The new replacement force has no moment
(200 g × 0 mm = 0 gmm). For equivalence, a couple of
+2,400 gmm must be added at the bracket (yellow curved
arrow). The red or yellow force systems do the same—the
tooth will not notice any difference.
Replacing a force with a couple is also useful in predicting tooth movement. A 200-g distal force is applied at
the mesial of the canine bracket (Fig 3-17). A replaced
equivalent force acting at the CR of a tooth produces
translatory movement, and a couple produces rotation
around the CR (see also chapter 9). If we replace the
original 200-g force (red arrow) at the bracket, the magnitude of the replacement force (yellow arrow) at the CR
(purple circle) is the same: 200 g. This CR is merely a
convenient point at which to sum the moments. The
original force (red arrow) moment is 200 g × 7 mm =
1,400 gmm (counterclockwise). The replacement moment
from the force (yellow arrow) is zero because there is no
distance between the force and the point of application,
so a couple of –1,400 gmm must be added at the CR. The
force system at the CR allows us to predict that the canine
will translate distally and rotate in a counterclockwise
direction.
Forces that are interchangeable are called equivalent. In
other words, the effect on teeth or arches is the same. If
the exchange rate is considered, an equivalence can be
established between the Euro and the American dollar.
This form of equivalence can change often and may not
always reflect buying power. Newtonian equivalence, on
the other hand, is more precise and always produces the
same effect, ignoring any local effects where the forces
are applied. Two force systems are equivalent if the sums
of their forces are equal and if the sums of their moments
around any arbitrary point are also equal (Fig 3-15).
∑F1 = ∑F2
∑M1* = ∑M2*
We have already considered some examples of equivalence. Resultants are equivalent to their component
forces. A single force can be replaced by a number of
forces or vice versa. The clinician can choose the best
equivalence for effective and convenient treatment. But
perhaps the most useful equivalence is the replacement
of a force with a force and a couple. Let us consider the
incisor in Fig 3-16. If a 200-g lingual force (red arrow) is
placed on the root, one might expect the root to move
lingually with a center of rotation on the crown (blue
dot). But it is not practical to place a force so far apically;
the appliance might have to be inserted by way of the
32
Equivalence of Forces
FIG 3-17 Any force can be replaced with a force and a couple. If the
replacement force and moment act at the CR, useful information on
how the tooth will move is obtained. The distal force at the bracket
causes distal-in or counterclockwise rotation of the canine.
FIG 3-18 A distal force and a moment translate the canine from the
occlusal view (red arrows). If the same thing is to be accomplished
using a lingual orthodontic appliance, a distal force of 200 g and a
–400-gmm moment are required (sum of moments shown by yellow
curved arrows). Note that moment direction on the lingual is opposite
to the moment applied on the labial.
But are the force systems different between labial and
lingual orthodontics? Equivalence can minimize any
surprises when working from the lingual. Let us consider
canine retraction as shown in Fig 3-18. A satisfactory
force system (red arrow) from the labial is 200 g with a
1,400-gmm clockwise moment (200 g × 7 mm = 1,400
gmm). We want to replace the labial force system at the
lingual bracket, which is 2 mm lingual to the CR. The
force (yellow arrow) is the same: 200 g. The moments are
conveniently summed around the lingual bracket (red
dot): 200 g × 9 mm = –1,800 gmm (counterclockwise).
The 1,400-gmm red couple is moved to the lingual
bracket. Why can this be done? It is a free vector. The
total moment acting at the lingual bracket is therefore
only –400 gmm (–1,800 gmm + 1,400 gmm = –400 gmm),
acting in a counterclockwise direction. The force system
working from the lingual is actually simpler than the labial
system. Less moment is needed, thereby preventing undesirable tooth rotation, and that moment is in the opposite
direction.
A two-dimensional (2D) plastic model of two teeth
was fabricated (Fig 3-19). Springs were attached to the
teeth to simulate the periodontal ligament. A chain elastic connects the two teeth (Fig 3-19a) and is angled so
that the right tooth tips and the left tooth translates. This
is one method for anchorage control to pit translation
against tipping. However, it is impossible to connect teeth
in this manner clinically because of tissue location. It is
possible, on the other hand, to place an equivalent force
system at the bracket, which is accomplished by a T-loop
(Fig 3-19b). A more detailed explanation of loop shape
for this purpose is discussed later. Figures 3-19c and 3-19d
show that the original force system from the elastic (red
arrows) is replaced by a force and a couple (yellow arrows)
at the bracket of each tooth. The anchorage tooth has the
greater moment. Equivalence allows us to practically
make an appliance that will work using brackets on the
crowns of the teeth.
Figure 3-20a is the initial frontal view of a patient with
a midline deviation. The patient is an extraction case
where the anterior segment needs translation to the left
side. A single force acting through the CR (purple circle
in Fig 3-20b) is very difficult to achieve due to anatomical limitations. The resultant oblique single force is therefore replaced with vertical and horizontal force components. Note that the horizontal force component can be
moved along its line of action so that the resultant can
be determined as a force acting on a point at the patient’s
right. The anterior segment translated to the left side
without tipping (Fig 3-20c). The law of transmissibility
and the equivalence principle allow us to practically make
an appliance that will work using brackets at crown level.
33
3
Nonconcurrent Force Systems and Forces on a Free Body
a
b
c
d
FIG 3-19 2D plastic model with teeth supported by springs. (a) An elastic is stretched at an angle between the roots,
causing the right tooth to tip and the left tooth to translate. (b) An activated T-loop achieves similar movement with
forces at the bracket. (c) Same elastic as in a, showing the forces. (d) Equivalent force systems: the original force from
the elastic (red arrows) and the replacement at the bracket with a T-loop (yellow arrows).
a
b
34
FIG 3-20 (a) Initial frontal view of a patient with a midline deviation. (b) A single force acting through the CR (purple circle) is
very difficult due to anatomical limitations. The oblique single force
resultant (yellow arrow) is replaced with vertical and horizontal
force components (red arrows). The horizontal component of
force acting at the patient’s left can be moved along its line of
action to intersect with the right vertical force to simply solve for
the resultant, since all forces act then at a point. (c) The anterior
segment translated to the left side without tipping.
c
2D Projections of 3D Force Systems
a
b
FIG 3-21 (a) 2D projection (xy plane) of a 3D body. The 100-g elastic is on the patient’s right side, and the 20-g elastic is on the left side. The
yellow arrow shows the 120-g resultant. (b) 2D projection (yz plane) of a 3D body. Note the location of the resultant (yellow arrow).
2D Projections of 3D Force
Systems
Describing a force system acting in a 3D space and how
it influences tooth movement is not a trivial pursuit.
Chapter 10 considers 3D tooth movement and questions
such as what is the meaning of the CR for asymmetric
teeth or groups of teeth. For now, however, we will handle
3D forces by projecting them onto three mutually perpendicular planes: x, y, and z. For simplicity, we will place
two vertical elastics. In Fig 3-21, the 100-g force acts
between the right lateral incisor and the canine on the
labial arch. The other 20-g force acts at the left tube of a
lingual arch. Figure 3-21a shows the xy plane (or the z
plane), and Fig 3-21b shows the yz plane (x plane). The
original component forces are the red arrows. In each
separate plane, a resultant is calculated using the principles explained in this chapter. The yellow arrow gives the
resultant force in each plane (120-g magnitude, direction,
and point of force application shown). The xz plane
(occlusal view) is shown in Fig 3-22c. The forces are depicted
in dots in the circle because the arrowhead is shown
looking from above. The force direction here is completely
vertical and parallel to simplify the discussion. The general
solution, of course, allows the handling of many forces
that are angled and not parallel with each other. This is
one way to evaluate forces in three dimensions.
The clinician can compress all forces in a given plane
to a 2D projection and evaluate what happens in each
plane. In Fig 3-21, what will happen in each plane? To
answer that question (Fig 3-22), the resultant force (red
arrow) must be replaced at the CR with a force and a
couple (yellow arrows). This applies once again the
concept of equivalence. From a lateral view (Fig 3-22a),
it can be seen that the cant of the occlusal plane will
steepen (increase) and the maxillary arch will extrude.
From the frontal view (Fig 3-22b), the resultant force
off-center will extrude the arch and cant the occlusal
plane, extruding the right side more than the left. Figure
3-23 depicts the 3D rendering of the forces. There are
two components to the calculations involved in three
dimensions projected into two dimensions. The first part
is defining a resultant (adding forces). That part is pure
Newton and is readily done. The second part involves a
concept—the center of resistance—which involves physics, biology, and unproven assumptions. For that reason,
the present discussion should only be considered an
introduction to the relationship of tooth movement to
forces, particularly in three dimensions.
Much of the basic knowledge has now been presented
about the scientific handling of forces from orthodontic
appliances. The following chapters consider the simplest
of all orthodontic appliances: appliances that deliver
single forces. These appliances are statically determinate,
which means that a single force measurement can
35
3
Nonconcurrent Force Systems and Forces on a Free Body
a
b
FIG 3-22 (a) The resultant is replaced with a force and a couple (yellow arrows) at the CR in the xy plane, which will extrude the maxillary
arch and steepen the occlusal plane. (b) The resultant is replaced
with a force and a couple (yellow arrows) at the CR in the yz plane,
which will extrude the maxillary arch and make it cant downward on
the patient’s right side. (c) The xz plane (occlusal view). The forces
are depicted in dots in the circle because the arrowhead is shown
looking from above.
c
FIG 3-23 3D view of the force system.
completely define the force system. Simple, however, does
not mean inferior. In many cases, these appliances may
be the most useful, practical, efficient, and predictable of
all appliances.
36
Recommended Reading
See the recommended reading in chapter 2.
Problems
1. A 100-g force is applied at the labial. Give the force
system at the lingual bracket that will do the same.
2. A 100-g force and a –400-gmm moment are
applied at the labial. Give the force system at the
lingual bracket that will produce the same effect.
3. A single force through the CR will translate the canine
distally. We can use a force through the CR; however,
this is not always practical. Find the two forces at points
A and B that will have the same effect.
4. We want to translate the canine from this view.
Find the two forces at points A and B that will do
the same as the single force shown. This is the
same problem as #3 except that the CR is further
lingually.
5. Replace the 200-g elastic with two forces at FA and FB.
6. Find the resultant of the two vertical elastics.
37
Problems
7. Find the resultant of the two Class II elastics.
8. For a given patient, 100 g of force and a 1,000-gmm
moment at bracket A produce the desired movement.
If bracket B is used, give the force system for the same
effect.
9. A wire extension is used at point A for canine retraction. Find the equivalent force system for point B.
10. Compare the effect on the molar of the 200-g force
if placed at the buccal (a) or lingual (b).
a
b
11. A tip-back spring is attached to the molar. The activation force on the spring is 100 g. Compare the molar movement in a, b, and c.
a
b
12. Compare the effect on the mandibular incisor of an
80-g force placed at the labial bracket (A) or on the
lingual attachment (B). The circle is the CR of the
incisor.
38
c
4
Headgear
“The fact that an opinion has been widely held is no
evidence whatever that it is not utterly absurd.”
— Bertrand Russell
The effect of a headgear on a tooth or a full arch is usually easy to
calculate. First the center of resistance (CR) of a tooth, segment, or
arch is determined. Deformation of the inner bow can be ignored
unless relevant. A line of action of the force through the CR produces
translation in the direction of the force. A line of action away from
the CR produces both translation and rotation around the CR. Older
terminology and descriptions of headgear usage, such as cervical or
occipital with a short or long outer bow, are too complicated and may
not be predictive. The clinician must first select a proper line of action
of the headgear force and design the headgear to deliver that force.
With a correct line of force, teeth can be translated, crowns tipped, or
roots moved distally. With a correct line of force, the occlusal plane
cant can also be maintained, increased, or decreased, thus increasing
or decreasing the vertical overlap (also known as overbite).
39
4
Headgear
FIG 4-1 Inner and outer bow headgear. The inner bow is inserted
into the molar tube, and a cervical elastic strap is connected at the
hook of the outer bow. The red arrow is the force from the cervical
elastic strap.
T
he headgear is the ideal appliance with which to
start a discussion of appliance design. Headgear is
a statically determinate appliance, which means that
a single measurement with a force gauge can provide the
clinician with most of the information required.
Headgear classification is usually determined by
anatomical geometry, such as where it is attached to the
head (cervical or occipital), the direction of the pull (high
or low), and outer bow length or position. However, these
classifications and the rules of thumb based on them have
led to serious errors in the use of headgear and have
certainly complicated the explanation of how headgears
work. In this chapter, emphasis is placed not on the geometry and shape but rather the force system of the headgear.
Two different designs, based on the manner of attachment to the teeth, are presented in this chapter. One type
has an inner archwire (bow) that attaches to a tube on
the molar and an outer bow where elastic force is placed
(Fig 4-1). This inner and outer bow headgear was developed by Oppenheim and taught at the University of Illinois to his student Kloehn, who helped popularize it in
America. This design is referred to as the Oppenheim
headgear, although historically, similar headgears have
been developed by others. It is commonly called the inner
and outer bow headgear. The second design presented in
this chapter is the J-hook headgear, in which separate
right and left outer bows attach at a hook near the canines
on the arch. Because the mode of force delivery is different for each of these two designs, they are considered
separately in this chapter. First let us consider the inner
and outer bow headgear and its force system in three
dimensions from the lateral, occlusal, and frontal views.
40
Inner and Outer Bow Headgear
from the Lateral View
Figure 4-1 shows a cervical headgear activated by an
elastic strap. The inner bow of the headgear is inserted
into a tube on the molar. For this analysis, we will assume
that there is no play between the inner bow and the molar
tube. In special conditions in which the inner bow is
round, as from the frontal view, we will consider play
because it is relevant. The relationship between the tube
and the tooth is considered rigid, but the bows of the
headgear are allowed to deform and deflect. The applied
force (red arrow) is the force acting at the attachment
hook on the outer bow. One could calculate the force
acting at the molar tube on the molar band, but this is
unnecessary because play normally does not exist in a
second-order direction (tip) between the inner bow and
the tube. The cervical elastic neck strap supplies the force.
Figure 4-2a is the equilibrium diagram of the neck strap.
Blue forces are equal and opposite and sum to zero
(Newton’s First Law). The forces are reversed in Fig 4-2b
(red force) and show the forces on the neck and the outer
bow (Newton’s Third Law). This force and its line of action
will determine the tooth movement. One can measure
the force with a force gauge, and the line of action is in
the direction of the stretched elastic.
Both inner and outer bows can be bent elastically
during force application. At point A in Fig 4-3, the inner
bow is placed into the molar tubes, and the appliance is
passive. But at point B, the strap is placed and the bows
deform elastically to their loaded shape and position. It
Inner and Outer Bow Headgear from the Lateral View
a
b
FIG 4-2 (a) The neck strap is in equilibrium (Newton’s First Law). The forces that stretch the
elastic strap (blue arrows) sum to zero. (b) The force on the outer bow (red arrow) is equal
and opposite to the force at the hook stretching the elastic (blue arrow). This is an example
of Newton’s Third Law.
a
1
2
3
b
1
FIG 4-3 The effect of the headgear is determined by its activation position. (A) The
headgear is placed in the molar tubes, but
the appliance and the cervical strap are not
yet engaged. This is the passive position. (B)
The strap is now placed, and the bows deform
elastically to their loaded shape and position.
This is the activated position from which the
force system effects are determined.
2
FIG 4-4 The force system from an occipital headgear. (a) The force from an occipital headgear is shown as a red arrow (1). It is replaced with
an equivalent force system at the CR by a force (2) and a couple (3) (yellow arrows). The force translates the molar’s CR apically and distally.
The couple rotates the tooth in a counterclockwise direction around its CR. (b) The combined movement of the molar (2). The root moves
apically and distally as the tooth inclination changes.
is from this shape that the line of action of the force is
determined by observing the direction of stretch of the
elastic. The new position of the outer bow hook is the
point of force application from which all calculations
start.
To analyze the effect from any headgear, first the line
of action of the force must be determined. In Fig 4-4a,
the line of action from an occipital headgear is shown.
The force (red arrow in 1) is directed upward and backward, anterior to the center of resistance (CR) (purple
circle in 1). The headgear force is then replaced at the CR
with a force and a couple (yellow arrows in 2 and 3). This
is based on the concept of equivalence (see chapter 3).
The yellow force produces translation along its line of
action at the CR, and the yellow couple will rotate the
tooth around its CR (crown forward and root back). We
assume that the displacements will be in the same direc-
tion as the force and moments at the CR and that all
displacements will be proportional to the forces and
moments acting at the CR. Chapter 9 will discuss this
concept in greater detail. For now, these simple assumptions are good enough for all clinical purposes. In our
current analysis, we also assume that the headgear and
the molar tube are rigidly connected and that there is no
significant play between the molar tube and the inner
bow of the headgear. Tipping from this play, as seen from
the lateral view, is ignored.
The molar in Fig 4-4a has two component movements
based on the force and couple at the CR: translation of
the molar upward and backward (2) and rotation of the
molar (3). The combined movement is shown in Fig 4-4b
(2). The headgear would cause the crown to tip mesially,
the root to move apically and distally, and the CR to move
upward and backward. This method does not tell us how
41
4
Headgear
a
b
FIG 4-5 A typical cervical headgear (design 1). (a) The line of action of the force (red arrow) from the neck strap is downward and backward
and lies inferior to the CR of the molar. The equivalent force system at the CR is a downward and backward force and a large clockwise couple
(yellow arrows). (b) The molar erupts and tips distally. The high moment at the CR produces significant tipping.
much molar rotation occurs in comparison with the
translation at the CR. This question will be discussed
later. For now, we have at least established the general
tooth movement of a given molar due to a headgear force.
Typical Headgear Designs
Because there are an infinite number of possible clinical
situations, let us analyze typical cases from the lateral
aspect.
Design 1: Typical cervical headgear
In Fig 4-5a, the line of action of the force (red arrow) from
the neck strap is downward and backward and lies inferior to the CR of the molar. To predict the tooth movement, we replace the red force with an equivalent force
system at the CR: a force and a couple. The replacement
force at the CR is 300 g (straight yellow arrow). If we sum
the moments around the CR, we find that the replacement
couple has a magnitude of +3,000 gmm (curved yellow
arrow).
300 g (red force) × 10 mm = 3,000 gmm (clockwise)
300 g (yellow force) × 0 mm = 0 gmm
For equivalence, a couple (curved yellow arrow) of
+3,000 gmm must be added at the CR. Note that the
same answer will be obtained even if the moments were
summed around a different point, such as the hook of
the outer bow.
42
The enlargement in Fig 4-5b shows the expected molar
movement. The CR moves downward and backward
(from the force shown by the straight yellow arrow). The
couple (curved yellow arrow) is clockwise and will rotate
the molar around its CR, moving the crown distally.
When we combine the two component movements, it is
apparent that most of the movement is distal tipping of
the molar. (Note that the amount of translation is exaggerated in the figure.) This much tipping occurs because
the line of action of the force is far away from the molar’s
CR. Relative to the occlusal plane, the molar moves
distally and also extrudes.
Design 2: Low cervical headgear
The design of the typical cervical headgear (design 1) has
the disadvantage that an extrusive force is placed on the
molar. But is it possible to use a neck strap and not
produce extrusion? Yes. Figure 4-6a shows a line of action
parallel to the occlusal plane. Because the perpendicular
distance from the applied force (red arrow) is greater than
that in design 1, force magnitude is reduced to 150 g to
deliver the same amount of moment. What is the effect
of the force on the molar? Again, we replace the 150-g
applied force with a force and a couple at the CR. At the
CR, the force is 150 g and the couple is +3,000 gmm. The
molar translates distally along a line parallel to the occlusal plane (Fig 4-6b). The crown tips distally, spinning
around the CR.
Note the differences in the force systems between
design 1 and design 2. The same amount of moment is
produced but with less force in design 2; therefore, less
Typical Headgear Designs
a
b
FIG 4-6 A low cervical headgear (design 2). (a) The force (red arrow) is placed much lower than in design 1 so that the force is parallel to the
occlusal plane. An equivalent force system at the CR produces the same amount of moment (3,000 gmm) with half the amount of force (150
g) as in Fig 4-5 because the length of the moment arm is doubled (20 mm). (b) Because the moment is so large relative to the force, the molar
primarily tips distally around the CR. Note that there is no extrusive force component relative to the occlusal plane.
a
b
FIG 4-7 A cervical headgear for translation (design 3). (a) The line of action of the force (red arrow) passes through the CR of the molar. It is
possible for a cervical headgear to produce distal translation; however, an extrusive direction is required. (b) The molar translates downward
and backward.
translation occurred. In addition, design 2 gives better
vertical control by not causing eruption of the molar.
What are the indications for a low cervical headgear?
A maxillary first molar may be tipped forward because
of early loss of primary molars or extraction of a premolar. If tipping is mainly needed around the CR, the force
system of the low cervical headgear is ideal. A force
distant from the CR will quickly tip the molar posteriorly.
A line of action closer to the CR will produce less tipping
and hence would be slower and less efficient. With this
design, less force is used at the neck strap; otherwise, the
moment would be too large for the tooth movement and
produce discomfort for the patient. If the line of action
of the force is placed too far apically, the root will move
distally (see Fig 4-9). Design 2 would be an ideal design
to correct this problem.
Design 3: Cervical headgear
for translation
It is possible for a cervical headgear to produce translation
of a maxillary molar. Note that the line of action of the
required force must go through the CR (Fig 4-7). The
300-g force from the cervical strap passes through the
CR of the tooth (red arrow), and therefore the tooth translates without any rotation. The direction of the translation
is both downward and distal. The distal force could be
useful for Class II correction, but the extrusion of the
molar could be problematic if the mandible rotates downward and backward. However, in some patients, occlusal
forces may prevent molar extrusion (small mandibular
plane angles); in growing patients, growth can allow for
molar eruption.
43
4
Headgear
a
b
FIG 4-8 An occipital headgear for intrusion and tipping of a molar distally (design 4). (a) An upward and distal force (red arrow) from an
occipital harness is applied distal to the CR. The equivalent force system at the CR is 300 g of force accompanied by a 3,000-gmm clockwise
moment (yellow arrows). (b) The molar’s CR moves upward and backward, and the molar tips backward.
a
b
FIG 4-9 An occipital headgear for moving the molar root distally (design 5). (a) The upward and backward force (red arrow) is placed anterior to the CR. The equivalent force system (yellow arrows) at the CR is in an upward and backward direction accompanied by a 3,000-gmm
counterclockwise moment. (b) The molar root moves backward and upward. The vertical dimension is controlled by the intrusive component
of the force.
44
Design 4: Occipital headgear
for tipping a molar distally
Design 5: Occipital headgear for moving
the molar root distally
Many different lines of action from a cervical headgear
are capable of tipping the molar distally. At best, this type
of headgear produces no occlusal erupting force. If better
vertical control is required or an open bite is present, it
may be necessary to create an intrusive force on the molar.
Figure 4-8a shows an upward and distal 300-g force (red
arrow) from an occipital harness applied distal to the CR
of a maxillary molar. The tooth feels 300 g of force accompanied by a 3,000-gmm clockwise moment at the CR.
The CR of the molar will move distally and apically, and
at the same time, the molar will tip back (Fig 4-8b).
During the correction of a Class II malocclusion, the first
molar may inadvertently tip back; in this situation, distal
root movement to correct the axial inclination is required.
How is this accomplished without erupting or intruding
the molar? The headgear design is similar to design 4,
except now the line of action is placed anterior to the
CR (Fig 4-9). The 300-g force is applied upward and
backward anterior to the CR by the elastic (red arrow).
The equivalent force system (yellow arrows) at the CR is
300 g in an upward and backward direction, accompanied
by a –3,000-gmm moment. Note that the moment is
Typical Headgear Designs
a
b
FIG 4-10 A headgear for molar translation parallel to the occlusal plane (design 6). (a) The force (red arrow) passes through the CR parallel
to the occlusal plane. (b) The tooth translates along the occlusal plane. Note that there is no extrusive or intrusive component.
a
b
c
d
FIG 4-11 (a to d) Many points of force application allow for versatility. The type of tooth movement is determined only by the relationship
between the CR (purple circles) and the force (red arrows).
counterclockwise, which will tip the root back and the
crown forward. Figure 4-9b shows the predicted tooth
movement from the headgear. The CR will move backward and upward. At the same time, the axial inclination
will be corrected by the couple acting at the CR.
Design 6: Headgear for molar
translation along the occlusal plane
Perhaps one of the most useful force systems is a distal
force through the CR parallel to the occlusal plane. Figure
4-10 shows design 6, a headgear that is capable of translating a molar distally without changing its vertical position. A connecting bar between the occipital harness and
the cervical strap bypasses the positioning limitations
imposed by the ear. Another method is the use of a separate occipital harness and cervical strap individually
connected to the outer bow of the headgear. In this case,
the resultant force must go through the CR and lie parallel to the occlusal plane. This type of headgear more
uniformly distributes force to the teeth because they are
not tipping. For that reason, it may be appropriate to use
heavier forces.
A range of headgear force directions and points of force
application are shown in Fig 4-11. Any method of headgear classification and proper headgear use depends on
an accurate definition of the force system.
45
4
Headgear
FIG 4-12 A headgear force applied at the CR (purple circle) of a full
arch can produce translation in many directions.
FIG 4-13 Forces (red arrows) with different directions produce a
clockwise moment of the maxillary arch.
FIG 4-14 Forces (red arrows) with different directions produce a
counterclockwise moment of the maxillary arch.
Headgears Acting on a Full Arch
clockwise moment, rotating the occlusal plane upward
in the anterior region (Fig 4-14). Similar appropriate
headgear designs as discussed for molar movement can
be selected to deliver the correct force system based on
treatment goals for the full arch. As before, the key to
success is the determination of a proper line of action for
the force.
Headgear is commonly used for nonextraction Class
II treatment. Figures 4-15 and 4-16 demonstrate the
extremes of clinical response in two patients. The patient
in Fig 4-15 had a full-cusp Class II malocclusion with
minimum growth during treatment. The headgear pull
to the entire arch was directed through the CR and parallel to the occlusal plane (Fig 4-15a). The occlusion is
shown before treatment (Figs 4-15b to 4-15d), after treatment (Figs 4-15e to 4-15g), and 2 years after retention
(Figs 4-15h to 4-15j). As seen on the superimposition,
maxillary distal translatory movement of the molars
occurred during treatment (Figs 4-15k and 4-15l). Some
uprighting of the maxillary incisors occurred along with
considerable translation of the incisors’ CRs.
Headgear forces can be applied not only to the first molar
but also to the full arch. For our analysis, we will assume
that all teeth are rigidly connected. A round wire into the
anterior teeth will allow tipping of the incisors and would
not accurately reflect the anticipated changes.
If more flexible wires are used, the wire can bend
between brackets, producing secondary tooth-to-tooth
effects (see chapter 14). Nevertheless, considering a full
arch as a unit is very useful in anticipating treatment
effects even if complete rigidity is not present.
The location of the CR of an entire arch can be estimated to lie between the roots of the two premolars
(purple circle in Fig 4-12). Therefore, a headgear force
applied at the CR of the full arch can produce translation
in many directions. A force away from the CR produces
a force and a couple. Many lines of action from these
potential forces can translate the CR and produce rotation
around the CR of the entire arch. Some forces could
produce a clockwise moment, steepening the occlusal
plane (Fig 4-13). Other forces could produce a counter46
Headgears Acting on a Full Arch
FIG 4-15 (a) The combination headgear. The resultant force (yellow
arrow) from the occipital and cervical headgears passes through the
CR of the full arch parallel to the occlusal plane. (b to d) Pretreatment
intraoral casts of a patient with a Class II malocclusion. (e to g) Intraoral casts after headgear treatment. (h to j) Intraoral casts 2 years
postretention. (k) Lateral superimposition of the maxilla, mandible,
and cranial base before (black) and after (red) headgear treatment.
The molar has translated distally in this nongrowing patient. (l) Occlusal superimposition of tooth positions before (black) and after
(red) headgear treatment.
a
b
c
d
e
f
g
h
i
j
k
l
47
4
Headgear
a
b
c
d
e
f
g
h
i
j
k
FIG 4-16 (a to c) Pretreatment intraoral casts of a patient with a Class II malocclusion. (d to f) Intraoral casts after headgear treatment. (g to
i) Intraoral casts 2 years postretention. (j) Cranial superimposition of tooth positions before (black) and after (red) headgear treatment. The
headgear prevented the maxillary molar from coming forward while the mandible grew normally. (k) Maxillary and mandibular superimpositions before (black) and after (red) headgear treatment. The maxillary molar did not translate distally, yet the root of the molar moved distally
from intraoral mechanics.
By contrast, the patient in Fig 4-16 had almost no distal
crown movement of the maxillary first molar. The root
of the molar moved distally by intraoral mechanics. The
maxillary incisors tipped to the palatal during closure of
the space between them.The headgear force was similar
to that used for the patient in Fig 4-15, but here the head48
gear served a different function. It held the maxillary arch
while the mandible grew normally to correct the Class II
malocclusion. This included maintaining the cant of the
occlusal plane and the expected vertical height during
growth.
How to Design a Headgear
a
b
c
d
e
f
FIG 4-17 Designing a headgear. (a) Step 1: Determine the location of the CR (purple circle). (b) Step 2: Establish the treatment goal. We want
the arch to translate upward and backward and to rotate in a counterclockwise direction (white dotted arrows). (c) Step 3: Replace the force
and the couple at the CR (yellow arrows) with a single equivalent force. The red arrow is correct, and the gray arrow is incorrect. (d) Step 4:
Adjust the harness and headgear so that the location of the two hooks (white circles) lies along the line of action of the force (dotted line). (e
and f) The length of the outer bow (e, short; f, long) does not affect the force system because both outer bows have the same line of action
(dotted line). The shape of the outer bow also does not alter the force system.
How to Design a Headgear
Much of the orthodontic literature has described the
fabrication and use of headgear based primarily on the
shape, length, and position of the outer bow: Outer bows
are to be short or long, high or low, and up or down. This
approach can often lead to improper designs and inaccuracies. Outer bows can be cut short when not required.
Moreover, these shape-based descriptions make a simple
design problem complicated. Headgear design should
instead be based on establishing a desired line of action
of the desired force.
The first step clinically is to determine the CR of the
tooth, segment, or full arch under consideration. We will
use a full arch for our example (Fig 4-17a). The center of
resistance is the geometric center of the roots through
which a force produces translation. A more thorough
discussion is covered in chapter 9. The CR is not a single
point, so it is usually represented in diagrams by a large
circle. Although there can be other confounding factors
in three-dimensional space, we can practically establish
a general area (large circle) for the CR. Even if the estimate
of the CR is somewhat incorrect, its position remains
constant on a tooth or group of teeth as treatment
progresses. This allows for estimation of the force direction for CR translation.
The second step is to establish our treatment goal for
both translation of the arch and rotation around the CR
(Fig 4-17b). Note that we want the arch to translate
upward and backward and to rotate in a counterclockwise
direction (ie, incisor moves superiorly).
The third step is to replace the force and the couple
(yellow arrows) at the CR with a single equivalent force
(red arrow) (Figs 4-17c and 4-17d). The force must be
parallel to the force at the CR and must lie apical to the
CR to produce the correct moment for equivalence. The
red arrow is correct, and the gray arrow is incorrect (see
Fig 4-17c). The correct red arrow in Fig 4-17d represents
the line of action and its sense of the headgear force. If a
greater moment is required at the CR, the force is moved
further away according to the laws of equivalence. The
ultimate design and fabrication of the headgear is now
simple. The line of action of the force is established (dotted
line). There are two hooks, one from the headgear harness
and one from the outer bow (white circles). Both will lie
along the line of action. The elastic is also stretched along
the line of action of the force. The outer bow can be longer
or shorter, and there is no difference in the effect of the
headgear on the maxillary arch provided the line of action
of the force is the same (Figs 4-17e and 4-17f ). The design
shown in Fig 4-17f may look ridiculous, but the line of
force is the same as that in Fig 4-17e, so the effect is the
same no matter how the outer bow is shaped.
Some fine-tuning of the inner and outer bows may be
required to compensate for any bending during placement
of the elastic. It should be remembered that the relationship between the elastic and the CR is determined after
the elastic is placed and the bows have elastically
deformed.
In short, the simple secret to headgear design is to
establish a line of action of the force and design the headgear around it. Note that in Fig 4-18 the three outer bows
49
4
Headgear
FIG 4-18 Three outer bows vary in length and position, yet the effect
is the same because the single force produced by each bow (red
arrow) shares the same line of action (dotted line).
a
b
FIG 4-19 (a) Altering the cant of the maxillary occlusal plane with a cervical-pull headgear. To understand the effect on the maxillary arch, the
headgear force (red arrow) is replaced with an equivalent force system at the CR (yellow arrows). The maxillary arch extrudes relative to the
occlusal plane and rotates around the CR in a clockwise direction. (b) The direction of rotation of the maxillary arch. This rotation compensates
for the extrusive component of the cervical headgear. The overall effect is to steepen the maxillary occlusal plane and reduce the open bite.
FIG 4-20 Altering the cant of the occlusal plane with an occipitalpull headgear. The headgear force (red arrow) is replaced with the
equivalent force system at the CR (yellow arrows). The maxillary arch
CR intrudes relative to the occlusal plane and rotates around the CR
in a clockwise direction. This type of headgear reduces the vertical
dimension and rotates the mandible forward more effectively than
the cervical headgear shown in Fig 4-19.
50
Clinical Monitoring and Corrective Action
vary in length and position, and yet the effect will be the
same because the line of action is constant. So here we
see that a force-driven rather than a shape-driven
approach is more effective and that shape must be subservient to force. Headgear descriptions also must be more
specific. For example, “high-pull occipital” is vague and
describes many different lines of force and effects. A
better description might be “45 degrees to the occlusal
plane (upward and backward) and anterior to the CR.”
This description is based on the force system.
Altering the Cant of the Occlusal
Plane with the Headgear
A Class II open bite patient with nonparallel occlusal
planes is shown in Fig 4-19. The goal is to move the maxillary arch distally to correct the Class II molar relationship
and to steepen the maxillary occlusal plane (clockwise
rotation of the arch) so that it parallels the cant of the
mandibular occlusal plane. Ideally, the vertical facial
height should be maintained or reduced. The maxillary
and mandibular occlusal planes must be parallel in order
to achieve normal vertical overlap (also referred to as
overbite).
Could a cervical headgear be used (Fig 4-19a)? The
equivalent force system at the CR produces two effects:
The maxillary arch extrudes relative to the occlusal plane
and rotates around the CR in a correct clockwise direction
(yellow arrows). The extrusion is undesirable because it
might increase the vertical dimension. This may not be
the case if sufficient rotation around the CR occurs
because of the equivalent moment at the CR rotating the
maxillary arch downward in the front and upward in the
back. Note that in Fig 4-19b, rotation of the maxillary
arch at the CR compensates for any downward and backward translation of the arch. Therefore, the cervical headgear line of force may be a reasonable choice. We could
not have necessarily assumed undesirable extrusion just
because the pull was downward. The relationship of the
maxillary incisor to the lip will also improve as the maxillary occlusal plane steepens, because the original incisor
position is superior to stomion.
If a reduction in vertical dimension and forward rotation of the mandible are the goals, the line of force is
changed to upward and backward, posterior to the CR
(Fig 4-20). The further posterior the force, the greater the
moment relative to the force that angles the maxillary
arch cant to fit the mandibular arch. Sometimes it is
advantageous to add extra wire to the outer bow to
achieve this added rotation effect. On the other hand,
moving the force closer to the CR (more anteriorly)
decreases the moment and reduces the occlusal plane
cant.
Clinical Monitoring and Corrective
Action
The headgear offers many possibilities if we first define
the proper line of action for our headgear force system.
Unfortunately, sometimes the clinical response can vary,
requiring corrective procedures. Perhaps difficulty in
identifying a CR or a deformed bow could be the cause.
Understanding the same principles that guided us during
design can also help during troubleshooting.
The goal for the patient shown in Fig 4-21 was to translate the first molar distally, primarily parallel to the occlusal plane. The direction of the line of force (red arrow)
was correct but lay too far apically (Fig 4-21a). Because
of the equivalent force system (yellow arrows) at the CR,
the root moved distally and the crown came forward
slightly (Figs 4-21b and 4-21c). To correct the problem,
the force was moved occlusal to the crown (Fig 4-21d),
and the final result was satisfactory (Fig 4-21e). The inclination correction primarily required rotating the molar
around its CR. One could argue that the pull in Fig 4-21f
might be more efficient with reduced force magnitude
because the outer bow is so far from the CR that the force
system approaches a couple. This design is ideal when no
further translation is needed during correction.
51
4
a
e
Headgear
b
f
c
d
FIG 4-21 (a) A headgear was applied to an end-to-end Class II
malocclusion. Note that the line of action of the force is too far
apical to the CR of the molar. (b) Pretreatment cephalometric
radiograph. (c) Cephalometric radiograph after initial headgear
treatment. Note that the root moved distally and the crown came
forward slightly. (d) The principle of equivalence is used to correct
the undesired tooth movement. The force was moved occlusal
to the CR. The headgear force (red arrow) is replaced with an
equivalent force system at the CR (yellow arrows). This force
system will move the root mesially and the crown distally. (e)
Cephalometric radiograph after headgear treatment. The molar
axial inclination has been corrected. (f) A low cervical headgear
may be more efficient if no further translation of the CR is needed and correction of the axial inclination is mainly by rotation
around the CR. The headgear force (red arrow) is replaced with
an equivalent force system at the CR (yellow arrows), with the
same amount of moment but less force.
Inner and Outer Bow Headgear
from the Occlusal View
FIG 4-22 Occlusal view of a symmetric headgear. The headgear
forces (red arrows) are replaced with a resultant (yellow arrow) at
the CR. Assuming that the inner bow of the headgear connects both
molars rigidly, the two molars are considered a single rigid body,
and the CR of the two molars lies at the midpoint (purple circle).
No rotation moment exists around the CR, and the molars equally
translate to the distal.
52
The occlusal view is important if a different molar distalization is needed on either side. It is possible to deliver a
greater force on one side. The reasoning and the solution
for how this is done require the application of equilibrium
laws. Equilibrium diagrams are described in detail later
in this book (see chapter 8); therefore, we will adopt a
simpler approach to understand asymmetric headgear
applications.
A symmetric headgear is shown in Fig 4-22. Let us
assume that the inner bow of the headgear is joined
(glued) into each buccal tube. This bow connects both
molars rigidly. The two molars are considered a single
rigid body, and the CR of the two molars lies at the
midpoint between them (purple circle). The red arrows
on the hook of the outer bow are the forces from the neck
strap or elastics from the harness. Their resultant is the
Inner and Outer Bow Headgear from the Occlusal View
Left
Right
a
Left
Right
b
FIG 4-23 An asymmetric headgear. (a) The forces from the headgear (red arrows) are replaced with a resultant (yellow arrow), which is
off-center. (b) The off-center force (red arrow) is replaced with an equivalent force system at the CR (yellow arrows). Note that it is comprised
of an oblique force and a counterclockwise couple. The left molar will move more distally than the right molar.
FIG 4-24 Clinical application of an asymmetric headgear. (a) The maxillary right
molar needs more distal movement than
the left molar, and both molars need to
move to the left side for axial inclination
decompensation. The resultant from the
asymmetric headgear forces is replaced
with an equivalent force system (yellow
arrows) at the CR, showing a desirable moment and a lateral force to the left. (b) The
lateral force for decompensation makes
the unilateral reverse articulation worse
before the surgery.
a
yellow arrow. As the resultant force passes through the
CR in the occlusal view, it will translate the CR. Because
there is no moment produced around the CR, no rotation
occurs.
Symmetric headgear will always have equal forces on
each side. Even if we apply different force magnitudes to
each side of the headgear, the neck strap will instantly slide
to find a new state of equilibrium with equal and opposite
forces. Therefore, maintaining different magnitudes of
force on each side of the outer bow in a symmetric headgear is impossible unless the neck strap is glued to the
neck.
Figure 4-23 shows an asymmetric headgear. The length
of the outer bow is asymmetric with the left side, which
is longer than the right side. The neck strap is still in
equilibrium but now lies off-center to the patient’s right.
This is the position that the neck strap will move to as
the patient functionally turns his or her head. The force
magnitudes applied to the outer bow end hooks are equal
(red arrows). The oblique resultant of the individual red
b
arrows (yellow arrow) is off-center toward the left molar
(closer to the longer outer bow side in Fig 4-23a). If we
replace this single force with an equivalent force system
at the CR of the molars (Fig 4-23b), it consists of an
oblique distal force with a counterclockwise couple (yellow
arrows). Therefore, the left molar will move more in a
distal direction than the right molar because of this counterclockwise rotation. The single force at the CR (yellow
arrow) has not only a posterior component of force but
also a lateral component that moves both molars to the
right side. This lateral component of the force is usually
considered an undesirable side effect.
Figure 4-24 shows a clinical application of asymmetric
headgear in which the right molar needs more distal
movement and both molars need to move to the left side
for buccolingual axial inclination decompensation. The
force system is the same as that in Fig 4-23 except that it
is a submental vertical view. The molars feel asymmetric
distal forces and, at the same time, a lateral (horizontal)
component of force. For this clinical situation, the lateral
53
4
Headgear
a
54
b
FIG 4-25 Vertically placed headgear tubes on the molars allow
freedom of rotation from a bilateral distal force (red arrows). The
equivalent force system at each CR (yellow arrows) shows that the
molars will move distally and rotate with their mesial sides outward.
FIG 4-26 (a and b) A patient with severe mesial-in rotation on the
maxillary right first molar only. A headgear tube was welded on the
right side, oriented vertically. Because the vertical tube allowed y-axis
rotation, the maxillary right first molar not only moved distally but also
rotated mesial out, distal in.
force is desirable to decompensate the buccoversion of
the maxillary right molar area for orthognathic surgery
and is not considered a side effect. Clinical cases should
be carefully selected with a complete understanding of
the possible limitations of asymmetric headgear force
systems. Narrowing of the inner bow width can occur
with the asymmetric design. Along with lateral force, this
can correct unilateral reverse articulation in properly
selected patients.
In Fig 4-25, vertical tubes (gray circles) are placed to
increase the freedom to rotate around the y-axis. Play
between the molar tube and the inner bow allows for
symmetric rotation of molars. A round inner bow bent
at 90 degrees is inserted into the vertical tube so that the
molars are free to rotate in the occlusal view. This might
be indicated if the first molars are badly rotated. The
forces from the headgear are the red distal arrows. As
each molar is free to rotate individually, the two molars
cannot be considered a single rigid body anymore. How
will the molars move? Let us replace the red distal forces
with an equivalent force system at the CR of each molar
(yellow arrows). The molar now feels both a distal force
at the CR and a mesial-out, distal-in rotation couple.
Clinically, when the molar has tipped forward for any
reason, such as following the early loss of primary molars
or extraction of the permanent premolar, the mesial
usually rotates inward and the distal outward. In this
situation, vertically placed headgear tubes on rotated
molars could help not only for the distal movement but
also to correct the rotation. Vertical tubes allow for easier
placement of the headgear. Figure 4-26 shows a patient
who presented with severe mesial-in rotation of the
maxillary right first molar only. Unparalleled horizontal
tubes on each molar make it difficult to insert the rigid
inner bow, so a headgear tube was welded on the right
side, oriented vertically. Because the vertical tube allowed
y-axis rotation, the maxillary right first molar not only
moved distally but also rotated mesially outward. Here,
asymmetric tooth movement was accomplished with a
symmetric outer bow shape.
Inner and Outer Bow Headgear
from the Frontal View
By varying the direction of the headgear force superiorly,
there is not only a horizontal force component but also
vertical components of force. Figure 4-27a shows the
frontal projection of such a force system from design 5
(see Fig 4-9). In the analysis of the force system from the
frontal view, the teeth and the headgear cannot be considered as a rigid body because the inner bow and molar
tube are round, so individual rotation of each tooth
around the tube is possible. The vertical component of
the headgear force (red arrow) is replaced with an equivalent force system at the CR of each molar (yellow arrows).
In Fig 4-27b, equivalence tells us that the molars will
intrude, the crowns will tip buccally, and the roots will
move lingually. The roots will tip more to the palatal side
if the inner bow is rigid enough to prevent buccal tipping
J-Hook Headgear
a
c
b
FIG 4-27 (a) Frontal view of an occipital headgear force system. The round wire of the inner bow allows freedom of rotation around the x-axis
of the molars. Any intrusive component of the headgear (red arrows) could cause buccal tipping. Note the equivalent force system (yellow
arrows) at each CR. (b) The molars intrude, the crowns tip buccally, and the roots move lingually. (c) If a passive transpalatal arch is placed,
individual molar tipping is not allowed, and intrusive forces (red arrows) can be replaced with a resultant (yellow arrow) at the midpoint between the molars (purple circle).
FIG 4-28 J-hook headgear. The eyelet of the
bow is inserted into a hook on the archwire
and can deliver only a single force at a fixed
point (the hook). No moment is produced.
FIG 4-29 Possible force systems from the
J-hook headgear. The red arrows show the
possible directions of force at the fixed point
of force application (black circle).
of the crowns of the molars. If a rigid passive transpalatal arch is engaged as shown in Fig 4-27c, individual molar
tipping is not allowed, and the two molars can be considered a single rigid body in which the CR lies at the
midpoint (purple circle). The resultant (yellow arrow) of
the two headgear forces (red arrows) passes through the
CR; therefore, the molar, will not tip but will translate in
an intrusive direction.
J-Hook Headgear
Figure 4-28 shows another type of headgear called a
J-hook headgear. It is shaped like a J and has an eyelet at
the anterior ends of separate right and left bows, which
FIG 4-30 The gray arrows show the forces
that are not possible because the lines of action do not pass through the fixed point of
application (green hook).
are inserted into right and left hooks (green) on the archwire. There is no inner bow connecting the left- and rightside outer bows of the headgear. Unlike the inner and
outer bow headgear, the force application point is limited
to the archwire hook. Only a single force can be applied
there because the connection at the anterior eyelet and
hook allows freedom of rotation, eliminating any
moments. Figure 4-29 shows the possible force systems
from the J-hook headgear. Red arrows show the possible
directions of forces with a fixed force application point—
the green hook (black circle). Only magnitude and direction can be varied in this type of headgear, so the types
of tooth movement are limited. For example, forces from
the gray arrows shown in Fig 4-30 are not possible because
their lines of action do not pass through the hook.
55
4
a
Headgear
b
e
10 y 2 mo
f
h
Protraction Headgear
Figure 4-31 shows a protraction (or reverse) headgear on
a full maxillary arch. It was designed to apply an extraoral
anterior force on a nonextraction patient. The elastics
are stretched from the face mask to the hooks on the
labiolingual appliance (Fig 4-31c). Figures 4-31b and
4-31d show the pretreatment and posttreatment intraoral
photos, respectively. This protraction headgear mechanism is similar to that of a J-hook headgear in that only
56
c
10 y 3 mo
10 y 2 mo
g
d
12 y 10 mo
12 y 10 mo
FIG 4-31 Protraction headgear acting on a full arch. (a) The elastic
is engaged between the hook (black circle in c) of the labiolingual
appliance and the hook of the face mask. (b to d) Intraoral view before
(b), during (c), and after (d) headgear treatment. (e) The force system
from the protraction headgear. A single force (red arrow) acts on the
hook of the labiolingual appliance, and its line of action (dotted line)
passes through the CR of the maxillary arch (lower purple circle). For
analysis of orthopedic effects, the equivalent force system on the CR
of the maxilla (upper purple circle) is shown with yellow arrows. (f)
Cephalometric radiograph prior to protraction headgear treatment.
(g) Cephalometric radiograph after protraction headgear treatment.
(h) Superimposition before (black) and after (red) protraction headgear treatment. A significant amount of forward translation of the
maxillary arch with a little possible forward orthopedic effect on the
maxilla is shown.
a single force is applied on each side and the force application point is fixed at the hook (black circle in Fig 4-31c).
Figure 4-31e shows the force system from the protraction headgear. Assuming that the labiolingual appliance
rigidly connects all of the teeth in the maxillary arch, the
CR of the maxillary arch lies between the roots of the
premolars (lower purple circle). The force from the
protraction headgear (red arrow) was designed so that
the line of action passes through the CR of the maxillary
dentition and the direction of force is forward and down-
Protraction Headgear
ward (dotted line). Therefore, the maxillary dentition will
translate forward and downward along the line of action.
In the analysis of the orthopedic effect, the force from
the protraction headgear was replaced with an equivalent
force system at the CR of the maxilla (upper purple circle).
The yellow arrows show the equivalent force system
acting on it. The maxilla will not only translate forward
and downward but will also rotate in a counterclockwise
direction. Figures 4-31f to 4-31h show the before and
after cephalometric radiographs and superimposition
tracings. The superimposition tracings show that the
Class III molar relationship and anterior reverse articulation were corrected mainly by forward translation of
the maxillary arch, accompanied by a possible slight
orthopedic forward movement of the maxilla.
There is debate about how much of an orthopedic effect
is achieved with protraction headgear. Typically, the
mandible can rotate downward and backward. Maxillary
anterior translation may be limited. Also, the CR of the
maxilla is not exactly known. Most of the correction with
a protraction headgear appears to be dental rather than
skeletal; however, more research is needed.
A protraction headgear on the first molar only was
used in a Class III extraction patient to help translate the
maxillary first molar forward rather than to translate the
entire arch (Fig 4-32). An extension bar was rigidly
connected to the molar tube (Figs 4-32a and 4-32c), and
the force (red arrow) was applied at the hook (black circle)
so that the line of action of the force passed through the
CR of the first molar (dotted line). A passive transpalatal
arch rigidly connected both molars to prevent individual
molar rotation (Fig 4-32e). Figures 4-32g and 4-32h show
the cephalometric radiographs before and after protraction headgear treatment; the superimposition tracings
are shown in Fig 4-32i. The Class III molar relationship
and anterior reverse articulation were corrected by
protraction of the maxillary molar and retraction of the
mandibular anterior teeth. Note that there were no significant orthopedic effects.
A common error is to position the line of action of the
protraction force at the level of the brackets, somewhat
parallel to the occlusal plane, shown in Fig 4-33. Because
the force is not acting through the CR, a moment is
produced that will reduce the occlusal plane angle, resulting in an open bite.
57
4
Headgear
a
14 y 11 mo
b
e
g
i
58
14 y 11 mo
c
15 y 2 mo
d
19 y 3 mo
f
14 y 11 mo
h
19 y 3 mo
FIG 4-32 Protraction headgear acting on a
molar. (a) An extension bar is placed at the
buccal tube, and the force (red arrow) acts on
the hook (black circle) of the extension bar.
The line of action (dotted line) passes through
the CR of the molar. (b) Intraoral view before
treatment. (c) The extension bar in place with
a hook for an elastic (black circle). (d) Intraoral
view after headgear treatment. (e) A passive
transpalatal arch is placed so that individual
rotation of the molars is prevented. (f) The
force system from the protraction headgear.
The line of action (dotted line) of the protraction headgear force (red arrow) passes
through the CR (purple circle) of the molar. (g)
Cephalometric radiograph before protraction
headgear treatment. (h) Cephalometric radiograph after protraction headgear treatment.
(i) Superimposition before (black) and after
(red) protraction headgear treatment. The
Class III molar relationship and the anterior
reverse articulation were corrected by protraction of the maxillary molar and retraction
of the mandibular incisors. Note in the cranial
base superimposition that the amount of orthopedic effect is insignificant.
Recommended Reading
FIG 4-33 An improper line of action of elastic force (red arrow) would produce
an anterior open bite because of undesirable rotation caused by a couple (replaced force system at the CR, yellow arrows). Ideally, force should be directed
through the CR of the maxillary arch to prevent unwanted rotation of the arch.
Recommended Reading
Jacobson A. A key to the understanding of extraoral forces. Am J
Orthod 1979;75:361–386.
Badell MC. An evaluation of extraoral combined high-pull traction and
cervical traction to the maxilla. Am J Orthod 1976;69:431–466.
Kloehn SJ. An appraisal of the results of treatment of Class II malocclusions with extraoral forces. In: Kraus BS, Reidel RA (eds). Vistas
in Orthodontics. Philadelphia: Lea & Febiger, 1962:227–258.
Baldini G. Unilateral headgear: Lateral forces as unavoidable side
effects. Am J Orthod 1980;77:333–340.
Baldini G, Haack DC, Weinstein S. Bilateral buccolingual forces
produced by extraoral traction. Angle Orthod 1981;51:301–318.
Barton JJ. High-pull headgear versus cervical traction: A cephalometric comparison. Am J Orthod 1972;62:517–539.
Drenker EW. Unilateral cervical traction with a Kloehn extraoral mechanism. Angle Orthod 1959;29:201–205.
Gould E. Mechanical principles in extraoral anchorage. Am J Orthod
1957;43:319–333.
Güray E, Orhan M. “En masse” retraction of maxillary anterior teeth
with anterior headgear. Am J Orthod Dentofacial Orthop 1997;112:473–
479.
Haack DC, Weinstein S. The mechanics of centric and eccentric cervical traction. Am J Orthod 1958;44:345–357.
Hershey HG, Houghton CW, Burstone CJ. Unilateral face-bows: A
theoretical and laboratory analysis. Am J Orthod 1981;79:229–249.
Hubbard GW, Nanda RS, Currier GF. A cephalometric evaluation of
nonextraction cervical headgear treatment in Class II malocclusion.
Angle Orthod 1994;64:359–370.
Kuhn RJ. Control of anterior vertical dimension and proper selection
of extraoral anchorage. Angle Orthod 1968;38:340–349.
Melsen B, Enemark H. Effect of cervical anchorage studied by the
implant method. Eur J Orthod 2007;29:i102–i106.
Nikolai RJ. Bioengineering: Analysis of Orthodontic Mechanics. Philadelphia: Lea & Febiger, 1985:322–371.
Perez CA, de Alba JA, Caputo AA, Chaconas SJ. Canine retraction
with J hook headgear. Am J Orthod 1980;78:538–547.
Tanne K, Hiraga J, Kakiuchi K, Yamagata Y, Sakuda M. Biomechanical
effect of anteriorly directed extraoral forces on the craniofacial
complex: A study using the FEM. Am J Orthod Dentofacial Orthop
1989;95:200–207.
Tanne K, Matsubara S, Sakuda M. Stress distributions in the maxillary
complex from orthopedic headgear forces. Angle Orthod 1993;63:
111–118.
van Steenbergen E, Burstone CJ, Prahl-Andersen B, Aartman HA. The
role of a high pull headgear in counteracting side effects from intrusion
of the maxillary anterior segment. Angle Orthod 2004;74:480–486.
Worms FW, Isaacson RJ, Speidel TM. A concept and classification of
centers of rotation and extraoral force systems. Angle Orthod
1973;43:384–401.
59
Problems
1. Compare forces FA, FB, and FC. Do length and position
of the outer bow make any difference?
2. The outer bow is the same as in problem 1, but the
direction of pull is different. Compare forces FA, FB, and
FC. Do length and position of the outer bow make any
difference?
3–8. Replace the 500-g headgear force with an equivalent force system at the CR. Describe how the molar will move.
60
3
4
5
6
7
8
Problems
9. Replace the combination headgear forces with a single
equivalent force. θ1 and θ2 are equal, and θ1 = 20
degrees. How will the molar move? Show the line of
action of the resultant force. What is the magnitude of
the resultant?
10. Two identical headgears are used on a Class II patient. What is the effect of moving the tube 4 mm gingivally?
Compare FA and FB.
a
b
11. A J-hook headgear is used. Draw the force vector at
the hook to translate the maxillary dentition. Is it possible to translate the maxillary dentition parallel to the
occlusal plane?
12–13. Replace the headgear force at the CR. Describe the movement of the maxillary arch.
12
13
61
5
The Creative Use of
Maxillomandibular
Elastics
“Truths are easy to understand once they are discovered;
the point is to discover them.”
— Galileo Galilei
Maxillomandibular elastics (or intermaxillary elastics) are commonly
used because of their simplicity; however, a lack of understanding of
their force system can lead to many serious problems. Elastics are
usually classified by the direction of the force (eg, Class II or Class
III elastics). Sometimes force magnitude is considered, but point of
force application is left out. Therefore, many different types of Class
II elastics can be applied. There are short or long elastics. Short elastics can be placed anteriorly or posteriorly to produce asynchronous
occlusal plane cant effects. Proper use of maxillomandibular elastics
requires consideration of the attachment point of the elastic (line
of action of the force) in respect to the center of resistance of each
arch. In this way, vertical dimension, occlusal plane cants, and vertical
overlap can be controlled and corrected if necessary. Often too many
elastics are used when a single resultant elastic at the correct location
would work better. However, sometimes more than a single elastic is
needed when the attachment point is not directly accessible. Vertical
elastics to cover up intra-arch mechanical side effects are rarely the
best solution. All maxillomandibular elastics and their actions should
be analyzed in three dimensions.
63
5
E
The Creative Use of Maxillomandibular Elastics
lastomeric rings, more commonly called elastics, are
routine in orthodontic treatment and are some of
the simplest appliances used. Unfortunately, however,
the mode of operation and the proper selection of maxillomandibular elastics (also referred to as intermaxillary
elastics) are not well known or practiced clinically.
Different types of elastics are used either within the
same arch (intra-arch elastics) or between arches (maxillomandibular elastics). One orthodontic sequence is to
accomplish intra-arch alignment early and then later
coordinate the arches that do not fit either because of
side effects during intra-arch alignment or because of
remaining original discrepancies. Commonly, maxillomandibular elastics are needed to correct problems introduced during early stages of treatment. Maxillomandibular elastics can also be used directly to enhance
anchorage or correct intra-arch or maxillomandibular
problems.
Because there are an infinite number of possible maxillomandibular applications (directions and points of force
application), for simplicity in this chapter, our discussion
will be limited to different types of bilateral Class II elastics, vertical elastics, and transverse or crisscross elastics,
followed by asymmetric elastics such as unilateral Class
II elastics and combined Class II/Class III elastics. This
approach should provide a sufficient explanation that can
be extrapolated to all maxillomandibular applications so
as to minimize undesirable effects.
Because of the infinite number of clinical situations that
surround treatment planning with maxillomandibular
elastics, some assumptions are made. Let us assume that
a rigid arch is placed in all of the brackets, allowing no
rotation or play in all planes. The theme of this chapter
is to show the effects of single forces to the maxillary and
mandibular arches (as rigid units without play), so play
is only mentioned briefly. The centers of resistance (CRs)
are best estimates, recognizing variation and threedimensional difficulties. These are our boundary conditions for studying the effects of maxillomandibular elastics; other conditions, such as the use of round wires
allowing rotation of the incisors or wire elastic deformation beyond our boundary, may still be partially valid.
Treating both arches as rigid bodies simplifies our analysis so that we may arrive at core concepts without losing
ourselves in the details. Although the effect of elastics on
full rigid arches is the main consideration, the effects on
segments and nonrigid wires are also discussed briefly.
Maxillomandibular elastics represent a simple appliance
that is not precise. As the patient moves the jaw and alters
the vertical dimension, the forces change. Variation in
elastic size and degradation caused by the fluids in the
64
mouth add to this variability. For standardization, we will
maintain the vertical dimension near the mandibular rest
position, because typically elastics are not worn during
chewing of food.
To develop the concepts in this chapter, Class II elastics
are discussed initially, with the understanding that similar principles can be used for other force directions. Not
all Class II elastics are the same, and different types of
Class II elastics may produce radically different effects.
Part of the problem is the traditional classification that is
based on force direction: Class II, Class III, vertical, crisscross, etc. Direction is insufficient to describe a force
system. Lacking in particular is the point of force application. Therefore, a Class II elastic is evaluated from three
separate views: the lateral view parallel to the midsagittal
plane, the frontal view, and the occlusal view. The coordinate system used is the occlusal plane.
What Does a Class II or Class III
Elastic Do?
If a Class II elastic is placed between the maxillary canine
and the mandibular second molar (Fig 5-1), the point of
force application is at the canine and molar hooks, and
the force acts along its line of action (green elastic). To
predict arch movement, we replace the applied force (red
arrow) with an equivalent force system (yellow arrows)
at the CR (purple circle) of each arch. Figures 5-1a and
5-1b are identical except that the force system is depicted
in the maxillary arch in Fig 5-1a and in the mandibular
arch in Fig 5-1b. Note that the perpendicular distances
from the CR of the maxillary and mandibular arches to
the elastic (D1, D2 ) are about the same, and hence the
equivalent moments (curved yellow arrows) are the same.
So how will the arches move? From the equivalent force
system at the CR, the maxillary arch moves downward
and backward and the mandibular arch moves upward
and forward. Simultaneously, the large moments cause
rotation of both arches around their CRs (Fig 5-1c). This
analysis is similar to how the headgear was studied in
chapter 4, except now two arches are being considered.
In short, to understand what a maxillomandibular elastic will do, the elastic force is replaced with an equivalent
force system at the CR of each arch by a force and a
couple. Two movements are noted: (1) the direction and
magnitude of the translation by the force and (2) the
direction and magnitude of rotation around the CR.
Figure 5-1 is an example of a long Class II elastic, where
the distances from the elastic to the maxillary and
mandibular CRs are the same, thereby causing the
What Does a Class II or Class III Elastic Do?
a
b
c
d
FIG 5-1 Long Class II elastic. (a and b) A single force from the elastic (red arrow) is replaced with an equivalent force system (yellow arrows) at
the CRs of the maxillary and mandibular arches. (c) Both arches rotate synchronously (dotted curved arrows) in the same clockwise direction
because of the same magnitude and direction of the moments (D1 = D2 ). (d) Lateral superimposition of cephalometric radiographs before (black)
and after (red) long-term use of Class II maxillomandibular elastics in an extraction case using round wire. Both the maxillary and mandibular
arches rotated in a clockwise direction, with extrusion of the maxillary anterior teeth and mandibular posterior teeth.
moments at the CRs to be equal and rotation around
each CR to be the same. In other words, the occlusal
plane of both the maxillary and mandibular arches will
rotate clockwise (steepen) by about the same amount.
Figure 5-2 shows an adult Class III patient treated nonsurgically with maxillary and mandibular premolar extraction
and long-term use of Class III elastics. The maxillary
posterior segment moved anteriorly, and the mandibular
anterior segment moved posteriorly, as shown in lateral
cephalometric radiographs before and after (see Figs 5-2c
and 5-2d). The superimposition shows that the occlusal
plane has been rotated counterclockwise as well (see Fig
5-2e).
If the occlusal planes rotate by the same amount, the
rotation is called synchronous. Ignoring other factors, the
vertical overlap (also known as overbite) will be maintained; otherwise, factors such as tooth extrusion and
rotation of the mandible could influence the amount of
vertical overlap.
65
5
The Creative Use of Maxillomandibular Elastics
a
b
c
e
66
d
FIG 5-2 A Class III adult patient treated with maxillary and mandibular premolar extraction and Class III elastics.
(a and b) Lateral views of models before and after treatment. (c and d) Horizontal tooth movement before and
after treatment. (e) Superimposition
showing that the occlusal plane has
been rotated counterclockwise.
Synchronous Class II Elastics
FIG 5-3 Short Class II elastic. A single force from the elastic (red
arrows) is replaced with an equivalent force system (yellow arrows)
at the CR of each arch. It is also synchronous because the CR is an
equal distance from the force in each arch (D1 = D2). The moment is
lower and the vertical component of force is greater than with the
long Class II elastic.
Synchronous Class II Elastics
The green elastic shown in Fig 5-3 is also synchronous,
because the CR is an equal distance from the force in
both arches. The point of attachment is between the distal
of the maxillary first premolar bracket and the mesial of
the mandibular second premolar bracket. Because the
distances (D1, D2 ) are equal, it can be described as a
synchronous short Class II elastic. Note that the length
of the red arrow is the same as that in Fig 5-1 because
the force magnitude is the same. How does the predicted
movement of a short elastic compare to that of the long
elastic? The direction is different, with the short elastic
having a greater vertical component of force in comparison with the horizontal force. Perhaps of more significance are the equivalent moments at the CRs (yellow
curved arrows), which are much smaller than those with
the long elastic.
Treatment goals can vary in respect to the level and
cant of the treated occlusal plane. Do we want to extrude
teeth and increase the vertical dimension by rotating the
mandible downward and backward? Do we want to maintain the original cant or increase its angle? The choice of
a long or short synchronous elastic is important in
achieving the patient-specific treatment objective. The
short elastic will not steepen the occlusal plane very
much. But what about the increased vertical force? We
cannot assume that this will always be deleterious.
Patients can grow vertically, and the eruption of teeth
can be masked with the increased vertical dimension.
Some patients with flatter mandibular planes are highly
resistant to an increase in vertical dimension from the
extrusive component of the elastic. Nongrowing adult
patents usually show little increase in vertical dimension
following treatment. Thus, the effect of a Class II elastic
is dependent not only on the elastic force but also on the
occlusal forces and function.
Traditionally, the orthodontist has evaluated the Class
II elastic primarily by its direction. It was assumed that
the more horizontal the force, the better. In most applications, direction may be modified some but not significantly; of greater importance are the equivalent moments
operating at the CRs of the arches. These moments are
determined by the points of force application of the elastic and the lines of action of the forces.
In some Class II patients, the goal is to minimize the
increase in vertical dimension. To determine whether a
short or long elastic is preferable in this situation, the
effect of large moments on the posterior end of the arches
should be considered. The long elastic (see Fig 5-1b) that
produces a large moment around the CR will extrude the
mandibular second molar (see Fig 5-1c). Long-term use
of Class II elastics and round wire led to clockwise rotation of both arches and extrusion of the maxillary anterior
and mandibular posterior teeth in the patient shown in
Fig 5-1d, producing excessive exposure of the maxillary
anterior teeth with relaxed lips. For every millimeter of
molar eruption, the mandible could rotate, significantly
reducing the vertical overlap in the incisor region by 2
to 3 mm. Because it is the moment that rotates the arch,
it could be argued that the short elastic (with the smaller
moments) is more conservative in preserving vertical
dimension even though the vertical component of force
is greater (see Fig 5-3). However, the horizontal component of force, which contributes to the displacement of
the CR, is much less. More clinical research is needed to
study the effect of varying the elastic force position and
magnitude on the control of vertical dimension. The
effects of short-term use of Class II elastics are very different from changes after considerable growth, which may
mask the initial tooth movement. This chapter only
describes the immediate force systems and their effects
on the teeth.
67
5
The Creative Use of Maxillomandibular Elastics
a
b
FIG 5-4 Short Class II elastic placed anteriorly. (a and b) A single
force from the elastic (red arrow) is replaced with an equivalent force
system (yellow arrows) at the CR of the maxillary arch. (c) The moment
in respect to the CR will be different for each arch; therefore, only
the maxillary arch rotates.
c
Asynchronous Class II Elastics
Let us now move the short elastic anteriorly so that its
line of action passes through the CR of the mandibular
arch (Figs 5-4a and 5-4b). The translatory effects at the
CR are the same as in Fig 5-3; however, the moment in
respect to the CR will be different for each arch. A large
moment (yellow curved arrow) at the CR of the maxillary
arch steepens the maxillary occlusal plane (rotates it
clockwise). Because the line of action of the force on the
mandibular arch goes through the CR of that arch, no
moment and hence no rotation are produced in the
mandibular arch. The arch rotations are asynchronous,
with only a maxillary arch rotation leading to a lack of
occlusal plane parallelism and an increase in vertical
overlap (Fig 5-4c).
If we move the same short elastic posteriorly so that
the force (red arrow) now goes through the CR of the
maxillary arch, the opposite effect will be observed
68
(Fig 5-5a and 5-5b). Replacing the force system at each
CR, it is observed that the maxillary arch will not rotate
and the mandibular arch occlusal plane will steepen
(rotate clockwise), leading to a reduction in vertical overlap (Fig 5-5c).
Finally, let us move the same short elastic further
forward so that the line of action is between the distal of
the maxillary central incisors and the distal of the mandibular lateral incisors (Fig 5-6). This is called an anterior
vertical elastic, but the distal translatory effect on the CR
from the distal (horizontal) component of force is the
same as the previous short elastics shown in Figs 5-3 to
5-5; only the rotation tendency will be different. The
maxillary arch will steepen, and the mandibular arch will
flatten, leading to an increase in the deep bite. Note that
the moment at the CR is greater on the maxillary arch
than on the mandibular arch because D1 is greater than
D2.
Asynchronous Class II Elastics
b
a
b
FIG 5-5 Short Class II elastic placed posteriorly. (a and b) A single
force from the elastic (red arrow) is replaced with an equivalent force
system (yellow arrows) at the CR of the mandibular arch. (c) The
moment in respect to the CR will be different for each arch; therefore,
only the mandibular arch rotates.
c
FIG 5-6 Anterior vertical elastic. The maxillary and mandibular
arches will rotate in opposite directions, leading to an increase in
vertical overlap. Most of the rotation will occur in the maxillary arch
(D1 > D2 ).
The various Class II elastics discussed so far are representative but not fully inclusive of all possibilities. Changing the magnitude and direction of the force is important.
Of even greater significance is the positioning (point of
force application) of the elastic force. It is the point of
force application that will determine the amount and
direction of arch rotation around the CR and the occlusal
plane cant effect.
69
5
The Creative Use of Maxillomandibular Elastics
FIG 5-7 Short Class II elastic force (red arrow) placed anteriorly in a
Class II open bite case. An equivalent force system at the CR of the
maxillary arch (yellow arrows) indicates that a large moment is only
produced in the maxillary arch, closing the open bite and reducing the
Class II malocclusion. The cant of the mandibular plane of occlusion
will not change.
a
b
FIG 5-8 Short Class III elastic placed anteriorly in a Class III open bite case. (a) The single force (red arrow) is through the CR of the maxillary arch. (b) An equivalent force system at the CR of the mandibular arch (yellow arrows) has a large moment, closing the open bite and
reducing the Class III malocclusion. The cant of the maxillary plane of occlusion will not change.
a
b
FIG 5-9 Short Class II elastic placed posteriorly in a Class II deep bite case. (a) The single force (red arrow) is through the CR of the maxillary
arch. (b) An equivalent force system at the CR of the mandibular arch (yellow arrows) produces a large moment, opening the bite and reducing
the vertical overlap. The cant of the maxillary occlusal plane will not change.
Let us now consider three clinical examples of how an
equivalence analysis can help us to select the best possible elastic pull. The patient shown in Fig 5-7 has nonparallel maxillary and mandibular occlusal planes, a severe
anterior open bite, and a Class II malocclusion. Patients
70
with this type of malocclusion are most likely candidates
for surgery and not just Class II elastic therapy. However,
let us assume that some Class II maxillomandibular elastics are needed. What would be the best line of action
for the force? A short elastic placed at the maxillary canine
Nonrigid Arches with Third-Order Play
FIG 5-10 Frontal view of the long Class II elastic shown in Fig 5-1.
The replaced equivalent force system (yellow arrows) at the CR shows
that the mandibular second molar (terminal molar) will move in a
superior direction; at the same time, the moment produced by the
elastic at the CR will tip the molar crown lingually.
bracket attached to the mandibular first premolar bracket
would produce a large moment at the CR of the maxillary
arch, which would steepen the maxillary occlusal plane,
close the open bite, and reduce the Class II malocclusion.
The cant of the mandibular occlusal plane would not
change.
In a similar manner, the best placement of the line of
force for a Class III elastic for the Class III open bite
shown in Fig 5-8a would be through the CR of the maxillary arch. Asynchronous rotation around the CR occurs
in the mandibular arch (Fig 5-8b), and no rotation is
produced in the maxillary arch. These two open bite cases
are shown as exaggerations to better explain proper maxillomandibular elastic use and not to suggest that maxillomandibular elastics are the best treatment modality.
Sometimes a patient with deep bite will still have excessive vertical overlap after leveling. A careful appraisal will
show two perfectly leveled arches that are not parallel
and are anteriorly converging (Fig 5-9). If Class II elastics
are needed, one must be careful not to increase the vertical overlap. The best elastic use is a short elastic placed
as far distal as possible on the second molar with a line
of action through the maxillary CR. Note that the maxillary arch does not rotate and the mandibular arch steepens to better parallel the cant of the maxillary occlusal
plane. A bite block placed anteriorly is helpful to disengage the posterior teeth during this corrective stage. It is
best to avoid this undesirable leveling side effect with
good mechanics during initial alignment, because correction may be difficult.
Nonrigid Arches with Third-Order
Play
So far this discussion of Class II elastics has outlined
many of the important principles of usage and has not
necessarily presented elastics as optimal treatment for
the correction of a specific Class II malocclusion. There
are other modalities, such as headgear, functional appliances, temporary anchorage devices (TADs), and loop
springs, for that purpose. It has been assumed that the
wires are rigid and that no play exists between the wire
and the bracket. If round wires are used, one might expect
a similar response; however, incisors could change their
axial inclinations. There would be less change in the cants
of the occlusal planes with full rigidity of wires and teeth.
Furthermore, with a round wire without third-order
control, maxillary molars can tip to the buccal, and
mandibular molars can tip to the lingual, resulting in a
crossbite (also known as a scissor bite).
Viewed from the frontal plane, the long Class II elastic
shown in Fig 5-1 produces a vertical component of force
away from the CR of the mandibular arch. The mandibular second molar (terminal molar) will move in a superior direction; at the same time, the moment produced
by the elastic at the CR will tip the molar crown lingually
(Fig 5-10).
Nevertheless, the determination of the effects of a
maxillomandibular elastic in a rigid, no-play system is
usually a good starting place in deciding proper maxillomandibular elastic use. In this chapter, this boundary
condition is applied to demonstrate the principle of
equivalence.
71
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The Creative Use of Maxillomandibular Elastics
FIG 5-11 Posterior crisscross elastic in proximal view. (a) Force magnitude and direction
can vary depending on jaw opening and hook
placement. (b) For simplicity, we will assume
the line of action to be an arbitrary line connecting the points at the brackets or hooks
where the elastic is attached.
a
b
a
b
FIG 5-12 Unilateral posterior crisscross elastic in a continuous arch. (a) The forces from the crisscross elastic (red arrows) are replaced with
equivalent force systems at the CRs (yellow arrows). (b) Asynchronous occlusal plane effects causing an open bite on the left side are anticipated.
Lateral or Crisscross Elastics
Crisscross elastics are commonly placed on individual
molars, on an entire posterior segment, or on a full arch.
Let us consider the effect of crisscross elastic placement
on a full arch, assuming again a rigid archwire without
any play. This may or may not be a good idea in clinical
situations, but it will serve as the basis for evaluating
lateral elastic force. A buccolingual crisscross elastic is
shown in Fig 5-11. What is the line of action of the force?
The answer is complicated, because the elastic contacts
the molar’s occlusal anatomy (Fig 5-11a). For simplicity,
we will assume the line of action to be an arbitrary line
connecting the points (brackets or hooks) where the
elastic is attached (Fig 5-11b).
Some clinicians might place a maxillary and mandibular passive continuous arch in the malocclusion presented
in Fig 5-12, which shows a crisscross elastic inserted only
on the right side. This may not be correct, but let us
72
analyze the force system. The equivalent force system
(yellow arrows) is calculated at the CRs (purple circles) of
the maxillary and mandibular arches. A large moment is
produced in the maxillary arch in a counterclockwise
direction. The opposing force (red arrow) on the mandibular arch produces a negligible moment at the CR of
that arch, and hence no rotation occurs around the CR
(Fig 5-12a). A large moment is produced in the maxillary
arch that cants the occlusal plane occlusally on the right
side and apically on the left side (Fig 5-12b). If this elastic
is used, an open bite will be produced on the patient’s left
side. It is common for a unilateral elastic to be placed on
a continuous arch to correct a posterior unilateral crossbite. Unfortunately, this can produce an asynchronous
effect with an open bite and dual cants of the occlusal
plane as seen from the frontal aspect.
The open bite on the left side could be reduced by
placing a vertical elastic so that a clockwise moment acts
at the maxillary CR and a counterclockwise moment acts
Lateral or Crisscross Elastics
a
b
c
FIG 5-13 (a) To balance the moment created by the unilateral posterior crisscross elastic in Fig 5-12, a vertical elastic force (red arrow) is
applied on the left side. The vertical elastic’s equivalent force system at the CR of the maxillary arch (upper yellow arrows) has an equal and
opposite moment as the crisscross elastic on the right side. (b) The yellow arrows are the resultants from the two elastics. (c) The resultant
force is replaced with the equivalent force system at each CR (yellow arrows).
a
b
FIG 5-14 Bilateral crisscross elastics. (a) The resultant (yellow arrows) of the two forces of the crisscross elastics (red arrows) lies an approximately equal distance (D1 ≈ D2) from the maxillary and mandibular CRs. (b) The equivalent force systems at the CRs of each arch have synchronous
moments (yellow arrows), rotating the maxillary and mandibular arches in the same direction so that no lateral open bite will be produced.
at the mandibular CR (Fig 5-13a). The vertical elastic
must be placed in the identical anteroposterior position
as the crisscross elastic. The resultant forces produced at
the maxillary and mandibular arches are depicted in
yellow arrows in Fig 5-13b. Once again, resultant single
forces are replaced with equivalent force systems at the
CRs of the maxillary and mandibular arches (Fig 5-13c),
showing that both arches will rotate in a counterclockwise
direction and no open bite will be produced on the left
side.
Another alternative to avoid the asynchronous effect
in Fig 5-12 is to use two bilateral crisscross elastics
(Fig 5-14a). In this application, the resultant (yellow
arrow) of the two forces (red arrows) of the crisscross
elastics lies an approximately equal distance from the
maxillary and mandibular CRs (D1 ≈ D2 ). Therefore, once
again resultants are replaced with equivalent force
systems on the CRs of the maxillary and mandibular
arches (Fig 5-14b). Figure 5-14b also shows the same
direction of action on each arch. The maxillary arch will
translate downward to the left and synchronously rotate
around the CR in a counterclockwise direction. The
mandibular arch will translate upward to the right and
synchronously rotate around the CR in a counterclockwise direction. Both the maxillary and the mandibular
occlusal planes will cant in the same direction so that no
lateral open bite will be produced. The difference between
Fig 5-13 and Fig 5-14 is the direction. Figure 5-13 shows
short maxillomandibular elastics with a more vertical
component of force, while Fig 5-14 shows long maxillomandibular elastics with a more horizontal component
of force in the frontal view.
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5
The Creative Use of Maxillomandibular Elastics
a
FIG 5-15 Changing the point of force application of a single force (red arrows) from
crisscross elastics in a continuous arch. Arch
rotation (occlusal view) will be produced
unless the force passes through the CR: A,
clockwise rotation; B, translation; C, counterclockwise rotation.
b
FIG 5-16 The location of a crisscross elastic force (red arrow) and arch rotation (occlusal
view). The equivalent force system at the CR is represented by yellow arrows. (a) An anterior
crisscross elastic rotates the arch in a clockwise direction. (b) An elastic placed at the first
molar rotates the arch in a counterclockwise direction.
FIG 5-17 The anterior crisscross elastic
rotates the maxillary arch in a clockwise direction and the mandibular arch in a counterclockwise direction around the center of
rotation (blue dot).
The application of a unilateral elastic to correct a
malocclusion with a unilateral crossbite may not be the
best mechanics according to this analysis. Better alternatives include asymmetric lingual arches, crisscross
elastics on only the offending segment, and TAD applications.
The use of a crisscross elastic on a continuous arch is
not trivial because many side effects can occur. This
requires us to look also from the occlusal view (Fig 5-15).
If the force is in alignment with the CR, the arch will only
translate to the patient’s left side. If the force is anterior
or posterior to the CR, both translation and rotation
around the CR will be observed. Figure 5-16a shows an
anterior crisscross elastic rotating an arch in a clockwise
direction; in Fig 5-16b, the elastic is placed at the first
molar and the maxillary arch rotates in a counterclockwise direction.
Therefore, anterior crisscross elastics produce a
discrepancy between maxillary and mandibular arch
coordination, particularly in the anterior region of the
arch. In Fig 5-17, note the discrepancy of anterior overjet
74
by clockwise rotation of the maxillary arch and counterclockwise rotation of the mandibular arch. Figure 5-18
shows a patient with a midline discrepancy and asymmetric overjet. It is anticipated that the crisscross elastic
will correct the midline and rotate the maxillary arch in
a counterclockwise direction and the mandibular arch in
a clockwise direction, which is desirable in this case.
An anteriorly placed crisscross elastic is sometimes
used to correct a midline discrepancy and an asymmetry
in buccal occlusion. But is this an effective method with
minimal side effects? Figure 5-19 shows the equivalent
force system (yellow arrows) at the maxillary and mandibular CRs. From the frontal view, the force is away from
the CR, tipping the maxillary arch to the left and the
mandibular arch to the right. This could aid in the midline
correction; however, the lateral tipping and the canting
of the maxillary and mandibular occlusal planes (counterclockwise) are undesirable. The canting of the occlusal
planes would be particularly unesthetic from the frontal
view (Fig 5-20).
Lateral or Crisscross Elastics
a
b
c
d
e
f
FIG 5-18 A patient with midline discrepancy
and asymmetric overjet. (a and b) Initial view.
(c and d) Frontal views before and after using
an anterior crisscross elastic. (e and f) Occlusal views before and after using an anterior
crisscross elastic. (g and h) After the treatment.
The midline and arch coordination problems
were solved simultaneously.
g
h
FIG 5-19 Anterior crisscross elastic (red arrows). Equivalent force
systems at the maxillary and mandibular CRs are represented by
yellow arrows. The force system would aid in midline correction; however, the canting from the moments of the maxillary and mandibular
occlusal planes is undesirable.
FIG 5-20 Predicted treatment result from an anterior synchronous
crisscross elastic. The canting of the frontal occlusal planes would
be particularly unesthetic.
What about the moment acting in the maxillary occlusal view? Is this moment useful (see Fig 5-16a)? In theory,
this should help correct maxillary midline movement to
the left and Class II correction on the maxillary left side.
The couple (yellow arrows) will produce rotation of the
arch en masse—both roots and crowns move distally on
the left side. This type of tooth translation is very slow,
and it is not a practical way to correct an asymmetry
because of side effects if the force is acting at bracket
level. Furthermore, note that if actual translation and
rotation of the full arch occurs, arch harmony is lost,
and a reverse articulation and crossbite are created (Fig
5-16b). Note the changes in the canine and molar regions;
even rotation alone around the CR can create crossbite
and reverse articulation. What can occur more rapidly
is a mandibular shift to the patient’s right side. In that
situation, the correction may only be temporary, unless
the original crossbite and reverse articulation are a
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The Creative Use of Maxillomandibular Elastics
a
b
FIG 5-21 Anterior crisscross elastic (red arrows) placed off-center. Equivalent force systems (yellow arrows) at the CRs show a maxillary
occlusal plane that rotates very little (a) and not at all (b) because D1 is very small. Therefore, the cant of the maxillary occlusal plane will
be maintained. An open bite may occur on the right side because of the counterclockwise rotation of the mandibular arch due to the large
moment at the mandibular CR (large D2).
functional shift. The moment around the CR from the
occlusal view can also produce an iatrogenic crossbite
and reverse articulation as a side effect. Note the changes
in the molar region after the rotation (see Fig 5-16a).
Unlike Figs 5-19 and 5-20, Fig 5-21 shows an off-center
anterior crisscross elastic producing an asynchronous
occlusal plane effect. In addition to the equal and opposite translatory movements, only the mandibular occlusal plane rotates in a counterclockwise direction, producing an open bite on the right side. If not carefully planned,
placement of anterior crisscross elastics can lead to many
undesirable side effects.
Anterior crisscross elastics placed for midline discrepancy correction may cause many side effects in three
dimensions. However, sometimes these side effects may
be desirable; once we completely understand the force
system, we can use these side effects to our advantage in
carefully selected cases.
Figure 5-22 shows an adult patient with a midline
discrepancy whose maxillary and mandibular occlusal
planes canted downward on the right side from the frontal view. The treatment objective was to correct the
midline and cant the occlusal planes synchronously in a
clockwise direction without surgery. An anterior crisscross elastic was placed. The replaced equivalent force
system on the maxillary and mandibular CRs shows that
the moments are produced to rotate both arches in a
clockwise direction. After prolonged use of the anterior
crisscross elastic, significant correction had occurred.
The forces and moments produced by the elastic
76
simultaneously corrected the midline discrepancy (see
Figs 5-22c and 5-22d) and reduced the occlusal plane
cant. The occlusal plane became parallel with the smile
arc by clockwise rotation of both arches (see Figs 5-22e
and 5-22f ).
So what does the anterior crisscross elastic do? It moves
the maxillary and mandibular arch in opposite directions
with rotations in three dimensions. The rationale for use
of the crisscross elastic is the added effect of altering the
cant of the frontal plane of occlusion and helping to
correct any lateral dental asymmetry that might be present. Therefore, careful case analysis is mandatory before
applying anterior crisscross elastics.
In some cases, the anterior crisscross elastic helps to
eliminate an occlusal interference so that the mandible
is allowed to reposition to centric relation to eliminate
the mandibular functional shift. Figure 5-23a shows a
patient with a midline discrepancy accompanied by a
unilateral buccal reverse articulation on the right side.
Particularly in some young patients, it is difficult to guide
a mandible into centric relation to diagnose a functional
shift or a true asymmetry. As a therapeutic treatment
(diagnosis with treatment), rapid palatal expansion was
performed on the maxillary arch (Fig 5-23b). Note that
the midline discrepancy was reduced because a functional
shift was resolved (Fig 5-23c). Blindly using an anterior
crisscross elastic to treat this patient for midline correction would not make any sense; rather, treatment was
specifically aimed at reducing the occlusal interferences
of the posterior teeth in reverse articulation.
Lateral or Crisscross Elastics
a
FIG 5-22 A patient with a midline discrepancy and occlusal plane canting. (a) Initial
head film. The maxillary and mandibular occlusal planes cant downward on the right side.
(b) Initial intraoral frontal view. (c and d) Models of the teeth before and after treatment. The
replaced equivalent force system (yellow
arrows) on the maxillary and mandibular CRs
shows that the forces and moments are produced to rotate both arches in a clockwise
direction. (e and f) Facial views before and
after treatment. Not only was the midline corrected, but the abnormal canting of the occlusal plane was also reduced.
a
b
c
d
e
f
b
c
FIG 5-23 A mixed dentition patient with a midline discrepancy and unilateral buccal reverse articulation. (a) Initial intraoral view. The midline
discrepancy and buccal reverse articulation on the right side are shown. (b) A rapid palatal expander was used in the maxillary arch. (c) Intraoral
view after rapid palatal expansion treatment. The midline discrepancy was reduced. The midline correction was the result of a mandibular shift
back to centric relation. In conjunction with proper diagnosis, it is better to remove occlusal interferences by planned tooth movement than
by blind use of an anterior crisscross elastic.
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a
b
d
c
e
FIG 5-24 An adult patient with a midline discrepancy and unilateral buccal reverse articulation. (a)
Initial intraoral view. The mandibular midline is off to the right side with a buccal reverse articulation on the right side. (b) A unilateral constriction lingual arch was placed. (c) Intraoral view after
unilateral constriction. (d) Frontal view after the buccal reverse articulation was resolved. The midline discrepancy is resolved. (e) Frontal view after retraction of the mandibular anterior teeth using
symmetric mechanics. Midline correction was produced by correcting a mandibular shift. Proper
diagnosis and specific tooth movement to eliminate occlusal interferences offered a better treatment
modality than an anterior crisscross elastic.
FIG 5-25 An untreated skeletal asymmetry
case with the chin and midline deviated to the
right side. The teeth have naturally compensated for the given skeletal asymmetry.
Figure 5-24a shows an adult patient with a midline
discrepancy and a buccal reverse articulation on the right
side. The midline discrepancy was due to a functional
shift of the mandible. An asymmetrically activated lingual
arch was inserted on the mandibular molars for unilateral
constriction of the mandibular right first molar (Figs
5-24b and 5-24c). Figure 5-24d shows the frontal view
after the buccal reverse articulation was resolved, and
Fig 5-24e shows the frontal view after retraction of the
mandibular anterior teeth. This case demonstrates an
important principle: Do not automatically place a
78
crisscross elastic if there is a midline discrepancy, but
rather diagnose the problem first. If there is a mandibular shift, select the best mechanics to move the offending
tooth or teeth; do not select an asymmetric elastic unless
indicated for other sound reasons.
Although Class II/Class III elastics or anterior crisscross elastics are not usually effective in correcting the
buccal occlusion in skeletal asymmetric subdivision
patients (unless they help to unlock a mandibular shift),
an anterior or posterior crisscross elastic might be helpful in the following type of asymmetry.
Subdivision Elastics
FIG 5-26 Various locations of vertical elastics. Canting of the occlusal plane will be produced unless the force is passing through the CR.
A, counterclockwise rotation of the maxillary arch and clockwise rotation of the mandibular arch; B, no rotation; C, clockwise rotation of the
maxillary arch and counterclockwise rotation of the mandibular arch.
FIG 5-27 Vertical elastic placed off-center. The equivalent force
system at the CR produces an open bite on the left side.
It is common for compensations for a skeletal asymmetry to occur in the axial inclination of anterior teeth
and the cant of the occlusal plane. Figure 5-25 shows a
skeletal asymmetry with the chin and midline to the
patient’s right. Note that the teeth have compensated for
this asymmetry—the maxillary right canines are tipped
to the buccal, and the mandibular right canines are tipped
to the left side. The occlusal plane has canted downward
on the left side. This is nature’s way of improving the
occlusion. There are some patients for whom this natural
compensation has not occurred or for whom more
compensation is indicated; here a laterally directed force
to the full arch or individual segments via maxillomandibular elastics may be indicated. In these applications,
the principles outlined in this chapter for general maxillomandibular elastic use should be most helpful in planning a special force system.
bite on the left side. Vertical elastics should only be used
after careful study that takes into account their rotational
effect in three dimensions.
Vertical Elastics
Vertical elastics are often used together with an archwire
to augment vertical alignment. However, one must be
careful in using vertical elastics because they may incorrectly cant an occlusal plane. We have seen how Class II
elastics can cant an occlusal plane. The same can occur
with vertical elastics in the lateral view (Fig 5-26). A force
near the CR will translate the maxillary arch vertically;
forces away from the CR will translate and rotate the
maxillary arch. In a similar manner, the off-center vertical force in Fig 5-27 could produce an unwanted open
Subdivision Elastics
When there is a difference in occlusion between the left
and right sides, such as Class II on one side and Class I
on the opposite side, unilateral Class II elastics on a
continuous arch are commonly used to attempt to correct
the asymmetry. However, this can be problematic for a
number of reasons. Figure 5-28a shows that the elastic
produces a moment in the occlusal view of the maxillary
arch that tends to rotate the entire arch around the CR.
Although the Class II molar relationship on the right side
may sometimes be improved with a unilateral Class II
elastic, correction is usually disappointing. Why? This
couple (curved yellow arrow) seen from the occlusal view
would have to rotate the dental arch en masse in each
arch. This would take a very long time; therefore, the
usual successes are mostly the result of initial mandibular shifts.
Unilateral elastics appear to correct asymmetric occlusions but later relapse because the correction involved a
shift into a temporary pseudo-centric occlusion. Therefore, this en masse movement is not typically recommended. Rather, individual tooth movement around the
arch is the proper goal, or more often, extraction of teeth
is indicated.
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The Creative Use of Maxillomandibular Elastics
a
b
FIG 5-28 Unilateral Class II elastic. (a) Occlusal view of the maxillary arch showing an equivalent force system at the CR (yellow arrows).
The maxillary arch tends to rotate around the CR and can produce a buccal reverse articulation. (b) Frontal view showing an equivalent force
system at the CR (yellow arrows), producing an open bite on the left side. The mandibular arch rotates more because D2 > D1.
In addition, a unilateral Class II elastic can produce
other side effects. The lateral forces can lead to a crossbite
on the right side and reverse articulation on the left side
(see Fig 5-28a). If we look from the frontal view
(Fig 5-28b), the unilateral Class II elastic force on the
patient’s right side (red arrow) will translate the maxillary
arch occlusally to the right, and the mandibular arch will
translate occlusally to the left. The yellow arrows are the
equivalent force system at each CR. In a similar manner
to a vertical elastic (see Fig 5-27), moments are produced
at both CRs in opposite directions that will cant both
the maxillary and the mandibular occlusal planes, producing a lateral open bite on the left side. Unlike with the
unilateral vertical elastic, however, the moment to the
mandibular arch CR is considerably greater than that of
the maxillary arch CR because the perpendicular
distance is larger in the mandibular arch (D2 > D1 ).
Can this open bite be avoided when a unilateral Class
II elastic is used? Many orthodontists will first try a
supplemental vertical elastic on the left side to close the
open bite (Fig 5-29a). As depicted, the magnitude of the
left vertical elastic force (red arrows) is set to balance the
maxillary and mandibular moments from the right Class
II elastic. The replaced equivalent force system of the left
vertical elastic at the CR is shown in shadowed yellow
arrows (Fig 5-29b). After all moments are summed to
their respective CRs from both the Class II elastic and
the vertical elastic, the problem still remains. If the vertical elastic force is carefully calibrated, no lateral open bite
will be observed; however, the cant of the occlusal plane
will move downward on the left side, though this canting
could be negligible because the magnitude of the moment
is so low. If one can accept the slight canting of the
80
occlusal plane side effect, this method could be desirable;
unfortunately, balancing the moments might be hard to
do clinically. In short, the vertical elastic might help close
the lateral open bite, but it is not an optimal solution.
The anteroposterior placement of the vertical elastic
is also very critical. It must be placed at the anteroposterior CR; otherwise, the action of the Class II elastic will
be changed. For example, if the left vertical elastic is
placed distal to the anteroposterior CR, an asynchronous
movement occurs, seen from the lateral view. From this
view, an anterior open bite will be produced. The maxillary arch will flatten, and the mandibular arch will steepen.
If this effect is understood by the clinician, this can be
either desirable or undesirable, depending on the circumstances. Care must be taken to always think in three
dimensions. Nonetheless, a lateral component of force
still exists (Fig 5-29c).
Perhaps the best solution for the lateral component of
force and the lateral open bite problem associated with
a unilateral Class II elastic is to use the crisscross elastic
(Fig 5-29d). This will prevent occlusal plane rotation from
the frontal view (Fig 5-29e). The occlusal view moment
will probably be ineffective, but it operates in the correct
direction. The replaced equivalent force system at the CR
is depicted with yellow (right) and shadowed yellow (left)
arrows in Fig 5-29f.
Another strategy for a subdivision patient is to use a
Class II elastic on the right side and a Class III elastic on
the left side (Fig 5-30). From the frontal view, all of the
elastic forces to both arches will produce moments at the
CR that will rotate the maxillary and mandibular arches
clockwise (each yellow arrow in Fig 5-30b is the resultant
of the two red elastic forces). The net lateral component
Subdivision Elastics
a
b
c
d
e
f
FIG 5-29 (a) Unilateral Class II elastic with a vertical elastic on the opposite side. (b) The magnitude of the moment at each CR from the left
vertical elastic force (shadowed arrows) is set to balance the moment (equal and opposite) from the right Class II elastic (yellow arrows). It may
prevent the rotation of the maxillary arch, but the mandibular moment still exists. (c) Unilateral Class II elastic with a posterior vertical elastic.
The horizontal component of force from the Class II elastic produces an unwanted lateral reverse articulation. (d) A unilateral Class II elastic
with a crisscross elastic from the maxillary palatal to the mandibular buccal. (e) There will be no occlusal plane rotation in the frontal view,
but extrusive forces are increased. (f) Occlusal view of the force system of the unilateral Class II elastic and crisscross elastic. There is no lateral
component of force. The moment will probably be ineffective to rotate the entire arch, but it operates in the correct direction.
FIG 5-30 (a) Class II elastic on the right side
and Class III elastic on the left side. (b) The
resultant forces (yellow arrows) show that both
arches will rotate synchronously in a clockwise
direction.
a
of force to the right on the maxillary arch and the canting
of both maxillary and mandibular occlusal planes are
usually undesirable side effects. In the occlusal view of
the maxillary arch (Fig 5-31a), the individual single force
from the Class II elastic on the right side (red arrow) is
replaced with a yellow equivalent force and moment at
b
the CR; the Class III elastic on the left side (red arrow) is
replaced at the CR by a shadowed yellow force and
moment. The resultant force system from both elastics
is given in Fig 5-31b. It is clear that not only anteroposterior forces but also a counterclockwise moment and
a significant lateral component of force are present
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The Creative Use of Maxillomandibular Elastics
a
b
FIG 5-31 Class II elastic on the right side and Class III elastic on the left side (occlusal view of the maxillary arch). (a) The shadowed yellow
arrows are the replaced equivalent force system of the Class III elastic on the left side, and the yellow arrows are the replaced equivalent force
system of the Class II elastic on the right side. (b) The final resultant of each force system at the CR is depicted in yellow arrows. It shows not
only a couple but also a lateral and posterior force.
FIG 5-32 Class II elastic on the right side and Class III elastic on
the left side (lateral view). In this special case, the resultants (yellow
arrows) are acting at the maxillary and mandibular CRs, and no
change in the occlusal plane cants is anticipated.
FIG 5-33 Posterior crisscross elastic on a rigid continuous arch
without play. The equivalent force system at the CR (yellow arrows)
shows lateral-occlusal translation and rotation around the CR of
the full arch. This type of movement has little contribution to the
crossbite correction.
(Class II/Class III elastics do not produce only a couple).
Just like with a unilateral Class II elastic, the value of the
CR moment in the occlusal view from Class II/Class III
elastics can be questioned as an efficient way to correct
an asymmetric occlusion. In the lateral view, it can be
seen that Class II/Class III elastics produce large extrusive
forces with minimal change of the canting of the occlusal
plane (Fig 5-32).
in the frontal view of the maxillary arch: (1) lateral and
occlusal translation and (2) rotation around the CR of
the full arch (Fig 5-33). The translation force could be
useful; however, let us concentrate on the moment at the
CR. The moment will cause the maxillary arch to rotate
around the CR. Note that the maxillary right molar moves
occlusally and the maxillary left molar moves apically,
with little contribution to the crossbite correction.
Let us compare this with the crisscross elastic to the
right buccal segment alone (Fig 5-34). For anchorage
purposes, a continuous archwire is placed in the mandibular arch. The maxillary buccal segment will also translate
laterally and downward. With the same crisscross elastic
force, more crossbite correction by translation and rotation around the CR is observed. The most important
difference between a crisscross elastic on a segment
rather than on a full arch is the effect of the moment at
Segmental Elastics
It can be advantageous to apply a maxillomandibular
elastic to a buccal segment rather than to a full arch for
crossbite correction. Let us consider, for example, a
unilateral crossbite on the right side. If crisscross elastics
are applied to the full arch, two effects will be observed
82
Segmental Elastics
a
FIG 5-34 A posterior crisscross elastic on
the right buccal segment only. An equivalent force system at the CR is depicted in
yellow arrows. The moment on the segment
at the CR produces more lingual crown
movement than in Fig 5-33.
FIG 5-36 (a and b) Bent wire lingual hook
using a hinge cap lingual bracket.
FIG 5-37 (a) Extended lingual hook. (b and c)
The hook can be slid anteriorly or posteriorly to
vary the location of the elastic force.
b
c
FIG 5-35 Various locations of forces and predicted tooth movements from posterior crisscross
elastics on the buccal segment (occlusal view). (a) Force placed at the CR. (b) Force placed
anterior to the CR. (c) Force placed posterior to the CR.
a
b
a
b
the CR. Note in Fig 5-34 that the moment on the segment
alone tips the maxillary molar around its CR and produces
more lingual crown movement for crossbite correction.
Segmental crossbite correction therefore minimizes the
vertical response and emphasizes the desired horizontal
effects. The full arch is less sensitive to tipping. Detailed
explanation can be found in chapter 9. It may be argued
that a segment might extrude more from the vertical
component of the elastic force; however, occlusal force
may minimize this effect. Just like with the use of maxillomandibular elastics on a full arch, three-dimensional
thinking is required to ensure full control in achieving
segmental crossbite correction. We are required to look
from the occlusal view as well as from the lateral and
frontal views.
c
A buccal force on the maxillary segment is placed at
the CR if translation is required in a buccal direction
(Fig 5-35a). If the force is placed anterior to the CR, the
canine end of the segment will move more than the terminal molar (Fig 5-35b); if the force is placed posterior to
the CR, the terminal molar will be displaced the most
(Fig 5-35c).
Crisscross elastics may require additional hooks on
the lingual of the teeth. An alternative is the use of a
lingual attachment, such as a hinge cap lingual bracket,
whereby a wire is placed that acts as a hook (Fig 5-36).
A short, segmental lingual wire with a hook can also be
placed in the hinge cap bracket and can be slid forward
or backward to alter the anteroposterior position of the
elastic force (Fig 5-37).
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The Creative Use of Maxillomandibular Elastics
a
FIG 5-38 Multiple maxillomandibular elastics. It would be difficult to get the patient’s
compliance to wear so many elastics. All
elastics can be replaced with an equivalent
single elastic.
FIG 5-39 Posterior woven up-and-down elastic. (a) The up-and-down elastic is placed on
a patient with a posterior lateral open bite. Is this the correct treatment? (b) The required
force system for closing the posterior open bite would be a single force (red arrows) as far
posterior as possible.
a
FIG 5-40 Anterior up-and-down elastic.
The up-and-down elastic is poorly designed
because a single elastic placed as far anteriorly as possible (green elastic) would close
the anterior open bite most efficiently.
b
FIG 5-41 Elastics for open bite closure. (a) A single elastic at the midline is not esthetic and
is problematic. (b) It is better to replace the single force at the midline with an equivalent force
system from two elastics placed bilaterally. Sometimes multiple elastics can be indicated for
practical purposes.
Elastic Redundancy
Orthodontic patients are sometimes seen wearing many
elastics in a complicated manner. It is therefore desirable
to simplify the use of maxillomandibular elastics, both
so the orthodontist can better understand the force
system and so the patient can more easily insert them.
Multiple elastics (Fig 5-38) can challenge patient compliance. It would be simpler to use only one elastic that acts
as the resultant of the many elastic forces. The best
approach is to first determine what type of arch movement (translation and rotation around the CR) is required
and to then establish the line of action of the elastic to
achieve that goal. Our patients will appreciate the fewer
elastics that must be placed and reciprocate with better
compliance. When many elastics are used, there is a good
84
b
probability that the force system has not been carefully
reasoned and that treatment errors will develop.
Another common error is the use of an up-and-down
elastic woven between many teeth to close an open bite.
The patient in Fig 5-39a has a lateral open bite that developed during the use of a Herbst-type appliance. The open
bite exists because the maxillary and mandibular planes
of occlusion are not parallel—they diverge at the posterior. Which force system is the best to make parallel the
occlusal planes with a vertical elastic? In order to obtain
the largest moment for parallelism, the force should be
placed as far posterior to the CRs as possible. The simple
elastic in Fig 5-39b achieves that objective. The added
elastic material in Fig 5-39a actually reduces the efficiency
because some of the vertical force is anterior to the maxillary and mandibular CRs.
Using Class II Elastics and Headgear Simultaneously
FIG 5-42 Anterior vertical elastics used to enhance canine extrusion; however, the force of the elastics is anterior to the maxillary
and mandibular CRs, so these elastics will increase the deep bite.
FIG 5-43 Elongated box-shaped vertical elastic (light green elastic).
Its long horizontal portion could irritate the gingiva, and the horizontal portion of the elastic force produces unnecessary mesiodistal forces that could potentially rotate the attached teeth. This elastic could be replaced with a simpler form (triangular elastic).
The woven up-and-down elastic in Fig 5-40 is also
poorly designed, because the goal is to place the elastic
force anterior to the CRs of the maxillary and mandibular arches. A single elastic on each side as far forward as
possible would be the simplest and optimal placement.
In lateral view, a vertical elastic at the midline would
give the largest moment to make parallel the two occlusal planes and close the open bite because an elastic
placed at the midline would have a high moment-to-force
ratio at the CR (Fig 5-41a). However, a single elastic placed
at the midline is not esthetic and is problematic. Very little
deviation of the line of action from the CR will result in
rotation of the maxillary or mandibular anterior segment.
Perhaps the angulations of the roots are not symmetric
so that the location of the CR could be off-center or placement of the elastic could be off-center by mistake. It is
better to replace the single force at the midline with an
equivalent force system from two elastics placed bilaterally (Fig 5-41b). The rotation of the maxillary and mandibular anterior segment is easier to control by adjusting the
force level of each side. Theoretically, redundant elastics
can be replaced by a single elastic, but sometimes multiple elastics can be indicated for practical purposes.
of the maxillary plane, and this might be desirable.
However, a unilateral high canine after leveling may show
a frontal cant to the plane of occlusion. In this situation,
a unilateral vertical elastic can overcompensate for the
occlusal plane canting. The vertical (Class II) elastic in
Fig 5-42 was used unilaterally to enhance the canine
extrusion and to close the lateral open bite. Because the
force of the elastic is anterior to the maxillary and
mandibular CRs, an adverse side effect would be an
increase in the deep bite of a patient who had too much
vertical overlap initially.
Indiscriminate vertical elastic use from elongated box
shapes can present problems other than loss of simplicity. The box elastic shown in Fig 5-43 can irritate the
gingiva with its long horizontal portion of the elastic. The
horizontal portion of the elastic also produces unnecessary mesiodistal forces that could potentially rotate the
attached teeth, particularly if wire engagement is not
complete.
Common Side Effects with
Maxillomandibular Elastics
Let us compare two modalities of Class II treatment. A
headgear with a force through the CR of the maxillary
arch and parallel to the occlusal plane could give good
correction with excellent vertical control, including the
cant of the occlusal plane. However, good patient compliance may be lacking. Class II elastics may be better
accepted by the patient, but the occlusal plane could
Vertical elastics are commonly used to augment the
forces from leveling archwires. It may be true that the
leveling of a high canine bilaterally could flatten the cant
Using Class II Elastics and
Headgear Simultaneously
85
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The Creative Use of Maxillomandibular Elastics
a
b
FIG 5-44 Combination of Class II elastics and headgear. (a) An
anterior, short Class II elastic force (red arrows) can be replaced
with an equivalent force system (yellow arrows) at the CR of the
maxillary arch, which would rotate the maxillary arch clockwise. (b)
A headgear force is applied to negate this moment. Any line of action
of the depicted headgear forces (shadowed red arrows) is possible.
(c) The overall resultant force (yellow arrow) acting on the CR of the
maxillary arch from both the Class II elastic force (red arrow) and the
occipital headgear force (shadowed red arrow).
c
steepen. One approach is to combine both the headgear
and the Class II elastics in the following way. A short
Class II elastic is anteriorly placed so that the cant of the
mandibular occlusal plane is maintained (Fig 5-44a). The
maxillomandibular elastic is worn at all times so that
continuous force is applied. This elastic will produce a
large moment at the maxillary CR in a clockwise direction
(curved yellow arrow). A headgear is used to negate this
moment (Fig 5-44b). Many different lines of action are
possible from both cervical and occipital headgears,
provided that the moment from the headgear is counterclockwise to the maxillary CR (shadowed red arrows in
Fig 5-44b). Most likely, the headgear moment to the
maxillary CR does not completely balance the elastic
force moment instantaneously. Because the headgear is
only worn part-time, the headgear moment must be
greater than the elastic moment. This requires careful
patient monitoring at each appointment. The overall
resultant force (averaged over time) acting on the maxillary arch, if an occipital headgear is used, is shown as a
yellow arrow in Fig 5-44c. The upward and backward
translation of the maxillary arch should occur if this
combination approach is used with minimal headgear
wear.
Recommended Reading
Adams CD, Meikle MC, Norwick KW, Turpin DL. Dentofacial remodeling produced by intermaxillary forces in Macaca mulatta. Arch Oral
Biol 1972;17:1519–1535.
Kim KH, Chung CH, Choy K, Lee JS, Vanarsdall RL. Effects of
prestretching on force degradation of synthetic elastomeric chains.
Am J Orthod Dentofacial Orthop 2005;128:477–482.
Dermaut LR, Beerden L. The effects of Class II elastic force on a dry
skull measured by holographic interferometry. Am J Orthod
1981;79:296–304.
Kuster R, Ingervall B, Bürgin W. Laboratory and intra-oral tests of the
degradation of elastic chains. Eur J Orthod 1986;8:202–208.
Hanes RA. Bony profile changes resulting from cervical traction
compared with those resulting from intermaxillary elastics. Am J
Orthod 1959;45:353–364.
86
Reddy P, Kharbanda OP, Duggal R, Parkash H. Skeletal and dental
changes with nonextraction Begg mechanotherapy in patients with
Class II division 1 malocclusion. Am J Orthod Dentofacial Orthop
2000;118:641–648.
Problems
1. Replace the two vertical maxillomandibular elastics
with a single elastic.
2. Replace the two anterior crisscross elastics with a
single elastic in a and b.
a
b
3. Two vertical elastics are placed at the maxillary arch
bilaterally between the canine and the first premolar. (a)
Frontal view. (b) Occlusal view. Find a resultant single
force that is equivalent. Is it possible to place this resultant on the archwire?
4. Two vertical elastics are placed at the maxillary arch
asymmetrically. (a) Lateral view. (b) Occlusal view.
Replace these elastics with an equivalent single elastic.
Do you have to know the location of the CR to solve this
problem?
a
a
b
b
87
Problems
5–7. A unilateral Class II elastic is attached to the maxillary arch. Replace this elastic with a force system at the CR.
(Note that problems 5 to 7 are the same elastic from different views.)
5
6
8. Maxillary and mandibular rigid archwires are placed
during finishing. A 100-g off-center force (green dot) is
needed to make the maxillary and mandibular occlusal
planes parallel. (From the frontal view, the maxillary left
side must move downward; from the lateral view, the
anterior open bite must be closed.) You are only allowed
to place two equivalent vertical elastics attached to the
maxillary archwire. Where are they located and what is
their magnitude?
7
9–10. Replace the Class II maxillomandibular elastic with
a force system at the maxillary and mandibular
CRs. Discuss occlusal plane synchrony and asynchrony.
9
10
88
Problems
11–12. Replace the anterior crisscross elastic with a force
system at the maxillary and mandibular CRs.
Discuss occlusal plane synchrony and asynchrony.
13. The treatment objective is to close the anterior open
bite and correct maxillary anterior protrusion by
moving the maxillary arch backward and rotating it
clockwise. Design the proper maxillomandibular elastic and briefly explain your design.
11
12
14. The maxillary midline is off-center, and the maxillary
occlusal plane is canted upward on the right side.
Design the proper maxillomandibular elastic for
correction and briefly explain your design (consider
the frontal view only).
15. The maxillary and mandibular midlines are off, and
both the maxillary and the mandibular occlusal
planes have a cant. Design the proper maxillomandibular elastic for correction and briefly explain your
design (consider the frontal view only).
89
6
Single Forces
and Deep Bite
Correction by
Intrusion
“Carve the peg by looking at the hole.”
— Korean proverb
Deep bite (or excessive vertical overlap) is best described as a symptom. Many variables are involved in this type of malocclusion, including
a small vertical dimension, extruded incisors, or posterior teeth in
infraocclusion. Hence, many different modalities are required for
correction. This chapter describes the principles, biomechanics, and
appliances for intruding incisors. In the past, intrusion was not considered a possibility because, with heavier forces, posterior extrusion was
mainly observed. The force system needed for incisor intrusion requires
proper force magnitude and force constancy (a low force-deflection
rate). Ideally, force direction should be somewhat parallel to the long
axis of the tooth. With flared incisors, the three-piece intrusion arch
has the advantage that the force is delivered more posteriorly. Initial
leveling with a continuous arch can be nonproductive if intrusion is
required. Anchorage considerations include using very small force
magnitudes, moving the posterior segment's center of resistance (CR)
as far anteriorly as possible, and controlling the large moment on the
posterior segment.
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6
Single Forces and Deep Bite Correction by Intrusion
FIG 6-1 Deep bite correction by posterior eruption.
Class II elastics and round archwires were used, and
the mandible was rotated backward.
A
FIG 6-2 Deep bite correction by genuine incisor intrusion. An intrusion arch was
used, and no backward mandibular rotation occurred.
deep bite is a common characteristic of many
malocclusions, particularly Class II patients.
Although continuous arches and many complicated
force systems can be involved in deep bite correction,
single forces by cantilevers or elastics can be most efficient in applying an optimal force system for either anterior intrusion or posterior extrusion.
Because deep bite is a symptom, there are many causes
and hence many solutions for correcting it. Sometimes
incisor intrusion is indicated, and in other patients the
extrusion of posterior teeth is required. Differential diagnosis must be followed by differential treatment mechanics. Some important indications for anterior intrusion
include excessive maxillary incisor exposure, large vertical dimension of the lower face, facial convexity, a Class
II skeletal discrepancy, little anticipated mandibular
growth, normal or small interocclusal space, periodontal
considerations, existing tooth alignment, and desirable
occlusal plane cant. The variability in treatment goals
must also differentiate between the amount of intrusion
of the maxillary and the mandibular incisors.
Figure 6-1 is an anterior cranial base superposition of
a patient treated with Class II elastics and round wires.
The deep bite was corrected by eruption of the mandibular posterior teeth, which produced downward and
backward rotation of the mandible. Undesirably, the lower
face vertical dimension is too great, and the total facial
convexity has worsened. The incisor is vertically exposed
too much in respect to the upper lip. Without further
growth, mandibular position may not be stable, which
will lead to counterclockwise jaw rotation and recurrence
of the deep bite. This result can best be described as
92
“opening the bite.” This is not a desirable goal. Furthermore, as the mandible rotated backward during treatment, the Class II malocclusion became worse and hence
harder to treat.
Compare these results with the Class II patient treated
by incisor intrusion in Fig 6-2, where the lower face vertical dimension has been controlled. The chin has moved
forward as a result of the horizontal mandibular growth,
and none of this growth has been lost by backward
mandibular rotation.
This chapter is devoted to the mechanics for successful
and efficient incisor and canine intrusion. Chapter 7
describes methods of posterior segment extrusion. Both
anchorage control and occlusal plane considerations are
discussed.
Can Teeth Be Intruded?
In the past, it was believed that intrusion of teeth would
lead to undesirable sequelae such as tooth devitalization
and loss of alveolar attachment. It is now established that
light continuous forces can intrude incisors without jeopardizing the gingival attachment and can perhaps in some
instances improve the attachment configuration. Early
cephalometric studies showed little incisor intrusion
because the heavy forces used primarily erupted the
posterior teeth. Root resorption can be associated with
incisor intrusion, but this can be minimized by reducing
the intrusive forces. It has been shown that as the forces
from intrusion increase, the rate of tooth movement does
not increase, but the amount of root resorption does
Continuous Intrusion Arch
FIG 6-3 Intrusion means the apical movement of the
CR of a tooth or group of teeth, not of the incisal edge
of the tooth. (a) True intrusion of incisors. The CR has
moved apically. (b) Pseudointrusion. The incisor edge
has tipped superiorly, but the CR has not intruded.
a
a
b
b
c
FIG 6-4 Continuous intrusion arch mechanism. (a) Deactivated continuous intrusion arch. The intrusion arch (green) is inserted into auxiliary tubes on the two first molars and is ready for activation. A rigid wire is placed in the buccal segment and in the anterior segment (gray).
Leveling can be carried out at the same time as intrusion; therefore, the material and the size of the wire may vary. (b) Activated continuous
intrusion arch (orange). The intrusion spring is activated and tied to the anterior segment. (c) A passive lingual arch (gray) is inserted to prevent
adverse side effects from an intrusion arch.
increase. Control of the vertical dimension or incisor
intrusion in the maxillary arch is often attempted using
a high-pull J-hook headgear. However, this may be contraindicated because heavy forces are directly and intermittently placed on the maxillary incisors.
Confusion often surrounds the term intrusion. In this
text, it has a strict meaning: the apical movement of the
CR of a tooth or groups of teeth. In Fig 6-3a, the CR
exhibits true intrusion. In Fig 6-3b, the deep bite is
corrected by tipping of the crown. The CR has not moved.
This is an example of deep bite correction without intrusion, or pseudointrusion.
There are two basic mechanisms for incisor intrusion:
the continuous intrusion arch and the three-piece intrusion arch. Both are based on the cantilever principle of
applying single forces without a moment in the incisor
region. The selection of each mechanism is dependent
on the treatment goals required for the individual patient’s
malocclusion.
Continuous Intrusion Arch
The classic continuous intrusion arch (spring) mechanism
is shown in Fig 6-4. A relatively rigid wire (gray) is placed
in the right and left buccal segments (usually at least 0.018
× 0.025–inch stainless steel [SS] wire). Anterior segment
leveling can be carried out at the same time as intrusion;
therefore, a sequence of wires from flexible to more rigid
can be placed in the anterior segment. A lingual or transpalatal arch is present to control widths and maintain
overall occlusal symmetry. The two buccal segments and
the lingual arch form the posterior anchorage unit. The
active intrusion arch (orange) is inserted into auxiliary
tubes on the two first molars and is attached anteriorly.
The anterior point of attachment can be in the midline
or even distal to the incisors (discussed later). The intrusion arch is a rectangular β-titanium (β-Ti) or SS wire,
rectangular so that the wire will not rotate in the molar
auxiliary tube, which ensures greater force accuracy and
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6
Single Forces and Deep Bite Correction by Intrusion
FIG 6-5 A rigid continuous incisor “bypass arch” (gray).
This archwire provides similar anchorage control to a
passive lingual arch.
a
b
FIG 6-6 Intrusion of all four maxillary incisors. (a) Pretreatment. (b) Posttreatment.
reproducibility. Its overall configuration shape (including
location of the bend), wire material (β-Ti or SS), and cross
section (which can vary from 0.016 × 0.022 to 0.018 ×
0.025 inch) are not critical because the intrusive force is
measured with a force gauge. The intrusion arch is bent
apically (see Fig 6-4a) and activated occlusally by a ligature
tie to the anterior segment (see Fig 6-4b). Note the occlusal step in the intrusion arch to prevent the arch from
contacting the canine bracket after activation.
The lingual arch is necessary because the arch width
and arch form may be difficult to maintain with the intrusion arch alone due to the highly flexible wires and large
activation forces required for intrusion. A lingual arch
gives the security of positive control while preventing
side effects from an intrusion arch (see Fig 6-4c). An
alternative method is the use of a continuous arch with
a stepped bypass around the teeth requiring intrusion
(Fig 6-5). The rigid continuous arch gives the anchorage
control, and the intrusion arch that delivers the forces is
the same as that shown in Fig 6-4.
Figure 6-6 shows intrusion of the four maxillary incisors. A rope ligature tie is applied in the midline during
activation. Note that as the deep bite correction has
occurred, the position of the maxillary incisor in respect
to the maxillary canine has changed (see Fig 6-6b). This
positional change does not occur in bite opening where
the mandible is hinged open.
94
Global Characteristics of the
Intrusion Force System
The overall intrusion force system is depicted in Fig 6-7.
The deactivation intrusive force (red arrow) acting on the
incisors is produced by the intrusion arch (orange). The
activation force (blue arrow) is the equal and opposite
force (Newton’s Third Law). The blue force is also the
force acting on the posterior segments and can be
replaced with an equivalent force system at the posterior
CR. If the posterior teeth move during anchorage loss,
they will erupt, and the posterior occlusal plane will
steepen (see Fig 6-7a). Let us look in more detail at the
incisors, where the intrusive force acts anterior to the CR
(Fig 6-7b). If we replace the red intrusive force with an
equivalent force system at the CR of the incisors, the
yellow moment shows that the incisors would tip labially.
One method of preventing the incisors from flaring is to
position the force further posteriorly so that it acts
through the CR of the incisors (discussed later). If we
want to keep the point of force application forward, it
will be necessary to tie the intrusion arch with a distally
directed tie. Note that the combination of an intrusive
force and a distal force produces a resultant that is
directed upward and backward (red arrows in Fig 6-8). If
properly applied, the resultant force will act through the
CR of the incisors, and no undesirable flaring will occur.
Global Characteristics of the Intrusion Force System
a
b
FIG 6-7 The force system of a continuous intrusion arch. (a) The anterior blue arrow is the force to activate the arch (activation force). An
equal and opposite force (red arrow) is acting on the tooth as the arch deactivates (deactivation force). The posterior yellow arrows are the
equivalent force system of the blue force acting at the posterior CR. The posterior segment will erupt, and the posterior occlusal plane will
steepen. (b) The force system of an intrusion arch at the anterior segment. The force (red arrow) from the intrusion arch is replaced with an
equivalent force system (yellow arrows) at the anterior CR. The incisor will not only intrude but also tip to the labial.
FIG 6-8 A distally directed ligature tie changes the line of action
(red arrows) through the CR, preventing undesirable labial flaring.
FIG 6-9 Measuring the intrusive force with a force gauge. An intrusion arch is basically a free-end cantilever, so the accurate force
system can be determined simply with a force gauge and a ruler.
The continuous intrusion arch is basically a free-end
cantilever. It can be tied anteriorly by a single tie or with
a single force; hence, the force system is easy to understand and measure. By comparison, an archwire tied into
multiple edgewise anterior brackets is much more
complicated (forces and moments) and unpredictable.
Figure 6-9 shows the direct measurement of an intrusion
arch with a force gauge. The advantage of a cantilever is
that accurate force systems can be measured and
predictably placed in the patient. Only a force gauge and
a ruler are required—not elaborate force-sensing equipment.
The key to successful intrusion involves particularly
good force control: force magnitude, force constancy, a
point contact, point of force application, and force direction. In addition, one should take advantage of segmental leveling and understand special anchorage considerations. Let us consider these points separately.
95
6
Single Forces and Deep Bite Correction by Intrusion
Force (g)
FIG 6-10 Recommended average
intrusive forces measured at the midline. Note that very light forces are
used for intrusion.
Central incisors
All four incisors
Canines and incisors
Optimal force magnitude
Early experiments on monkeys by Dellinger1 established
the importance of controlling force magnitude for incisor
intrusion. No advantage is gained by increasing the
magnitude of force; higher force levels only increase root
resorption and potential anchorage loss without increasing the rate of tooth movement. Figure 6-10 provides
some recommended force averages. These values represent both studies and clinical experience.2 Overall, lighter
forces are used for intrusion today than were used 50
years ago. Variation in the applied force is recommended
based on root size and other biologic considerations.
Force values are also given for intrusion of all six anterior
teeth. This is usually not recommended unless a temporary anchorage device (TAD) is used, because the heavier
forces needed for more teeth will disturb posterior
anchorage.
With the continuous intrusion arch, the force can be
measured with a force gauge in the mouth or simulated
outside of the mouth. Standardized tip-back bends are
not recommended for the individual patient because the
length, width, and cross section of the intrusion arch can
be quite variable.
Force constancy
An important characteristic of any orthodontic appliance
is the constancy of the force delivered during tooth movement. Figure 6-11 shows a cantilever by which a 100-g
force is applied at the free end. The free end deflects 10
mm before it comes to rest. The force-deflection (F/∆)
rate measures the constancy of the force.
96
F/∆ =
100 g
= 10 g/mm
10 mm
This F/∆ rate denotes that for every millimeter of activation within the elastic range of the wire, 10 g of force is
produced. For example, 5 mm of activation will produce
50 g; 7 mm will give 70 g. If there is a linear relationship
between force and deflection, this is referred to as Hooke’s
law—a law of physics stating that the force required to
extend or compress a spring (or archwire in this case) by
a certain distance is proportional to that distance. Not
all orthodontic appliance components show a linear relationship between force and deflection, but many are close
to linear, so the concept of the F/∆ rate, even if it is only
an approximation, is very useful. The F/∆ rate also tells
us about the constancy of the force as the teeth move.
Let us activate the cantilever to a full 10 mm. The attached
tooth will feel 100 g of force initially. The tooth moves 1
mm; force magnitude drops 10 g as given by the F/∆ rate.
After 5 mm of movement, the force is reduced to 50 g.
Figure 6-12 compares an appliance with a high F/∆ rate
to an appliance with a low F/∆ rate. Based on Hooke’s
law, the greatest force is produced after the initial activation, and the force reduces during deactivation. Three
force zones are depicted: red, excessive; green, optimal;
and yellow, suboptimal. Let us follow the appliance with
the low F/∆ rate (blue line). The force magnitude remains
in the optimal green zone over a wider range of deactivation (∆2 ). On the other hand, the appliance with the
high F/∆ rate (red line) abruptly changes force level during
deactivation (∆1 ) and produces too excessive a force (red
range) to be practical. The appliance with the low F/∆
rate delivers a relatively constant optimal force and also
Global Characteristics of the Intrusion Force System
FIG 6-11 The force-deflection rate (F/∆) of a cantilever. If the rate is
linear, it follows Hooke’s law.
FIG 6-12 High versus low F/∆ rate appliances. The slope of the
line represents the F/∆ rate (red line, high; blue line, low). The force
zones are depicted in colors: red, excessive; green, optimal; yellow,
suboptimal. The horizontal ∆1 and ∆2 represent the range of action of
each appliance in the optimal force zone.
FIG 6-13 F/∆ rate of a 0.016-inch SS continuous full archwire. To
obtain 80 g of intrusive force, 0.05 mm of activation is required, which
is practically unrealistic.
requires fewer activations because it works over a larger
distance (∆2 > ∆1 ).
Leveling archwires even of so-called light wire have
high F/∆ rates. For example, consider a 0.016-inch SS
wire used to level two incisors in a Class II, division 2
patient (Fig 6-13). Stiffness can vary depending on many
factors, including bracket width and interbracket distance,
but these numbers are representative. Suppose that we
want to deliver 80 g of force in the midline. The F/∆ is
1,600 g/mm. How much activation is required? Only 0.05
mm. This small of an activation is not realistic. No orthodontist can see or activate 0.05 mm. Even if a correct
0.05-mm activation is achieved, once the tooth moves
0.05 mm, the force will drop to zero. The high F/∆ wire
would require frequent reactivations. Because of this,
typically with appliances with high F/∆ rates, no effort is
made to deliver optimal force levels. The tooth is initially
jolted with excessive force, and a healing process is
allowed between appointments.
Typical F/∆ values of a continuous intrusion arch are
given in Fig 6-14. The F/∆ rate is 4 g/mm at the midline.
Activation of 10 mm will obtain 40 g of force, a reasonable
magnitude for two maxillary incisors. The low F/∆ rate
has several advantages. Accuracy is enhanced in that an
error of 1 mm in activation leads to only a 4-g force error.
The force is also relatively constant, changing only 4 g for
every 1 mm of intrusion. The large deflection required
also allows little reactivation and longer intervals between
appointments. Note that an intrusion of 3 mm or more
could occur within an optimal magnitude force zone
without reactivation (green zone).
Sometimes there is confusion about how much to activate or deflect an orthodontic appliance. In the traditional
shape-driven concept, deflection was dictated by how far
97
6
Single Forces and Deep Bite Correction by Intrusion
FIG 6-14 F/∆ rate for a 0.017 × 0.025–inch β-Ti continuous intrusion
archwire. The force zones are depicted in colors: red, excessive; green,
optimal; yellow, suboptimal. The low F/∆ rate intrusion archwire is
activated in the green zone. Note that the F/∆ rate is dramatically
reduced due to large interbracket distance.
FIG 6-15 The amount of activation of the intrusion spring is determined by the amount of force required; therefore, it varies with F/∆,
which is dependent on shape, material, and cross section of the wire.
Suppose F/∆ = 8 g/mm; then 10 mm of activation provides 80 g of
intrusive force for four incisors.
FIG 6-16 The intrusion archwire is tied to the anterior segment, but
it is not placed in the bracket slot. Therefore, it does not produce side
effects from unpredictable forces and/or moments.
a tooth should move. For example, 2 mm of intrusion is
required; therefore, the activation is 2 mm. In a forcedriven appliance, however, activation is determined by
the required force level. For example, a stiffer material
like stainless steel rather than β-Ti was used in the intrusion arch shown in Fig 6-15; in this case, 10 mm of activation was required for 80 g of force for intrusion of four
incisors. The amount of intrusion required has nothing
to do with the amount of activation, much like the
distance we are driving on a highway has nothing to do
with the speed we drive. The force-driven appliance uses
the amount of deflection to control force magnitude,
which varies with F/∆, which in turn is dependent on
shape, material, and cross section of the wire.
Force application at a point
The intrusion arch is not placed in the brackets of the
incisors so as to avoid common side effects from extra98
neous forces and moments acting between the incisor
brackets (Fig 6-16). A separate wire is placed in the anterior segment that can be either active or passive. A separate ligature tie is placed from the intrusion arch to the
anterior segment. This tie delivers a single force (the force
acts at a point) or a series of single forces if more than
one tie is used. The intrusion arch is tied incisally or gingivally (Fig 6-17) to the incisor brackets to avoid gingival
impingement.
If an intrusion arch is placed into the incisor brackets,
second- and third-order side effects are commonly
produced. The second-order tip effect is shown in Fig
6-18. If an archwire is placed in the molar auxiliary tubes
and deflected by a force in the incisor region, a curvature
is produced (see Fig 6-18a). This curvature will cause the
roots of the incisors to displace to the mesial (see Fig
6-18b). The patient in Fig 6-19 had good mesiodistal incisor axial inclinations at the start of treatment (see Fig
6-19a). After the intrusion arch was placed in the incisor
Global Characteristics of the Intrusion Force System
a
b
FIG 6-17 (a) A separate wire is placed in the four-tooth segment, and the intrusion arch is tied labially. (b) A wire is placed in the two-tooth
segment, and the intrusion arch is tied gingivally.
a
b
FIG 6-18 Second-order side effects from the intrusion arch placed into the bracket slot. (a) The intrusion arch (green) is activated with the
force (blue arrow), and a curvature is produced (orange). (b) The curvature will produce convergence of the incisor roots.
a
b
FIG 6-19 An example of treatment with an intrusion arch placed into the incisor brackets. (a) Before treatment. (b) After
treatment. Note the second-order side effects of root convergence; furthermore, there is little intrusion.
brackets, the roots moved to the mesial (see Fig 6-19b).
Furthermore, little intrusion occurred. If the intrusion
arch is not placed in the brackets, this side effect produced
by secondary curvature in the intrusion arch can be
avoided.
Even more troubling can be third-order side effects.
Any torque placed in the incisor region will alter the
intrusive force. The green wire in Fig 6-20a is passive with
no active vertical force. The incisor position of the wire
is twisted (inset) so that lingual root torque is produced
during insertion. In order to place the wire into the incisor brackets, a clockwise activation moment (curved blue
arrow) is required to twist the wire so that it is parallel
to the bracket. Note that not only does this moment cause
twist of the incisor portion of the wire, but the entire wire
bends elastically to the occlusal; thus, the placing of a
localized twist in the wire (incisor torque or third-order
moment) produces a vertical extrusive force. The clinical
99
6
Single Forces and Deep Bite Correction by Intrusion
a
b
FIG 6-20 Third-order side effects from an intrusion arch placed into the bracket slot. (a) Placement of a twist in the wire (crown labial,
root lingual direction) produces a vertical extrusive force. (b) The opposite direction of twist in the wire (crown lingual, root labial direction)
produced an additional vertical intrusive force. Additional vertical intrusive force may jeopardize the posterior anchorage unit because of the
increased moment.
a
b
FIG 6-21 Simultaneous intrusion and leveling. (a) Before leveling. A 0.016 × 0.022–inch braided SS archwire was placed in the anterior segment for leveling. (b) After leveling. A 0.017 × 0.025–inch SS archwire was placed for en masse intrusion.
significance of torque producing an extrusive force is
obvious. During intrusion, we carefully measure the
magnitude of the intrusive force. If we purposely or accidentally place any lingual root torque in the intrusion
arch, this can reduce or completely overwhelm the intrusive force. The incisors may actually extrude.
A similar effect can occur if labial root twist is placed
in the incisor portion of the wire (Fig 6-20b). In this case,
an activation torque (moment) is needed to twist the
anterior part of the intrusion arch into the incisor brackets in a counterclockwise direction (blue arrow). The
intrusion arch deflects apically and hence increases the
intrusive force. But is this acceptable since the added
vertical force is in the correct direction? Unfortunately,
any increase in intrusive force can tax the anchorage.
The increased intrusive force produces an increase in
extrusive force on the posterior teeth and, even more
significant, a moment on the posterior teeth, potentially
steepening the occlusal plane. Therefore, because incisor
intrusion requires low force magnitudes that must be
carefully measured, one should avoid placing the intrusion
100
arch into the incisor brackets due to the deleterious effects
of torque.
Is it possible to use round wire for an intrusion arch to
avoid the side effect of vertical forces associated with
torque? Most anterior segments of at least four incisors
exhibit some form of a curvature in the occlusal view, so
third-order anterior moments can be created even with
round wire. In a Class II, division 2 case, however, a round
wire might be satisfactorily inserted into the two central
incisor brackets with minimal side effects from the
moment.
Another advantage of not placing the intrusion arch
directly into the incisor brackets is the convenience and
efficiency of simultaneously aligning and intruding the
incisors. One can begin with flexible, large deflection
wires and sequentially replace them with stiffer wires in
the incisor segment. This avoids the need for a separate
leveling phase, and the same intrusion mechanism is left
in place over a long time frame. Only the incisor segment
requires changing (Fig 6-21).
Changing the Point of Force Application
FIG 6-22 Changing the point of force application, frontal view. Intrusive force is applied off-center for frontal canting correction.
The primary reason for not inserting the intrusion arch
into the incisor brackets is to maintain the predictability
of the force system. A single force can be measured with
a force gauge, and with that information the force system
is statically determinate. Statically determinate means
that the law of static equilibrium is sufficient to solve the
given problem. It may be theoretically true that small
deviations from this condition can occur if the anterior
segment is active or if the intrusion arch twists to deliver
torque to the molars; however, most of the time these
deviations are not relevant and can be practically ignored.
Changing the Point of Force
Application
If a continuous arch applies an intrusive force on an incisor bracket, the force is most likely anterior to the incisor’s
CR so that the incisor may flare or the root apex may
move lingually. It is often desirable during intrusion to
move the force further posteriorly. From the frontal view,
the intrusion force can be applied at the midline or
off-center if canting of the frontal occlusal plane is
required (Fig 6-22).
Let us consider three incisors with identical and typical axial inclinations; an intrusive force is applied to each
incisor at 90 degrees to the occlusal plane (Fig 6-23). If
the force is applied at the incisor bracket (see Fig 6-23a),
an equivalent force system at the CR tells us that the
incisor will intrude and the moment will cause the crown
to flare and the root to move lingually. This intrusive force
is not completely efficient for two reasons. Relative to the
long axis of the tooth, there are two components (shadowed yellow arrows)—a labial force and an intrusive
force—and thus some of the force is lost in an unwanted
labial direction. Furthermore, the moment at the CR that
would flare the incisor requires the arch to be tied back.
Although difficult to be delivered exactly, the ensuing
lingual root movement could be desirable for many Class
II, division 2 patients. It might also be useful in some
extraction Class II, division 1 patients to help minimize
lingual crown inclinations after space closure.
In Fig 6-23b, the intrusive force is applied posterior to
the incisor bracket with a line of action through the CR.
This is possible by extending the wire distally on the anterior segment, forming the posterior extension. Typically
the force is applied near the center of the canine crown.
(If the canine has erupted, the posterior extension is
stepped apically to avoid the canine.) Intrusion can be
efficient because no moment side effect in respect to the
CR is produced. The force direction, however, may not
be ideal if parallel intrusion along the long axis of the
incisor is intended.
The intrusive force in Fig 6-23c is applied lingual to the
incisor’s CR. Here the moment at the CR is in a direction
to tip the incisors lingually. This can be most useful in
the treatment of flared incisors, where intrusive forces
are required.
In Class II, division 2 patients, the force can be successfully applied labially at the bracket (Fig 6-24). The direction of the intrusive force, almost parallel to the long axis
of the tooth and the line of action, is very near to the
CR—the most favorable intrusion configuration for teeth
with parabola-shaped roots. The direction of the CR
moment will flare the crown and improve the incisor
inclination. If lingual brackets are used, intrusion would
still be efficient, but the desirable flaring moment effect
would be lost.
The flared mandibular incisor in Fig 6-25 is problematic
if the force is applied labially at the bracket (see Fig 6-25a).
A very large moment is present that will flare the incisor
more. The intrusive force is best applied distally so that
the line of force is passing through the incisor’s CR (see
Fig 6-25b). It is common to find mandibular arches with
excessive curves of Spee that require more extrusion of
posterior teeth than incisor intrusion. Even with these
patients, one must control the reciprocal vertical forces
on the incisors to avoid unwanted incisor flaring.
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Single Forces and Deep Bite Correction by Intrusion
a
b
c
FIG 6-23 (a to c) Changing the point of force application, lateral view.
The applied intrusive force (red arrow) is replaced with an equivalent
force system at the CR (yellow arrows). Labial or lingual tipping of the
incisor is produced unless the force is through the CR.
a
102
b
FIG 6-24 Intrusive force applied at the maxillary central incisor in a
Class II, division 2 case. The intrusive force (red arrow) at the bracket
is helpful. The replaced force system at the CR (yellow arrows) shows
both a favorable intrusion force and a moment, improving the tooth
inclination.
a
b
FIG 6-25 Intrusive force applied to labially flared mandibular incisors. (a) Force applied at the bracket causes a large moment that will
flare the incisors more labially. (b) The intrusive force is best applied
distally so that the line of force passes through the incisor’s CR.
FIG 6-26 The effect of tying back the archwire. (a) Tying back the
archwire may prevent labial movement of the incisor bracket; however, the incisor still changes its axial inclination. (b) Even if the force
is through the CR, undesirable lingual root movement is inevitable.
Can this incisor flaring be avoided by tying back the
archwire? Unfortunately, that could produce lingual
movement of the root apex, and the root is already too
far lingual. In Fig 6-26a, the tying back of the archwire
may prevent labial movement of the incisor bracket;
however, the incisor still changes its inclination by lingual
movement of the apex. Moreover, the CR moment is so
great that it is difficult to tie back the archwire sufficiently
to prevent actual labial flaring of the incisor bracket. Even
if the distal force is exactly correct as in Fig 6-26b, with
the force through the CR, undesirable lingual root movement still occurs. Generally, teeth that are flared are not
efficiently intruded because the force is not primarily
directed along the long axis of the incisor and a great
horizontal distance from the CR is present. For that
reason, in major intrusion cases the force is redirected
parallel to the long axis of the teeth. This concept is
discussed later in this chapter.
The best solution for intruding a protrusive incisor is
to position the force through the CR or any position
posterior to the incisor bracket by tying bilaterally the
continuous intrusion arch on the posterior extension of
the anterior segment. Another possibility is the use of
two separate intrusion arches (placed bilaterally). This
mechanism, called the three-piece intrusion arch, is more
versatile in treating asymmetries, handling space opening
and closing requirements, and changing force direction.
Three-Piece Intrusion Arch
FIG 6-27 Components of a three-piece intrusion arch.
a
b
c
FIG 6-28 Shape of the three-piece intrusion arch. (a) Shape when deactivated. (b) Shape when activated. (c) The hook on the mesial end
allows free sliding of either the posterior or anterior segment.
FIG 6-29 The force system of the threepiece intrusion arch. (a) Deactivation force
system from the arch (red arrows). (b) Replaced equivalent force system at each CR
(yellow arrows). The intrusive force is easily
applied at the CR by using a distal extension.
a
Three-Piece Intrusion Arch
Figure 6-27 shows the three-piece intrusion arch. It
consists of (1) right and left posterior segments, (2) right
and left intrusion springs, and (3) an anterior segment
with a posterior extension. In addition, a lingual or transpalatal arch is inserted for arch width and symmetry
control. The deactivated and activated shapes of the
three-piece intrusion arch are shown in Figs 6-28a and
6-28b. Separate intrusion springs allow for independent
intrusion activations on each side of the arch. Note the
hook on the mesial end of the intrusion spring, which
allows free sliding of either the posterior or anterior
segment to open or close space (Fig 6-28c). For example,
if posterior teeth are allowed to tip back, the hook would
b
slide distally on the posterior extension of the anterior
segment. The force system acting on the teeth of the
three-piece mechanism is shown in Fig 6-29 (activated
spring). Deactivation forces and moments (red arrows)
are acting on the teeth at the hook and the molar tube.
The equivalent force systems at the anterior and posterior
CRs are shown in yellow. Because the force acts through
the CR of the incisor segment, the incisors will exhibit
translatory intrusion. No flaring will occur. The posterior
segment will extrude, and the posterior occlusal plane
will tend to steepen. Nevertheless, anchorage control can
be very good.
Consider the forces to activate the spring to intrude
four maxillary incisors in Figs 6-30a (deactivated intrusion spring) and 6-30b (activated intrusion spring). For
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6
Single Forces and Deep Bite Correction by Intrusion
a
b
FIG 6-30 Anchorage control in a three-piece intrusion arch. (a) Shape when deactivated. Force is applied to activate the arch (blue arrow).
(b) Shape when activated. There is less reciprocal posterior moment due to the decreased moment arm (30 mm) compared with a continuous
intrusion arch (50 mm).
a total of 60 g, 30 g are applied on each side. The deactivation force (red) to the buccal segments is also 30 g,
which produces a 900-gmm moment (30 g × 30 mm) at
the posterior CR. The primary concern in anchorage loss
is the moment steepening the plane of occlusion. Here
the moment is smaller because of the light 30-g intrusive
force and also because of the shortened distance from
the hook to the posterior CR. (The horizontal distance
from the incisor bracket to the posterior CR is greater
with the continuous intrusion arch.) Added to the promising anchorage potential are five posterior teeth on each
side. (Including a second molar can be helpful.) Note that
the posterior extension has been stepped apically to avoid
the canine bracket. The lingual arch contributes to the
overall anchorage control because it prevents individual
movements of the posterior segments; if any tooth movement does occur, an en masse movement of all posterior
teeth would be required.
Altering Force Direction
During intrusion, it can be desirable to alter the direction
of the force. Intrusion of single-rooted teeth is most efficiently accomplished with forces parallel to the long axis
of the tooth. The intrusion spring in Fig 6-31 acts at 90
degrees to the occlusal plane. To change its direction so
that it is parallel to the long axis of the incisors, a distally
directed chain elastic can be added. Not much force is
needed. If the intrusive force from the intrusion spring
is 30 g, the force can be similar—not hundreds of grams.
A force from a chain elastic that undergoes degradation
due to the fluids in the mouth is not very accurate, and
a metal spring might be more predictable (Fig 6-32), but
the simplicity and comfort of the chain elastic, if compensated for and carefully observed, allow for its use.
104
Another method to redirect the force is to use a simple
cantilever (Fig 6-33a). The angle of the ligature tie in Fig
6-33b denotes the force direction. To facilitate the use of
a cantilever for this application, the position of the helix
can be varied so that the direction of the intrusive force
can be kept relatively constant (Fig 6-34).
The direction of the intrusive force from the spring
may also be altered by bending the posterior extension
on the anterior segment at a suitable angle (Fig 6-35).
This can only work if there is minimal friction between
the hook and the wire extension. Note that the intrusive
force (red arrow) is altered because the line of force is
perpendicular to the distal extension. With minimal friction, the force is always perpendicular to the distal wire
extension so that the force is acting along the long axis
of the tooth. Note that the magnitude of intrusive force
along the axis is larger than the measured vertical force
(Fig 6-36a). Imagine a large friction between the wires
such as a hook placed on the wire; then the force direction
(red arrow) does not change because the frictional force
provides the component of force parallel with the posterior extension of the wire (Fig 6-36b). If friction is low,
the horizontal force component along the wire is lost,
and the force can be redirected. Of course, some friction
must be present, so the force redirection will be unpredictable.
A common clinical challenge is to have a line of action
parallel to the long axis of an incisor and through its CR.
Can this be accomplished if the point of force application
is at the incisor bracket? We can add a horizontal force
component to the vertical intrusive force shown in red
in Fig 6-37a. The resultant force (yellow arrow) acts
through the CR, but its direction is not parallel to the
long axis (dotted black line) of the incisor. In order to
have the proper direction, the point of force application
must be changed to lie posterior to the lateral bracket on
Altering Force Direction
FIG 6-31 Altering the force direction in a three-piece intrusion arch
using an elastic chain. A distally directed chain elastic can be added
to redirect the intrusive force parallel to the long axis of the incisors.
a
FIG 6-32 Altering the force direction in a three-piece intrusion arch
using a metal coil spring. A force from a metal coil spring might be
more predictable.
b
FIG 6-33 Altering the force direction in a three-piece intrusion arch by changing the angle of the ligature tie. (a) Deactivated shape. (b)
Activated shape. The length of the cantilever arm is adjusted so that the angle of the ligature tie coincides with the intended force direction.
a
b
FIG 6-34 Altering the force direction in a three-piece intrusion arch by changing the position of the helix. (a) Deactivated shape. (b) Activated
shape. The position of the helix can be varied so that the direction of the intrusive force can be kept relatively constant.
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6
Single Forces and Deep Bite Correction by Intrusion
b
a
FIG 6-35 Altering the force direction in a
three-piece intrusion arch by bending the
distal extension. The intrusive force will act
perpendicular to the distal extension, provided that there is no friction between the hook
and the distal extension.
a
FIG 6-36 (a) With minimal friction, the force is always perpendicular to the posterior extension of the wire. Therefore, the measured intrusive force is a vertical component of force
accompanied by a horizontal component of force. (b) With high friction, the direction will
remain perpendicular to the occlusal plane.
FIG 6-37 Control of the line of action in each type of
intrusion arch. (a) In a continuous intrusion arch, if the
intrusive force is applied at the bracket, the resultant
force can be through the CR by adding a horizontal
force component. However, its line of action (dotted
black line) is not parallel to the long axis of the incisor.
(b) In a three-piece intrusion arch, the point of force
application is moved posterior to the lateral bracket on
the extension so that the resultant is not only through
the CR but also parallel to the long axis of the incisor.
b
a
b
c
d
the posterior extension of the anterior segment (Fig
6-37b). This is one application of a three-piece intrusion
arch.
Thus, force control involves not only force magnitude
and force constancy but also point of force application
(line of action) and direction. Many other possibilities
are summarized in Fig 6-38. All forces are parallel to the
long axis of the incisor except Fig 6-38b. Parts a and d
have opposite CR moment effects, while parts b and c
produce pure intrusional translation without a moment
(rotation). Translation along the long axis of the tooth is
only possible in c by changing the force angle and point
of force application so that its line of action goes through
106
FIG 6-38 (a to d) Various possibilities of intrusive
force application replaced at the CR. All forces are
parallel to the long axis of the incisor except b. Parts
a and d have opposite CR moment effects, and parts
b and c have pure intrusive translation without a
moment. Translation along the long axis of the tooth
is only possible in c by changing the force angle so
that its line of action goes through the CR; b also
translates, but the force is no longer parallel to the
long axis of the tooth.
the CR; b also shows translation, but the force is no longer
parallel to the long axis of the tooth.
The importance of force direction and point of force
application is demonstrated in Fig 6-39. The young adult
patient shows considerable bone loss, extrusion, and flaring of the maxillary right central incisor associated with
localized periodontitis (Figs 6-39a to 6-39d). Leveling
with a continuous edgewise archwire is limited because
the forces are too great and unknown. Also, it would be
difficult to direct the force along the long axis of the incisor. Therefore, a three-piece intrusion arch with right and
left chain elastics was used to redirect the force along the
long axis of the tooth. At the start, a single force of 20 g
Altering Force Direction
FIG 6-39 The patient shows bone
loss, extrusion, and flaring of the maxillary right central incisor. (a to d) Before
treatment. (e) Lateral view. The resultant
is made parallel to the long axis of the
tooth by adding a distal force. (f) Frontal view. (g) Occlusal view. (h) Frontal
view after treatment. (i) Occlusal view
after treatment. (j) Pre- and posttreatment radiographs. Note the improved
bone level and attachment. The tooth
is still functioning in the mouth after
30 years.
a
b
c
d
e
f
h
i
parallel to the long axis of the tooth and posterior to the
CR of the maxillary right central incisor was planned.
This resultant single force was resolved into 14 g of intrusive and distal components of force (Fig 6-39e). From the
frontal view, 14 g of intrusive component forces were
resolved again bilaterally into 7.5 g on the right side and
6.5 g on the left side of the intrusion spring (Fig 6-39f ).
The 14 g of distal force was also resolved into 7.5 g on
the right and 6.5 g on the left elastic chains (Fig 6-39g).
Thus, in 3D, the resultant force acted through the CR
g
j
from the frontal view and lay slightly posterior to the CR
from the lateral view. Also note that the point of force
application (located at the hook of the intrusion spring)
is distal to the canine bracket (see Fig 6-39g). After intrusion, the tooth was stabilized with a continuous arch (Figs
6-39h and 6-39i). Note the improved bone architecture
and attachment (Fig 6-39j). After treatment, no pocket
was evident during periodontal probing. The tooth is still
functioning in the mouth after 30 years.
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Single Forces and Deep Bite Correction by Intrusion
a
b
c
d
FIG 6-40 The patient has a large interlabial gap and 9 mm
of incisor showing at rest. A continuous intrusion arch was
used. (a and b) Before treatment. (c and d) After treatment.
(e) Superimposition shows the dramatic intrusion of the
maxillary incisor.
e
The patient in Fig 6-40 had a large interlabial gap and
a maxillary incisor that was positioned 9 mm below the
upper lip at rest (Figs 6-40a and 6-40b). The maxillary
and mandibular premolars were extracted. To correct
the vertical incisor exposure, a continuous intrusion arch
was used. The mandibular arch curve of Spee was maintained, and a curve of Spee was built into the maxillary
arch to fit the mandibular curvature (Figs 6-40c and
6-40d). Note the dramatic intrusion of the maxillary incisor (Fig 6-40e). Unlike hinging open a mandible during
deep bite correction, intrusion is a slow movement. One
should not expect over 1 mm of movement per month.
In this patient, no growth was present, so the deep bite
correction was accomplished mainly by maxillary incisor
intrusion.
The force system of simultaneous intrusion and retraction of four maxillary incisors with the three-piece intrusion arch is shown in Fig 6-41. This is particularly
108
appropriate if a flared maxillary incisor is present (Figs
6-42a to 6-42c). A force lingual to the CR produces an
intrusive force at the CR and a moment that tips the
incisors lingually. The patient in Fig 6-42 was treated with
a three-piece intrusion arch with a distal force from a
chain elastic. Initially the two incisors were retracted,
and this was followed by retraction of the four maxillary
incisors as a unit (Figs 6-42d and 6-42e). During the en
masse movement, the point of force application was
moved posteriorly to about the position of the center of
the canine (Fig 6-42f ). Note that considerable retraction
of the maxillary incisor occurred with minimal anchorage
loss (Figs 6-42g to 6-42k). This was to be expected because
the only distal forces on the anterior region were approximately 20 g per side. These forces were used not for direct
retraction but rather to redirect the force of the intrusion
arch so that the resultant was parallel to the long axis of
the incisors.
Altering Force Direction
FIG 6-41 The force system of simultaneous intrusion and retraction
with the three-piece intrusion arch. The yellow arrows show the replaced force system at the CR.
a
b
c
d
e
f
g
h
i
j
k
FIG 6-42 The patient has a flared maxillary incisor. (a) Lateral cephalometric radiograph before treatment. (b and c) Frontal and lateral intraoral views before treatment. (d and e) A three-piece intrusion arch was used to produce intrusive force and a moment that tipped the incisors
lingually. (f) Occlusal view after intrusion and retraction. (g to i) Frontal and lateral intraoral views after treatment. (j and k) Superimposition of
lateral tracings and occlusogram. Note that considerable retraction of the maxillary incisor occurred with minimal anchorage loss.
109
6
Single Forces and Deep Bite Correction by Intrusion
FIG 6-43 Leveling of a Class II, division 2
case with a continuous full archwire. (a) Before leveling. (b) After leveling. Note that little
intrusion has occurred. The lateral incisors
erupted, and the posterior plane of occlusion
was steepened.
a
b
a
b
c
d
a
b
FIG 6-44 (a to d) The two central incisors
are intruded to the level of the lateral incisors first.
FIG 6-45 Leveling of a Class II, division 1 case. (a) In many cases, four maxillary incisors
can be intruded as a unit to the level of the canine. (b) Mandibular arches can display an
excessive curve of Spee, with the four incisors stepped as a unit occlusally. This step can be
used advantageously if intrusion is required.
Avoiding Initial Leveling Arches
It is common practice to use high-deflection and lowerforce wires to align and level at the beginning of treatment. This may not be a good idea if genuine intrusion
of incisors is a treatment goal. If a Class II, division 2
malocclusion is leveled, for example, with a nickel110
FIG 6-46 Deep bite may not be apparent
with flared incisors. Note after lingual tipping
that the CR must be moved apically (∆).
titanium wire, little intrusion occurs (Fig 6-43). The lateral
incisors will erupt, and the posterior plane of occlusion
will steepen because of the large moment at the posterior
CR from the incisor intrusive force. It is better to take
advantage of the original anatomical geometries in tooth
arrangement. Thus, in the Class II, division 2 malocclusion, the two central incisors are intruded to the level of
Avoiding Initial Leveling Arches
a
b
c
FIG 6-47 Patient with severely flared incisors. (a) The vertical overlap looks normal before treatment. (b) A three-piece
intrusion arch was used for retraction of the maxillary incisors
and leveling of the mandibular arch. (c) After treatment. (d)
Visual treatment objectives. Note that significant intrusion of
the maxillary incisors was anticipated. (e and f) Lateral cephalometric radiographs before and after treatment. The threepiece intrusion arch achieved the planned treatment objectives.
d
e
the lateral incisors (Fig 6-44). Similarly, in many Class II,
division 1 patients, the four maxillary incisors can be
intruded as a unit to the level of the canine (Fig 6-45a).
Some mandibular arches can display an excessive curve
of Spee with four incisors stepped as a unit occlusally.
This step can be used advantageously if intrusion is
required (Fig 6-45b).
Flared incisors can give the illusion that no vertical
discrepancy exists. In Fig 6-46, a straight wire from the
posterior teeth lines up with the incisors (dotted line at
the brackets). Note that when the incisor is tipped lingually
to a correct axial inclination, the CR moves apically (∆).
Early recognition of the potential step between the incisors and the canine allows for the most efficient treatment. This is not necessarily force redirection along the
flared tooth’s long axis. If space is available, the incisors
first can be retracted by tipping. Later, with the better
inclination, the incisor can be intruded.
f
Because intrusion of the incisors may be necessary in
many patients, it is obvious that the original malocclusion
alignment is most important. The patient in Fig 6-47 has
normal vertical overlap (or overbite) at the start of treatment but potential deep bite after retraction of the maxillary incisors by tipping (Fig 6-47a). Intrusion and retraction of the maxillary anterior segment are simultaneously
accomplished by the three-piece mechanism (Figs 6-47b
to 6-47d). Some incisor intrusion occurred in the mandibular arch as the posterior curve of Spee was leveled. Note
the significant intrusion of the maxillary incisors in Figs
6-47e and 6-47f.
The anatomical tooth arrangements with their steps
between segments discussed here for intrusion may also
be suitable for resolution with extrusive mechanics (see
chapter 7).
111
6
Single Forces and Deep Bite Correction by Intrusion
a
b
FIG 6-48 Special anchorage control. (a) The reciprocal force system in the posterior segment shows extrusive force and tip-back moment.
(b) Occlusal chewing forces may help negate the vertical extrusive force, but the tip-back moment remains.
FIG 6-49 Mean difference before and after the treatment in degrees
(with standard deviation shown in parentheses), showing the effect of
the number of teeth on occlusal plane cant in the buccal segment. If a
posterior segment has only a first molar in comparison with a canine
to first molar, anchorage is poor, and the first molar will tip back.
Special Anchorage Considerations
Historically, the major limitation in achieving genuine
incisor intrusion has been anchorage control. In response
to intrusive forces, extrusive forces operate on the posterior teeth. Note in Fig 6-48 that, in addition to the extrusive force, a large moment at the CR steepens the occlusal plane of the posterior segment; it is this moment that
is the most important aspect of intrusion anchorage
control. Occlusal chewing forces may help negate the
vertical extrusive force, but the tip-back moment remains.
Sound anchorage principles such as increasing the
number of teeth in the buccal segment, increasing the
rigidity of the posterior segment wire, and placing a
lingual arch are beneficial. If a posterior segment has only
a first molar in comparison with a canine to first molar,
anchorage is poor, and a steepening of the posterior
segment will occur (Fig 6-49). One should look beyond
the number and root size of the posterior teeth. Consider
112
the distance between the anterior point of attachment
and the posterior CR. Figure 6-50a shows a flared incisor.
A good strategy might be to retract the incisors (even
partially) before intrusion (Fig 6-50b). This reduces the
perpendicular distance to the CR (L2 < L1) and thereby
reduces the unwanted steepening moment to the posterior teeth. Adding teeth to the buccal segments also influences the position of the posterior CR. Adding a canine
moves the CR forward, and adding a second molar moves
it backward. Although the canine root mass is smaller,
its addition to the posterior segment may be more significant for anchorage than the addition of the second molar
(Fig 6-51). The steepening moment is reduced by shortening the distance to the posterior CR. It is the same
principle as moving the attachment hook posteriorly with
a three-piece intrusion arch, which has already been
discussed.
The use of a headgear can negate the steepening occlusal moment from any intrusion arch. A number of
Special Anchorage Considerations
a
b
FIG 6-50 The effect of length of the intrusion arch. (a) A flared incisor. It is better to retract the incisors even partially before intrusion. (b) Retracting the incisors reduces the length of the intrusion arch (L2 < L1 ) and thereby reduces the unwanted steepening moment to the posterior teeth.
a
b
FIG 6-51 The location of the posterior CR in accordance with the
addition of teeth. (a) Adding a canine to the anchorage unit moves
the CR anteriorly. (b) Adding second molars to the anchorage unit
moves the CR posteriorly.
FIG 6-52 The use of a headgear with an intrusion arch. A number
of different directions (shadowed red arrows) can negate undesirable
tip-back moments (curved red arrow) from the intrusion arch. The
headgear forces are much larger; therefore, minimal headgear wear
is sufficient.
different directions (shadowed red arrows) are possible,
as shown in Fig 6-52, with both occipital and cervical
headgears. Because intrusion forces are relatively small
and headgear forces are much larger, minimal headgear
wear is required. Usually little or no headgear is required
to back up anchorage if the mechanics described in this
chapter are used (Fig 6-53).
The use of implants and TADs can certainly simplify
anchorage considerations for incisor intrusion. However,
the improved anchorage should not be used as an excuse
to employ excessive intrusive forces. Figure 6-54 shows
a patient in whom the first premolars were used as
anchorage to intrude all the maxillary incisors. Note the
genuine intrusion of incisors relative to the canine as a
reference. Later the first premolar was extracted; therefore, this tooth served like an implant for the incisor
intrusion stage of treatment.
Sometimes a high-pull headgear anterior to the CR of
the maxillary arch (or anterior to the CR of the posterior
segment to back up the anchorage) is used to intrude
incisors and to reduce the cant of the occlusal plane of
the maxillary arch. A high-pull J-hook headgear anterior
to the CR is commonly used to intrude incisors along
with a full archwire (Fig 6-55a). However, the direct application of intermittent, fluctuating high-level forces in the
incisor region may lead to root resorption. Because the
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6
Single Forces and Deep Bite Correction by Intrusion
FIG 6-53 The effect of headgear on occlusal plane cant (with
standard deviation in parentheses). Usually little or no headgear is
required to back up anchorage if intrusion force levels are kept low.
a
a
b
b
maxillary archwire may not be sufficiently rigid, it is not
a good idea to place heavy forces in the incisor region
that could lead to root resorption. If a headgear is used,
it is preferable to use an inner bow inserted into the molar
tube for improved force control and distribution of forces
away from the incisors (Fig 6-55b) than the J-hook headgear. It is better to use the headgear only to help control
the posterior segments during incisor intrusion, with the
headgear force on the posterior teeth and not the incisors.
Typically, in patients who require both incisor and
canine intrusion, it is considered too challenging to
114
FIG 6-54 The use of a tooth to be extracted
as an anchorage unit. (a) Before intrusion.
The first premolars were used as anchorage
to intrude the four maxillary incisors. (b) After
intrusion. Note the genuine intrusion of the
incisors relative to the canine as a reference.
Later the first premolar was extracted.
FIG 6-55 (a) A high-pull J-hook headgear
anterior to the CR is commonly used to intrude incisors along with a full archwire. The
direct application of intermittent fluctuating
high-level forces in the incisor region may
lead to root resorption. (b) Headgears with
inner and outer bows better distribute force
to the posterior teeth.
achieve intrusion of six anterior teeth by dental anchorage alone. These patients may require orthognathic
surgery or the use of plates and TADs. If canine intrusion
is needed, it might be best to gain full control with a
continuous edgewise arch stepped around the canine (a
canine bypass). A separate cantilever can be employed
for the canine intrusion (Fig 6-56). Another possibility is
a bypass arch with a rectangular loop welded to vertically
align the canine (Fig 6-57). The loop can also be used to
simultaneously rotate the canine or change mesiodistal
axial inclination.
Special Anchorage Considerations
FIG 6-56 Separate canine intrusion using a
cantilever. A continuous archwire bypassing
the canine provides the best anchorage for
separate canine intrusion.
FIG 6-57 Separate canine intrusion using a
loop. A rectangular loop welded to a bypass
archwire is used to simultaneously rotate the
canine or change mesiodistal axial inclinations during intrusion. (a) Before treatment.
(b) After treatment.
a
b
a
b
a
b
FIG 6-58 The use of a reciprocal tip-back moment. (a) Wire is not inserted into the full posterior segment so that the continuous intrusion
arch produces a tip-back moment on the first molar only. The incisors will intrude, and reciprocally the molar will tip back, helping to correct
the Class II malocclusion. (b) A three-piece intrusion arch can be more efficient because it allows an unrestrained molar tip-back.
The last anchorage consideration is to accept and practically use the reciprocal tip-back moment from the intrusion arch. Wire is not inserted into the entire posterior
segment, so that the continuous intrusion arch shown in
Fig 6-58a produces a tip-back moment on the first molar
only. The incisors will intrude, and reciprocally the molar
will tip back, helping to correct a Class II malocclusion.
A three-piece intrusion arch can be more efficient
because it allows an unrestrained molar tip-back effect
(Fig 6-58b). Tip-back mechanics from cantilevers and
three-piece arches are further discussed in the next chapter on posterior extrusion mechanics.
The Class II malocclusion in Fig 6-59 can be improved
using a continuous intrusion arch because the reciprocal
moment steepens and tips back the posterior teeth. If
posterior segments are tipped forward, one can take
advantage of this inclination problem during incisor
intrusion for acceptable molar tip-back.
115
6
Single Forces and Deep Bite Correction by Intrusion
a
b
c
d
e
f
FIG 6-59 (a to f) A patient with deep bite and a Class II malocclusion. Note not only that the incisors are intruded but also that the Class II
malocclusion is improved as the reciprocal moment tips back the posterior teeth.
References
1. Dellinger E. A histologic and cephalometric investigation of premolar intrusion in the Macaca speciosa monkey. Am J Orthod
1967;53:325–355.
2. van Steenbergen E, Burstone CJ, Prahl-Andersen B, Aartmand
IH. Influence of buccal segment size on prevention of side effects from incisor intrusion. Am J Orthod Dentofacial Orthop
2006;129:658–665.
Recommended Reading
Burstone CJ. Applications of bioengineering to clinical orthodontics.
In: Graber TM, Vanarsdall RL (eds). Orthodontics: Current Principles
and Techniques, ed 4. Philadelphia: Mosby, 2005:293–330.
Burstone CJ. Biomechanics of deep overbite correction. Semin Orthod
2001;7:26–33.
Burstone CJ. Deep overbite correction by intrusion. Am J Orthod
1977;72:1–22.
Burstone CJ, Marcotte MR. Problem Solving in Orthodontics: GoalOriented Treatment Strategies. Chicago: Quintessence, 2000.
116
Choy K, Pae EK, Kim KH, Park YC, Burstone CJ. Controlled space
closure with a statically determinate retraction system. Angle Orthod
2002;72:191–198.
Romeo DA, Burstone CJ. Tip-back mechanics. Am J Orthod 1977;
72:414–421.
Shroff B, Lindauer SJ, Burstone CJ, Leiss JB. Segmented approach to
simultaneous intrusion and space closure: Biomechanics of the threepiece base arch appliance. Am J Orthod Dentofacial Orthop 1995;107:
136–143.
Shroff B, Yoon WM, Lindauer SJ, Burstone CJ. Simultaneous intrusion
and retraction using a three-piece base arch. Angle Orthod 1997;67:
455–462.
van Steenbergen E, Burstone CJ, Prahl-Andersen B, Aartman IHA.
The role of a high pull headgear in counteracting side effects from
intrusion of the maxillary anterior segment. Angle Orthod 2004;74:480–
486.
Vanden Bulcke M, Sachdeva R, Burstone CJ. The center of resistance
of anterior teeth during intrusion using the laser reflection technique
and holographic interferometry. Am J Orthod 1986;90:211–219.
Vanden Bulcke M, Sachdeva R, Burstone CJ. Location of the center
of resistance of anterior teeth during retraction using the laser reflection technique. Am J Orthod 1987;90:375–384.
Problems
For problems 1 through 7, a 60-g force (30 g per side) is placed in the middle of a four-tooth anterior segment
by a continuous intrusion arch.
1. Replace the force system at the CR of the anterior and
posterior segments.
2. The incisors are flared. Replace the force system at
the CR of the anterior and posterior segments.
3. The posterior CR has moved anteriorly because the
canine was included in the anchorage unit. Replace
the force system at the CR of the anterior and posterior segments.
4. The posterior CR has moved further anteriorly because
the maxillary second molar was excluded from the
anchorage unit. Replace the force system at the CR
of the anterior and posterior segments.
5. Find the magnitude and sense of the headgear force
(FHG ) on the given line of action to keep the occlusal
plane from canting. Assume an exact balance.
117
Problems
118
6. Replace the force system at the CR of the anterior
and posterior segments.
7. Compare the anchorage using only one molar with
problem 1.
8. A 30-g force is placed on the distal extension of the
anterior segment (on each side) by a three-piece
intrusion arch. Replace the force system at the CR of
the anterior and posterior segments.
9. A 30-g force is placed on the distal extension of the
anterior segment (on each side) by a three-piece
intrusion arch. Compare the effects of FA, FB, and FC.
Replace the force system at the CR of the anterior
and posterior segments for each force.
10. A continuous intrusion arch is tied off-center to the
left. What effect does it have on the cant of the occlusal plane of the incisors and the posterior segments
in the frontal view? Posterior segments are rigidly
connected by a lingual arch. Replace the force
through the CR for the answer.
11. An intrusive force is directed parallel to the long axes
of the flared incisors by a three-piece intrusion arch.
A coil spring (FH ) and intrusion arch (F V) are the
components of the 30-g intrusive force. Find the
magnitude of FH and FV. What will happen to the incisors and the posterior segment?
7
Deep Bite
Correction by
Posterior
Extrusion
“He who loves practice without theory is like the sailor
who boards ship without a rudder and compass and
never knows where he may cast.”
— Leonardo da Vinci
Most patients with deep bite (excessive vertical overlap) require some
extrusion of posterior teeth. Along with facial growth, this can lead to
stable deep bite correction. Archwires with an intrusive force to the
incisors can produce posterior extrusion that is a combination of rotation and translation. In the mandibular arch, translatory extrusion and
reduction in the occlusal plane angle are described as Type I extrusion.
Type II extrusion is produced by anterior arch disengagement with
posterior vertical elastics. Combining Type I arch mechanics with an
extrusive headgear offers an additional method for parallel posterior
extrusion. Leveling arches may lead to posterior extrusion, but control
is usually lacking. The placement of exaggerated curves of Spee in
the maxillary arch or reverse curves of Spee in the mandibular arch
produces unwanted moments. The completely flat arch is an orthodontic construct that is not necessarily valid biologically.
119
7
Deep Bite Correction by Posterior Extrusion
b
FIG 7-1 Type I posterior extrusion. An intrusive force to the anterior
teeth produces a tip-back moment and an extrusive force on the
posterior segments.
FIG 7-2 Type II posterior extrusion. A single force at the center of
resistance (purple circle) produces parallel extrusion of the posterior
segment without rotation.
b
a
b
FIG 7-3 Force system from straight-wire leveling. The posterior teeth tip back and extrude (a), and an anterior moment displaces the canine
root to the mesial (b).
N
ot every patient with deep bite requires intrusion
of the incisors. Because intrusion requires demanding mechanics, it is desirable to identify the situations where posterior extrusive mechanics can be
employed. Patients who exhibit good mandibular growth
or have a large interocclusal space may be successfully
corrected by increasing the vertical dimension. Let us
consider three treatment possibilities by extrusion: (1)
intra-arch tooth movement where initial alignment is
poor; (2) fully aligned arches where respective occlusal
planes converge anteriorly; and (3) synchronous occlusal
plane changes associated with Class II or Class III maxillomandibular elastics.
Type I Posterior Extrusion
Some patients exhibit a mesially tipped posterior segment
that requires a combination of extrusion and rotation
(Fig 7-1). An intrusive force to the canine or anterior
teeth produces a tip-back moment on the posterior
segments. This combination of rotation and extrusion is
classified as a Type I posterior extrusion for deep bite
120
correction. It is the easiest extrusion to produce during
leveling because intrusive forces to the canines or the
incisors will create a rotational moment (tip-back) on the
posterior segment (see chapter 6).
Other patients require extrusion of the occlusal plane
of the posterior segment without changing its cant. In
Fig 7-2, parallel extrusion (Type II extrusion) is shown.
The mechanics of Type II extrusion require special methods to eliminate the undesired tip-back moments.
Type I extrusion is most commonly seen in the mandibular arch and is associated with an exaggerated curve of
Spee. Although a straight wire can create a desirable
moment on the posterior segment, anteriorly a straight
wire could displace the canine roots to the mesial (Fig 7-3).
Sometimes a reverse curve of Spee is placed in a continuous mandibular archwire. This might help alleviate some
of this canine root displacement; however, it is too imprecise to solve the problem and may create other unwanted
effects.
The three-piece intrusion arch discussed in chapter 6
can be used to extrude and tip back (rotate) the posterior
segment (Fig 7-4). The appliance and force system look
similar to the mechanism for intrusion, but there are
Type I Posterior Extrusion
a
b
FIG 7-4 Three-piece tip-back (or intrusion) arch. (a) The force system rotates and extrudes the posterior segment. (b) A larger intrusive force
than that used for intrusion is applied to all six anterior teeth so that intrusion is kept to a minimum.
a
b
FIG 7-5 Tip-back of posterior teeth. (a) When indicated, a posterior stabilizing archwire is not inserted, and a tip-back spring is engaged on
the first molar only. (b) The premolars will tip individually, not en masse, due to the transseptal fibers or a ligated ligature wire. Space (∆) distal
to the canine can be gained without any distal force.
FIG 7-6 The tip-back is accomplished without flaring of the mandibular incisors by placing the intrusive force distal to the CR of the
anterior segment (red arrow). The replaced equivalent force system
(yellow arrows) at the CR indicates that the anterior segment will
intrude and tip slightly to the lingual.
significant differences. During intrusion, the forces are
distributed to the incisors only. For extrusion, on the
other hand, the intrusive force is applied to all six anterior teeth. Also, larger forces are used—perhaps 100 g
per side instead of 30 to 40 g per side. Emphasis is placed
on delivering a large enough moment to efficiently tip
back and extrude the posterior segment. Figure 7-4a
shows a –2,000-gmm tip-back moment. Note that the
intrusive force acts at the center of resistance (CR) of
the anterior segment to prevent incisor flaring. If indicated, space can be effectively regained by individual
tip-back of posterior teeth in selected cases without any
distal force. The premolars will spontaneously tip distally
due to transseptal fibers or ligature wire tying together
the posterior teeth. The hook on the tip-back spring,
which can slide distally on the anterior segment, allows
for distal movement of the posterior segment (Fig 7-5).
Thus, space can be opened to alleviate moderate crowding in the mandibular arch. Unlike most continuous
archwires, this is accomplished without flaring of the
mandibular incisors by placing the intrusive force distal
to the CR of the anterior teeth (Fig 7-6). This further
ensures that no incisor flaring will occur and allows a
longer range of distal sliding on the distal extension of
the anterior segment. A lingual arch or a transpalatal
arch (TPA) can be useful to maintain arch form, arch
width, and molar buccolingual axial inclinations.
121
7
Deep Bite Correction by Posterior Extrusion
a
b
c
d
FIG 7-7 The components of a three-piece tip-back (intrusion) arch. (a) Posterior segment, anterior segment, and tip-back
extrusion spring. (b) Left and right posterior segments are stabilized by a passive lingual arch. (c) Deactivated shape.
(d) Activated shape.
FIG 7-8 The continuous tip-back spring is indicated if distal movement of the posterior teeth is not required. Anterior force position
variation can still be achieved.
a
b
c
FIG 7-9 Varying the anterior point of force application. Intrusive force acting at the CR (a), anterior to the CR (b), and distal to the CR (c). The
position of the force influences not only the flaring tendency of the incisors but also the rotation of the posterior segments—incisor intrusive
forces applied further forward produce more posterior rotation.
122
Type II Posterior Extrusion
FIG 7-10 (a) The deep bite makes it difficult to
bond brackets. (b) A fixed bite plate was inserted (not shown) in the maxillary arch temporarily,
and a continuous tip-back arch was placed to
extrude and rotate the posterior teeth.
a
a
b
b
FIG 7-11 A mandibular bite plate opens the FIG 7-12 (a and b) A maxillary bite plate is easily bonded to a palatal horseshoe arch. Fabribite and creates occlusal space for the posterior cation can be either directly in the mouth or in a laboratory procedure.
teeth to extrude.
The three-piece tip-back (Type I extrusion) arch (or
intrusion arch) consists of (1) a posterior segment (right
and left segments connected by a lingual arch), (2) an
anterior segment, and (3) right and left tip-back and
extrusion springs (Fig 7-7). The design can be simplified
if distal movement of the posterior teeth is not required
by using a continuous tip-back spring (without sliding
hooks) from the right to the left molar auxiliary tubes
(Fig 7-8). The point of force application to the incisors
can also be moved closer to the CR or further anteriorly
to the incisor brackets (Fig 7-9). The position of the force
influences not only the flaring tendency of the incisors
but also the rotation of the posterior segments—incisor
intrusive forces applied further forward produce more
posterior rotation relative to the extrusive posterior translation.
It might be difficult to level a mandibular arch using
wires and brackets in a patient with deep bite, because
occlusal forces can shear off the mandibular incisor brackets. Figure 7-10 shows a Type I continuous intrusion arch
placed in the auxiliary molar tubes. An intrusive force
(150 g) was placed at the incisor midline, causing the
posterior teeth to extrude and rotate. A fixed bite plate,
a temporary appliance, was inserted into the maxillary
arch to prevent the occlusion from debonding the
mandibular anterior brackets. The bite plate was removed
once the mandibular arch was leveled by extrusion (after
approximately 10 weeks).
Type II Posterior Extrusion
It is much easier to extrude and rotate a posterior segment
during intra-arch alignment than to extrude it in a parallel manner, maintaining the posterior plane of occlusion.
The mechanics of Type II (parallel) extrusion require vertical forces, and intra-arch moments are to be avoided.
Undesirable arch moments can come from a step apically
to the incisors, mandibular curves of Spee, and almost
any type of incisor intrusion arch.
The appliance for Type II extrusion relies on disengaging the posterior teeth with an anterior bite plate. Posterior segments are then free to spontaneously erupt or,
more likely, to be extruded using vertical elastics. The
mandibular bite plane opens the bite and creates occlusal space for the posterior teeth to extrude (Fig 7-11). It
is usually preferable not to have any continuous archwires
in place so that the separate maxillary and mandibular
posterior segments can be extruded as a unit. A bite plate
in the maxillary arch can be attached to a precision palatal horseshoe arch (Fig 7-12). A growing patient in Fig
7-13 shows anterior deep overbite (Figs 7-13a to 7-13c).
An anterior bite plate and vertical elastics were used.
Deep overbite was corrected by parallel extrusion of the
maxillary posterior segments (Figs 7-13d to 7-13f ). The
maxillary archwire was cut to facilitate this parallel extrusion. Figures 7-13g and 7-13h show the lateral cephalometric radiographs before and after treatment.
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7
Deep Bite Correction by Posterior Extrusion
a
b
c
d
e
f
FIG 7-13 (a to c) A patient with deep overbite.
(d to f) An anterior bite plate and vertical elastics were used to extrude the maxillary posterior segments in parallel. (g and h) Cephalometric radiographs before and after treatment.
g
h
When the bite plate is in place, the posterior teeth are
separated. Depending on the malocclusion, they can be
joined together as a unit (Fig 7-14) or handled as individual teeth (Fig 7-15). Vertical elastics are applied through
the CR of a segment if parallel eruption is required. The
elastic will flatten the maxillary and steepen the mandibular posterior occlusal plane cant (see Fig 7-14). Sometimes, the direction of the elastics will be slightly Class II
or Class III or not exactly at the CR; hence, care should
be taken to minimize any problems by carefully evaluating the progress of the tooth movement. In summary, for
deep bite correction via Type II extrusion, undesirable
moments are avoided by avoiding mechanics from the
archwire and relying directly on vertical forces from vertical maxillomandibular elastics.
Undesirable side effects are possible with these
mechanics. The elastic force is buccal to the CR, which
means that the posterior teeth could tip toward the
lingual. For the short period of time in which the elastics
are used, no major problem should be observed; for full
control of arch width, lingual arches can be used.
124
Figure 7-16a shows a deep bite in which the discrepancy
is in both arches. Note the step relationship between the
canine and the first premolar brackets. The posterior
teeth are disengaged by a bite plate (pink in Fig 7-16b).
Space is now present between the maxillary and mandibular posterior teeth. The resultant of the box elastic
(green) is close to the CR of both posterior segments. The
posterior segments now erupt without changing the
occlusal plane cants (Fig 7-16c).
It could be advantageous to use a fiber-reinforced
composite (FRC) ribbon instead of a stabilizing archwire
in the anchorage arch (Fig 7-16d). Here the discrepancy
is in the maxillary arch, with the step between the canine
and the first premolar. Only the maxillary teeth are bracketed. For some patients, the deep bite should be corrected
at the initial stage of treatment. One approach is to place
the FRC ribbon before brackets and vertical elastics are
placed (Figs 7-16e to 7-16g). The FRC can then be
removed and brackets placed for the continuation of
treatment.
Type II Posterior Extrusion
FIG 7-14 Posterior teeth are separated by a bite plate and extruded
en masse using a stabilizing archwire. The resultant forces (red arrows) from the elastic pass distal to the CR for extrusion and rotation.
FIG 7-15 Posterior teeth are extruded individually without a stabilizing archwire when individual movement is required for maximum
intercuspation and alignment.
a
b
c
d
e
f
FIG 7-16 (a) A deep bite that requires Type II extrusion of posterior teeth in both arches. (b)
The posterior teeth are disengaged by a bite plate (pink), and a rectangular elastic (green) is
placed. The resultant forces (red arrows) are close to the CR of both posterior segments. (c)
The posterior segments are extruded without changing their occlusal plane cants. (d) The
discrepancy is in the maxillary arch only, and an FRC ribbon is used for full mandibular anchorage. (e) The FRC can be used before full bracketing to correct the deep bite in the early
stages of treatment. (f) After deep bite correction, connecting wire must be used to maintain
the extrusion. (g) The discrepancy is in the mandibular arch, and the same principle of placing
an FRC ribbon is applied.
g
125
7
Deep Bite Correction by Posterior Extrusion
a
b
FIG 7-17 The leveling of a deep bite by a continuous archwire. (a) Red
arrows show the forces on the teeth at the brackets. (b) The replaced
equivalent force system at each CR. Note the large moment acting on
the posterior segment because the canine extrusive force has a large
perpendicular distance to the posterior CR. (c) The occlusal plane will
steepen as a result of posterior rotation and extrusion accompanied
by possible intrusion of the anterior segment.
c
126
FIG 7-18 Deep bite leveled by a continuous
archwire. Note the steep maxillary occlusal
plane and flat mandibular occlusal plane
(black lines). The anterior vertical overlap (or
overbite) was little improved (circle).
FIG 7-19 Parallel eruption of posterior teeth can be accomplished
by adding an extrusive force (FB ) distal to the CR of the posterior
segment. As L1 is limited, FB has to be problematically large to balance
the moment produced anteriorly by an intrusive force (FA ).
If an intrusive force is placed on the incisors with a
continuous archwire (Fig 7-17a), the response of the
posterior teeth is both extrusion and rotation. Rotation
is inevitable because an extrusive force has a large perpendicular distance to the posterior CR. Therefore, the
replaced equivalent force system at the posterior CR
shows a large moment (Fig 7-17b). As a result, the occlusal plane will be steepened by posterior rotation and
extrusion accompanied by possible intrusion of the anterior segment (Fig 7-17c). It still may be advantageous to
Type II Posterior Extrusion
a
b
FIG 7-20 A cervical headgear with a force as far back as possible
and mainly directed downward can be a very efficient way to balance the steepening moment from the anterior force. (a) A resultant
force (yellow arrow) passes through the CR. (b) The moments of the
replaced force systems of the headgear (yellow arrow) and the tipback spring (shadowed yellow arrows) at the CR cancel each other
out. (c) The resultant force on the posterior segment (red arrow)
acting on the CR.
c
use a wire connecting the anterior and posterior teeth so
that some incisor intrusion will occur even if the objective is primarily posterior extrusion. At a minimum, it is
desirable to prevent the incisors from erupting further.
But how can we prevent the undesirable side effects of
maxillary occlusal plane steepening and mandibular
occlusal plane flattening (Fig 7-18)? Parallel eruption of
posterior teeth can be accomplished by adding an extrusive force (FB ) distal to the CR of the posterior segment
(Fig 7-19). In order for the resultant force to go through
the CR (required for posterior translation), the moments
from both anterior and posterior forces must sum to zero,
measured around the CR. The use of a vertical elastic is
problematic because the force (FB ) should be larger due
to the limited length of the moment arm (L1 in Fig 7-19).
One of the solutions to this problem could be a cervical
headgear (Fig 7-20) with a force as far back as possible
and mainly directed downward; this can be a very efficient
way to balance the steepening moment from the anterior
force. Of course, the force system from the headgear is
not absolutely balanced with the intrusive force to the
incisors, because the intrusion arch acts continually and
the headgear is worn only part-time. Note in Figs 7-20b
and 7-20c that an exact balance for parallel posterior
extrusion requires a line of action though the maxillary
arch CR that is downward and backward. This is not an
instantaneous balance but a balance over time, requiring
careful monitoring of the patient’s progress.
The patient in Fig 7-21 was treated with a continuous
intrusion arch and a cervical headgear. The headgear force
was directed downward and backward and produced a
large moment to balance the intrusive force. The patient
had favorable mandibular growth related to his puberty
growth spurt, which accounted for most of the Class II
malocclusion and deep bite correction (see Fig 7-21i).
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7
Deep Bite Correction by Posterior Extrusion
a
FIG 7-21 A patient with deep bite treated with a continuous intrusion arch and a cervical
headgear. (a to d) Before treatment. (e to h) After treatment. (i) The maxillary posterior teeth
were significantly extruded, and the occlusal plane cant was maintained. Favorable mandibular
growth aided the correction of the Class II malocclusion and deep bite.
b
c
d
e
f
g
h
i
Curves and Reverse Curves
of Spee
A method that is commonly suggested for deep bite
correction is the placement of an exaggerated reverse
curve of Spee in the mandibular archwire or an exaggerated curve of Spee in the maxillary archwire. However,
128
both diagnostically and biomechanically, this does not
make much sense.
Let us start by considering the occlusal plane or occlusal curvature goals of orthodontic treatment. Naturally,
the occlusal surfaces do not line up in a plane; some curve
of Spee is present. As shown in Fig 7-22, little curvature
may be evident anterior to the first molars. Curvature
distal to the first molar can be marked because of the
Curves and Reverse Curves of Spee
FIG 7-22 The natural occlusal plane shows the curve of Spee. Little
curvature is present anterior to the first molars. The curvature distal
to the first molar can be marked because of the axial inclinations of
the second molars as they erupt.
FIG 7-23 Leveling the curve of Spee could make a Class II malocclusion worse as the maxillary second molars move forward and
the mandibular second molars move backward due to the leveling
moments.
FIG 7-24 Misconception of the mandibular curve of Spee in which
the teeth are analogous to spokes on a wagon wheel.
immature axial inclinations of the second molars as they
erupt. Later in development, the maxillary second molar
crowns can exhibit a similar distoaxial inclination. This
posterior curvature is natural and should not be leveled.
Leveling moments could make a Class II malocclusion
worse as the maxillary second molars move forward and
the mandibular second molars move backward (Fig 7-23).
Why then do some orthodontists believe that a finished
dental arch should have teeth aligned along a flat occlusal plane? Perhaps it is an inability to easily correct deep
bites or the desire to have a wire slide through a molar
tube during sliding mechanics. There are other sound
biomechanical methods to solve deep bite and space
closure problems while still allowing a normal occlusal
plane curvature.
Another misconception relates to patients with an
excessive mandibular curve of Spee. It is assumed that
the brackets are aligned along a curvature with roots
diverging from the front to the back of the arch. It has
been suggested that the teeth are analogous to spokes on
a wagon wheel (Fig 7-24). In the typical deep bite patient,
however, axial inclinations of the teeth may be relatively
normal because the deep bite is caused by vertical position and not tooth angulation. The patient in Fig 7-25 has
a deep bite, but the tooth angulations are relatively
normal; the mandibular incisor roots are not labially
positioned through the labial plate of the bone as in the
“wagon wheel” model. What is required is primarily vertical tooth movement either by extrusion or intrusion with
little change of axial inclination. As discussed previously,
any posterior eruption should be by translation (Type II
extrusion). Because a curvature in a wire produces
moments, a reverse curve of Spee is not indicated for the
mandibular arch in this type of malocclusion.
129
7
Deep Bite Correction by Posterior Extrusion
FIG 7-25 A patient showing a severe curve
of Spee and deep bite. (a and b) Incisor angulations are relatively normal. (c and d) The
incisors have overerupted.
a
b
c
d
FIG 7-26 The effects of a reverse curve of Spee in
the mandibular arch. Note that no vertical change
occurred on individual teeth but that the roots of
all teeth converged toward the center of the arch.
(Reprinted from Kojima and Fukui1 with permission.)
The effects of placing a continuous archwire with a
reverse curve of Spee were studied by Kojima and Fukui1
using a numerical analysis (Fig 7-26). Brackets were
aligned in a straight line, and a reverse curve of Spee was
incorporated into the mandibular arch. Note that no
vertical change occurred on individual teeth, but the roots
of all teeth converged toward the center of the arch. This
is to be expected because uniform curves that are part
of a circle basically deliver moments, not vertical forces.
The vertical overlap (or overbite) might improve, but
favorable axial inclinations are sacrificed.
Let us follow what would happen if an archwire with
an exaggerated reverse curve of Spee was placed in a
mandibular arch (Fig 7-27a), contributing to a deep bite.
Unlike the patient in Fig 7-25, there is an axial inclination
problem that needs correction. A first premolar was
extracted, and the anterior and posterior segments tipped
during space closure. Even here, where moments are
130
needed, a reverse curve of Spee arch is a mistake. The
tooth (bracket) discrepancy is only between the canine
and the second premolar. To the mesial of the canine and
distal of the second premolar, all brackets are in good
alignment (Fig 7-27b). Now let us do a tooth by tooth
analysis, starting with the problem area—canine to
second premolar. A straight wire between these two
brackets (Class VI geometry; see chapter 14) will produce
equal and opposite couples (Fig 7-27c). This would be
appropriate to move roots together, provided the arch is
tied back or these teeth are tied together. Now let us add
a curvature to the entire archwire, placing it only between
these two teeth (Fig 7-27d). Again, this is not a problem
because the desirable moments moving the roots together
are augmented. A smooth curvature or a segment of a
circle produces equal and opposite couples.
The difficulty arises when a curvature is placed along
the entire arch in all the brackets. Note in Fig 7-27e that
Curves and Reverse Curves of Spee
a
b
c
d
e
f
FIG 7-27 The force system on the mandibular teeth from an archwire with a reverse curve of Spee. (a) After extraction of
the first premolar, the anterior and posterior segments tipped into the extraction site, causing an axial inclination problem. An
archwire with a severe reverse curve of Spee was placed. (b) There is an intersegmental discrepancy only between the canine
and the premolar. (c) If a straight wire is placed between these two brackets, the Class VI geometry will produce favorable
equal and opposite couples. (d) A curvature between these two brackets will augment the favorable force system. (e) The
same force system is produced between the premolar and the first molar. Note that the moment in the wrong direction moves the root of the
second premolar distally. (f) Similarly, the force system between the first and second molars will produce unwanted distal root movement of
the first molar. If all moments are added, the first molar will feel no moment because moments from the adjacent teeth (second premolar and
second molar) cancel each other out. The second molar will feel an unwanted tip-back moment.
the curvature still produces equal and opposite moments,
but now the root of the second premolar moves distally.
It can also be seen that any curvature between the first
and second molars (Fig 7-27f ) will produce unwanted
tooth movement of the first molar. If all moments are
added, the first molar will feel no moment because
moments from the adjacent teeth (second premolar and
second molar) cancel each other out. The second molar
will feel an unwanted tip-back moment. Thus, even in a
patient requiring axial inclination change, using wire
moments can be incorrect if delivered from a continuous
archwire with a reverse curve of Spee. Therefore, if a
continuous archwire is to be used for this malocclusion,
the reverse curvature should only be placed between the
canine and the second premolar.
In an extraction case in which loss of control has led
to tipping, a continuous archwire (either straight or
augmented with a localized curvature) can be used to
correct the problem. This mechanism has a small range
of action because the wire activation is over a small
interbracket distance between the canine and the second
premolar. A segmental approach can lead to a superior
force system allowing for a simple root spring. Figures
7-28a and 7-28b show the required force system to correct
the axial inclinations and to eliminate the excessive curve
of Spee. The arch is divided into two segments: an anterior segment and a posterior segment. A 0.018 × 0.025–
inch β-titanium wire with a curvature (a reverse curve
of Spee) connects the auxiliary tubes on the first molar
and on the canine (Figs 7-28c and 7-28d). The interbracket distance has been markedly increased from 5
mm to 14 mm, decreasing the force-deflection rate and
producing a more constant moment magnitude that acts
over a longer range. The good axial inclinations within
131
7
Deep Bite Correction by Posterior Extrusion
a
b
c
d
FIG 7-28 Segmental approach to the malocclusion in Fig 7-27 with separate, passive anterior and
posterior segments. (a) The correct force system from a root spring (red arrows). (b) The response to
the force system. (c) Deactivated shape of a root spring. The curvature delivers the correct required
force system continuously. Blue arrows show the activation force system to activate the spring. (d)
Activated shape. Red arrows show the deactivation force system acting on the tooth segments.
a
b
each segment are maintained as an efficient en masse
movement is achieved.
The patient in Fig 7-29 had excessive vertical overlap
during the finishing phase of orthodontics, and it was
decided to correct this overlap by placing a localized
exaggerated curve of Spee in the maxillary arch. This
resulted in the roots of the canine and the first premolar
contacting. This localized curvature arch produced the
wrong force system for a vertical displacement discrepancy mainly caused by poor incisor bracket height. The
correct force system requires mainly vertical forces.
It has been commonly said that leveling a curve of Spee
requires added arch length. In fact, a general rule of
thumb is that for every millimeter of depth of curve of
Spee, one millimeter of added arch length is needed.
However, this oft-repeated idea is incorrect for many
reasons. First, there are different types of excessive curves.
132
FIG 7-29 (a) A patient showing excessive
vertical overlap during finishing. (b) A localized exaggerated curve of Spee was placed
in the maxillary arch, which resulted in the
roots of the canine and the premolar contacting.
The tooth arrangement with normal axial inclinations
(Fig 7-30a) does not require more space in the arch
because the teeth and brackets need to move only vertically. The confusion regarding added length arose because
formerly orthodontists would take a flexible brass wire
and contour to the occlusal surface of the dental cast.
When the wire was flattened out, it became longer. Unfortunately, this flattening and lengthening has nothing to
do with the amount of space required. We can see this
same effect if a continuous archwire is placed into the
brackets of an arch with an excessive curve (Fig 7-30b).
The inserted orange archwire is shorter than the extended
flat archwire. This is an appliance problem that can lead
to flaring of the incisors and possible space opening but
tells us nothing about the space needed. If the wire is free
to slide distally, this lengthening effect disappears.
Curves and Reverse Curves of Spee
a
b
FIG 7-30 Leveling the curve of Spee. (a) Normal axial inclinations do not require added space in the arch. (b) The misconception that added
space is required came from the fact that there is a difference (∆) between the projected lengths of a curved wire and a straight wire. This is
one limitation of leveling with a straight continuous archwire.
FIG 7-31 Tipped posterior teeth with abnormal axial inclinations
may require added space, but the amount may not be significant.
a
b
FIG 7-32 Many curves of Spee appear similar, but tooth position
and the required biomechanics are different for each situation.
(a) Parallel eruption with no moments. (b) Mandibular posterior segment tip-back. (c) Root movement with equal and opposite moments
between the canine and the first premolar.
c
Some added arch length may be required for tipped
teeth with abnormal axial inclinations. The posterior
segments in Fig 7-31 are tipped mesially and hence
require more space. But even with this type of occlusal
curvature, the diagram shows that little added space is
needed for small angular changes. In short, the need for
added arch length to compensate for an excessive curve
of Spee has been exaggerated.
Some curves of Spee may be too large, and reduction
is indicated. The biomechanics will differ because the
original anatomical basis (tooth position) varies widely
(Fig 7-32). Note that parts a, b, and c have similar occlusal curvatures but structurally are very different if one
looks at the axial inclinations of the teeth.
133
7
134
Deep Bite Correction by Posterior Extrusion
Reference
Braun S, Hnat WP, Johnson BE. The curve of Spee revisited. Am J
Orthod Dentofacial Orthop 1996;110:206–210.
1. Kojima Y, Fukui H. A numerical simulation of tooth movement by
wire bending. Am J Orthod Dentofacial Orthop 2006;130:452–459.
Germane N, Staggers JA, Rubenstein L, Revere JT. Arch length considerations due to the curve of Spee: A mathematical model. Am J Orthod
Dentofacial Orthop 1992;102:251–255.
Recommended Reading
Roberts WW III, Chacker FM, Burstone CJ. A segmental approach to
mandibular molar uprighting. Am J Orthod 1982;81:177–184.
Andrews FL. The six keys to normal occlusion. Am J Orthod 1972;
62:296–309.
Romeo DA, Burstone CJ. Tip-back mechanics. Am J Orthod 1977;
72:414–421.
Baldridge DW. Leveling the curve of Spee: Its effect on the mandibular arch length. J Pract Orthod 1969;3:26–41.
Woods M. A reassessment of space requirements for lower arch
leveling. J Clin Orthod 1986;20:770–778.
Problems
1. A three-piece tip-back (intrusion) arch is used to
upright and extrude the posterior segments. A moment
of –2,000 gmm is needed on each posterior CR. How
much force should be applied anteriorly between the
central and lateral incisors?
2. A three-piece tip-back (intrusion) arch is used to
upright and extrude the posterior segments. A
moment of –2,000 gmm is needed on each posterior
CR, and force is applied on a distal extension. How
much force is applied anteriorly? Compare the extrusive force on the posterior segment between problem
1 and problem 2.
3. To upright the second molar, –2,000 gmm is required. How much vertical force (FA , FB ) is applied in a and b? Which
is better? Discuss.
a
b
4. The three-piece tip-back arch is activated with a 50-g force. What happens to the anterior segment in a ? What
happens to the anterior segment in b ? What happens to the posterior segment in each?
a
b
135
Problems
136
5. An incisor bite plate bonded to the mandibular incisors
disengages the posterior occlusion. A vertical elastic
is used to erupt the maxillary posterior segment by
translation (Type II extrusion). Are there side effects
on the mandibular arch? If so, what is their effect on
the deep bite correction?
6. The maxillary arch right posterior segment requires
Type II extrusion with unilateral vertical elastics. The
mandibular arch was stabilized by a full continuous
archwire for rigid anchorage. Replace the force system
on the maxillary and mandibular CRs. Discuss the
desirable and undesirable effects.
7. The three-piece tip-back spring on the mandibular
arch and a Class II elastic are placed. Two anterior
forces (100 g) are measured. Find the equivalent force
systems at the CR of the maxillary arch, the mandibular posterior segment, and the mandibular anterior
segment.
8. A uniform reverse curve of Spee is placed in the
mandibular arch. The measured moment values
between each of the two brackets are given sequentially (eg, first molar and second molar). Moments
vary because of different angulations and interbracket
distances. Add the total moments for the canine,
second premolar, first molar, and second molar. Ignore
any extraneous vertical and horizontal forces and the
moments anterior to the canine. Are the moments
uniform on each tooth? Discuss the effects.
8
Equilibrium
“Every body persists in its state of being at rest or of
moving uniformly straight forward, except insofar as it is
compelled to change its state by force impressed.”
— Isaac Newton
The important concept of equilibrium is based on Newton’s First Law.
In a body at rest or at uniform velocity, the sum of all the forces and
moments is zero. When an archwire is inserted into all of the brackets
with activation forces, the wire is elastically deformed yet is in equilibrium. The archwire does not accelerate or exert any force to move
the patient. Therefore, the activation force diagram is in equilibrium. A
free-body (equilibrium) diagram and the laws of equilibrium are very
useful in solving for unknowns in an orthodontic appliance. It aids in the
selection of the best appliance or adjustment. Clinicians always have
treatment goals for their therapy, and unless the archwire or appliance
can be placed in equilibrium, the goals are not possible. Deactivation
force diagrams depicting forces that act on the teeth can be obtained
from the equilibrium diagram using Newton’s Third Law by reversing
force and moment direction. The equilibrium equations are not only
useful for the understanding of appliances but are also applicable to
understanding the biomechanics of tooth movement and physiologic
stresses in the temporomandibular joint.
137
8
Equilibrium
a
c
T
he most important concept from physics that can
be applied to orthodontics is the equilibrium principle. It is based on Newton’s First Law, which states
that a body remains at rest or in motion with a constant
velocity unless acted upon by an external force. The overall mechanism that delivers orthodontic forces to teeth
and bones is a spring, and this energy-storing device is
the active component of all orthodontic appliances. The
spring, which can be fabricated from many materials,
including metal, rubber, or polymer, stores the mechanical energy charged by the orthodontist during activation
and releases it slowly. The orthodontic spring comes in
many different applications and shapes: archwires, coil
springs, elastics, aligners, and loops, among others.
Free-Body Diagram
It may be surprising that orthodontic appliances are in
equilibrium, because they are required to deliver forces
to move teeth. Equilibrium means that no resultant forces
are acting on the orthodontic appliance. To understand
the equilibrium, we need to compose a valid free-body
diagram of an appliance. Let us consider the simplest
appliance: an elastic. Imagine that we isolate this object
of our interest (an elastic) and eliminate other objects. If
the elastic is hooked on the mandibular second molar
(Fig 8-1a), we can bring it forward with an applied force
of 100 g (Fig 8-1b). The free-body diagram shows only
the forces acting on the elastic (Fig 8-1c). Especially in
orthodontics, the weight of appliances is ignored. An
138
b
FIG 8-1 Newton’s First Law. (a) An elastic is hooked on the mandibular second molar and is ready to be activated. The orthodontist
brings it forward with a force of 100 g. (b) Once the elastic is engaged,
it remains stretched and keeps the state of rest in equilibrium. (c) In
a free-body diagram, only the forces acting on one item are shown.
Here, the elastic is simplified as a simple rectangle.
elastic can be shown as a simple rectangle. The next step
is to draw all the forces and moments acting upon the free
body and to check if the sum of all depicted forces and
moments is zero, making it a valid free-body diagram. In
this case, two blue forces are responsible for stretching the
elastic, and they are considered activation forces because
they are applied to the appliance to activate it.
The activated (stretched) elastic is also in equilibrium,
because the sum of all forces is zero (100 g + [–100 g] =
0 g). Clinically, we know that the elastic is in equilibrium;
it is not accelerating. Most importantly, it is not exerting
an unbalanced resultant force on the patient. It does not
push the patient up to the ceiling, out the front door, or
out of the window. Once it is recognized that the appliance is in equilibrium (not pushing, pulling, or rotating
the patient), the laws of equilibrium can help us solve for
unknown forces. In Fig 8-1c, the force on the right (100
g) was measured with a force gauge. We do not need to
measure the left force because the elastic has to be in
equilibrium; therefore, the left force must be equal in
magnitude and opposite in direction (–100 g). The activated elastic demonstrates Newton’s First Law of a body
at rest or at uniform velocity. Figure 8-2, on the other
hand, demonstrates Newton’s Third Law: For every action
(or force), there is an equal and opposite reaction (or
force). In this figure, Sisyphus pushes the rock up (action,
blue arrow), and the rock pushes down on Sisyphus (reaction, red arrow).
Now let us take a closer look at the canine hook (Fig
8-3). Two forces (blue arrow and red arrow) are present:
The hook holds the stretched elastic forward; the blue
Free-Body Diagram
FIG 8-2 Newton’s Third Law. Sisyphus
pushes the rock up (action, blue arrow),
and the rock pushes down on Sisyphus
(reaction, red arrow) with equal and opposite forces to each other.
FIG 8-3 Two equal and opposite forces from an appliance and the
tooth are action and reaction. Clinically the force required for activation of the appliance is called activation force, and the force on the
tooth as the appliance deactivates is called deactivation force.
FIG 8-4 Deactivation force diagram
showing separate unbalanced deactivation forces on both arches. This is not an
equilibrium diagram, although all of the
forces sum to zero.
activation force deforms the appliance, and the red deactivation force acts to move the teeth during deactivation
of the spring. Newton tells us that there are always two
equal and opposite forces when two bodies interact: in
this case, the activation force and the reactive deactivation
force. It is the deactivation force (red arrow) that will
move teeth (see Fig 8-3). Note that Fig 8-4 is the same as
Fig 8-1b except that the force directions are reversed.
This force diagram, showing only the force system on the
objects of interest, such as forces from the spring acting
on the teeth (and excluding forces from mastication,
gravity, etc), is called a deactivation force diagram.
Because the activated appliance is always in equilibrium,
its force diagram should always be in equilibrium.
Figure 8-1c is the free-body diagram of the elastic;
conceptually, it correctly depicts the elastic (the appliance) in a state of rest. Figure 8-4, however, shows separate unbalanced forces on the maxillary and mandibular
arches. Strictly speaking, it is not a free-body diagram
because more than one object (tooth) with force acting
upon it is depicted. Each tooth is not shown in equilibrium
because the diagram does not show all forces (eg, stresses
at the periodontal ligament [PDL]) acting on the object
(ie, the teeth). Figure 8-5 shows two valid force diagrams
in equilibrium. Figure 8-5a demonstrates that the appliance is in equilibrium from the activation forces. Figure
8-5b shows that the deactivation force and the restraining forces (stresses) from the periodontium are also in
equilibrium. Thus, teeth as well as appliances can have
suitable and correct equilibrium diagrams if the PDL
support is considered.
An open coil spring is shown pushing both canines
distally in Fig 8-6a. Because only distal forces acting on
the canine are depicted in the figure, it might imply that
using an open coil spring between the canines can move
the canines distally without anchorage loss. However,
this implication is wrong. In this case, two appliances
such as the open coil spring and the archwire are used.
The open coil is compressed by the canine bracket in the
mesial direction, and further buckling is prevented by
the archwire running inside the lumen of the coil spring.
The archwire produces distributed forces to the open coil
139
8
Equilibrium
FIG 8-5 Two equilibrium diagrams. (a) The
spring with blue activation forces. (b) The
tooth (blue) with a red deactivation force
and stresses (forces) in the periodontium.
Compression-side and vertical components
of the stresses in the PDL are not depicted
because they also sum to zero.
a
b
FIG 8-6 It is not possible to produce an appliance with distal force only. (a) An open coil
spring is shown pushing both canines distally. This implication is wrong. (b) The open coil
spring is in equilibrium. (c) The archwire is
in equilibrium. (d) Overall deactivation force
system acting on the teeth.
a
b
c
d
spring posteriorly so that the open coil spring is in equilibrium (Fig 8-6b). The archwire is also in equilibrium by
distal forces at each free end and distributed forces from
the open coil spring (Fig 8-6c). Therefore, the resultant
forces from two appliances would be distal forces at the
ends of the archwire and mesial forces acting on each end
of the open coil spring to compress it. The deactivation
force system acting on the teeth is depicted in Fig 8-6d.
No matter how noble and sophisticated an appliance is,
it cannot overcome the laws of physics; a single distal
force is not possible. What if we don’t cinch back the wire,
to eliminate mesial forces? The wire will fly out of the
patient's mouth. Remember: There is no such thing as a
free lunch.
In our simple example of a 100-g maxillomandibular
elastic attached to the maxillary arch, what is the force
on the mandibular arch at the molar? The answer comes
from applying both Newton’s First and Third Laws. Step
1 is to place the appliance in equilibrium (see Fig 8-1b).
140
Because the measured force to the right is 100 g, the
unknown force to the left must also be 100 g in magnitude
(First Law). Step 2 is to reverse the direction of the forces
from the appliance activation force diagram (see Fig 8-4).
This gives us the deactivation forces acting on the teeth
(the dental arches); hence, the conclusion is that the elastic gives equal and opposite forces to the dental arches.
Some orthodontists have believed that this conclusion
comes solely from Newton’s Third Law, but that is incorrect. It relies on the equilibrium application of the First
Law. Although the Third Law is involved in reversing
force direction, no calculation is involved in Step 2
because there is always an equal and opposite force
between bodies.
Some orthodontists have also incorrectly assumed that
it is Newton’s Third Law that explains anchorage. For
example, this assumption would state that during space
closure, an appliance (eg, a loop) delivers equal and opposite force systems to the anterior and posterior segments.
Types of Support and Number of Reactions
a
b
c
FIG 8-7 Types of support in an intrusion spring. (a) Fixed support. (b) Roller support. (c) Pinned support. Orthodontic appliances
are connected by one of these three basic types of support.
Newton’s Third Law seems to support this supposition,
but it is not a correct application. The First Law, on the
other hand, provides the correct answer that some forces
may be equal while the force system can differ between
the anterior and posterior teeth.
Types of Support and Number of
Reactions
An activated spring or appliance is elastically deformed
and engaged in attachments such as brackets, hooks, or
eyelets. The appliance is in equilibrium by the activation
force systems from these attachments. The interface
between the appliance and the attachment is called the
support. The allowable force systems are different based
on the types of support. It is crucial to identify the behavior of supports in order to compose a valid equilibrium
diagram to analyze the force system from the appliance.
There are basically three types of support in orthodontics:
fixed support, roller support, and pinned support. Figure
8-7 shows each type.
Fixed support
Fixed supports restrain the appliance in any direction of
translation or rotation. It can resist horizontal forces (Fx),
vertical forces (Fy), and moment (Mz), which means there
are three reactions in two dimensions. The distal end of
an intrusion spring that is inserted into the auxiliary tube
or slot of an edgewise bracket with no play and infinite
friction is a fixed support (Fig 8-7a). A soldered joint in
an appliance is a typical fixed support.
Roller support
Roller supports allow the appliance freedom to slide and
rotate over the roller. The surface can be in any angle and
it can resist a single force (Fy) perpendicular to the surface
(Fig 8-7b). It has only one reaction—a single force that
has a direction perpendicular to the contact point. A
hook at the mesial end of an intrusion spring on a flat
wire can be a roller support if we ignore the friction. A
Begg bracket is a roller support because it cannot exert
horizontal force or moment.
141
8
Equilibrium
a
b
c
FIG 8-8 The equilibrium diagram of the intrusion spring. (a) The isolated activated intrusion spring is depicted in orange. (b) Types of support
at each end, and unknowns. (c) The unknowns are solved using static equilibrium. The sum of forces and moments is zero.
Pinned support
Pinned supports allow only rotation, and they resist both
horizontal and vertical forces (Fx, Fy). The direction of a
force can be changed, but there is no moment. A hook
with infinite friction (see Fig 6-36b) or a connection
between an elastic and a hook or eyelet is an example of
a pinned support (Fig 8-7c). The J-hook headgear is a
pinned support as well (see Fig 4-29).
Let us now consider an intrusion spring where both
forces and moments are present. Figure 8-8 shows the
activated intrusion arch; the activated arch is depicted in
orange, and the deactivated arch shape is depicted in
transparent green (Fig 8-8a). The activated spring is
isolated and replaced with a black rectangular box shape.
The distal part of the spring is inserted into the tube, the
anterior hook is engaged in the stabilizing archwire, and
they are replaced with corresponding types of supports.
The allowable force systems on the supports are depicted,
yet magnitude and directions are not determined. A
downward activation force is present anteriorly, while
upward and horizontal forces and a counterclockwise
couple are depicted at the posterior end of the spring (Fig
8-8b). A downward activation force (100 g) is measured
and present anteriorly. The activated intrusion spring is
in equilibrium because the sum of the vertical forces (100
g + [–100 g]) is zero and the sum of the horizontal force
is zero (an anterior roller support cannot produce any
horizontal force; therefore, Fh = 0). The sum of the
moments, measured from the molar tube (red dot), is
also zero (100 g × 25 mm = +2,500 gmm and a moment
at the molar tube of –2,500 gmm). The free-body diagram
of the appliance is completed (Fig 8-8c).
In Fig 8-9, the forces are reversed in red to show the
deactivation forces that are in the direction of the tooth
movement. The distal extension arm in the anterior
142
segment feels intrusive force, and the molar bracket feels
an extrusive force and clockwise moment that tips the
posterior segment backward and the root forward.
The forces acting on the molar can be depicted in a
number of ways (Fig 8-10): (1) a force and a couple at the
center of the bracket (see Fig 8-10a); (2) a downward force
and two single forces producing a couple at the edge of
the bracket (see Fig 8-10b); or (3) two forces with the
larger force at the mesial (see Fig 8-10c). They are all
equivalent.
Figure 8-8c is the free-body diagram showing the intrusion spring in equilibrium. If a 100-g force is measured
in the incisor region, the activation force and moment
on the molar bracket need not be measured. Here
Newton’s First Law is used to calculate any unknowns.
Figure 8-9 reverses the direction of all forces and moments
based on the Third Law to show the forces acting on the
teeth; no calculations are needed. It is the no-brainer part
of the analysis. Clinicians are usually more interested in
this deactivation force diagram—forces acting on the
teeth (see Fig 8-9)—than the activation force diagram
from which it was derived. Although all forces and
moments sum to zero, the deactivation force diagram
does not show any single object in equilibrium and should
not conceptually be called an equilibrium diagram. Misinterpretations of deactivation force diagrams are common;
the forces and moments in these diagrams are independent and act separately at the molar tube and at the incisor bracket. These forces are what move the teeth. It is
not an equilibrium diagram because the teeth are in
equilibrium by the deactivation force and the stresses in
the PDLs, which are not depicted. By contrast, in the
equilibrium diagram, the activation forces and moments
act on the entire spring and not on separate entities.
However, once we understand the process above, the
intermediate process is omitted so that the teeth and
Basic Concepts and Formulas of Equilibrium
a
FIG 8-9 The deactivation force diagram shows the forces acting on
the tooth. The directions of the deactivation force system (red) are
the reverse of the activation force system shown in Fig 8-8c (based
on Newton’s Third Law).
b
c
FIG 8-10 The force system acting on the molar can be depicted in a
number of ways: a force and a couple at the center of the bracket (a), a
downward force and two single forces producing a couple at the edge
of the bracket (b), or two forces with the larger force at the mesial (c).
FIG 8-11 Incorrect expression. The force systems acting on the
incisor and molar are not action and reaction. This is a misuse of
Newton's Third Law.
appliance are depicted together with reverse directions
of force on one diagram. This final diagram is a deactivation force diagram of the teeth or a free-body diagram
of an appliance with reversed force systems.
Note that it is not correct to state that the molar feels
the equal and opposite reaction of extrusive force and
tip-back moment to the action of an intrusive force on
the incisor in accordance with Newton's Third Law, which
is frequently found in many literature sources (Fig 8-11).
This example demonstrates and emphasizes why the
principle of equilibrium—Newton's First Law—is applied
on a single object (ie, the intrusion spring), not between
two objects such as the incisors and molar. On the other
hand, Newton's Third Law, or the law of action and reaction, is always between two objects (ie, between the spring
and the tooth).
Basic Concepts and Formulas of
Equilibrium
The formulas used in equilibrium calculation are very
simple. Because the appliance is at rest and not accelerating, the following information is known:
1.
∑F = 0
2.
∑M = 0
where F is force and M is moment. These two conditions
can then be written as six equations, one for each component in 3D space: ΣFx = 0, ΣFy = 0, ΣFz = 0, ΣMx = 0,
ΣMy = 0, and ΣMz = 0 (Fig 8-12a). In this chapter,
143
8
Equilibrium
FIG 8-12 Degrees of freedom. The object in 3D (a) has six degrees of freedom,
whereas the object in 2D (b) has three
degrees of freedom.
a
b
a
b
FIG 8-13 Equilibrium tells us that the sum of all forces and moments should be zero. Any unbalanced force (a) or moment (b) is impossible
on the archwire.
a
b
FIG 8-14 No matter how diverse and complicated the forces on
the teeth may be, the archwire is always in equilibrium. (a) Deactivation forces (red arrows) act independently on each tooth. (b) The
activation force system (blue arrows) acts on the entire archwire.
Moments and forces sum to zero. All forces and moments are artistic
approximations.
144
we will consider only two dimensions, so there are
only three equations needed for equilibrium: ΣFx = 0,
ΣFy = 0, and ΣMz = 0 (Fig 8-12b).
Imagine that we are inserting an archwire into all of
the brackets of a malocclusion and we must apply forces
to elastically deform the wire or the PDL. When the
patient leaves the office, there is no resultant acting on
the archwire. The equilibrium formula tells us that any
unbalanced force is impossible on the arch (Fig 8-13a).
The balanced forces are possible because they sum to
zero. The same is true with any moments (Fig 8-13b).
One cannot deliver only a moment from an arch to rotate
a molar (see Fig 8-13b). Balanced moments produced by
a couple and a force or a couple alone are possible to meet
equilibrium requirements. Thus, in Fig 8-13, the unbalanced force system to the archwire is impossible—no
amount of wire bending or bracket placement can make
it happen. A simple equilibrium diagram can keep us out
of trouble by quickly identifying what is impossible from
a given archwire.
Basic Concepts and Formulas of Equilibrium
a
b
FIG 8-15 The equilibrium principle provides boundary conditions
to solve for unknown forces. (a) The activation moments at both
the incisors (500 gmm) and the molar bracket (–3,500 gmm) are
measured, and FA and FB are unknowns. (b) The spring is in static
equilibrium by vertical forces of 86 g. The vertical forces do not need
to be measured. (c) The deactivation force system acting on the teeth
(red arrows). The direction of the force system is reversed from the
equilibrium diagram (b).
c
No matter how diverse, complicated, and independent
the forces on the teeth may be, the archwire or the appliance is always in equilibrium. In Fig 8-14a, a flexible
nickel-titanium (Ni-Ti) wire is placed into all of the brackets, which are badly malaligned. Many forces and
moments acting on the teeth (red arrows) can be produced
at the brackets, as artistically depicted in the diagram.
The teeth move in response to these deactivation forces,
and yet the forces that activate the archwire (Fig 8-14b)
are in equilibrium—the sum of the forces and moments
on the wire is zero. The force system from this straight
wire does not always deliver desired tooth movement,
but at least we can be assured that the archwire and the
bracket geometry will produce a force system that is in
equilibrium. It is better for us to do the thinking than to
let the appliance do it.
The primary value of the equilibrium principle is that
it gives us a powerful tool to solve for unknown forces,
so that we can better design our appliances. The tool is
called boundary conditions, which provide simple equations to solve for unknown forces. The deactivation force
diagram from the intrusion spring shown in Fig 8-9 is
very easy to determine. We can measure the force on the
arch at the incisor and replace it with an equivalent force
system at the molar bracket or some posterior center of
resistance (CR). An activation force diagram of the intrusion spring will give the same result but is not necessary.
A similar deep bite scenario is depicted in Fig 8-15a. Let
us measure the activation moments at both the incisors
(500 gmm) and the molar bracket (–3,500 gmm). All of
the other forces do not need measurement and can now
be calculated by boundary conditions.
∑M = 0
500 gmm + (–3,500 gmm) + M = 0
M = 3,000 gmm
M will come from vertical forces. There are no more
couples at the incisor and molar because all moments
have been measured there. Any horizontal forces are
ignored.
FB × 35 mm = 3,000 gmm
FB molar = +86 g
∑F = 0
FA incisor = –86 g
The full answer is given in Fig 8-15b, showing the activated spring in equilibrium. To obtain the deactivation
forces on the teeth, all forces and moments have their
directions reversed (Fig 8-15c).
145
8
Equilibrium
a
b
FIG 8-16 The object of equilibrium does not need to be a rigid body. (a) An activation force is applied to a
flexible coil spring in equilibrium. (b) The shape of the coil spring is continually changing as it moves from
point A to point B, but the spring is always in equilibrium with equal and opposite activation forces.
The principle of equilibrium applies not only to rigid
bodies but also to nonrigid deformed bodies and even to
bodies with a constant velocity. An activation force is
applied to a coil spring (Fig 8-16a). Although it changes
shape, it is continually in equilibrium with the activation
forces. The spring in its original shape before activation
is the passive shape, and the blue arrows show the activation force system (Fig 8-16b). Movement can occur
between points A and B in Fig 8-16b, but still the net
force is zero. Other examples of objects in equilibrium
are a squeezed balloon (where distances between points
change) and a flexible seesaw that changes its shape.
Step 2
Solving Problems Using
Equilibrium
Step 3
Let us now follow the steps in applying the equilibrium
principle to clinical situations. We decide to use a 2 × 4
appliance from the first molars to the incisors (Fig 8-17).
We would like to place a 40-g intrusive force on the incisors. To prevent flaring of the incisors, labial root torque
of 300 gmm is also applied. The question is: What will
happen to the first molar?
Step 1
Figure 8-17 shows the deactivation forces acting on the
teeth, from which we must determine all other forces. In
this step, we start the force diagram (Fig 8-18) by reversing the direction of the known forces so they now become
activation forces—forces acting on the wire (Newton’s
Third Law). All activation forces are in blue.
146
Apply formula 1.
∑F = 0
Add any forces at the molar that will be required for
equilibrium. Because there is a downward force acting
at the incisors, place an upward arrow at the molar (Fig
8-19a). The diagram is now in equilibrium from the forces
except for the magnitude of the molar force. According
to formula 1, the force magnitude is 40 g (Fig 8-19b).
Apply formula 2.
∑M = 0
In order for equilibrium to exist, the sum of the
moments around any point must equal zero. Let us sum
the moments around the red point on the incisors in Fig
8-20a. This point is selected as convenient because it
simplifies calculation, eliminating the force at the incisors.
The equation holds true, however, for any arbitrary point.
40 g × 30 mm + (–300 gmm) + Mmolar = 0
Mmolar = –900 gmm
The moment acting on the arch in the molar region is
a counterclockwise 900 gmm (Fig 8-20b). The equilibrium
diagram is now complete, and the vertical forces balance.
Couples on the wire at the molar and at the incisors
balance with the couple produced by the 40-g vertical
forces.
Solving Problems Using Equilibrium
FIG 8-17 The red force system is planned on the incisor segment for intrusion without flaring.
a
FIG 8-18 Starting the free-body diagram for the activated intrusion spring. The direction of the known force
system on the incisors is reversed in blue.
b
FIG 8-19 The vertical upward force at the molar is added to maintain the equilibrium of force (a). It is calculated
as 40 g (b).
a
b
FIG 8-20 (a) A moment is necessary at the molar for the appliance to be in equilibrium. (b) It
is calculated as –900 gmm. The intrusion spring is now in equilibrium with balanced forces
and moments.
Step 4
Of course, clinically we want to know the force acting on
the teeth. As explained before, this is a simple step in
which all forces and moments on the force diagram are
reversed. The red arrows in Fig 8-21 denote the deactivation force system on the teeth. Now we can answer the
question as to what will happen to the molar. The molar
will extrude, the crown will move distally, and the root
will move forward. The forces and moments are low, so
FIG 8-21 We are interested in the force system acting on the teeth during deactivation
of the arch. The direction of the force system
is reversed in red. The molar will extrude, the
crown will move distally, and the root will
move forward.
anchorage is relatively good for incisor intrusion. Anchorage could be augmented by adding more teeth to the
posterior segment.
The force diagram could be more complicated (eg,
more horizontal forces), but we have kept it simple to
develop the equilibrium principle and how it is applied.
To better define molar movement, the force at the molar
tube should be replaced with an equivalent force system
at the CR. The CR is so close to the center of the tube
that this step can be ignored in this case.
147
8
Equilibrium
a
b
c
d
FIG 8-22 (a) The red force system is planned on the incisor segment
for intrusion and lingual root movement. (b) The activated arch in
the equilibrium diagram. The direction of the known force system
is reversed in blue. (c) The equilibrium of forces is used to solve for
the unknown forces at the molar. (d) The equilibrium of moments is
used to solve for the unknown moment at the molar. An arbitrary red
dot is selected for calculation of moment. Now the activation force
diagram of the spring can be completed with all known forces and
moments. (e) All forces and moments are reversed in direction to
produce the deactivation force diagram of the forces (red arrows)
acting on the teeth.
e
Important considerations
Because Fig 8-21 shows the forces on the teeth, the question is asked: Should we use this diagram only without
first putting the archwire (the appliance) in equilibrium?
This can lead to error unless the orthodontist understands
what the diagram means. All of the red forces and
moments sum to zero, but each force and moment acts
separately on individual teeth (the molar and two incisors). The blue force equilibrium diagram in Fig 8-20b
has the forces acting on the entire archwire. Sometimes
beginning students look at the deactivation force diagram
(see Fig 8-21) and think that the vertical forces will flatten
the occlusal plane of the whole arch or that the moments
will steepen the occlusal plane. But this is not correct.
148
The +900-gmm moment and 40-g downward force act
only on the molar, primarily tipping the molar crown
distally. The incisors are acted upon by a different force
and moment, which will cause them to primarily intrude.
The blue forces act on the entire archwire, but the red
forces are independent and do not interact.
The second error in starting with a deactivation force
diagram of the unbalanced forces on individual teeth is
concluding improper force and moment direction. Some
forces may be erroneously placed from other appliance
components, or maybe activation forces are used instead
of deactivation forces. It is wise to first put the appliance
in equilibrium to avoid mistakes and to allow full understanding and only then to reverse force direction to obtain
the forces on the teeth. Later, with experience, one can
Solving Problems Using Equilibrium
a
b
FIG 8-23 (a) A 2,100-gmm moment will excessively tip the molar back. (b) The deactivation force diagram tells us that eliminating the 40-g
intrusive force will give a 900-gmm moment at the molar.
use only the deactivation force diagram if fully understood. It is certainly true that for one wire, activation and
deactivation forces are equal and opposite. Each appliance
component should be separately studied with an equilibrium diagram if unknowns are present.
Working example
In another patient with deep bite, the maxillary incisors
are in linguoversion (Fig 8-22a). We decide to apply a
40-g intrusion force with a 100-g tie-back force. To
improve the incisor axial inclinations, a moment (torque)
of –1,200 gmm is needed on the incisor. Again we ask
the question: What is the reciprocal force system on the
molar?
First, we start to compose the equilibrium diagram (Fig
8-22b), which shows the known forces and moments on
the archwire. The blue activation forces are the same as
in Fig 8-22a, except their directions have been reversed.
In Fig 8-22c, based on the formula ∑F = 0, the horizontal and vertical forces (blue arrows) are placed on the
wire at the molar tube. The vertical upward force is 40 g,
and the distal force is 100 g.
The formula ∑M = 0 is then used to calculate the
unknown moment at the molar. Arbitrarily, a convenient
point is selected at the incisor brackets (red dot), and all
moments are summed in respect to this point (see Fig
8-22d).
40 g × 30 mm + (–100 g × 3 mm) + 1,200 gmm
+ Mmolar = 0
Mmolar = –2,100 gmm
The equilibrium diagram is now complete (Fig 8-22d);
all forces and moments are reversed in direction to show
the forces (red arrows) acting on the teeth (Fig 8-22e).
However, although all the forces in Fig 8-22e sum to zero,
this is not an equilibrium diagram because unbalanced
independent forces are acting on the teeth.
If we want to place this force system on the incisors,
there exists a major anchorage problem on the molar. A
2,100-gmm moment could tip back the molar (Fig 8-23a).
With the aid of the activation force diagram (forces on
the wire) and the deactivation force diagram (forces on
the teeth), we can now do some creative thinking to
improve the force system. Maybe 40 g is not needed to
intrude the incisors. Let us eliminate this intrusive force
and recalculate the moment in the activation force
diagram (Fig 8-22d).
(–100 g × 3 mm) + 1,200 gmm + Mmolar = 0
Mmolar = –900 gmm
Now the moment on the molar (in the deactivation
force diagram) becomes 900 gmm (Fig 8-23b). This is
much better. Perhaps anchorage can be reinforced by
including more teeth or by using a headgear if necessary.
Another possibility is to start the incisor root movement
at a later stage of treatment. Notice that the tie-back
horizontal forces (100 g) produce a counterclockwise
rotation on the equilibrium diagram (see Fig 8-22d). If
we can increase these forces, the molar will feel a smaller
moment. This is a possibility, but the incisor crown will
retract further to the lingual, which might not be an
acceptable treatment objective. Therefore, the tie-back
solution to the large molar moment is limited. We can
try out many possibilities and choose the best force
system; however, unless our diagram is in equilibrium, it
will not be possible. The thinking sequence in Figs 8-22
and 8-23 can be helpful even without calculating the
149
8
Equilibrium
a
b
c
d
FIG 8-24 (a) An intrusion arch with an intrusive force of 100 g is
placed. The entire buccal segment is used for anchorage. (b) The
activation force system on the arch, reversing the direction of the
intrusive force. (c) The equilibrium of forces is used to calculate the
upward force of 100 g at the molar. (d) The equilibrium of moments
is used to calculate the counterclockwise moment of 3,000 gmm. The
equilibrium diagram is completed with the complete force system.
(e) The reversed force system shows the deactivation force system
acting on the brackets.
e
actual numbers. It tells us, for example, that if we elect
not to measure the intrusive force and the lingual root
torque, it is possible to obtain much higher magnitudes,
leading to significant anchorage loss.
Summary
The use of a deactivation force diagram with the direction
reversal on the teeth gives the clinician a powerful tool
for treatment planning. The clinician can try out many
different strategies on paper to optimize the treatment.
Trial and error on a patient using an appliance is timeconsuming and could lead to permanent harm. Trial and
error on a diagram, however, is not only friendly to both
the orthodontist and the patient but is also more versatile.
150
The equilibrium principle has been used to determine
unknown forces acting from an orthodontic appliance
when the forces are known at the bracket positions.
Another step is necessary, however, when the forces and
moments at a tube or bracket are replaced at a relevant
CR. In the examples given so far, the molar CR (purple
circle) and molar tube center were close enough that
replacement was not necessary (see Fig 8-23).
Equilibrium and Equivalence
In this text, two important principles of physics have been
applied to clinical orthodontics: equivalence and equilibrium. In this section, both principles are needed. Figure
Equilibrium and Creative Biomechanics
a
b
FIG 8-25 (a) The movement of the posterior segment is predicted with a replaced equivalent force system (yellow arrows) at the CR of the
posterior segment. (b) The direct replacement of anterior force by equivalence, not using the equilibrium principle, gives the same result.
8-24a shows an intrusion arch with a separate anterior
segment and a solitary intrusive force of 100 g. For
anchorage, an entire buccal segment from canine to
second molar is used. We want to know what will happen
to the posterior segment and how good its anchorage
will be. Figure 8-24b is the activation force diagram showing a downward force to activate the arch. Applying the
ΣF = 0 formula, an upward force of 100 g is placed on the
arch at the center of the molar tube (Fig 8-24c). Switching to the ΣM = 0 formula, the moment at the molar tube
on the arch is calculated to be –3,000 gmm (Fig 8-24d).
In this relatively simple situation, force direction is
reversed from the equilibrium diagram to give the forces
and moments acting on the molar. Our answer is the
100-g extrusive force at the center of the molar tube and
a 3,000-gmm clockwise moment (red arrows in Fig 8-24e).
To predict posterior segment movement, it is necessary
to replace this force system at the CR of the posterior
segment (Fig 8-25a). Here the equivalence equations are
applied. At the posterior segment CR, there is a 100-g
occlusal force on the teeth with a moment of 2,200 gmm
(yellow arrows are the replaced equivalent force system
of the red arrows). In the chapters on equivalence (see
chapter 3) and deep bite correction (see chapter 6), we
solved similar problems without using the principle of
equilibrium. We directly replaced the 100-g downward
force at the incisors with a 100-g extrusive force at the
CR and a moment of +2,200 gmm (100 g × 22 mm =
+2,200 gmm) (Figure 8-25b). The answer, of course, has
to be the same, even if only the principle of equivalence
is applied in this special case. Therefore, in a statically
determinate force system using mechanisms like an intrusion arch in which all forces are known on the incisors,
it is not necessary to develop an equilibrium diagram.
Equilibrium and Creative
Biomechanics
Equilibrium and other principles from physics allow both
simulation of treatment stages and a scientific basis for
appliance selection and use. Let us consider a few examples by which we can try out different treatment modes
with simple diagrams rather than subjecting the patient
to trial and error procedures.
A patient whose first premolar will be extracted presents with a high mesially tipped canine (Fig 8-26). The
goal is to tip the canine distally. Note how the CR of the
canine will move only slightly distally with a marked
occlusal displacement. To eliminate any friction considerations, let us use a simple vertical loop to bring the
canine down. Figure 8-27 shows the force system acting
on the tooth that we want, including horizontal forces of
200 g to close the space (any moments from the vertical
loop are ignored for simplicity). To prevent the molar
from tipping forward, we will apply a 2,000-gmm moment
at the molar tube. With a 10-mm distance to the molar
CR, this might give a 10-mm moment-to-force ratio and
hence reasonable molar anchorage. Is this a possible force
system for the patient? Let us set up a few valid examples
of force diagrams in equilibrium to answer this question.
All forces and moments have their direction reversed
in Fig 8-28. The sum of the forces is zero, but the sum of
the moments is not zero because there is an unbalanced
–2,000-gmm moment acting at the wire; therefore, this
force system is not possible. We decide to apply two vertical forces (a couple) at the canine and the molar to put
the loop into equilibrium (Fig 8-29a). The deactivation
force diagram (Fig 8-29b) with force directions reversed
151
8
Equilibrium
FIG 8-26 A patient whose first premolar was extracted presents with
a high mesially tipped canine. The goal is to tip the canine distally.
Note how the CR of the canine will move only slightly distally with a
marked occlusal displacement.
FIG 8-27 A simple vertical loop spring is planned to bring the canine
downward into occlusion. The force system (red arrows) that we want
acts on the canine. The moment-to-force ratio of 10 mm was placed
at the molar for reasonable anchorage. Is this force system possible?
FIG 8-28 The activation force system on the spring with a reversed
direction of force. The diagram is not in equilibrium, because there is
an unbalanced –2,000-gmm moment acting at the wire.
a
b
FIG 8-29 (a) The spring in the diagram can be put in equilibrium by
adding two vertical forces (100 g). (b) The deactivation force diagram
for simulation and evaluation. The forces acting on the teeth are
shown, reversing the force directions from a. (c) The horizontal and
vertical forces (red arrows) are replaced with single forces (yellow
arrows). The forces on the molar are reasonable, with sufficient moment to keep the molars from tipping forward. The yellow resultant
force on the canine is slightly intrusive, and the line of action of force
is closer to the CR, which is not desirable. This simulation procedure
avoids a lengthy clinical error in treatment. Note that the canine
intrudes instead of extruding.
c
152
Equilibrium and Creative Biomechanics
FIG 8-30 The deactivation force diagram
of forces on the teeth after placing a couple
at the canine. This is also unsatisfactory because the canine may translate or even flare.
FIG 8-31 The deactivation force diagram of
forces on the teeth after an extrusive force is
placed at the canine. This is desirable for the
canine, but the favorable moment is lost on
the molar, and it will tip forward.
is now valid and can be used to discuss the loop appliance
and its desirability.
The force system on the molar is reasonable. The resultant forces are depicted as yellow arrows in Fig 8-29c.
There should be sufficient moment to keep the molars
from tipping forward from the red mesial force on the
molar. The canine is more problematic. The desired movement of the CR is downward and backward, but the intrusive component of force on the canine is inadequate. This
is not a good outcome to have because the canine is
already in infraocclusion. Also, the line of action of the
force is moved closer to the CR to produce less effective
mesial root displacement. The canine will probably tip
around a center of rotation near the apex level, and the
apex is too far to the distal already (note the predicted
canine position after movement by this force system [in
blue] in Fig 8-29c). There is no reason to place this appliance in the patient because we know the force system is
undesirable.
Now let us work with the red forces on the teeth from
the deactivation force diagram in Fig 8-29b; practically,
it is not necessary to go back to the original equilibrium
diagram of the forces on the loop. We will try out different deactivation force systems from the loop to see if we
can improve the outcome.
Perhaps if the horizontal magnitude of the force is
reduced to 100 g, less molar moment and hence less
vertical force will be needed. However, an intrusive force
is still present on the canine, and the line of action of the
FIG 8-32 A straight leveling wire will erupt
the canine but significantly intrude the incisors, leading to an anterior open bite and
tipping the molar forward as a major loss
of anchorage occurs. A single-wire appliance has inherent limitations to obtaining a
desirable force system in this malocclusion.
resultant forces remains the same, so this is not a substantial improvement.
Another possibility is to remove the vertical forces and
to place a counterclockwise couple (crown forward and
root backward) on the canine. But alas, this is also unsatisfactory because the canine may translate or even
flare—a movement opposite to what we want (Fig 8-30).
Let us consider placing an extrusive force on the canine.
This is better for the canine, but our deactivation force
diagram tells us that the favorable moment is lost on the
molar, and it will tip forward (Fig 8-31). In short, because
we cannot change the laws of physics, there is no good
solution if we want to use the vertical loop appliance. It
is better to find this out by an equilibrium analysis than
by any unexpected side effects observed after the appliance is inserted into the patient’s mouth.
In fact, any configuration of a single-wire appliance
will not have the proper solution for this problem. A
straight-wire arch with a coil spring or a chain elastic
with a tip-back bend to preserve posterior anchorage will
also intrude the canine or at least prevent its eruption;
even a straight leveling wire will erupt the canine but
significantly intrude the incisors, leading to an anterior
open bite and tipping the molar forward as a major loss
of anchorage occurs (Fig 8-32).
A better solution to obtaining desirable and consistent
forces is the use of dual wires for this highly displaced
canine case. The molar and the incisors are connected
by a rigid bypass wire to stabilize the arch (Fig 8-33a). A
153
8
Equilibrium
a
b
c
d
e
separate elastic or spring is attached to the canine, originating from the buccal hook of the molar. The spring is
in equilibrium (Fig 8-33b). The 200-g deactivation force
from the spring has a downward and backward pull on
the canine (Fig 8-33c). It is far enough from the canine
CR that the canine crown will move distally, and the root
will come forward. As an independent force, the spring
force magnitude can be varied. If desired, the spring can
be attached to a hook on the lingual of the canine to minimize canine rotation. Unlike the single-wire solution, the
retraction force has an extrusive component of force (Fig
8-33d). The rigid bypass wire can be replaced with
fiber-reinforced composite for better stabilization.
The mesial force from the spring attached at the molar
tube acts on the entire bypass arch (Fig 8-33e). Because
154
FIG 8-33 Dual-wire system for a superiorly displaced canine. (a)
The molar and the incisors are connected by a rigid bypass wire to
stabilize the large anchorage unit. (b) A separate elastic or spring
is attached to the canine and a hook at the molar. The spring is
in equilibrium by activation forces (blue arrows). (c) The reversed
deactivation force system on the canine (red arrows) shows that the
200-g force is far enough from the canine CR that the canine will
easily tip. (d) The replacement components of the resultant force
(yellow arrows) on the canine show that the CR will move occlusally
and distally, which is desirable. (e) The replaced equivalent force
system (yellow arrows) at the CR of the total anchorage segment by
the entire bypass arch. With a rigid bypass arch and light force, the
posterior anchorage should be maintained.
the bypass arch is rigid, any anchorage loss would be seen
primarily as a flattening of the occlusal plane. With a light
force, this might be adequate to maintain posterior
anchorage.
The flattening of the entire occlusal plane is helpful if
the incisors require significant intrusion, in which case
an intrusion arch can be placed instead of a bypass archwire (Fig 8-34a). A 100-g intrusion force acting over 20
mm can produce a 2,000-gmm moment on the molar
(Fig 8-34b). That moment could be adequate with the
200-g mesial force (Fig 8-34c) to translate the molar
forward and serve as excellent anchorage. Note how the
dual wires allow good control over the force system, force
magnitudes, force direction, and points of force application, which is not always possible with a single wire. At
Equilibrium and Creative Biomechanics
FIG 8-34 (a) An intrusion arch can be placed for needed intrusion
of incisors. (b) Deactivation force system. A 100-g intrusion force
acting over 20 mm can produce a 2,000-gmm moment on the molar. (c) Combined force system for the spring and intrusion arch. A
2,000-gmm moment could be adequate with the 200-g mesial force
to translate the posterior segment and serve as excellent anchorage.
a
b
the same time, there is the beneficial effect of reducing
deep bite if indicated.
Important note to remember
In applying equilibrium, we have gone back and forth
between the activation force diagram and the deactivation
force diagram of forces acting on the appliance and on
the teeth. The deactivation force diagram can certainly
be used to plan orthodontic strategies as long as it is
understood that, conceptually, it is the appliance—not
the teeth—that is in equilibrium in the activation force
diagram. The teeth are also in equilibrium, but the forces
depicted on the teeth in the deactivation force diagram
are incomplete.
Common misconceptions
A patient has a unilateral constriction on the right side.
The treatment goal is to expand the right side of the
mandibular arch more than the left side (Fig 8-35a). What
is the best way to deliver greater force on the right side
c
with a lingual arch in the mandibular arch? Do we step
the arch unilaterally as in Fig 8-35b? Or do we bilaterally
expand and yet reduce the stiffness on the left side unilaterally with a loop to reduce the force (Fig 8-35c)? This is
actually a trick question. Equilibrium requires that the
ΣF = 0; therefore, no matter how cleverly an appliance is
fabricated, the forces must be equal and opposite (Fig
8-35d).
Is it possible to do differential tooth movement using
a lingual arch? Yes, because instead of horizontal forces,
differential moments are applied on each tooth. The red
arrows in Fig 8-35e show the forces acting on the teeth.
Buccal crown torque is placed on the right molar. The
side effects of vertical forces are small due to the large
distance between the brackets at each molar. Figure 8-35f
(blue arrows) demonstrates the valid equilibrium diagram,
which has a moment on the right molar along with vertical forces on the arch at each molar. The mandibular right
molar primarily tips to the buccal (Fig 8-35g). Asymmetric applications of lingual arches will be discussed in more
detail in chapter 12.
155
8
Equilibrium
a
b
c
d
e
f
g
FIG 8-35 (a) A patient with a unilateral constriction on the right side. What is the best way
to deliver greater force on the right side with a lingual arch in the mandibular arch? (b) The
lingual arch with a step-out bend at the right side. This appliance is not feasible because the
lingual arch is not in equilibrium. (c) The lingual arch stiffness is reduced on the left side, but
this appliance is still not feasible. (d) No matter how cleverly an appliance is fabricated, the
forces must be equal and opposite. We cannot change the laws of physics. (e) The deactivation force system acting on the teeth (red arrows) shows that buccal crown torque is placed
on the right molar. The side effects (vertical movements) are small due to the large distance
between the vertical forces at each molar. (f) The activation force diagram in equilibrium, with
forces reversed in blue. (g) The mandibular right molar primarily tips to the buccal. Some
bilateral buccal expansion is needed.
FIG 8-36 An example of a confusing diagram. Diagrams should be clearly labeled as to whether the
shown forces act on the appliance or on the teeth. Here
some forces act on the teeth, and some are reacting
from the archwire insertion. Forces from the headgear
and elastics can have different effects if the archwire is
elastically deformed or acts as a rigid body. The shown
elastic acts on the archwire but is not a reactive force
from archwire insertion.
156
Recommended Reading
Conclusion
Some therapy attempts the impossible. Equilibrium
theory lets us know what is scientifically possible. Freebody diagrams published in journals or presented at
meetings are helpful in presenting force systems of new
ideas. The object in question—the appliance, not the teeth
(unless they are in equilibrium with their periodontal
support)—should always be in equilibrium. Diagrams
should be clearly labeled as to whether the shown forces
act on the appliance or on the teeth. Diagrams like Fig
8-36 should be avoided, where some forces act on the
teeth and some act on the archwire without explanation.
Moreover, the wire (its depicted force system) is not in
equilibrium.
This chapter has applied the equilibrium concept to
appliance design. Other applications are equally useful
and valid. The biomechanics of tooth movement or orthopedics can start with placing the teeth or bones into a
state of equilibrium. Functional movements of the mandible and stresses in the temporomandibular joint are studied or modeled based on equilibrium. Not confined to
orthodontic appliances, the equilibrium concept is
universal and has broad applications.
Recommended Reading
Burstone CJ. The biomechanics of tooth movement. In: Kraus B, Riedel
R (eds). Vistas in Orthodontics. Philadelphia: Lea and Febiger,
1962:197–213.
Burstone CJ, Koenig HA. Force systems from an ideal arch. Am J
Orthod 1974;65:270–289.
Halliday D, Resnick R, Walker J. Fundamentals of Physics, ed 8. Hoboken, NJ: Wiley, 2008.
Smith RJ, Burstone CJ. Mechanics of tooth movement. Am J Orthod
1984;85:294–307.
157
Problems
The problems below use the concept of equilibrium. First make an equilibrium diagram in which all given forces
are acting on whatever is in equilibrium—an arch or appliance, a tooth or a mandible. You must check the force direction carefully in the diagrams below. For example, if the given direction is for deactivation force, you must reverse the
direction on your diagram. In short, first get the correct results from your activation equilibrium diagram; then reverse
the force direction if forces on the teeth are required.
158
1. A 30-g force is applied anteriorly on each side (60 g
at the center) from an intrusion arch. Find all forces
and moments acting at the molar tube.
2. A 200-g force is applied from a lingual arch on the
left molar. Find all forces and moments acting on the
lingual tubes. (a) Assume that right and left couples
are equal. (b) Assume that no couple exists on the
right side. The direction of force and moments must
be determined.
3. A 2,000-gmm couple is applied from a lingual arch
to rotate the left molar clockwise. No other couples
or horizontal forces are applied. Find all other forces
acting on the molar tubes.
4. A vertical loop placed off-center for space closure
produces unequal couples. Find all forces acting on
the canine and premolar. Given forces and moments
are on the teeth.
Problems
5. A straight wire is used to level the canine and premolar brackets. Solve for unknown forces and moments.
Ignore horizontal forces, and assume that couples on
each bracket are equal. What are the undesirable side
effects?
6. A lingual arch is used to tip the right molar to the
buccal by an applied couple. Ignore horizontal forces;
no couple is used on the left molar. Find all forces
acting on both molars. What are the undesirable side
effects?
7. A root spring acts to move the canine root to the distal.
Give the forces and moments acting on the canine and
molar bracket.
8. The mandible is in equilibrium as the subject clenches
his teeth with 1,000 g of muscle force. Determine the
force at the condyle. This is one of many simple
temporomandibular joint models. Some believe the
condyle is not a stress-bearing region.
9. Give all resultant forces and moments acting at the CR
of the posterior segment from all appliances—headgear, loop arch, and maxillomandibular elastic. Ignore
the vertical discrepancy of the tubes of molar attachment. Rectilinear components for the resultant are
satisfactory.
10. The Herbst appliance, activated by the muscle force,
is in equilibrium. Give all forces and moments acting
at the CR of each molar.
159
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9
The Biomechanics
of Altering Tooth
Position
“Prediction is very difficult, especially if it’s
about the future.”
— Niels Bohr
This chapter considers the correct force systems and optimal force
magnitudes to produce different types of tooth movement. Before the
required forces are established, an accurate method of describing
tooth movement is necessary. Concepts such as an axis (center) of
rotation or a “screw axis” are applicable. Primary tooth movement is
produced by a couple or the line of action of force acting through the
center of resistance (CR). Derived tooth movement is produced by a
combination of primary movements where the line of force is away from
the CR. In theory, all axes of rotation can be produced with a single
force whose line of action lies on or off of the tooth. The clinician must
estimate where that force is and may replace it with a moment and
a force at the bracket (moment-to-force [M/F ] ratio). Force systems
change over time following tooth displacement; this consideration is
part of an optimal force system. Stress and strain in the periodontal
ligament and bone are more relevant than force alone in determining
the force level to use clinically.
163
9
The Biomechanics of Altering Tooth Position
FIG 9-1 A force acting through the CM of a free body causes linear
acceleration by translation. If the force is moved away from the CM,
the body will undergo a combination of linear and angular acceleration. Each CM point in a given plane projected at 90 degrees forms
an axis, and all axes—no matter the plane—will intersect at a point.
T
he core of orthodontic treatment involves tooth
movement, bone displacement, growth, and overall
remodeling by force systems. This chapter discusses
the relationship between the force system and the change
in tooth position.
What is an optimal force system for tooth movement?
To answer that question, a number of factors must be
considered. What level of force magnitude should be
delivered? Should the force be intermittent, constant, or
pulsating? Will the force move the tooth to the target
position correctly? Here the concern is the quantitative
explanation of tooth-displacement patterns—how to
move a tooth from position A to position B and what
force system is required for this movement. How is the
force system different if an incisor is to be tipped lingually,
translated lingually, or the crown held in place as the root
is moved lingually?
Free and Restrained Bodies
Imagine a square body floating in space. Force acting
through its center of mass (CM) will produce linear acceleration (Fig 9-1). If the force is moved away from the CM,
the body will undergo a combination of linear and angular acceleration. A pure moment or couple produces only
angular acceleration around the CM.
A tooth is not a free body. It is a body restrained by the
periodontal ligament (PDL) and other periodontal and
164
FIG 9-2 The block is held in place by springs attached to supports.
It moves but does not accelerate. The force acting on a point called
the CR will cause the rectangular block to move by translation. Each
CR point projected at 90 degrees forms an axis, but
all red axes may not intersect at a point. Therefore,
because of unknowns, the CR is usually depicted
as a circle (in 2D) or a sphere (in 3D) rather than
a point.
bony structures. Now imagine the block from Fig 9-1 as
a restrained body held in place by attached springs (Fig
9-2). We push on it with a force, but it does not accelerate. The rectangular block suspended by springs can be
considered a structure undergoing deformation. Force
acting on a point called the CR (center of resistance) will
cause the rectangular block to move by translation. The
CR is analogous to the CM in that it is a point at which
translation occurs, but they have entirely different locations. For one thing, a CM is a point that can be found
in three-dimensional (3D) space. Each point projected
at 90 degrees forms an axis, and all axes—no matter which
plane they are in—will intersect at a point. This is not
necessarily true for a restrained body (note in Fig 9-2 that
the red axes do not intersect at a point). The location of
the CR of a tooth or a group of teeth is influenced by
many factors. The direction of force may alter the location
of the CR because the tooth is not morphologically
symmetric or the PDL has asymmetric properties (anisotropy). Biologic changes during tooth movement over
time can alter root length and periodontal support along
with PDL properties. Therefore, the CR (or, in 3D, an axis
of resistance) may change somewhat during treatment.
The tooth with a single-point CR in three dimensions
is a special case with an ideal isotropic PDL. Fortunately,
substantial asymmetry and variations are not the rule, so
practically the concept of CR is both useful and valid. It
is still better to think of the CR of a tooth not as a point
but rather as a circle (a sphere in three dimensions).
Methods to Describe Change of Tooth Position
a
b
c
FIG 9-3 A simplified tooth model with a spring support. (a) Passive state. (b) A force is applied at the CR that causes the tooth to translate.
The tooth moves, yet it is still in equilibrium as one spring extends and the other compresses. (c) A realistic tooth model. The applied force is
balanced by the sum of all the compressive, tensile (small red arrows), and shear stresses in the PDL.
Although chapter 10 discusses these axes of resistance
and rotation in three dimensions in greater detail, this
chapter introduces and develops these concepts in two
dimensions.
In chapter 8, the concept of equilibrium was applied
to the archwire and the orthodontic appliance, but the
tooth is also in a state of equilibrium. In that chapter, a
two-dimensional (2D) model of a tooth with attached
elastics demonstrated the relationship between forces
and tooth displacement. Here, for further simplification,
a model of a canine with springs attached at the root is
shown (Fig 9-3a). A force is applied at the CR of the tooth
that causes the tooth to translate (Fig 9-3b). The tooth
moves, and yet it is still in equilibrium as one spring
extends and the other compresses. The equilibrium
diagram (Fig 9-3c) shows the tooth in equilibrium—the
applied force is balanced by the sum of all the compressive and tensile stresses (small red arrows) in the PDL.
One should not confuse the appliance equilibrium
diagram with the tooth equilibrium diagram; they are
two distinct systems.
Methods to Describe Change of
Tooth Position
Before we can accurately relate forces to tooth movement,
it is necessary to have an accurate method to describe a
change in tooth position. For example, to say that the
central incisor should tip lingually is too vague. An incisor could tip around an axis at its apex or at the center
of the root, and each would require an entirely different
force system. Other qualitative descriptions such as “flaring,” “aligning,” and “opening the bite” are too imprecise
to serve as a basis for establishing a proper force system.
There are many possible and valid methods to describe
tooth position or movement. Let us consider some of the
advantages of each method.
Method 1
The vertical incisor in Fig 9-4a is to be moved lingually
and intruded, with its axial inclination corrected. So far
the description is qualitative and too vague. Method 1 is
to describe translation and rotation around the CR of the
incisor. The purple circle is the initial CR, and a dotted
line is connected to the CR before and after the planned
movement (Fig 9-4b). This is its translatory path, and the
x and y coordinates can therefore be plotted. The incisor
not only translates but also must rotate around its CR
(Fig 9-4c) to reach its final desired position. The total
rotation angle and the translation magnitude and direction define the tooth movement (Fig 9-4d). The rotation
occurs around an axis at 90 degrees to the plane in which
the force acts.
There are advantages and disadvantages to this method.
The advantage is the direct relationship between applied
forces and the CR (a force acting at the CR translates a
tooth, and a couple rotates a tooth around the CR). The
disadvantage is that the position of the CR is a calculated
estimate, not a real anatomical point. Particularly with
asymmetric teeth and in three dimensions, an actual
point may not exist. However, any estimated CR point
can be copied accurately from an initial position to a final
position tracing of a tooth and reliably gives the direction
of the desired force.
The line of action of an applied force is not necessarily
identical to the CR path, but any small deviation that is
not clinically relevant will be ignored.
165
9
The Biomechanics of Altering Tooth Position
a
b
c
d
Method 2
The most common method of describing tooth movement
is based on bracket position change (method 2). In a
sense, a local coordinate system is built into the bracket.
If the bracket is correctly bonded to the tooth, in addition
to its occlusogingival level, three rotational axes are established (Fig 9-5a). E. H. Angle classified these axes as first,
second, and third order. Currently, clinicians speak of
“torque,” “rotation,” and “tip.” To avoid confusion, this
chapter describes rotation around an x-axis (Fig 9-5b), a
y-axis (Fig 9-5c), and a z-axis (Fig 9-5d). Terms like torque
used to describe tooth movement are misleading and
incorrect. The term torque will be used in this chapter
only to describe the force system consisting of a couple
or a pure moment. Inclination angles or change of axial
inclination should not be called “torque.” Z-axis rotation
around the bracket changes the mesiodistal axial inclination, x-axis rotation changes the labiolingual (faciolingual) inclination, and y-axis rotation rotates the tooth
(see Fig 9-5). Various slot angles known as prescriptions
are built into the bracket. Imagine a bag filled with many
loose brackets; one could describe it as a bag of individual coordinate systems. In three dimensions, three trans166
FIG 9-4 Description of tooth movement
(method 1). (a) Translation of the CR and
rotation around the CR. The vertical incisor
is to be moved lingually and intruded, with
its axial inclination corrected. (b) The purple
and blue circles indicate the initial and final
CRs, while the dotted line is the CR path from
the planned movement. (c) The incisor incrementally rotates around its CR (curved
dotted arrow). (d) The total rotation angle
(θ), the translation magnitude (D), and the
force directions (dotted arrows) define the
tooth movement.
lations and three rotations of any bracket (tooth) are
possible. The potential to move in a given direction is
quantified as the degrees of freedom, and a tooth in space
has six degrees of freedom for full control. An edgewise
appliance using round wire may have only five degrees
of freedom.
Tooth displacement at the bracket can be described in
two ways. The change of the coordinate system from
position 1 to position 2 can be based on the coordinate
system of the initial tooth (bracket) position, or a global
coordinate system away from the individual tooth under
consideration can be used. Let us consider method 2,
where the only coordinate system is on the bracket-tooth.
In Fig 9-6, an incisor is moved forward and intruded;
displacement is exaggerated for demonstration purposes.
Because only translation occurs, the sequence is not
important for describing tooth movement. Three paths
are possible (1, 2, and 3), and all have the same end point
(Fig 9-6a). The clinician may look at the bracket path from
start to finish to determine the direction (line of action)
of the force. This is incorrect except in the special case
of tooth translation (without rotation) in which the CR
and bracket move parallel to each other. In Fig 9-6b, a
line connecting the CR of position 1 and position 2 is
Methods to Describe Change of Tooth Position
a
b
c
d
FIG 9-5 Description of tooth movement using change in bracket position (method 2). (a) A local individual coordinate system for each tooth
is built into the bracket with three rotational axes. (b) X-axis rotation: third-order rotation, or torque. (c) Y-axis rotation: first-order rotation, or
rotation. (d) Z-axis rotation: second-order rotation, or tipping.
a
b
FIG 9-6 (a and b) An incisor is moved forward and intruded. Because only translation occurs, the sequence of movement is not important.
Three paths are possible (1, 2, and 3), but all have the same end point. The clinician typically looks at the bracket to determine the desired tooth
movement and to evaluate the result (a). The motion path of the bracket and the line of action of the force that is required for that movement
are identical if the tooth translates only. The paths of the CR and the bracket are parallel with the force direction (b).
parallel to the bracket path. Note that the three paths in
Fig 9-6a are all possible. The clinician, however, must
select the most advantageous path. This method of
describing tooth movement is simpler for tooth translation only; however, it becomes more complicated if we
must also rotate the bracket around any axis or multiple
axes, because the sequence of the rotations leads to different end points for tooth position.
The use of angles to describe movement is nothing
new. The famous physicist Euler described in detail the
proper manipulation of descriptive angles—Euler angles.
The interesting thing about using angles for describing
motion is that they are not cumulative (ie, they cannot
be added irrespective of sequence). For example, a bank
account is cumulative. The total balance will be the same
regardless of the sequence of deposits and withdrawals.
It is the same for the translated bracket in Fig 9-6; the
sequence makes no difference. The incisor can first be
moved to the labial and then intruded, or it can first be
intruded and then moved to the labial; either way, it will
still end up in the same position. But this is not true for
angles—angles are noncumulative.
Consider the lingually tipped incisor in Fig 9-7. The
coordinate system is drawn at the bracket. Let us try two
different paths. In Fig 9-7a, the incisor will first be translated. After translation, the tooth is shown in white. Now
the tooth is rotated around the x-axis at the bracket, and
its final position is shown in blue. In Fig 9-7b, the sequence
is changed: The incisor is first rotated around the x-axis
and then translated. Remember that the coordinate
system is fixed on the tooth. Note that the blue tooth in
Fig 9-7b ends up in an entirely different position than in
167
9
The Biomechanics of Altering Tooth Position
a
b
FIG 9-7 Rotation and translation using the individual coordinate
system at each bracket. (a) The incisor will first be translated (white
tooth) and then rotated to its final position (blue tooth). (b) The sequence is changed: The incisor bracket is first rotated (white tooth)
and then translated (blue tooth). The Euler angles used for describing
motion are not cumulative. The end result depends on the sequence.
a
b
Fig 9-7a. This example is in 2D, and adding additional
rotations on other planes will only compound the
sequence problem.
In short, any three given bracket angles are insufficient
to describe tooth movement to an end point if the bracket
itself is used as the only coordinate system. Sequence
must be given. On the other hand, if an additional outside
reference or global coordinate system is used, three axis
angles measured to the common reference can be sufficient. A modification of method 2 uses a global coordinate
system based on the occlusal plane (Fig 9-8). Now it
makes no difference if the incisor is translated first and
then rotated around the x-axis (Fig 9-8a) or rotated
around the x-axis first and then translated (Fig 9-8b).
Figure 9-9 shows the proposed central incisor tooth
movement in 2D: an x-axis displacement to the left and
a z-axis rotation. Without a global reference coordinate,
168
a
b
FIG 9-8 Rotation and translation at each bracket measured to a
global coordinate system. It makes no difference whether the incisor is
translated first and then rotated (a) or rotated and then translated (b).
FIG 9-9 The end result of x-axis displacement and z-axis rotation
depends on the sequence. (a) Displacement and then rotation. (b)
Rotation and then displacement. The final position of the tooth (blue
tooth) is sequence dependent, and the movements are noncumulative.
the end point is indeterminate because it depends on the
sequence. It is important not only to have a meaningful
bracket prescription and correct positioning of the
bracket on the tooth but also to know the relationship
between the bracket coordinate system and the global
coordinate system. A straight wire or series of wires
between the two brackets could eventually align the teeth.
The sequence of alignment is determined by the biomechanics of the wire-bracket system and the biologic
response and is not dictated by any determined sequence
of translations and rotations. Furthermore, the final
occlusal plane cant may not be parallel to any original
global occlusal plane reference.
The bracket position in method 2 has the advantage of
supplying a definitive landmark during treatment,
provided that it is not changed. This method is easy for
the clinician to understand and use because it relates well
Methods to Describe Change of Tooth Position
FIG 9-10 (a) When a couple is applied at the
bracket, the tooth rotates around the CR. (b)
To rotate around the bracket, a lingual force
and a couple must be applied. (c) Some clinicians may believe that a twisted wire at the
bracket produces a force system that rotates
the tooth around the bracket if a couple is
applied, but that is incorrect. (d) Incorrect
bracket path for a couple.
a
a
b
c
d
b
FIG 9-11 A single force on the crown will tip an incisor to the labial with a center of rotation
near the CR. (a) Correct force system. (b) Observing only the change in bracket position
(bracket path) does not directly give the correct force system. Note the wrong direction of an
intrusive force with an unnecessary moment.
to the appliance (eg, “I must step the tooth up; therefore,
I must bend the wire with an upward step”). The global
reference plane should also be kept constant. This may
be difficult if most teeth are malaligned. Unfortunately,
the relationship between the change in tooth position
and the force system to produce such a change according
to bracket position is complicated. A good example is a
twist placed in an archwire, where the force system in an
ideal situation could be a couple (Fig 9-10a). When a
couple is applied at the bracket, some clinicians believe
that the tooth will tend to rotate around the bracket. The
rotation of the tooth around the bracket also requires a
force; hence, possible anchorage loss and an unexpected
final tooth position may be expected (Fig 9-10b). This
fallacy is often based on looking at the bracket position
before and after and assuming that any translation means
a force in that direction, while any rotation implies a
couple (Figs 9-10c and 9-10d). The force system in Fig
9-11a is correct; a single force on the crown will tip an
incisor to the labial with a center of rotation (CRot) near
FIG 9-12 An incisor is moved lingually and
intruded, and its axial inclination is changed.
A force along the path of the CR and rotation
around the CR give good information about
the tooth movement and the force system
(dotted arrows). A line connecting the brackets before and after movement (gray arrow)
is not the correct line of action of the force.
the CR. Observing only the change in bracket position
(Fig 9-11b) gives the wrong conclusion: an intrusive force
with the wrong direction and an unnecessary moment.
In Fig 9-12, an incisor is moved lingually and intruded,
and its axial inclination is changed. Following the path
(dotted arrow) of the CR and rotation around the CR
gives good information about the tooth movement and
relates directly to the force system. Note that a line
connecting the brackets before and after (gray arrow) is
not the correct line of action of the force. The bracket
path parallels the CR path only in the special case of
translation (without rotation).
Let us consider a bracket with the latest and best slot
prescription so that if the teeth were aligned on a straight
wire with a correct occlusal plane cant, a perfectly treated
occlusion would be observed. Here is the problem: If we
start with a malocclusion, which rotations should we do
first (x-, y-, or z-axis rotation)? Because the angles of the
axes are sequence-sensitive and a good reference plane
could be lacking, the best sequence is not obvious.
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The Biomechanics of Altering Tooth Position
a
b
a
c
d
b
FIG 9-14 The occlusogingival level of the bracket path does not change in a and b. It parallels
the CR path only during pure translation (a).
With identical bracket angle prescriptions, the final
result may vary depending on the sequence of tooth
movement. There are an infinite number of possibilities;
the final result is not determined by a prescription of
angles alone. The orthodontist should decide on the best
path for a tooth to move from its initial position to its
final position at the end of treatment. Perhaps one stage
of movement is the best, not a series of independent
rotations. The bracket and a straight wire based on the
forces delivered by an appliance may eventually align
teeth, but the sequence may not be the best for the particular situation, and extraneous tooth movement may occur
in the interim.
In another example, the goal is to tip the maxillary
incisor lingually and to maintain the same occlusogingival level of the bracket. One possibility (Fig 9-13) is to tip
the incisor lingually by a single force, displacing the root
forward using round wire (Figs 9-13a and 9-13b). No
intrusive force is applied, so the position of the CR does
not change much. In the next stage (Fig 9-13c), the CR
must be intruded and moved distally to correct the
170
FIG 9-13 The goal is to tip the maxillary
incisor lingually and to maintain the same
occlusogingival level of the bracket. (a and
b) Tipping the incisor lingually by a single
force displaces the root forward using round
wire (clockwise rotation from a to b). (c) In
the next stage, the CR must be intruded and
moved distally with rotation around the CR
to correct the faciolingual axial inclinations
(counterclockwise rotation). (d) The most
direct movement with clockwise rotation.
Note that changes in the bracket path do
not follow changes in the location of the CR.
FIG 9-15 A mandibular molar tipped to the
lingual. A single force from a crisscross elastic could directly bring the molar into good
alignment in one stage of movement.
faciolingual axial inclinations. Alternatively, the most
direct movement is depicted in Fig 9-13d. Note that the
path of the bracket deviates widely from the CR path.
Figure 9-14 shows another possibility: lingual translation
of the bracket followed by x-axis rotation around the
bracket. In both Figs 9-13 and 9-14, following the path
of the CR and rotation around the CR is more useful than
planning mechanics only from the change in bracket
position. Building in bracket angulations in three dimensions is certainly helpful during finishing, when it is
simpler to have a straight wire hold and finely tune the
occlusion, but it is fallacious to think that this method
can correctly dictate the sequence of tooth movement or
provide the correct force system required during treatment. Once again, the orthodontist must do the thinking,
not the appliance.
Method 2 of describing tooth movement can therefore
lead to an inefficient or even wrong path of tooth movement; moreover, moment and force direction are not
directly related to bracket positioning. For example,
consider a malocclusion in which a mandibular molar is
Methods to Describe Change of Tooth Position
a
b
c
FIG 9-16 (a) If an archwire is used, a round wire is adequate because no moments are needed. (b) Torque is not indicated if rotation around
the CR is desired because a buccal force is more efficient. (c) If a CRot at the apex is wanted, the direction of torque is reversed, accompanied
with a single force at the bracket. The rotation direction of the tooth (dotted arrow) is the same with each force system.
FIG 9-17 Pilots control aircraft considering 3D rotations: roll (x),
yaw (y), and pitch (z). They cannot blindly use a series of rotation
commands unrelated to an outside coordinate system. A series of
harmonized maneuvers such as banking to the right by roll followed
by a slight nose-up pitch is needed for a smooth turning to the right
while maintaining altitude. The sequence of control is very important.
Simply controlling the rudder by yaw only will crash the plane. Can we
expect a straight wire to automatically give a harmonized maneuver?
tipped to the lingual. A single force from a crisscross
elastic could directly bring the molar into good alignment
in one stage of movement (Fig 9-15). If an archwire is
used, a round wire is appropriate because no moments
are needed (Fig 9-16a). Torque is not indicated if rotation
around the CR is desired because a buccal force is more
efficient (Fig 9-16b). If root apices are to be maintained,
the reverse direction of torque accompanied with a single
force at the bracket is needed (Fig 9-16c). Note that some
types of tooth movement require torque, while others do
not. It is all too easy to place improper torque after simply
looking at bracket angles without understanding the
forces and moments involved.
Orthodontists are not the only professionals that use
Euler angles with strange terminology (tip, torque, and
rotation). Pilots control planes considering 3D rotations
(Fig 9-17): roll (x-axis), yaw (y-axis), and pitch (z-axis).
They cannot blindly use a series of rotation commands
unrelated to an outside coordinate system. A series of
harmonized maneuvers such as banking to the right by
roll followed by a slight nose-up pitch is needed for a
smooth turning to the right while maintaining the altitude. The sequence of control is very important.
Controlling the rudder by yaw only will crash the plane.
Method 3
Method 3 of describing tooth displacement is the establishment of a center of rotation (CRot) in two dimensions
or an axis of rotation in three dimensions. The procedure
is as follows:
1. Identify an arbitrary landmark on the tooth. It is at
the apex (red dot) in this case (Fig 9-18a).
2. Place the landmark on the tooth before and after
movement.
3. Connect the two points (Fig 9-18b).
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The Biomechanics of Altering Tooth Position
FIG 9-18 Description of tooth movement using a CRot (method 3).
(a) Identify an arbitrary landmark (red dot) on the apex of the tooth.
(b) Connect the two points before and after tooth movement. (c) Bisect
the line and drop a perpendicular line from it. (d) Place a second arbitrary
landmark (incisal edge) on the tooth before and after movement. (e)
Connect the two second landmarks (incisal edges) and drop a perpendicular line from the midpoint. (f) The CRot is at the intersection of the
two perpendicular lines, marked as a blue dot. (g) Every point on the
tooth or any extension from the tooth rotates around the CRot. (h) The
beginning and terminal tooth positions can be correct while the intermediate tooth positions can differ, and the movement of the tooth does
not necessarily follow the path of the arc.
a
b
c
d
Center of rotation
Center of rotation
f
g
4. Bisect the line and drop a perpendicular line from it
(Fig 9-18c).
5. Place a second arbitrary landmark (incisal edge) on
the tooth before and after movement (Fig 9-18d).
Theoretically the first and second landmarks can be
placed anywhere; however, it is recommended to place
the first (apex) and second (incisal edge) landmarks as
far from each other as possible to enhance the accuracy
of intersection.
6. Connect the two second landmarks (incisal edges) and
drop a perpendicular line from the midpoint (Fig
9-18e). The CRot is at the intersection of the two
perpendicular lines, marked as a blue dot (Fig 9-18f ).
172
e
Center of rotation
h
Every point on the tooth or any extension off the tooth
(such as an attached bracket or lever) is rotating
around this center of concentric circles (Fig 9-18g).
Note that the sequence of all the intermediate tooth
displacements follows an arc, while tooth movement
between any two positions is a straight line. In reality, the
beginning and terminal tooth positions can be correct
while the intermediate tooth positions can differ, and the
movement of the tooth does not necessarily follow the
path of the arc (Fig 9-18h). For this reason, strictly speaking, the CRot is called an instantaneous center of rotation.
Primary Tooth Movement
FIG 9-19 Universal description of tooth movement in 3D; “screw
movement” along the axis of rotation. Note that it is a left-handed
screw, which means that a counterclockwise rotation will advance
the screw.
There are a number of limitations to the CRot concept.
First, it is two-dimensional. To describe tooth movement
in 3D, an axis of rotation is used. A perpendicular line to
a 2D plane can form an axis of rotation. The tooth can
rotate around the axis in 3D space. An axis of rotation
does not have to lie inside the tooth; for example, an axis
of rotation could be any place away from the tooth (x, y,
and z), with any angle to a tooth’s usual coordinate system.
This, of course, makes it difficult for the clinician to visualize tooth movements and relate them to an appliance.
It may be easier for the orthodontist to use perpendicular projections that are more familiar and more intuitive
to visualize. Although it may seem that an axis of rotation
in 3D space could define most kinds of tooth movement,
there are special situations that are not covered. Nägerl
et al have suggested a more universal approach using the
concept of a screw movement1 (Fig 9-19). The screw
movement incorporates an axis of rotation with simultaneous translation along the axis. The screw movement
along the axis of rotation can describe any movement in
3D space. The axis of rotation is also instantaneous,
describing before and after positions only and not the
path of the intermediary tooth movement. For simplicity,
and because 3D orthodontics with 3D biomechanics is
in its infancy, this book uses 2D representations for the
CRot concept in most of the discussion. Chapter 10
considers 3D displacements and biomechanics.
Force Systems and Tooth
Movement
The following sections discuss the relationship between
force systems and the pattern of tooth displacement. These
sections are based on research using theoretical analysis,
numerical techniques like finite element analysis, experimental studies on humans and animals, and direct
measurement using transducer, laser reflection, and holographic interferometry. Studies include a range of quantitative measures, from macroscopic cephalometric
measurement to microscopic wavelengths of light.
Primary Tooth Movement
The two primary tooth movements are translation and
rotation around the CR. Theoretically, in an ideal isotropic model, a force with a line of action acting through the
CR (red arrow in Fig 9-20) should produce translation
only (dotted arrow in Fig 9-20). Every part of the tooth
should move parallel with the force vector. In many orthodontic texts, this movement is called bodily movement.
A pure moment or a couple will rotate a tooth around its
CR (Fig 9-21). For simplicity, it is assumed in practice
that the tooth displacement (dotted arrow in Fig 9-20) is
parallel to the applied force (red arrow). This is probably
a good estimate; however, recent research shows that this
is not exactly true in special situations.2,3
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The Biomechanics of Altering Tooth Position
FIG 9-20 A force (red arrow) with a line of
action acting through the CR should produce translation (dotted arrow). Every part
of the tooth should move parallel with the
force vector. In many orthodontic texts, this
movement is called bodily movement.
a
FIG 9-21 A pure moment (couple) will rotate a tooth around its CR.
b
FIG 9-23 (a) The centroid (CM) of a paraboloid of revolution is also located at a
one-third distance, a calculation that is independent from most CR studies. (b) Shell
theory, which may better model the PDL as a sum of several thin 2D shells (centroid
at two-fifths the distance each, purple dots) covering the paraboloid of revolution,
also gives one-third the distance (red dot in a).
Research shows that the CR of a single symmetric,
parabola-shaped root (in 3D) is positioned approximately
one-third of the distance from the alveolar crest to the
apex, measured from the alveolar crest4 (Fig 9-22). It is
interesting to note that the centroid (CM) of a paraboloid
of revolution is also located at a one-third distance, a
calculation that is independent from most CR studies
(Fig 9-23a). Shell theory (Fig 9-23b), which may better
model the PDL as a sum of several thin 2D shells
(centroid at two-fifths the distance each, purple dots)
covering the paraboloid of revolution, also gives one-third
the distance (red dot in Fig 9-23a). It has been estimated
that the CR of a molar is near its trifurcation or bifurcation5 (Fig 9-24).
174
FIG 9-22 Research shows that the CR of a
single symmetric, parabola-shaped root (in
3D) is positioned approximately one-third
of the distance from the alveolar crest to the
apex, measured from the alveolar crest.4
FIG 9-24 The CR on a mandibular molar is near
its bifurcation. (Reprinted from Burstone4 with permission.)
The location of the CR is relatively independent of the
applied force magnitude if the PDL strain is small (Fig
9-25). However, if the magnitude of the force is large
enough, not only the PDL but also the alveolar bone or
even the tooth itself will undergo nonlinear deformation;
hence, the location of the CR can change with force
magnitude. This chapter considers the CR and CRot independent of force magnitude and only considers the tooth
displacement in the PDL space, not in the surrounding
bone. This simplifies the understanding and presentation
of the biomechanics of tooth movement. Future research
is needed to better define the relationship between force
or couple magnitude and the CRot. Note that in a study
in which the couple magnitude was increased, the CRot
Primary Tooth Movement
FIG 9-25 The location of the CR is relatively independent of the
applied force magnitude if the PDL strain is small.
FIG 9-27 Clinical implications of the CR in three situations. (a) A
typical given M/F ratio of 10 mm at the bracket will be equivalent to a
force acting through the CR (D1 ). (b) With significant root resorption, a
lower M/F ratio is required (D2 ). (c) The adult patient with significant
alveolar bone loss requires the highest M/F ratio to translate the
tooth lingually (D3 ).
moved horizontally; ie, the tooth extruded more as the
couple increased (Fig 9-26). This study did not differentiate tooth movement produced by PDL strain from that
produced by alveolar bone deformation.2
More research is still needed to locate the CR of all
teeth in all planes, of groups of teeth (segments), and of
full arches. This knowledge is needed even without
considering the complications of a CR in 3D space. It
should also be recognized that significant variation exists
among patients with differing tooth morphologies, periodontium, and periodontal changes during treatment.
Consider the clinical implications of the three central
incisors shown in Fig 9-27. In Fig 9-27a, an arbitrary but
typical M/F ratio of 10 mm at the bracket will be equivalent to a force acting through the CR. In Fig 9-27b, the
amount of root resorption means that a lower M/F ratio
is required. The adult patient in Fig 9-27c, with much
alveolar bone loss, requires the highest M/F ratio to translate the tooth lingually. Mesiodistal sliding mechanics
would create the highest friction in part c. As mentioned
FIG 9-26 When the couple magnitude is increased, the CRot moves
horizontally; ie, the tooth is extruded more as the couple is increased.
a
b
c
previously, a CR should not be considered a point but
rather a circle in 2D because of variations. Even with
these limitations, a CR is a useful concept for practical
clinical application.
A pure moment or a couple rotates a tooth around its
CR. A couple applied at the various positions of the incisor bracket in Fig 9-28 produces rotation around the CR
and not—as some orthodontists may believe—around
the bracket. If couples of the same magnitude are applied
at different points on the tooth or even on extensions
away from the tooth, the action is the same. Rotation
around the bracket requires a couple and a force. Couples
are free vectors, and unlike forces, their point of force
application does not change how a tooth will move. Note
also in Fig 9-29 that the directions of the two equal and
opposite forces comprising a couple make no difference
as long as the magnitudes of the couple moments are the
same.
The visualization of a couple as a free vector rotating
a tooth around its CR may be difficult, so instead let us
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The Biomechanics of Altering Tooth Position
FIG 9-28 A couple applied at different bracket positions on an incisor; however, both actions produce rotation around the CR.
FIG 9-29 The directions of the two equal and opposite forces that
comprise a couple make no difference in the force system. All teeth
will rotate around the CR.
a
b
c
d
FIG 9-30 (a to d) The visualization of a couple as a free vector rotating a tooth around its CR. See text for explanation.
calculate the effect of identical couples at a tooth. In Fig
9-30a, two equal and opposite forces of 100 g are applied
to a canine. The moment is equal to one force times the
perpendicular distance to the other force (Fig 9-30b).
100 g × 10 mm = +1,000 gmm
Adding the two moments gives +1,000 gmm, which is
the same answer we got when we multiplied the force
times the perpendicular distance to the other force.
Now let us move the couple occlusal to the crown (Fig
9-31a). The moment at the CR (Figs 9-31b and 9-31c)
from the upper force (left red arrow) is
Now let us calculate the moment in respect to the CR.
100 g × 22 mm = +2,200 gmm (clockwise)
100 g × 5 mm = +500 gmm (Fig 9-30c)
The moment from the lower force (right red arrow) at the
CR (Figs 9-31d and 9-31e) is
100 g × 5 mm = +500 gmm (Fig 9-30d)
176
100 g × 12 mm = –1,200 gmm (counterclockwise)
Derived Tooth Movement
a
d
b
e
c
f
FIG 9-31 (a to f) The couple is moved occlusal to the crown; however, the CR feels the same moment as from the more apically placed
couple. See text for explanation.
Adding the two moments still gives the same answer:
+1,000 gmm (Fig 9-31f ). The CR feels the same moment
from the occlusally placed couple as from the more
apically placed couple because couples are free vectors.
A pure moment or a couple delivers no force to a tooth.
The sum of all forces is zero. It may seem confusing to
the clinician that a tooth will move without a resultant
force. The special case of a couple produces rotation
around the CR by a moment alone.
Derived Tooth Movement
When the line of action of a force is away from the CR,
the displacement is called derived tooth movement. Various lingual forces in Fig 9-32 have lines of action occlusal
and apical to the CR, and each of them produces a different axis of rotation. The forces occlusal to the CR produce
varying degrees of lingual tipping (clockwise rotation),
and the forces apical to the CR produce lingual root
movement (counterclockwise rotation). Let us select one
of the forces at the level of the alveolar crest (Fig 9-33)
that will tend to tip the incisor lingually (rotate it clockwise) around an axis at the apex. Because the force is not
at 90 degrees to the long axis of the tooth, a small intrusive component is created that will be ignored for now.
The lingual force can be replaced with an equivalent
lingual force and a couple at the CR (yellow arrows in Fig
9-33b). The tooth will translate from the force, and the
couple will rotate the tooth around its CR. In this particular case, the balance of each primary displacement
produces the CRot near the apex. In Fig 9-33b, the translatory tooth movement is shown in the transparent tooth
and the rotation around the CR in the blue transparent
tooth.
The starting point in determining what force system is
needed for an orthodontic appliance is to locate the single
force that will produce the desired CRot. Any desired
CRot can be accomplished with a single force either on
or away from the tooth; the challenge is to find the line
of action of force application. The exception, of course,
is rotation around the CR, where a couple is required. To
help in determining force position, a “stick” diagram is
most useful. Let us use the example of lingual crown
tipping around a CRot at the apex (blue dot in Fig 9-34).
The tooth is represented as a gray stick with a purple
circle as the CR (see Fig 9-34). A force (red arrow) is
applied at the alveolar crest region that typically tips an
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9
The Biomechanics of Altering Tooth Position
a
FIG 9-32 Various lingual forces have lines
of action occlusal and apical to the CR, and
each of them produces a derived tooth
movement.
b
FIG 9-33 (a) A force at the level of the alveolar crest will tend to tip the incisor lingually
(rotate it clockwise) around an axis at the apex. (b) The lingual force can be replaced with an
equivalent force and a couple at the CR (yellow arrows). The tooth will translate from the force,
and the couple will rotate the tooth around its CR. The balance of each primary displacement
produces the CRot at the apex.
FIG 9-34 Stick diagram of controlled
tipping. (a) A force (red arrow) is applied
at the alveolar crest region that typically tips an incisor around the apex. The
equivalent force is replaced at the CR
with a force (yellow arrow in b) and a
couple (yellow arrow in c). (b) The stick
(tooth) is translated to the lingual from
the force component. (c) The clockwise
moment (yellow arrow) rotates the stick
around its CR so that the CRot is at the
apex (blue dot).
a
a
b
b
c
c
incisor around the apex. The force is replaced at the CR
with a force (yellow arrow in b) and a couple (yellow arrow
in c). In Fig 9-34b, the stick (tooth) is translated to the
lingual from the force component. Finally, after the translation (Fig 9-34c), the clockwise moment rotates the stick
(tooth) around its CR so that the CRot is at the apex (blue
dot). In this example, we already knew the correct position
of the red force to produce rotation around the apex based
on research6; our analysis only serves to explain why the
CRot could be at the apex. But suppose we did not know
178
d
FIG 9-35 Stick diagram of root movement. (a) Suppose we did not know where
to position the force to rotate an incisor
around its incisal edge. (b) It is readily
seen that the CR moves along the line
of action of applied force. Therefore, the
direction of the force must be lingual (yellow arrow). (c) The stick must be rotated
so that the CRot is at the incisal tip. A
counterclockwise couple can achieve this
(yellow arrow). (d) Only a force apical to
the CR (red arrow) can achieve this result;
thus, the location of the force must be
apical to the CR.
where to position the force to tip an incisor around the
apex. Here the stick diagram is especially useful. If we
draw the before and after tooth positions, it is readily
seen that the CR moves along the line of action of applied
force. This tells us that the direction of the force must be
lingual (see Fig 9-34b). The remaining question is the
location of the force. The stick must be rotated so that
the CRot is at the apex. A clockwise couple can achieve
this (see Fig 9-34c). On which side of the CR could a force
produce a clockwise couple? Only a force occlusal to the
Derived Tooth Movement
a
b
c
d
e
f
g
h
FIG 9-36 The effect of changing force position in the simple geometry of horizontal forces at 90 degrees to the long axis of a tooth. The position of force will be moved apically from the crown sequentially. (a) The force acts at the level of a typical bracket, and the CRot (blue dot) is
about a millimeter or so apical to the CR (purple circle). (b) The force is moved apically. The tooth tips, with the root still moving in the opposite
direction as the crown, but less so than in part a. (c) The force is placed at the alveolar crest. The CRot is at the apex. No part of the tooth is
moving in the opposite direction as the applied force. (d) The force is slightly occlusal to the CR. Both the crown and the root apex move in
the same direction. The CRot moves off the tooth. (e) As the force is approaching the CR to produce translation, the CRot is diverged to lower
infinity. (f) The force is placed slightly apical to the CR. The CRot is moving downward from upper infinity toward the crown. (g) The force is
placed more apically. Root movement with a CRot at the incisal edge could occur. (h) Moving the force further apically places the CRot near
but occlusal to the CR, and the tooth movement begins to approach the movement of a couple.
CR can achieve this result; thus, the location of the force
must be occlusal to the CR, and its exact location must
be determined by additional calculation.
Let us now consider another example (Fig 9-35) of an
incisor that requires root movement (a CRot around the
incisal edge, blue dot). Unlike Fig 9-34, all that is known
at the start of our determination is the force direction.
In Fig 9-35a, the CR moves lingually; therefore, a lingual
yellow force at the CR is required (Fig 9-35b). What is
the direction of the couple to rotate the stick (tooth) so
that the CRot is at the incisal edge? The correct direction
is counterclockwise (Fig 9-35c). So where does the single
force go? It is positioned somewhere apically to the CR
(red force in Fig 9-35d). Additional research tells us that
it is approximately 2 to 4 mm apical to the CR. But what
is the force system at the bracket, where an appliance
delivers the force that produces the incisor root movement? That is the easy part: an equivalent force system
to the single force on the root. A more detailed discussion
is given later in this chapter.
Figure 9-36 describes the effect of changing force position in the simple geometry of horizontal forces at 90
degrees to the long axis of a tooth. The graphics are
general in nature and do not necessarily reflect exact
positions. Within a moderate range, force magnitude
does not seem to influence the location of the CRot, as
shown in Fig 9-38. The old idea that heavy forces tip teeth
more than light forces is certainly incorrect.
In Fig 9-36a, the force acts at the level of a typical
bracket, and the CRot (blue dot) is about a millimeter or
so apical to the CR (purple circle). The force is far enough
from the CR that the effect from the moment overwhelms
the effect of the force so that tooth movement is similar
to that produced by a couple. Now let us sequentially
move the force apically (Fig 9-36b). The CRot is moving
apically, with the root still moving in the opposite direction as the crown, but less so than in part a. If the force
is placed further apically at the alveolar crest (Fig 9-36c),
the CRot is moving to the apex. The CRot moves off the
tooth in Fig 9-36d, where the force is slightly occlusal to
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9
The Biomechanics of Altering Tooth Position
FIG 9-37 Stick diagrams can be used in the occlusal view. A molar
is to be moved around a CRot near the mesial contact area. A force
(red arrow) placed off-center to the distal will translate the stick
buccally and rotate it around its CR in a counterclockwise direction.
The CRot could be at the mesial contact area, depending on how far
distal the force is placed.
the CR; both the crown and the root apex move in the
same direction. As the force is approaching the CR to
produce translation, the CRot is diverged to lower infinity (Fig 9-36e).
Moving the force slightly apical to the CR (Fig 9-36f )
abruptly moves the CRot downward from upper infinity
toward the crown. In Fig 9-36g, root movement with a
CRot at the incisal edge could occur. Moving the force
further apically places the CRot near but occlusal to the
CR, and the tooth movement begins to approach the
movement produced by a couple (Fig 9-36h).
Note in Figs 9-36a and 9-36h that when the force is far
away from the CR, there is a part of the tooth that moves
in the opposite direction to the applied force. In other
words, the tooth movement becomes similar to pure
rotation by a couple.
The same “stick” diagrams as in Figs 9-34 and 9-35 can
be used in the occlusal view. A molar is to be moved
around a CRot near the mesial contact area (Fig 9-37). A
force (red arrow) placed off-center to the distal will translate the stick (tooth) buccally and rotate it around its CR
in a counterclockwise direction. The CRot (blue dot)
could be at the mesial contact area depending on how
far distal the force is placed. The further distally the force
is placed, the closer the CRot will be to the CR.
Eight physical 2D models demonstrate the effect of
force position and magnitude in Fig 9-38. The teeth are
180
suspended by a series of elastics simulating the PDL. The
green line is drawn on the transparent tooth and the red
line on the background so that they coincide at rest. As
the tooth is displaced by a force (red arrows), the amount
of displacement and rotation is seen by the gap and the
angle between the two lines. The location of intersection
(blue circle) of the blue and red lines represents a CRot
such as that in Figs 9-34 and 9-35. The first thing to notice
is that the CR moves parallel to the applied force, no
matter where it is applied. In some cases, as in Figs 9-38a
and 9-38b, some part of the tooth may move in the opposite direction as the force, but the CR never does. As the
force position moves downward from the bracket to the
apex (Figs 9-38a to 9-38d), the intersection (blue circle)
moves as explained by the curves in Fig 9-43, discussed
later in this chapter. Note in Figs 9-38a and 9-38b how
the force magnitude does not affect the location of intersection (CRot, blue circles) and that only the amount of
rotation is increased as the force magnitude is increased.
Also note the amount of tooth displacement of the teeth
(∆) in Figs 9-38a and 9-38f. These two force systems show
a similar amount of tooth displacement at the crown, yet
the amount of force used is totally different. Based on the
elongation of the chain elastic, very light force is used in
Fig 9-38a and very heavy force is used in Fig 9-38f;
however, the maximum amount of stress the PDL feels
would be the same. In other words, very light force can
induce very high stress in uncontrolled tipping (see Fig
9-38b). Even though the force magnitude is very low,
repeated light faciolingual forces at the crown by the
muscle of the cheek and the tongue induce physiologic
mobility of uncontrolled tipping. As a result, the width
of the PDL is most narrow near the middle of the root
and widens coronally and apically from the middle.
Empirically, clinicians have learned that the most efficient
way (the least force used) of extracting a tooth is repeated
uncontrolled tipping, which can induce very high stress
on the PDL and bone. Also note that the location of the
CRot changes abruptly near the CR.
In summary, in derived tooth movement (1) the CR
moves parallel to the applied force; (2) a single force with
a correct line of action can produce any required center
or axis of rotation; (3) the CRot is independent of force
magnitude; and (4) the CRot changes abruptly when the
force is near the CR. Even the exception—rotation around
the CR—can be obtained by a force applied away from
the CR, and that distance does not have to be great. Figure
9-39 reminds us that there are infinite force positions in
3D and that they do not necessarily have a line of action
through the tooth itself.
Derived Tooth Movement
FIG 9-38 Eight physical 2D models
demonstrate the effect of force position
and magnitude. The teeth are suspended
by a series of elastics simulating the PDL.
The green line is drawn on the transparent tooth, and the red line is drawn on
the background so that they coincide at
rest. As the tooth is displaced by a force
(red arrows), the amount of displacement
and rotation is seen by the gap and the
angle between the two lines. The location
of the intersection of the green and red
lines (blue dot) is like a CRot in the stick
diagram. (a and b) Uncontrolled tipping.
(c and d) Controlled tipping. (e and f )
Translation. (g and h) Root movement.
Comparing a and b, c and d, e and f, and
g and h, the force position is the same
but the magnitude is greater on the right
(b, d, f, and h). The force magnitude does
not affect the location of the CRot. Note
that the amounts of displacement of the
bracket in a and f (∆) are similar even
though the applied force magnitude is
different.
a
b
c
d
e
f
g
h
FIG 9-39 There are infinite force positions and corresponding axes of rotation
in 3D.
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9
The Biomechanics of Altering Tooth Position
FIG 9-40 (a) A force (red arrow) apical to
the CR that could produce root movement
with a CRot around the incisal edge. (b) The
yellow force system is equivalent to the single
red force.
a
b
Force Systems at the Bracket
An obvious question is: What is the force system that an
appliance must place at the bracket to produce the
required CRot? It is nice to know that a force placed 2
mm apical to the CR could produce root movement
around the incisal edge, but it may not be possible to
place the force so far apically. The use of wire extensions
(lever arms) sometimes allows a force to be placed apically
on the root, but there are limitations because of gingival
impingement. The most common approach is to place
the forces at a bracket. It is therefore necessary to replace
the correct single force with an equivalent force system
at the bracket. Figure 9-40a shows a force (red arrow)
apical to the CR that could produce root movement with
a CRot around the incisal edge. The formulas for equivalence discussed in chapter 3 can now be used to replace
the 100-g force on the root with an equivalent force
system at the bracket (Fig 9-40b).
∑ F1 = ∑ F2
Therefore, 100 grams must be applied at the bracket.
∑ M1 = ∑ M2
A point on the bracket (red dot) is selected to sum the
moments. A couple of –1,200 gmm must be placed at
the bracket (curved yellow arrow). The yellow force system
is equivalent to the single red force. Other force characteristics, such as the force-deflection (F/∆) rate and
moment-deflection (M/θ) rate, may change during deactivation of the spring so that an equivalent M/F ratio may
not be constantly maintained. If a ratio is made between
the moment (couple) and the force, the force system at
the bracket is defined. Here the ratio is 12:1. The CRot
of a tooth is determined by the M/F ratio and is mainly
independent of force magnitude. If the M/F ratio is used,
182
the point of force application must be given. The unit of
measurement for the M/F ratio is millimeters, and the
M/F ratio simply denotes how many millimeters away
from the bracket a single force must be placed.
One might think that if a lot of effort or work is required
to insert a wire into the bracket, then the tooth will feel
excessive stress in the PDL, which could be harmful and
could lead to undesirable side effects like root resorption.
However, this is not always true. One should not confuse
the effort or work that is required to insert an appliance
with the stress the PDL will feel. For example, consider
a walrus with a very long canine (Fig 9-41). Arbitrarily,
let us apply 100 g at the CR (Fig 9-41a). We will replace
this force system at two different levels for comparison:
(1) at bracket A, which is 10 mm away from the CR, and
(2) at bracket B, which is 60 mm away from the CR at the
tip of the tooth (Fig 9-41b). Because bracket A is 10 mm
from the CR, a force of 100 g and a moment of –1,000
gmm is needed at the bracket (yellow arrows in Fig 9-41c).
The distance from the CR to bracket B is 60 mm, so a
moment of –6,000 gmm is needed at that bracket along
with the 100 g force (yellow arrows in Fig 9-41d). What
does the orthodontist feel when he or she inserts the wire
into each bracket position? If a bracket existed at the CR,
the 100 g would feel light and the wire would be easy to
insert because no moment would be necessary. If the
orthodontist inserts the wire at bracket A, more effort
would be needed because of the 100-g force and torque
of –1,000 gmm. Yet the tooth itself will feel the same as
if the appliance delivered the 100 g through the CR. If
the orthodontist inserts the wire at bracket B, he or she
will need to be very strong to place it because of the
–6,000-gmm torque. However, even though it would
require hard work to activate the appliance, the force
system acting on the tooth at bracket B would have no
different effect on the incisor than the 100 g applied
through the CR.
Force Systems at the Bracket
a
b
c
d
FIG 9-41 A walrus with a very long canine demonstrates how the effort required to place a wire does not affect the force system acting on
the tooth. (a) A force of 100 g is applied at the CR. (b) Bracket A is placed 10 mm away from the CR, and bracket B is placed 60 mm away from
the CR at the tip of the tooth. (c) The replaced equivalent force system at bracket A (yellow arrows). (d) The replaced equivalent force system
at bracket B (yellow arrows). All force systems are equivalent, and hence the tooth feels the same stress.
FIG 9-42 LVDT with infinite resolution is
used to detect and trace the micromovement of the maxillary canine and the silicone
periodontium model under various loading
conditions.
With this general concept in mind, let us investigate
further about the location of the CR and CRot based on
some experiments. Because the amount of displacement
or rotation of the tooth within the PDL space is very
small, sophisticated technology like laser hologram or a
linear variable displacement transducer (LVDT) with
infinite resolution are required to detect and trace the
movement of the tooth under various loading conditions.
An experiment in which an in vitro maxillary canine was
placed in a silicone periodontium model (Fig 9-42) shows
the normal sequential change in the CRot as a horizontal
force is moved vertically. When the force position is plotted against the CRot, a typical hyperbolic curve is
produced (Fig 9-43). The horizontal axis gives the force
position (a) from the CR. The vertical axis gives the
position of the CRot (b) in respect to the CR. The
corresponding M/F ratios are depicted on the secondary
vertical axis. The absolute numbers of the CRot position
are less important than the general shape of the graph,
because there can be much variation in tooth morphology and PDL constitutive behavior.
Let us start with a primary tooth movement—the application of a pure moment or a couple in Fig 9-43. The
force at a large distance from the CR coronally (force A,
a = 25 mm or M/F ratio = –13 mm), the CRot (blue dot)
starts to moves from the CR to A'. For all practical
purposes, force so far from the CR acts like a couple
because the effect from the moment overwhelms the
effect of the force. Let us now move the force apically to
the level of the bracket (force B, a = 12 mm or M/F ratio
= 0 mm). The CRot is still very close to the CR (B'), only
a few millimeters further apically. The clinician should
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The Biomechanics of Altering Tooth Position
FIG 9-43 The force position from
the CR (a) versus the distance between the CRot and the CR (b).
CRots produce a typical hyperbolic
curve. Each position of the force (a)
or M/F at the bracket (A to E) corresponds with a CRot (A' to E'). When
the force is near the CR (from C to
E), the CRot abruptly changes from
(C') to –infinity and from +infinity to
cusp tip (E') (gray area outside of the
tooth).
be aware that either a couple or a single force at the
bracket will produce about the same effect. Next the force
is moved apically to a level at slightly below the alveolar
crest (force C, a = 4 mm or M/F ratio = 8 mm). Here the
CRot is at the apex (C'). If the force level is moved to the
CR (force D), translation occurs, and the CRot diverges
to lower infinity (D'). When the force is moved slightly
apical to the CR (force E, a = –3 mm or M/F ratio = 14
mm), root movement is observed with a CRot from upper
infinity (D') to the cusp tip (E'). The shape of the curve
shows us that great accuracy is required in a narrow range
of points of force application (gray area between forces
C and E) for CRots around commonly required points
above the incisal edge and below the apex of the root,
where important CRots for controlled tipping, translation, and root movement occur.
During anterior retraction, most clinicians prefer to
think of the force system that must be applied at the
bracket. The M/F ratio at the bracket is the equivalent
moment and force that must be applied at the bracket.
An increasing number of orthodontists are beginning to
use extensions and therefore require a sound biomechanical protocol for determining the position of the single
force away from the bracket.
Let us look in greater detail at the force system at the
bracket based on the data and concepts described in Fig
184
9-43 from the clinical point of view. A round archwire
with a force from an elastic or a spring delivers only a
force (Fig 9-44a). The crown moves lingually and the root
labially with a CRot slightly apical to the CR. This is
referred to as simple tipping or uncontrolled tipping. To
avoid movement of the apex in the opposite direction,
perhaps to achieve more desirable axial inclinations, the
CRot must be placed near the apex; this requires both a
lingual force (straight red arrow) and a counterclockwise
moment (curved red arrow in Fig 9-44b). The direction
of the moment is commonly called lingual root torque.
If the incisor requires lingual translation only and the
force remains the same, the magnitude of the moment
must be increased (Fig 9-44c). Finally, for root movement
(rotation around an axis at the incisal edge) when the
force is kept constant, the moment must be further
increased (Fig 9-44d).
Note that controlled tipping (rotation around the apex),
lingual translation, and root movement all require
moments or torques in the same direction. This may not
be intuitive because in parts b and d the teeth are rotating in opposite directions. As the hyperbola-shaped
tooth-movement graph demonstrated (see Fig 9-43),
most of the important rotation centers are found in a
narrow range of force positions. This narrow range of
force positions (gray area near CR in Fig 9-45) is the
Force Systems at the Bracket
FIG 9-44 The force system at
the bracket and CRot. (a) A single force at the bracket results
in uncontrolled tipping. (b) Controlled tipping. (c) Translation.
(d) Root movement.
a
b
c
d
“critical zone”; note that CRots from infinity to the incisal
edge or from infinity to the apex are possible with single
forces acting in this zone (see Fig 9-45). By contrast, rotating a tooth around the CR or close to the CR is not challenging because many force positions outside the gray
zone can achieve that result. Nägerl et al developed a
theory of proportionality (Fig 9-46) by which the distance
from the applied force to the CR (a) multiplied by the
distance from the CR to the CRot (b) gives a constant
(σ2).1 The value of σ2 may vary in accordance with the
direction of force; however, this formula is valid threedimensionally:
a×b=σ2
σ2 is a measure of the variance of the support of the
tooth by the periodontium and is called the center of
rotation constant of a tooth at a given direction of force.
Schematically visualized PDL support in accordance with
varying σ2 values is depicted in Fig 9-47. The higher the
sigma (σ), the wider the “critical zone” (Fig 9-47a). A high
σ2 makes the predictability of achieving any CRot much
easier, and a low σ2 means a greater sensitivity for locating a force to acheive an exact CRot (Fig 9-47b). Theoretically, if σ2 = 0 (Figs 9-47c and 9-47d), no matter where
the force is placed (except at the CR), rotation around
the CR will occur (red arrows in Figs 9-47c and 9-47d).
The translation is theoretically possible but extremely
difficult to achieve practically (see Fig 9-47c). Just a small
FIG 9-45 The important rotation centers from the apex to upper
infinity and from lower infinity to the incisal tip are found in a narrow
range of force positions near the CR. This narrow range of force
positions is the “critical zone.”
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The Biomechanics of Altering Tooth Position
FIG 9-46 Nägerl et al developed a theory of proportionality by which
the distance from the applied force to the CR (a) multiplied by the
distance from the CR to the CRot (b) gives a constant (σ2) known as
the center of rotation constant. σ2 is a measure of the variance of the
support of the tooth by the periodontium.1
a
b
c
d
FIG 9-47 (a) A high σ2 value provides a wider critical zone. (b) A low σ2 value provides a narrower critical zone. (c) When σ2 = 0, it is impossible
to produce various types of tooth movement. A single force acting at the CR produces translation. (d) No matter where the force is placed
(except at the CR), rotation around the CR will occur, and translation is very difficult.
amount of deviation from the CR or a force placed
anywhere except at the actual CR results in pure rotation.
It is beyond the scope of this book to describe σ2 in
detail; however, longer roots obviously have greater σ2
186
values. Also, wider roots and teeth splinted together as
a unit have greater σ2 value. σ2 is determined by root
morphology and periodontal behavior and not just by
root length.
A Couple or Single Force at the Bracket for Rotation Near the CR
a
b
c
FIG 9-48 A Class II, division 2 nonextraction case. (a) The maxillary central incisors are to be flared and the arch length increased. Either
labial force at the bracket (b) or lingual root torque (c) can achieve that goal.
A Couple or Single Force at the
Bracket for Rotation Near the CR
Twist in an orthodontic wire produces a couple or torque.
A couple anywhere or a single force at the level of a
bracket produces about the same CRot; the tooth tips,
with the crown moving in one direction and the root
apex moving in the opposite direction. Some orthodontists historically were taught that torque in a wire or
“torque” in the bracket will spin a tooth around the
bracket. To spin a tooth around a bracket, however, both
a moment (couple or torque) and a force are required. A
force tips a tooth around an axis about 1 to 2 mm apical
to its CR, and torque tips it around the CR. Clinically, it
is impossible to tell the difference in how the tooth moves
between the two approaches. Which is more efficient
when this type of tipping movement is indicated? Because
the end result from two completely different force systems
is similar, the choice depends on the feasibility of clinical
application and the best possible equilibrium diagram.
Let us consider a few examples in which a force is the
better choice over torque placement. Chapter 12 shows
some cases in which a couple is a better choice over a
single force.
Figure 9-48a shows a Class II, division 2 nonextraction
case in which the maxillary central incisors are to be
flared and the arch length increased. The CRot needed
at the incisor is near the CR; the crown comes forward
and the root moves back. Either labial force at the bracket
(Fig 9-48b) or lingual root torque (Fig 9-48c) can achieve
that goal. But which is more efficient? The better choice
is the labial force, which allows simultaneous leveling
with a round flexible archwire and is very simple. A labial
force can be introduced at the beginning of treatment.
To deliver torque, on the other hand, a full-size edgewise
wire is needed, and much tooth alignment must be
accomplished before a rectangular archwire with a twist
can be placed. Friction is more of a problem with a twisted
archwire because the wire must slide anteriorly to allow
for flaring. Moreover, inserting a twisted edgewise wire
could cause the bracket bond to fail or the bracket to
fracture if it is ceramic. The single force is also more
favorable because the slightly more apical CRot allows
less lingual movement of the apex.
It is common to finish treatment of a Class II patient
before the second molars fully erupt. Figure 9-49 shows
a frontal view of second molars. The mandibular molars
are in a correct position, but the maxillary molars are
erupting too far to the buccal. Spinning the molar around
an axis near the CR will improve its axial inclination and
reduce the buccal overjet (horizontal overlap). One possibility is to bond tubes on the second molars and place
lingual crown torque in a full-size edgewise wire (Fig
9-49a). Theoretically, such a force system is excellent, but
this is not a good choice practically for many reasons.
The localized tensile stress at the bonding surface is very
high, so that the newly bonded molar tubes may not reach
sufficient bond strength to withstand the torque. Practically it is difficult to place a full-size wire with torque
through the first and second molar attachments. Some
leveling may first be required. Also, it is difficult to deliver
a constant couple because unpredictable forces are associated with deflection of an edgewise wire. As the molar
tips lingually, the geometry of the wire to the bracket
changes, and pure torque no longer presents itself. Overall indeterminacy increases.
A single lingual force applied to the crown is much
simpler and is accomplished with a light round archwire
or a finger spring (Fig 9-49b). The force system is much
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The Biomechanics of Altering Tooth Position
a
b
FIG 9-49 The maxillary molars are erupting to the buccal side. Rotating the molar around an axis near the CR by a couple (a) or applying a single
lingual force to the crown (b) will reduce the buccal overjet. A lingual force (b) is the best choice. Placing torque may require preliminary leveling.
Pure continuous torque is difficult to practically achieve because as the molar moves lingually, the force system changes and is indeterminate.
FIG 9-50 In a simple anterior reverse articulation, a round wire
appliance delivering a single force (red arrow) without any torque
would flare the maxillary incisors while correcting their axial
inclination. Note that the CR only moves horizontally.
more predictable because no moment is associated, yet
the end result will be similar. Many times torque is
selected for the wrong reason; “torque” is often incorrectly
used to mean axial inclination rather than a couple force
system. “The molar has improper torque” does not make
sense semantically.
Sophisticated appliances like the edgewise arch were
developed to simultaneously deliver forces and moments,
and it is sometimes thought that more primitive appliances with less control and hence fewer degrees of freedom always produce compromised results. But this is not
necessarily the case because not all treatment requires
six degrees of freedom. For example, in a simple anterior
188
reverse articulation (also sometimes referred to as crossbite), a round wire appliance (or even a tongue depressor
as a lever) could flare the maxillary incisors while correcting their axial inclination (Fig 9-50). No “torque” is necessary.
There are many examples of a single-force appliance
(one degree of freedom) being inserted and beneficial
tooth movement occurring in all planes of space. This
involves both translation and rotation around all axes (x,
y, and z) through the CR or another relevant point. Even
removing an archwire from the tube or bracket to eliminate unwanted moments from the wire can further
enhance this simple “single force only” appliance therapy
if indicated. This text also discusses the use of wire extensions (cantilever arms), by which single forces can be
placed near or at the CR or away from the CR to produce
a different CRot. This is a separate application of the
concept of applying a simple force without a couple to
the tooth’s crown; if properly positioned and feasible, a
single force can produce any needed CRot.
When a Force and a Couple Are
Required for Tipping
It has already been discussed how a single force at the
bracket can produce satisfactory tipping if the goal is to
allow the root to move in the opposite direction. In most
malocclusions requiring significant retraction of incisors,
it is preferable to prevent the root apex from being
displaced forward. Figure 9-51a compares tipping an
incisor lingually around the CR with tipping it around
the apex. The end point for both is a normal incisor axial
inclination. Tipping around the apex, however, allows for
more retraction and hence is indicated for extreme over-
When a Force and a Couple Are Required for Tipping
a
b
c
d
FIG 9-51 In most malocclusions requiring significant retraction of incisors, it is preferable to prevent the root apex from being displaced
forward. (a) Compare the displacement of the bracket in uncontrolled tipping (D1 ) and controlled tipping (D2 ). The end point for both is a normal
incisor axial inclination. Controlled tipping allows more retraction. (b) Both a lingual force and a moment are required (red arrows). A force and
a moment are indicated for extreme overjets and for extraction cases. The CR (dotted arrow) moves in the same direction as the force (straight
red arrow). (c) The bracket moves downward and backward and rotates in a clockwise direction (dotted arrows). Some clinicians erroneously
think that the required force system is in the same direction as the bracket displacement (gray force system). (d) A patient showing an increase
in deep bite. The root is not displaced forward, and the CR must translate upward and backward. The correct force system is shown with red
arrows. The displacement of the bracket (gray arrows) is not the correct force system.
jets and for extraction cases. Maintaining the position of
the apex is called controlled tipping. No part of the tooth
is displaced in the opposite direction to the applied force.
As previously discussed, a moment and a force are
required. For simplicity, the force direction has been
ignored in this chapter up to now. The M/F ratios of the
graph in Fig 9-43 refer to forces at 90 degrees to the long
axis of the tooth. Figure 9-51b shows an incisor rotating
around the apex (blue dot). Note that both a lingual force
and a moment (lingual root torque) are required. The CR
moves occlusally and lingually. The direction of the force
is an average reflecting the displacement direction of the
CR (dotted arrow). The CR must move in the same direction as the force (with some exceptions and considerations that have been discussed and are here disregarded).
Many orthodontists describe the tooth movement path
at the bracket, as discussed earlier in this chapter. Note
in Fig 9-51c how the bracket moves downward and backward and rotates in a clockwise direction. Some
orthodontists erroneously think that the necessary force
system is found in this path (gray arrows). The correct
force system to produce the desired CRot, however, is
shown in red (see Fig 9-51b). Chapter 12 discusses the
relationship between different bracket positions and the
force system produced; this is an entirely different question, and even here we must relate our answers back to
the CR of the tooth. The clinician cannot reason that any
angle deviation means a corresponding couple or that
any x, y, z discrepancy gives the correct direction or
amount of force. Looking at the change in bracket position in this way is letting the bracket do the thinking.
Connecting the CR at position 1 with the desired position
2 gives the average direction of the force; the angle of
rotation around the CR gives the direction and is related
to the magnitude of the required moment. It is true that
we may not accurately know the position of the CR as
properly defined; however, we can accurately copy any
CR point from the first to the next tooth tracing. Even if
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The Biomechanics of Altering Tooth Position
there is some error, an estimate of the CR is better than
a bracket landmark, which is only indirectly influenced
by a force system because it is far from the CR. The closer
our estimate is to the CR, the more reliable is the interpretation of force and moment direction, because we are looking at primary displacements. In short, the change in position of the bracket should be called the bracket path and
is not directly related to the force system of that path
(except in the rare case of pure translation). The path of
the CR (translation and rotation) is directly related to the
direction and magnitude of the moment and force needed
for that CR path.
In some patients, an increase in the vertical overlap
(also known as overbite) is undesirable; for these patients,
a different approach is necessary (Fig 9-51d). The root is
not displaced forward, and the CR must translate upward
and backward. The direction of force is now upward and
backward, and the moment is clockwise, which is the
same direction as the bracket rotation path (gray arrow).
The correct force system is shown in red. If the bracket
does the thinking instead of the orthodontist, the force
is placed in the wrong direction (gray arrow). If a straightwire appliance is used in this case, the force system is also
incorrect.
In Fig 9-52, a recent experimental study using LVDT
shows that the displacement vector of the tooth (dotted
arrow) does not necessarily parallel the applied force
vector (red arrows).2,3 The small blue circles are the CRots
under varying horizontal forces (red arrows), which are
perpendicular to the anatomical long axis of the canine
(not drawn). The varying CRots have a linear asymptote
(black line) with an angle of 11 degrees to the long axis
of the tooth. The CR of the tooth moves perpendicular
to this line along the straight dotted arrow. The black line
formed by varying CRots is called a functional axis of the
tooth under given applied forces. A little horizontal
dispersion of the CRots near the CR is due to minor
extrusive vertical displacement by a couple, as explained
earlier in Fig 9-26. The angulation of the anatomical long
axis to a given force direction, the morphology and curvature of the root, and the anisotropic property of the PDL
may affect the angle of the functional axis. More research
is indicated to investigate the relationship between force
direction and tooth displacement; the best and most
practical assumption for now is that the CR moves parallel to the line of action of the force. We should expect
that, with new knowledge, some modification of this
principle may be required.
It is often said that a “light” force producing uncontrolled tipping can also create an optimal rate of tooth
movement. This is correct in that the tooth movement
190
measured at the crown of a tooth can be rapid. Look at
the maxillary superimposition in Fig 9-53, in which case
the incisors were retracted using a round wire and an
elastic. As predicted, the CRot is close to the CR (Fig
9-53a). Let us look at the compression side of the root,
which determines the rate of tooth movement. The stress
at the compression side is depicted in small red arrows
in Fig 9-53b; the same analysis is valid for the tension
side. The goal is to move the incisor lingually, and progress is measured by the amount of bone resorption on
the lingual surface of the root. The only useful resorption
is a small blue area of bone lingually at the alveolar crest.
The bone resorption is in the wrong direction on the
labial root surface (gray area). Why does tooth movement
appear so rapid? It is partly caused by the tipping; the
angle created by the tipping gives an optical illusion as it
augments the amount of tooth displacement at the incisal
edge. Note that the CR has moved little. Based on the
bone remodeling, little of the modification is useful in
reaching the final tooth position.
More importantly, even with very light force, localized
stress in the PDL may be excessively high (see Figs 9-38a
and 9-38b). Numerical analysis tells us that the localized
high stress in uncontrolled tipping is five times greater
than the uniform stress in translation produced by the
same magnitude of force.7,8 The radiograph and autopsy
model in Figs 9-54a and 9-54b show a patient whose
orthodontic treatment tipped the incisors lingually; the
root apices became very prominent. It is not proven that
the root resorption was caused by this tooth movement,
but possible root resorption (Fig 9-55) could occur
because of higher stresses at the apex and the large
amount of root displacement through the cortical plate
of bone (or subsequent to moving the root lingually).
Figure 9-56 shows a series of lateral cephalometric
radiographs from a case in which an incisor was moved
lingually by uncontrolled tipping followed by root movement (see Figs 9-44a and 9-44d). The apices of the incisor
went through so-called “round tripping,” moving forward
and backward, which is undesirable. The patient showed
relatively normal axial inclinations at the start of treatment (Fig 9-56a); uncontrolled tipping occurred, and the
apex apparently penetrated the labial cortical plate of the
bone (arrow in Fig 9-56b). Later, root movement was
performed to bring the apices to the lingual, and new
bone formation was seen at the labial side (Fig 9-56c).
Uncontrolled tipping by a single force at the crown is
biomechanically very easy, but it occurs too quickly for
the clinician to notice its side effects. The force system
required for root movement (CRot at the crown) may
look simple because it can be performed by a single force
When a Force and a Couple Are Required for Tipping
FIG 9-52 The functional axis of a tooth. The blue dots are the CRots
under varying vertical placements of horizontal forces (red arrows).
The black line is the functional axis of the tooth, an asymptote of
the CRots. The CR moves perpendicularly to the functional axis, as
shown by the dotted arrow. The forces coronal to the CR rotate the
tooth counterclockwise, and the forces apical to the CR rotate the
tooth clockwise.
FIG 9-53 The incisors were retracted by
uncontrolled tipping using a round wire and
an elastic. (a) As predicted, the CRot is close
to the CR. (b) The stress at the compression
side is depicted by small red arrows. Note
that the maximum stress occurs at the apex
and alveolar crest. The only useful resorption
is a small blue area of bone lingually at the
alveolar crest. Bone resorption is in the
wrong direction on the labial root surface
(gray area).
a
a
b
b
FIG 9-54 Radiograph (a) and autopsy model (b) showing posterior roots protruding through
bone.
FIG 9-55 Note the possible root resorption
due to high stress at the apex.
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9
The Biomechanics of Altering Tooth Position
a
b
c
FIG 9-56 The incisors moved lingually by uncontrolled tipping followed by root movement. The apices of the incisors went through socalled “round tripping,” which is undesirable. (a) The patient showed relatively normal axial inclinations at the start of treatment. (b) The apex
penetrated the labial cortical plate of the bone (arrow). (c) Root movement was performed to bring the roots back, and new bone formation
is seen at the labial side (arrow).
a
b
c
FIG 9-57 Methods of closing a diastema between the central incisors. (a) A straight wire placed into the brackets would produce equal and
opposite couples to tip the crowns together. Although the force system seems correct, pure rotation around the CR may not be manifested
because of high friction. (b) A simple elastic without a wire along with button hooks on the lingual of the teeth would be better than a straight
wire. The CRot would be closer to the apex for better inclination and stability. Rotation of the teeth will be reduced because D1 is smaller than
D2 in the lateral view. (c) A force with a moment from a wire segment with a curvature could produce the required equal and opposite couples
needed for controlled tipping.
apical to the CR or its equivalent force system at the
bracket; however, it needs a sophisticated clinical appliance to avoid the adverse side effects of anchorage loss.
Figure 9-57 shows a diastema between the central incisors. But what is the best way to treat it? A straight wire
placed in the brackets would produce equal and opposite
couples (Fig 9-57a). The direction of the moments will
tip the crowns together. Although the force system seems
correct, pure rotation around the CR may not be manifested because of the initially high frictional forces in a
distal direction on the incisors. Figure 9-57b, on the other
hand, is better because a force without a couple is used
without a wire. Using button hooks on the lingual of the
teeth will further reduce the tendency for incisor rotation
from the occlusal view. Ideally, the CRot should be closer
to the apex for better inclination and stability. A force
with a moment from a wire segment with a curvature
could produce the required equal and opposite couples
(whose directions are crowns apart, roots together)
needed for controlled tipping (Fig 9-57c). (Note that the
192
proper moment direction is opposite of the direction
given by a straight wire with a normal prescription.)
However, because friction is unpredictable, force system
control is very difficult. Frictionless mechanics with loops
delivering both forces and moments may provide a more
predictable force system.
Characteristics of an Optimal
Force System
What is an optimal force? Of course, it depends on treatment goals. It includes the traditional aspects of a force
vector: magnitude, direction, and point of application.
This chapter has considered mainly the point of force
application (ie, M/F ratios). Most evidence suggests that
the M/F ratio at the bracket determines the CRot and
that force magnitude is not a major factor in determining
the CRot.9
Characteristics of an Optimal Force System
FIG 9-58 The rate of tooth movement versus time. The graph shows
three distinct phases: initial, lag, and post lag.
FIG 9-59 A 3D stress diagram of a tooth can be a starting point
for understanding the subsequent biologic changes that occur. It
shows that for different CRots, if the applied force is kept constant,
maximum stress levels can vary significantly. Translation produces
the lowest stress levels because the stress-strain distribution is more
uniform (green line).
What is the best force magnitude to use with any given
M/F ratio? The goal is usually defined as rapid tooth
movement, minimal pain response, minimal tissue
damage (root resorption and alveolar bone loss), and
minimal anchorage loss. On a clinical level, the rate of
tooth movement has been studied and related to force
magnitude. A typical graph is shown in Fig 9-58. There
may be three distinct phases of response over time—the
initial phase, the lag phase, and the post lag phase. In the
initial phase, very rapid tooth displacement is observed
instantly. It is due to the nonbiologic, purely mechanical
deformation of the PDL. This mechanical displacement
is also known as physiologic mobility. The stress induced
in the PDL during this initial phase initiates the boneremodeling cascade. This bone remodeling requires time,
hence the lag phase. High stresses producing necrosis
also increase the lag phase. The post lag phase involves
a biologic response to bone remodeling: resorption and
apposition. Most significant is the biologic displacement
in the post lag phase. With lower stresses, sometimes no
lag phase is observed.
More force does not necessarily produce faster tooth
movement. In this text, some relative force magnitudes
for different types of tooth movement are given based on
the maximum stress level in the PDL. This may be state
of the art, but a better approach is needed. It is the stress,
not the force, that is distributed to the cementum, PDL,
and bone; the biologic response is to these stresses.
Therefore, more emphasis should be put on the stress
and strain in the PDL rather than the absolute magnitude
of force applied to the tooth. The following chapters focus
on the relationship between applied force on a tooth and
the distributed forces (stress) and deformations (strain)
in the PDL and the alveolar bone. These discussions will
provide a more basic understanding of optimal force
levels for tooth movement and anchorage control, both
physically and biologically. A 3D stress diagram of a tooth
can be a starting point in understanding the subsequent
biologic changes that occur (Fig 9-59). It shows that for
different CRots, if the applied force is kept constant,
maximum stress levels can significantly vary. Translation
produces the lowest stress levels because the stress-strain
distribution is more uniform (green line in Fig 9-59).
Figure 9-60 is a “working hypothesis” graph that shows
the relationship between compressive stress in the PDL
and the rate of bone resorption. The dimensions and
slopes are conceptual; however, future studies of the
graph’s concepts at stress-strain and molecular levels will
be important. The graph shows that with no added stress,
no bone resorption occurs. The stress is increased to a
magnitude where bone resorption will start to occur
(threshold). The idea of threshold force has been suggested
to explain a rationale for anchorage control. How low is
low enough? There is an interesting classical study by
Weinstein.10 He showed that 2 g of force can initiate
tipping movements of a tooth and concluded that the
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The Biomechanics of Altering Tooth Position
FIG 9-60 The hypothetical relationship between
compressive stress in the PDL and the rate of bone
resorption.
Heavy force
Heavy force
threshold, if it exists, is less than 2 g. Future studies of
tooth translation with lower stresses might demonstrate
a threshold more definitively.
As the stress is increased, there is a proportional
increase in the rate of bone resorption to the point of
optimal stress. Further increase in stress does not increase
the rate of bone resorption. Excessive stress levels leading
to undesirable tissue changes reduce the rate of bone
resorption. How high is excessively high? One 2D study
showed that more than 1.56 g/mm2 of stress can collapse
the capillary artery at the compression site of the PDL.
In canine retraction, such stress is produced from more
than 147 g for translation, 74 g for controlled tipping, 20 g
for uncontrolled tipping, and 83 g for root movement.
Therefore, the excessive force magnitude would be larger
than these force values.8 However, this study was not
intended to suggest specific force magnitude, because
the oversimplified 2D mathematical model assumed the
PDL to be homogenous and isotropic, with a linear stressstrain relationship. It also ignored variation in PDL
194
Light force
FIG 9-61 Microscopic view of
the compression side in an experimental animal. Aseptic necrosis and hyalinization (arrows)
are shown with heavy force.
Frontal bone resorption without
hyalinization is observed with
light force.
thickness as well as irregularity in the bone and tooth.
Nevertheless, it has been useful in delineating some
fundamental relationships between the magnitude of
force and the types of tooth movement. The magnitude
of the force needs to be increased or decreased depending on type of tooth movement desired.
Our general understanding of the response of a tooth
to an applied force tends to fit the graph, but more
research is needed. Histologic studies (Fig 9-61) on experimental animals show that heavy force can collapse the
capillary arteries and block the blood flow in the PDL,
leading to aseptic necrosis and hyalinization9 (white
arrows in Fig 9-61).Necrotic areas that need to be
removed can temporarily slow down the rate of tooth
movement, although rapid undermining resorption can
follow. Frontal bone resorption without hyalinization is
observed with lighter forces.
Any definition of an optimal force must include force
continuity. Some appliances store energy and deliver force
over a long time period, while others are relatively
Characteristics of an Optimal Force System
a
b
FIG 9-62 A wire with a high F/∆ rate is used to move a premolar located to the lingual. (a) Too much force is exerted at the initial activation.
(b) The tooth is within a very limited range of the optimal force zone (narrow green zone) and can hardly reach the final ideal position because
the force level is very low (suboptimal zone).
a
b
FIG 9-63 A wire with a low F/∆ rate is used to move a premolar located to the lingual. (a and b) The wire is overbent to provide a wide range
in the optimal force zone (wide green zone), spanning the entire movement process to the buccal.
intermittent in nature. Currently, continuous forces are
in vogue, using supposedly biologic magnitudes of force
or stress. Rapid palatal expansion is an example of an
intermittent force application using a screw. How is force
continuity measured? One measure is change in force
per unit time (g/day). The most common method is to
relate the change of force to the appliance deflection or
the change in tooth position (g/mm). In other chapters,
the force constancy of orthodontic appliances has been
described in terms of force deflection—the F/∆ rate
describes the change in force as a tooth moves, as the
deflection of the appliance is reduced by deactivation.
However, not all appliance components follow Hooke’s
law, and they may not have a linear relation between force
and deflection.
Let us consider a situation in which a premolar is far
to the lingual (Fig 9-62a). If an archwire is inserted that
has a high F/∆ rate, too much force is exerted at the initial
activation, and the tooth is exposed to the excessive force
zone (red). As the tooth continues to move, the optimal
and suboptimal force zones are passed through in rapid
sequence; the tooth is within a very limited range of the
optimal force zone and can hardly reach the final ideal
position because the force level is very low in the suboptimal zone (Fig 9-62b). Therefore, the overall force level
is not optimal. By contrast, a wire with a low F/∆ rate (ie,
not a straight wire) that is overbent or uses an overformed
loop could provide a wide range in the optimal force zone
(green zone in Fig 9-63), spanning the entire movement
process to the buccal. For more details about F/∆ rates,
see chapter 6. Not only can forces change as an appliance
deactivates, but other parameters such as M/F ratios can
also change (see chapter 13).
The influence of dynamic forces, such as rapidly changing forces (both high and low frequencies), may offer an
interesting possibility for optimizing orthodontic force
delivery. Hopefully, future research will focus on orthodontic force system optimization.
195
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The Biomechanics of Altering Tooth Position
References
Coolidge ED. The thickness of the human periodontal membrane. J
Am Dent Assoc 1937;24:1260–1270.
1. Nägerl H, Burstone CJ, Becher B, Messenburg DK. Center of rotation with transverse forces: An experimental study. Am J Orthod Dentofacial Orthop 1991;99:337–345.
2. Choy KC, Kim KH, Park YC, Han JY. An experimental study on the
stress distribution in the periodontal ligament. Korean J Orthod
2001;31:15–24.
3. Choy K, Kim KH, Burstone CJ. Initial changes of centres of rotation of the anterior segment in response to horizontal forces. Eur
J Orthod 2006;28:471–474.
4. Burstone CJ. The biomechanics of tooth movement. In: Kraus
B, Riedel R (eds). Vistas in Orthodontics. Philadelphia: Lea and
Febiger, 1962:197–213.
5. Burstone CJ, Pryputniewicz RJ, Weeks R. Center of resistance
of the human mandibular first molars [abstract]. J Dent Res
1981;60:515.
6. Tanne K, Koenig HA, Burstone CJ. Moment to force ratios
and the center of rotation. Am J Orthod Dentofacial Orthop
1988;94:426–431.
7. Tanne K, Koenig HA, Burstone CJ, Sakuda M. Effect of moment
to force ratios on stress patterns and levels in the PDL. J Osaka
Univ Dent Sch 1989;29:9–16.
8. Choy KC, Pae EK, Park YC, Kim KH, Burstone CJ. Effect of
root and bone morphology on the stress distribution in the
periodontal ligament. Am J Orthod Dentofacial Orthop 2000;
116:98–105.
9. Burstone CJ. Application of bioengineering to clinical orthodontics. In: Graber LW, Vanarsdall RL, Vig KWL (eds). Orthodontics:
Current Principles and Techniques, ed 5. Philadelphia: Elsevier
Mosby, 2012:345–380.
10. Weinstein S. Minimal forces in tooth movement. Am J Orthod
1967;53:881–903.
Dathe H, Nägerl H, Kebein-Meesenburg D. A caveat concerning center
of resistance. J Dent Biomech 2013;4:1758736013499770.
Recommended Reading
Tanne K, Nagataki T, Inoue Y, Sakuda M, Burstone CJ. Patterns of initial
tooth displacements associated with various root length and alveolar
bone height. Am J Orthod Dentofacial Orthop 1991;100:66–71.
Burstone CJ. The biophysics of bone remodeling during orthodontics—Optimal force consideration. In: Norton LA, Burstone CJ (eds).
Biology of Tooth Movement. Boca Raton, FL: CRC Press, 1989:321–333.
Burstone CJ, Every TW, Pryputniewicz RJ. Holographic measurement
of incisor extrusion. Am J Orthod 1982;82:1–9.
Geramy A. Alveolar bone resorption and the center of resistance
modification. Am J Orthod Dentofacial Orthop 2000;117:399–405.
Nikolai RJ. On optimum orthodontic force theory as applied to canine
retraction. Am J Orthod 1975;68:290–302.
Nikolai RJ. Periodontal ligament reaction and displacements of a
maxillary central incisor loading. J Biomech 1974;7:93–99.
Smith R, Burstone CJ. Mechanics of tooth movement. Am J Orthod
1984;85:294–299.
Soenen PL, Dermaut LR, Verbeeck RMH. Initial tooth displacement
in vivo as a predictor of long-term displacement. Eur J Orthod 1999;
21:405–411.
Steyn CL, Verwoerd WS, Merwe EJ, Fourie OL. Calculation of the
position of the axis of rotation when single-rooted teeth are orthodontically tipped. Br J Orthod 1978;5:153–156.
Synge JL. The tightness of teeth, considered as a problem concerning
the equilibrium of a thin incompressible elastic membrane. Phil Trans
R Soc Lond 1933;231:435–470.
Vanden Bulcke MM, Burstone CJ, Sachdeva RC, Dermaut LR. Location
of the center of resistance for anterior teeth during retraction using
the laser reflection technique. Am J Orthod Dentofacial Orthop
1987;91:375–384.
Burstone CJ, Pryputniewicz RJ. Holographic determination of center
of rotation produced by orthodontic forces. Am J Orthod 1980;77:396–
409.
Vanden Bulcke MM, Dermaut LR, Sachdeva RC, Burstone CJ. The
center of resistance of anterior teeth during intrusion using the laser
reflection technique and holographic interferometry. Am J Orthod
Dentofacial Orthop 1986;90:211–219.
Burstone CJ, Pryputniewicz RJ, Bowley WW. Holographic measurement
of tooth mobility in three dimensions. J Periodontal Res 1978;13:
283–294.
Yettram AL, Wright KWJ, Houston WJB. Center of rotation of a maxillary central incisor under orthodontic loading. Br J Orthod 1977;4:
23–27.
Christiansen RL, Burstone CJ. Centers of rotation within the periodontal space. Am J Orthod 1969;55:353–369.
196
Dermaut L, Kleutghen J, Clerck H. Experimental determination of the
center of resistance of the upper first molar in a macerated, dry human
skull submitted to horizontal headgear traction. Am J Orthod Dentofacial Orthop 1986;90:29–36.
Problems
1. In a to e, the perpendicular distance between the single force and the bracket is given. The black circle is the CR.
Replace each of the forces with an equivalent force system at the bracket by giving the M/F ratio required at the
bracket. Draw the correct direction of the force system at the bracket, a dot at the approximate CRot, and the
direction of rotation with a dotted curved arrow around the CRot.
a
b
c
d
e
2. Give the approximate force system at the lingual bracket for the buccally inclined maxillary left molar to rotate
around the green dot in a to c. Denote the correct direction of moments and forces. The black tooth outline is before
the movement, and the gray outline is after the movement.
a
b
c
197
10
3D Concepts
in Tooth Movement
Rodrigo F. Viecilli
“Seek simplicity, but distrust it.
We think in generalities, but we live in detail.”
— Alfred N. Whitehead
The scientific understanding of physical tooth-movement references
evolved over time, shifting from a two-dimensional (2D) model to a
three-dimensional (3D) model. However, the classic 2D concepts of
tooth movement may not always work in 3D. This chapter discusses the
evolution of concepts of tooth movement and outlines the differences
among the concepts of fulcrum, pivot, center of mass and centroid,
center and axis of rotation, and center and axis of resistance.
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3D Concepts in Tooth Movement
FIG 10-1 A primitive appliance to control crown and root tipping,
invented by Calvin Case in 1916. (From A Practical Treatise on the
Technics and Principles of Dental Orthopedia.)
T
he center of resistance of a tooth was originally
conceptualized as an analogy to the center of mass
of a free body. Initially idealized in two dimensions,
it could be determined by the iterative trial application
of a force until translation was obtained or by the application of a couple. In 2D, the center of resistance coincides
with the center of rotation when a couple is applied,
because the resultant force is zero. The center of resistance, as the center of mass, does not translate in the
absence of a resultant force; hence, it coincides with the
center of rotation only when a couple is applied.
This chapter discusses the evolution of the idea of
center of resistance, shifting from a simplified 2D model
to a generalized 3D understanding of the biomechanics
of tooth movement.
Origins of a Tooth-Movement
Reference
In 1916, Calvin Case used individualized tooth movement
appliances to record lines of action of the forces with the
intention of controlling the tipping tendencies of the teeth
(Fig 10-1). In so doing, he revealed some understanding
of where the tooth-movement reference should be to
obtain predominant crown or root movement. However,
it took over 40 years for this idea to evolve into the
concept of center of resistance, developed by Charles
Burstone and James Baldwin at Indiana University in the
1950s.
Such a gap in science and the abundant orthodontic
literature with little scientific rigor often led to confusion
200
FIG 10-2 A primitive experimental model of tooth movement depicted by Calvin Case in 1921, using a wood stick to determine the
“fulcrum,” or center of rotation. (From A Practical Treatise on the Technics and Principles of Dental Orthopedia.)
among the concepts of center of rotation, center of resistance, fulcrum, and pivot. Part of the problem was that
fulcrum and pivot were historically mentioned by early
orthodontists in an attempt to primitively describe the
biomechanics of tooth movement.
Fulcrum versus pivot
In 1921, Calvin Case included a figure in his textbook to
try to explain how a tooth would move in response to a
force; he named what we know today as the center of
rotation a fulcrum (Fig 10-2).
Physically, a fulcrum is defined as the support of a lever,
and a pivot is the point around which the lever pivots
(rotates). When a force is applied to a supported lever, it
typically pivots around the fulcrum, so the terms are often
used interchangeably. However, if the total load applied
to the lever is a couple (ie, net force of zero), then it would
rotate around its center of mass, so the pivot (center of
rotation) could technically be different from the fulcrum.
Because these concepts are more applicable to simplified
lever mechanics, they are not ideal to describe tooth
movement.
Center of rotation versus axis
of rotation
Geometrically, a 2D body rotates around a center of rotation, and a 3D body rotates around an axis of rotation.
In a 2D orthogonal projection, the body will appear to
rotate around a point. In 2D, a center of rotation is sufficient to describe tooth movement from position A to
Scientific Development of the Concept of Center of Resistance
B
Midpoint
Midpoint
Centroid
A
Midpoint
C
FIG 10-3 The centroid of a triangle (barycenter) can be determined
by the intersection of the median lines.
FIG 10-4 The centroid of the area under a parabola represents the
projected resistance of the root or PDL and is located at 40% of the
height, closer to the base.
position B. In 3D, however, the only comprehensive way
to describe the movement of a body is through screw
axis theory, which involves more refined mathematics
compared with center of rotation descriptions. Chasles’
theorem states that in Euclidean 3D space, any movement
of any object can be described by rotation around an axis
and translation along this same axis. For a rigorous 3D
theory of tooth movement, this is the only method available to describe any type of movement. However, in
orthodontics, because our tooth movement planning is
often simplified, we typically think in terms of 2D projections of the 3D teeth, and thus projection points of the
axes of rotation (centers of rotation) have been sufficient
for clinical applications. On the other hand, describing
tooth movement as a single 3D movement instead of
movement combinations is technically more appropriate
because it prevents ambiguous situations related to the
order of movements, as combining different rotations in
different orders in 3D can lead to a different final position.
Scientific Development of the
Concept of Center of Resistance
Origins of the concept of center
of resistance
The term center of resistance was first used by Leonardo
da Vinci in his 1505 book, Codex on the Flight of Birds.
The concept of center of resistance in orthodontics originated from the scientific discussions between Charles
Burstone and James Baldwin, who were responsible for
establishing biomechanics in orthodontics as a science
at Indiana University. Burstone and his research group
were involved in the evolution and formalization of all
2D and 3D scientific models of a tooth-movement reference, which culminated in the concepts of axes of resistance and volume of resistance, which are explained later
in the chapter.
As with any scientific model, the models for a reference
to tooth movement have been refined over time. The
following sections describe the rationale for each model
and how it was established.
Centroid, barycenter (center of gravity),
and center of mass
The centroid is the average geometric center of an object
(Fig 10-3). In free homogenous bodies, the center of mass
(center of gravity or barycenter) and centroid are located
at the same point. The inertia or resistance to movement
of a volumetric body is the same in all directions, and
hence a point is sufficient to describe this resistance.
Because the process of orthodontic mechanotransduction
(ie, where and how mechanical stimuli translate into tooth
movement) was incompletely understood, the first mathematical models of center of resistance utilized centroids
of a 2D root or periodontal ligament (PDL) projections
to represent resistance to tooth movement.
2D projection model
The first model was based on the centroid of an approximate 2D projection of a root or PDL, with the rationale
that the greatest resistance to tooth movement would be
represented by these. For instance, the model of a singlerooted parabolic root or PDL projection can be appreciated in Fig 10-4. With this model, the centroid that
represents the center of resistance is located at 40% of
the length of the root, closer to the alveolar ridge than
the apex.
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3D Concepts in Tooth Movement
FIG 10-5 Determination of the centroid of a
paraboloid of revolution. (Reprinted from Burstone and Pryputniewicz1 with permission.)
Centroid of a thin section
Centroid of a paraboloid of
revolution
Thin
section i
y
× 1
3 H
i
y
× 2
5 H
Locus of
centroids of
thin section
3D symmetric model
With this model, the entire root, represented by a paraboloid of revolution (constructed by rotation of the area
of a parabolic section along its long axis), is considered
the element of resistance to tooth movement1 (Fig 10-5).
The centroid is located at 33% of the long axis, within the
root half that is closer to the alveolar ridge.
It could be argued that if 3D PDL resistance should be
modeled, perhaps the centroid of a surface area (or a thin
volumetric shell) of a paraboloid of revolution could be
a more reasonable model for the center of resistance. This
model has never been published, but the author calculated it to be at 34% for a surface, which is clinically insignificantly different from the original volumetric model
published by Burstone and Pryputniewicz.1 It is also
worth noting that because the paraboloid of revolution
is a symmetric entity, if we assume PDL resistance is
uniform, this allows for the resistance to be represented
as a point, like the center of mass.
202
3D asymmetric models and the axes of
resistance
Recently, the reference model for tooth movement has
been revisited in light of recent research findings with
regard to orthodontic mechanotransduction. Because
stress measures the internal resistance (force per infinitesimal area), it could be an ideal scientific measure of
“resistance” of the PDL to instantaneous tooth displacement. However, if the clinical purpose of the concept of
center of resistance is to predict future tooth movement
after bone modeling takes place, perhaps the deformations of bone and tooth should not necessarily be
accounted for in the model of PDL stress, because these
are mostly recoverable.
We have shown that the PDL stresses are not uniform
in naturally shaped teeth. This is indeed very logical,
because tooth PDLs do not have axisymmetric morphology, and the PDL is histologically heterogeneous and
anisotropic (ie, material properties vary in magnitude
Scientific Development of the Concept of Center of Resistance
z
y
z
x
y
y
x
a
x
z
b
c
FIG 10-6 Axes of rotation determined by three perpendicular couples in the buccal (a), mesial (b), and apical (c) directions.
y'
z'
y'
x'
x'
z'
z
x
z'
y'
y
a
x'
y
x
b
z
c
FIG 10-7 (a to c) 3D locations of the axes of resistance. In each field of view, the correct reference for translation is the intersection of the
two axes that can be seen as lines.
depending on direction of the load). Furthermore,
because the stress-strain curve of the PDL is nonlinear,
the PDL may become stiffer in one direction than another
if resistance due to morphology is different in different
directions. The biologic reactions to PDL stress vary with
stress thresholds and, over time, add even more potential
differences in references to tooth movement in different
directions.
In 1991, Nägerl et al demonstrated large differences in
the positions of the centers of resistance for each direction
of tooth movement.2 In 2009, the author confirmed this
finding, noting that axes of resistance could be more
adequate references in 3D because a couple causes a rotation around an axis that could then be used as a reference
for translation.3 In 2010, it was shown in dog teeth that
the centers of resistance were statistically different in
different directions.4
The author recently used finite element analysis in a
model of a maxillary first molar to determine possible
differences in locations of axes of resistance solely due to
lack of PDL axisymmetry.5 It was found that the 3D axes
of resistance indeed do not intersect at a 3D point; in the
maxillary first molar study, the axes of resistance missed
each other by a maximum of 0.6 mm. Hence, it is reasonable to believe that, considering all possible causes of
differences in the PDL stress fields for each movement
direction, the center of resistance as a point does not exist
as a realistic physical entity.
If there are three axes of resistance for each possible
orthogonal direction of movement (Fig 10-6), how do we
know which of the three should be used as references for
translation in each direction? Well, the reference for
translation in the direction perpendicular to the field of
view is always at the intersection of the two axes that can
be seen as lines on that view (Fig 10-7).
Another question that arises is whether the axes of
resistance and rotation are the same when a couple is
applied. However, this question is revealed to be
203
10
3D Concepts in Tooth Movement
FIG 10-8 The projection of the z-axis of rotation (obtained by a
buccally oriented couple) is different than the projected center
of resistance for translation in the buccal direction.
y'
Center of resistance for
translation along z
x'
z'
Center of rotation for
a moment parallel to z
y
x
meaningless because a combination of two axes must be
used to determine a 2D projected point used as a translation reference (so that the body does not rotate in any
direction). So there is at least one axis of rotation that is
never coincident with this point. This can be better understood in Fig 10-8.
It is important to note that in segments of teeth, the
morphologic asymmetries that lead to noncoincident
axes of resistance will build up, and distances between
the axes will likely increase. This is a subject for future
studies.
3D volume of resistance
Dathe and Nägerl formalized the mathematics for a 3D
volumetric center of resistance,6 which perhaps could be
thought of as all possible locations of intersections of
axes of resistance for different tooth translation directions. Clinically, one can think that there is a 3D volume
of resistance (a volumetric center, not a point), with a
certain level of uncertainty that is dependent both on
morphology and other sources of asymmetric behavior
discussed earlier.
Practical limitations of the
concepts of center of resistance and
axes of resistance
It is important to note that the axes of resistance can vary
their positions during tooth movement because the PDL,
root, and bone are subject to constant biologic modeling.
204
Hence, the clinician should not see the position of the
axes as a static feature. Clinically, if using a system for
controlled tooth movement, we should start with our
best guess based on scientific literature and then continually correct the appliance and force system, depending
on the movement that was achieved with the previous
configuration and the desired future outcome.
References
1. Burstone CJ, Pryputniewicz RJ. Holographic determination of
centers of rotation produced by orthodontic forces. Am J Orthod
1980;77:396–409.
2. Nägerl H, Burstone CJ, Becker B, Kubein-Meesenburg D. Centers of rotation with transverse forces: An experimental study.
Am J Orthod Dentofacial Orthop 1991;99:337–345.
3. Viecilli RF, Katona TR, Chen J, Hartsfield JK Jr, Roberts WE.
Three-dimensional mechanical environment of orthodontic
tooth movement and root resorption. Am J Orthod Dentofacial
Orthop 2008;133:791.e11–791.e26.
4. Meyer BN, Chen J, Katona TR. Does the center of resistance
depend on the direction of tooth movement? Am J Orthod Dentofacial Orthop 2010;137:354–361.
5. Viecilli RF, Budiman A, Burstone CJ. Axes of resistance for tooth
movement: Does the center of resistance exist in 3-dimensional
space? Am J Orthod Dentofacial Orthop 2013;143:163–172.
6. Dathe H, Nägerl H, Kubein-Meesenburg D. A caveat concerning center of resistance. J Dent Biomech 2013;4:
1758736013499770.
11
Orthodontic
Anchorage
Rodrigo F. Viecilli
“Nature uses only the longest threads to weave her
patterns, so that each small piece of her fabric
reveals the organization of the entire tapestry.”
— Richard P. Feynman
This chapter explains the biomechanical basis for orthodontic anchorage. The intensity of the biologic response relates to mechanical
stimulus, and this stimulus, when combined with the biologic environment, leads to the clinical perception of anchorage value. Certain
appliances have the potential to change the degrees of freedom for
tooth movement and selectively enhance anchorage potential. These
appliances and the scientific rationale for typical clinical strategies to
improve anchorage are discussed in the chapter.
205
11
Orthodontic Anchorage
Definition and Clinical Perception
of Anchorage
The speed of tooth movement is the result of interaction
among many intertwined basic scientific variables. The
effects of many of these variables have not yet been quantified, but some have recently begun to be better understood.
attract fewer osteoclasts so that tooth movement is
slower. This is the main reason a molar has more anchorage value than an incisor.
It is important to note that different types of movements lead to different patterns of stress fields in the PDL,
and hence the anchorage value of a tooth will also depend
on the type of movement desired. Translation typically
has more anchorage value than controlled tipping, which
in turn has more value than uncontrolled tipping. For
instance, in translation, the compressive stresses can be
one-third of those for uncontrolled tipping for the same
tooth with the same force applied to the bracket because
the total load acting on the tooth is reduced.
The total load acting on a tooth is the equivalent force
system at the axis of resistance. The total load for a tooth
being tipped with 100 cN at the bracket is 100 cN (force)
+ 100 cN × d (moment), where d is the distance to the
axis of resistance. With translation, the load is smaller
because the total moment is zero, and a countermoment
is applied to the bracket to cancel out the moment of the
tipping force. This explains why, in tipping, larger peak
stresses affect the PDL.
Mechanical variables
Biologic variables
Periodontal ligament stress and the total tooth
load at the axis of resistance
Biologic variables are intrinsic factors that can vary locally
(from tooth to tooth) or between individuals. The two
main categories are the inflammatory response and the
bone quantity and quality.
In its broadest sense, orthodontic anchorage is resistance
to tooth movement. Hence, the anchorage value of an
appliance or dentoalveolar complex relates to its capacity
to resist movement. In intraoral anchorage, the clinical
perception of anchorage directly relates to the difference
in relative speed of tooth movement between units. The
following section discusses the variables that could potentially influence the anchorage value of dentoalveolar units.
Rationale for Anchorage from
a Basic Science Perspective
Orthodontic tooth movement is initiated when stress is
applied to the periodontal ligament (PDL). The number
of resorbing osteoclasts is initially directly proportional
to the third principal (“most compressive”) stress. If the
stress is low enough for the tissue to remain viable, direct
bone resorption occurs, which can quickly result in tooth
movement as the PDL space naturally widens after bone
resorption. If the stress is too high, hyalinization can
occur, which can delay tooth movement because the
necrotic tissue has to be removed after undermining
resorption. Depending on the magnitude of the intraoral
load, it is possible for teeth with less support to undergo
delayed tooth movement following excessive stresses,
especially when compared with larger teeth in which
necrosis did not occur. Direct bone resorption occurs in
the large tooth so that it can start moving more rapidly,
at least initially.
Larger teeth naturally have more PDL support, and
hence the stress magnitudes in the PDL are smaller than
those in little teeth when the same force is applied. Consequently, the anchorage value of larger teeth is also greater.
Everything else being equal, smaller stresses should
206
Inflammatory response
For a given equal mechanical stimulation, there can be
variations in the intensity of inflammatory response from
one periodontal or bone site to another, which can affect
the attraction of osteoclasts and the speed of tooth movement. These differences can affect the anchorage value of
a tooth from individual to individual as well as within
the same individual. One cause of this type of intraindividual variation in response is vascularity. Decreased
vascularity in one dentoalveolar site compared with
another provides less opportunity for cellular recruitment
and may promote ischemia and necrosis, which can delay
tooth movement. Furthermore, between different individuals, there can be differences in the genetic profile
that lead to differences in the performance of biologic
mediators such as prostaglandins, cytokines, leukotrienes, and growth factors. These can result in different
levels of inflammation and thus influence anchorage
potential.
Anchorage Values According to PDL Stress
FIG 11-1 Loads applied to the teeth (shown here in only one direction)
used to calculate the anchorage values. The force was fixed at 8 cN
and the moment at 50 cNmm.
Bone quantity and quality
Density. If the trabecular bone in the alveolus surrounding the tooth has decreased bone volume fraction, it
follows that there is increased porosity in the bone.
Hence, osteoclasts need to resorb less bone to result in
space for tooth movement. Decreased bone density may
also facilitate the work of osteoclasts. Thickness of cortical bone, trabecular bone volume fraction, and bone
density vary among dentoalveolar sites and individuals,
possibly affecting the speed of tooth movement and
anchorage value.
Bone remodeling rate or turnover. High bone turnover, or a quick cycle of renewal, means that large
numbers of osteoclasts are rapidly resorbing bone and
osteoblasts are rapidly reforming it. It is a natural process
of bone repair. The larger the number of cells already at
work performing bone remodeling, the easier it may be
for bone modeling associated with tooth movement to
occur. Hence, the anchorage value of a dentoalveolar site
will decrease in rapidly remodeling bone. The rate of
turnover may change depending on the jaw, dentoalveolar site, or individual. It is also important to remember
that bone morphology and turnover directly relate to
overall bone metabolism, including nutritional deficiencies; abnormalities of the kidney, gut, or parathyroid
function; or local pathologies. These factors can all alter
the anchorage potential of dentoalveolar sites in different
individuals or at different periods for the same individual.
Clinical application of injury to the bone can cause a
decrease in bone volume and density and an increase in
bone turnover due to increased inflammation. Different
methods to apply bone injury and increase tooth movement have been used over the last 100 years. The concept
is nothing new, but originally the effect was explained
only by a reduction in the volume of cortical bone, so
procedures were more aggressive and involved dramatic
decortications. The effects of frequent injuries to the bone
in the alveolar bone level and in root resorption still have
not been fully explored.
Anchorage Values According
to PDL Stress
The author has conducted finite element analyses to
calculate the load ratios necessary to promote equal PDL
stresses in teeth of average size and ideal occlusion (except
third molars) for different types of movement that occur
during the orthodontic alignment phase.1 At this point,
a linear model was used that is applicable to small loads.
The methodology consisted of the application of the same
load (force, 8 cN; moment, 50 cNmm) for different movements (intrusion/extrusion force at the bracket, buccal/
lingual crown tipping at the bracket, distal/mesial crown
tipping couple, and couples with the vector perpendicular to the occlusal plane) to all teeth studied. The simulated loads are shown in Fig 11-1. The PDLs were then
divided into thirds longitudinally (so that one of the
207
11
Orthodontic Anchorage
Mandibular central incisor = –3.7 kPa
Maxillary canine = –2 kPa
Stress normalized to MC = 1
Stress normalized to MC = 0.54
FIG 11-2 Example of third principal stress analysis for labial crown tipping, comparing standardized PDL regions for the maxillary canine and the mandibular central incisor (MC). The
same force (8 cN) causes stresses on the maxillary canine that are 54% of the stresses on the
MC. Hence, the anchorage value of the maxillary canine for this particular movement is 1.85
times that of the MC.
regions contained the third with the highest stress) and
in fourths transversally (so that one of the regions
contained the fourth with the highest stress). The intersection of these two regions determined the region of
analysis where stresses were averaged. The ratio between
the stresses determined the anchorage values for each
tooth. An example calculation is shown in Fig 11-2.
Clinical Intraoral Anchorage
Strategies
Number of teeth and segment units
The easiest strategy to enhance the anchorage of a unit
is to add more teeth to the unit. This expands the overall
support of the unit, decreasing the peak stresses and,
hence, the number of osteoclasts. What is the clinical
gain in posterior anchorage of adding second molars to
a maxillary posterior anchorage unit that contains the
second premolar and first molar, if space is closing against
the incisors and canine? The anchorage ratio of anterior
to posterior is nearly 1:1 without the second molar, which
means we could expect the space to close 50% by each
unit. By adding the second molar, this ratio changes to
1.6:1, which means that in a 7.8-mm space closure, we
could expect 4.8 mm of closure from anterior movement
and 3 mm from posterior movement (saving roughly
2 mm of anchorage loss).
208
Differential moments to attain
differential stress (anchorage)
The second major strategy to reinforce anchorage is to
apply differential moments by achieving a system in equilibrium for the two units. To understand this, let us examine the example in Fig 11-3. In order to tip one of the
molars with the intent of correcting a reverse articulation
(also known as a crossbite), the reactive system is planned
so that the contralateral molar translates. As explained
earlier, one can expect stresses at least three times lower
for translation compared with tipping, because during
tipping the total load acting on the tooth includes the
force and a moment. In translation, the total load acting
on the tooth consists only of the applied force. Vertical
forces in the system are equal and opposite and tend to
maintain the stress differential achieved by the application
of differential moments. This strategy is also used effectively for space closure (Fig 11-4), and it has been considered effective in clinical studies.
In this section, it is also relevant to discuss whether the
inclination of a tooth promoted by a distal crown tipping
bend affects its anchorage potential. Consider a molar
that is vertical compared with a molar that has its crown
angulated distally 10 degrees. If a tipping force is applied
to the vertical molar at the tube, the force system at the
axis of resistance will be the tipping force plus a moment
of 10 × d, where d is the distance to the axis of resistance
(see chapter 3). In the molar that is angulated distally, the
Clinical Intraoral Anchorage Strategies
FIG 11-3 Differential moment load system set to correct a reverse
articulation due to a lingually tipped maxillary molar. The intent of
the force system is to apply a translation load to the patient’s right
anchorage molar and a tipping load to the left molar, which can be
achieved by activation of a transpalatal arch. The total moment at the
CR from the red horizontal force (F) acting on the anchorage tooth
plus the applied orange couple (M) from the vertical force (F1 ) is zero,
while the tipped tooth will suffer large corrective tipping moments
from both F1 and F. This will allow the PDL of the reactive tooth to be
under significantly less stress, thus reinforcing its anchorage.
M = F 1d
F1
F1
d
F
F
M/F 10
If d = 40 mm and 2,000 gmm is required on the patient’s
right molar, the vertical forces are 50 g.
FIG 11-4 Example of T-loop mechanics utilized to close spaces in a 16-year-old hyperdivergent
patient with 12 mm of horizontal overlap (or overjet) (a). (b) The T-loop has classic Burstone
preactivation curvatures and is activated at 4.5 mm to deliver a force of approximately 250
cN. It was displaced posteriorly 3 mm to achieve an initial moment-to-force ratio of 5 mm in
the anterior teeth and 8 mm in the posterior teeth. (c) As the anterior unit tipped, V-bends
were added to the anterior portion of the loop to maintain the desirable force system on the
anterior unit. Note that intrusion of the mandibular incisors and canines had to be performed
to normalize the vertical overlap or overbite and avoid anterior interferences when space
closure was completed. (d) After root movement of the anterior unit, realignment was performed. No extraoral, elastic, or implant anchorage was used to close spaces in this case.
Differential moments to achieve differential stress is a powerful anchorage strategy with a
sound biomechanical basis.
b
c
distance to the axis of resistance is reduced. The cosine
function of 10 can be used to calculate the perpendicular
distance, which is 98.4% less (see Fig 2-10).Hence, there
is only a 1.6% reduction in the distance to the axis of
resistance, which means the moment of the force will be
1.6% smaller. This is a negligible effect with no clinical
consequence. To decrease the moment acting on the
molar by 30%, it would have to be angled back over 45
degrees, which is clinically impractical. Hence, it is possible to change the anchorage potential of a tooth by changing its angulation. However, minor angulation changes
of less than 10 degrees, as historically proposed in orthodontic techniques and prescriptions, are essentially
useless as anchorage-enhancement strategies. Active
a
d
application of moments during space closure can better
enhance anchorage because they have a proven scientific
rationale, as explained earlier.
It is interesting to speculate as to whether the anchorage gain effect originally observed by Tweed in his original technique occurred because bends were applied
during space closure (and not before, as later proposed).
As we have shown, applying tip-back moments during
space closure does not have the same effect as bends that
are applied before space closure and tip the tooth before
space closure. Some advantage could be gained if the
bends were applied during movement because differential moments would be delivered, hence decreasing the
total load at the axis of resistance.
209
11
Orthodontic Anchorage
a
b
c
d
Increasing the force at the active unit or
decreasing the force at the reactive unit
to obtain a moment-to-force differential
It is also possible to increase the movement of the active
unit by adding more force to the active unit or by canceling some of the force on the reactive unit, with the force
and moment magnitudes planned so that one unit translates and the other tips. Intraorally, this can be accomplished by the use of maxillomandibular elastics or fixed
functional appliances (such as Forsus, Herbst, and others).
An example of this strategy is shown in Fig 11-5.
Occlusal interlocking and interferences
The third major strategy to reinforce intraoral anchorage
is based on occlusal interlocking. Orthodontists have
long noticed that it is often more difficult to move teeth
or close spaces in patients with strong masticatory muscle
patterns. Although little data is available on this subject,
it is logical that a patient that maintains occlusal interlocking and pressure on the surfaces of the teeth will
add mechanical resistance to tooth movement if the
target tooth interlocks with opposing teeth. If a patient
maintains occlusal interlocking most of the time, the
loads acting on the tooth will also be distributed to the
opposing dentition, thus lowering the stresses acting
in the PDL. Although this is a natural occurrence,
210
FIG 11-5 Stages of space closure using a
custom calibrated 8 × 16–mm T-loop made
of 0.017 × 0.025–inch β-titanium wire. (a) The
T-loop initial load system delivers a momentto-force (M/F) ratio of roughly 6 mm to each
unit and a force of 300 cN at 8 mm of activation. After approximately 4 mm of space
closure, when a Class I canine relationship
was achieved, the M/F ratio delivered by
the loop to each unit changed to roughly 10
mm. (b) At that point, a maxillomandibular
Class III elastic was added. (c) A calibrated
α-β root spring made of 0.019 × 0.025–inch
β-titanium wire was added to deliver an M/F
ratio of 12 mm to the posterior unit and 10 mm
to the anterior unit. (d) The activations of this
spring resulted in a force of roughly 250 cN
at the ligature connecting the anterior and
posterior segments.
orthodontists have tried to enhance occlusal interlocking
by adding acrylic occlusion rims that are adapted to the
occlusion on both arches and promote enhanced occlusal interlocking to the target teeth to preserve anchorage.
Even in situations where occlusal interlocking is intermittent, the constant occlusal loading may disturb the
pattern of stresses in the PDL that would promote an
organized cellular response to achieve the intended
movement. Hence, occlusal loading is an unpredictable
factor in tooth movement patterns and speed, especially
in posterior teeth. Lack of horizontal overlap or excessive
vertical overlap can also lead to more difficulty controlling
tooth movement in the anterior region (Fig 11-6). Occlusal interferences, especially in patients with strong muscle
patterns, are a critical factor to consider when planning
orthodontic mechanics.
Soft tissue loads and growthrelated changes
The effects of soft tissue loading on the teeth cannot be
ignored. Parafunction of the perioral tissues can have
dramatic effects on tooth position. The effect of parafunctional habits or invading soft tissue space must be
considered in the anchorage plan because these tissues
are able to produce loads on the teeth. For example, a
patient with a forward tongue posture will probably
increase the apparent anchorage value of the anterior unit,
because the loads applied by the tongue will cancel out
Degrees of Freedom and the Biomechanical Basis of Intraoral Anchorage Devices
FIG 11-6 (a and b) A Burstone three-piece
intrusion arch used to intrude the mandibular incisors along their long axis and
eliminate the anterior interference prior to
space closure to correct the Class II canine
relationship. After enough horizontal overlap
and minor vertical overlap were achieved,
the minor spaces were closed using a 0.017
× 0.025–inch β-titanium base arch with a
tip-back bend to enhance anchorage and
distal-pull activation with cinch-back. Closing spaces with an anterior interference
could have caused a Class II molar relationship or distal displacement of the condyle.
Evaluating occlusal interferences is a critical
part of a sound anchorage plan.
a
some of the appliance-generated force on the anterior
teeth.
Growth displacement of bones (differential growth of
the mandible) can also change the apparent anchorage
value of teeth. Clinically, it may appear that maxillary
posterior teeth have more anchorage because the mandible is not a stable reference during the peak growth
period.
Degrees of Freedom and the
Biomechanical Basis of Intraoral
Anchorage Devices
When an anchorage unit is being designed, it is sometimes convenient to connect teeth in opposite sides of
the arch. This is often done with appliances such as the
transpalatal arch (TPA), lingual arch, Nance arch, and
horseshoe arch. The horseshoe arch is similar to a lingual
arch but is used in the maxillary arch.
Besides connecting teeth to establish a new anchorage
unit with more value, these appliances change the way
tooth movement occurs for these teeth. This section
discusses how these types of appliances can alter the
degrees of freedom for tooth movement and how this
can affect or assist in treatment outcomes. There are six
degrees of freedom in 3D, consisting of three translation
and three rotation components. Each of them can be
affected in different ways depending on the appliance
chosen.
b
A TPA connects first or second maxillary molars by
means of a stiff wire (typically 0.036-inch stainless steel
for stabilizing purposes). Because the wire is very stiff, it
modifies the movement of the molars in the six degrees
of freedom as follows:
1. Rotation of the teeth perpendicular to the occlusal
plane: This is useful in cases where the orthodontist
wants to control molar rotation (eg, during space
closure). The position of the molar tube can also be
used as a guide to determine the shape of the maxillary
arch during alignment. The molars are unable to rotate
independently but can rotate as a unit.
2. Buccolingual translation: Both molars are partially
constrained to translate lingually as determined by the
stiffness of the TPA. When used in the orthodontist’s
favor, this also helps in the control of arch form. For
a molar to translate lingually, the contralateral molar
needs to translate buccally. This adds some level of
stabilization to the arch form and width when an archwire is used to align adjacent teeth that are rotated,
such as premolars and second molars.
3. Occlusoapical translation: The palate and the tongue
are possible constraints to tooth movement in this
direction. Addition of acrylic to the center of the TPA
could theoretically enhance this, but the effect on
extrusion control is unpredictable due to the intermittent and variable nature of tongue forces. In any
case, the molars must translate together in this degree
of freedom.
211
11
Orthodontic Anchorage
a
b
FIG 11-7 (a) Diagram demonstrating the forces acting on the wires connecting the teeth. If the miniscrew is stable, it is able to stabilize the
wire by generating a load that is equal and opposite to the one applied to the posterior teeth. (b) Stainless steel ligature wire tied to the first
molar in an attempt to stabilize the posterior segment. This mechanical design will actually force the posterior segment to rotate around the
miniscrew. A tension load will develop on the ligature wire so that the resultant force system (hypothetical blue force) matches the constraint
(ie, the axis of rotation has to be located at the miniscrew). The blue force will cause a moment around the axis (center) of resistance. This
mechanical configuration will cause flattening of the occlusal plane and ultimately opening of the occlusion (open bite).
4. Lingual or buccal crown rotation: Most TPAs have
connection wire terminals with tooth inclination
control (eg, the original Burstone or Atkinson [universal] types). Hence, the teeth cannot have independent
changes in inclination; for a molar crown to rotate
lingually, the contralateral crown would have to rotate
buccally. This can also help to align the inclinations of
all teeth after insertion of the rectangular alignment
archwires, using the molar as a reference. Naturally,
this is only true if alignment is performed with a TPA
when the molars are already ideally positioned. Other
TPA designs such as the Burstone Precision TPA
system can fit a round wire at the hinge cap bracket
and keep the teeth free to incline. This also allows for
use of a rectangular wire that is round only on one
side, which can be useful for planning certain movements.
5. Mesiodistal translation: Both molars are constrained
to translate together. This means that the molars can
still move in this direction but must do it together. If
space exists in front of only one of the molars, then
there could be some value in prevention of molar
mesial drift. Otherwise, there are no restrictions.
6. Distal or mesial crown rotation: Both molars are
constrained to rotate together. The same rationale for
mesiodistal translation applies here.
For some time, it was suggested that a TPA could
enhance anteroposterior (mesiodistal) anchorage because
212
it would force the molars to move into cortical bone when
they would normally follow the normal line of the arch.
Studies have shown that the TPA does not seem to add
any anchorage value to prevent mesial drift.2 The conclusion of the clinical studies is logical, because any kind of
mesial molar movement, with or without a TPA, would
involve cortical modeling because the alveolar process is
always thicker at the molars. However, some anchorage
enhancement probably exists if space is present only
mesial to one of the molars, because the contralateral
tooth would be constrained by its adjacent tooth.
Another method of modifying the degrees of freedom
for movement of teeth is the use of orthodontic miniscrews. It is important to note that complete restriction
of tooth movement in all directions is achieved only if
the miniscrew is solidly connected to a tooth by means
of a stiff, rigid material—the classic indirect anchorage
method. For instance, inserting a rectangular wire in the
miniscrew slot and bonding it to a tooth results in solid
anchorage that depends only on the stability of the miniscrew (Fig 11-7a).
On the other hand, it has often been proposed that
stainless steel ligatures be used in an attempt to obtain
indirect anchorage (ie, a tooth connected to a miniscrew
by ligating it to the bracket [Fig 11-7b]). This strategy will
not result in solid anchorage; rather, it will add a constraint
that will modify tooth movement and perhaps lead to
dramatic clinical side effects if not carefully planned.
Recommended Reading
References
1. Viecilli RF, Katona TR, Chen J, Hartsfield JK Jr, Roberts WE. Orthodontic mechanotransduction and the role of the P2X7 receptor.
Am J Orthod Dentofacial Orthop 2009;135:694.e1–694.e16.
2. Zablocki HL, McNamara JA Jr, Franchi L, Baccetti T. Effect of
the transpalatal arch during extraction treatment. Am J Orthod
Dentofacial Orthop 2008;133:852–860.
Recommended Reading
Burstone CJ. The segmented arch approach to space closure. Am J
Orthod 1982;82:361–378.
Burstone CJ, Koenig HA. Optimizing anterior and canine retraction.
Am J Orthod 1976;70:1–19.
Kawarizadeh A, Bourauel C, Zhang D, Gotz W, Jager A. Correlation of
stress and strain profiles and the distribution of osteoclastic cells
induced by orthodontic loading in rat. Eur J Oral Sci 2004;112:140–147.
Viecilli RF. Self-corrective T-loop for differential space closure. Am J
Orthod Dentofacial Orthop 2006;129:48–53.
Viecilli RF, Kar-Kuri MH, Varriale J, Budiman A, Janal M. Effects of initial
stresses and time on orthodontic external root resorption. J Dent Res
2013;92:346–351.
Viecilli RF, Katona TR, Chen J, Hartsfield JK Jr, Roberts WE. Threedimensional mechanical environment of orthodontic tooth movement
and root resorption. Am J Orthod Dentofacial Orthop 2008;133:791.
e11–791.e26.
Xia Z, Chen J, Jiang F, Li S, Viecilli R, Liu S. Clinical changes in the
load system of segmental T-loops for canine retraction. Am J Orthod
Dentofacial Orthop 2013;144:548–556.
Hart A, Taft L, Greenberg SN. The effectiveness of differential moments
in establishing and maintaining anchorage. Am J Orthod Dentofacial
Orthop 1992;102:434–442.
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12
Lingual Arches
“New opinions are always suspected, and usually
opposed, without any other reason but because
they are not already common.”
— John Locke
Precision lingual arches can be quite versatile if used independently
or in combination with a facial archwire. Many malocclusions have
discrepancies that are best solved by bilateral mechanics rather than
the use of adjacent teeth as in a continuous arch. The lingual arch
that connects only two attachments bilaterally is a simple system
for understanding bracket-wire interactions. New designs allow for
insertion of a horseshoe or transpalatal lingual arch in the maxilla. The
traditional “ideal” arch shape may not be the correct shape as defined
for a correct force system. Active applications unique from the lingual
include unilateral tip-back and unilateral and bilateral molar rotation.
This chapter describes in detail how to shape a lingual arch to produce
a desired force system.
217
12
Lingual Arches
a
b
FIG 12-1 Two lingual arch designs connecting first molars. (a) Lingual transpalatal arch (TPA). (b)
Horseshoe lingual arch.
A
lingual arch can refer to many different things.
Lingual archwires can be placed in multiple brackets on the lingual surfaces of the crowns. This chapter considers only lingual arches that connect two teeth
across the arch, usually at the first molar (Fig 12-1).
Lingual arches can be used for passive applications to
preserve tooth position or for active applications to move
the teeth. Passive applications include space maintenance,
anchorage reinforcement to minimize side effects, and
serving as a base structure for attaching auxiliary springs.
Active applications include molar rotation, arch width
expansion and constriction both symmetrically and
asymmetrically, and unilateral tip-back.
Limitations of a Labial Appliance
The lingual arch can be used by itself or inserted to
complement a labial appliance (Fig 12-2). An additional
lingual arch is sometimes necessary because the labial
archwire has two major limitations: adjacent tooth
anchorage considerations and posterior width instability.
The arch has been used in architecture for thousands
of years because structurally it is very stable and can resist
vertical loads. It is found in many cathedrals, bridges, and
triumphal arches (eg, the Gateway Arch in St Louis).
However, the arch as a structure is very unstable laterally
at its free ends, so it requires strong support at these ends.
The same is true in a dental archwire. Even the stiffest
full-size 0.022 × 0.028–inch stainless steel labial archwire
may have a very low force-deflection (F/Δ) rate at the
free ends if loaded with a lateral force (Fig 12-3). Therefore, loss of terminal molar width can be frequently
observed after application of Class II and Class III elastics,
from headgear forces, and during interarch alignment
218
because the wires may have been subjected to some lateral
components of force. Consider the use of a low-stiffness
nickel-titanium (Ni-Ti) wire that is straight without any
arch form. If placed from terminal molar to terminal
molar, it may effectively align the teeth; however, the low
stiffness of the wire probably will not change the arch
form. Related to width stability is molar buccolingual
axial inclination control. An edgewise arch fully engaged
in the molar tube or bracket in theory might actively
control or passively keep the buccolingual molar inclination; in practice, however, the wire “play” allows for molar
inclination and potential width changes as a result of
vertical or horizontal components of force from an elastic or a headgear.
A more significant limitation of a labial archwire is its
inherent anchorage selection, where adjacent teeth determine the anchorage and force system produced. In Fig
12-4, first molars are positioned bilaterally and symmetrically to the buccal. A labial archwire uses second molars
and second premolars as anchorage (red forces), most
likely leading to side effects on the teeth adjacent to the
molars (Fig 12-4a). A lingual arch, however, works across
the arch to utilize reciprocal anchorage (Fig 12-4b). Many
useful possibilities exist for applying cross-arch anchorage selection either with symmetric or asymmetric force
systems. The labial arch gives only limited options for
using adjacent teeth for anchorage to move molars. Molar
anchorage for posterior tooth movement opens up more
useful possibilities for sound mechanics.
The cross-arch distance between two molars is one of
the largest interbracket distances available in the oral
cavity. Increased interbracket distance provides many
advantages, such as low F/Δ rate, increased range of
action, increased moment arms, and ease of evaluating
wire-bracket geometry. Because of this, the lingual arch
Attachments
FIG 12-2 The lingual arch can be used alone, or it can be
inserted to complement a labial appliance, because any
labial appliance has inherent limitations.
a
FIG 12-3 Even a full-size 0.022 × 0.028–inch stainless steel labial
archwire may have a very low F/∆ rate if loaded with a lateral force
at its free ends.
b
FIG 12-4 (a) A labial archwire uses second molars and second premolars as anchorage for moving a first molar, most likely leading to side
effects on adjacent teeth. (b) A lingual arch working across the arch can utilize reciprocal anchorage without any side effects.
can be one of the simplest fixed appliances where wires
are inserted into brackets. Methods of fabrication, insertion, and removal of lingual arches are found elsewhere,1,2
while this chapter places emphasis on the biomechanics
of the lingual arch.
Attachments
For passive applications, the lingual arch can be securely
soldered to a band; however, attachments for removable
wires allow for frequent active and passive adjustment
changes when indicated. A folded 0.036-inch (0.9-mm)
stainless steel wire fits snugly into a lingual sheath in Fig
12-5. There can still be some play, and the sheath
commonly deforms, changing shape so that full control
with six degrees of freedom is lacking. Therefore, a robust
and more precision fit lingual bracket (Fig 12-6) or a hinge
cap bracket (Fig 12-7) is preferable. In a labial archwire,
there is always a little play needed between the bracket
and the wire, even with a full-size wire, because sliding
mechanics is commonly required. By contrast, a 0.032 ×
0.032–inch square wire accurately fits in the slot of a
precision lingual bracket or hinge cap bracket so that full
three-dimensional control with six degrees of freedom
is assured. In very special applications, a 0.032-inch round
wire is used to allow some rotational movement around
the x-axis of the bracket, eliminating one degree of freedom to remove unnecessary torque.
Lingual precision brackets are preangulated (–12
degrees for maxillary teeth and +6 degrees for mandibular teeth) for ease of use (Fig 12-8). Orienting the bracket
slot parallel to the occlusal plane at the end of treatment
simplifies any twisting of the lingual arch. If buccolingual
axial inclinations are initially favorable, flat wires can be
easily inserted with minimal adjustment devoid of torque.
219
12
Lingual Arches
FIG 12-5 A folded 0.036-inch (0.9-mm)
stainless steel wire snugly fits into a lingual
sheath; however, there can still be some
play so that full control with six degrees of
freedom is lacking. Torque can easily deform
the sheaths.
a
a
FIG 12-6 A precision fit lingual bracket uses an O-ring
or metal ligature ties.
b
b
Lingual Arch Configurations
In this chapter, all lingual appliances connecting just two
brackets across the arch are called lingual arches. Typically, it is first molars that are connected; however, second
molars can also be connected, and even canine to canine
bars can comprise a lingual arch. In the maxillary arch,
two designs are basic: the transpalatal arch (TPA) and
the horseshoe arch. Although a TPA is usually inserted
from the mesial of the hinge cap (Fig 12-9a), it sometimes
is desirable to insert it from the distal (Fig 12-9b). This
can avoid impingement if a torus palatinus or a lingually
positioned second premolar is present. Placing a TPA
further distally can also influence the force system to
220
FIG 12-7 The precision lingual hinge cap
bracket. A 0.032 × 0.032–inch square wire
accurately fits in the slot so that full threedimensional control with six degrees of
freedom is assured. (a) Cap opened. (b) Cap
closed.
FIG 12-8 Lingual precision brackets are
preangulated in the third order for ease of
use. (a) Maxillary arch, –12 degrees. (b) Mandibular arch, +6 degrees.
produce association (described later in the chapter). The
maxillary horseshoe arch has the advantage of simplicity
and ease of fabrication because minimal palatal contouring is needed (Fig 12-10). Because wire orientation is at
90 degrees to a TPA, the force system is uniquely suited
to special types of tooth movement, which are discussed
later in this chapter.
Because of the tongue, mandibular lingual arches must
have the horseshoe configuration. Two types are commonly
used. The high mandibular lingual arch (Fig 12-11a)
touches the incisor cingulum and is used for space maintenance or for added incisor anchorage. It can also be
used to prevent the mandibular incisors from tipping
lingually in extraction therapy. The low mandibular
Lingual Arch Configurations
a
b
FIG 12-9 A maxillary TPA. It is usually inserted from the mesial of the bracket (a); however,
sometimes it is desirable to insert it from the distal (b).
a
FIG 12-10 The maxillary horseshoe lingual
arch has the advantages of simplicity and
ease of fabrication. Because the wire orientation is at 90 degrees to a TPA, the force
system is uniquely suited to special types
of tooth movement.
b
FIG 12-11 (a) A high mandibular lingual arch touches the incisor cingulum and is used for
space maintenance or for added incisor anchorage. (b) The low mandibular lingual arch is
placed below the tongue and does not touch the mandibular incisors. It is used passively to
prevent side effects or actively for reverse articulation, arch width control, molar rotation, and
molar tip-back applications.
lingual arch (Fig 12-11b) is placed below the tongue and
does not touch the mandibular incisors. It is more universal in its applications, including control of posterior
width, molar buccolingual axial inclinations, reverse
articulation mechanics, and serving as a base for finger
springs. The low mandibular lingual arch should be fabricated as far apically as possible so that the tongue will
not exert any vertical or forward force on it. Its low position has the added advantage of a smooth curvature that
is easy to fabricate and fit because contouring around
irregular teeth is not required (Fig 12-12).
Wire size and material
The F/Δ rate of the lingual arch can be varied by altering
the overall configuration of the arch, the wire cross
section (size and shape of the lingual arch), and the
material. When a high F/Δ rate is needed for a passive
FIG 12-12 The low mandibular lingual
arch is placed below the tongue so that its
low position has the added advantage of a
smooth curvature that is easy to fabricate
and fit because contouring around irregular
teeth is not required.
application, full-size 0.032 × 0.032–inch stainless steel
wire is used. For active applications, 0.032 × 0.032–inch
β-titanium alloy is better because the modulus of elasticity is only 0.42 that of stainless steel; thus, the force
magnitude is 0.42 times that of stainless steel for an
identical appliance, and the range of action is twice that
of stainless steel. If undesirable torque is to be avoided,
0.032-inch stainless steel or β-titanium round wires are
used. Table 12-1 summarizes the relative stiffness of different lingual archwires by shape and dimension of different
wire cross sections and materials. For simplicity, the relative stiffness of a 0.036-inch round stainless steel wire
is denoted as a base value of 1.0. Note that a full range
of wire stiffness and forces can be obtained (with and
without third-order torque) using a 0.032 × 0.032–inch
bracket; these wires have a precision fit with minimal
play.
221
12
Lingual Arches
Table 12-1 Relative stiffness of lingual archwires
Material
Wire size (inch)
Relative stiffness*
Stainless steel
0.036
1.0
Stainless steel
0.032 × 0.032
1.06
Stainless steel
0.032
0.62
β-titanium
0.032 × 0.032
0.45
β-titanium
0.032
0.26
FIG 12-13 One of the simplest applications of a
lingual arch is as a space maintainer. The molar is
prevented from tipping forward by the arch contact on the cingulum of the mandibular incisor.
*The relative stiffness of a 0.036-inch round stainless steel wire is denoted as a base
value of 1.0.
a
b
c
d
FIG 12-14 An extraction case treated without a lingual arch. (a) Before space closure. The anterior force from the space closure spring is
buccal to the CR. The replaced force system at each CR is shown with yellow arrows. (b) After space closure. The mandibular buccal segments
not only displaced mesially but also rotated mesial in after space closure (dotted line). (c) The force system at the mandibular buccal segment
is replaced by lateral and anterior components of the forces. (d) If a rigid passive lingual arch is placed, reciprocal equal and opposite forces
and moments will cancel each other out and will be effectively avoided; however, the anterior component of force (yellow arrows) still exists.
Passive applications
An important function of a passive lingual arch is to stabilize the posterior teeth together as a unit and to maintain
arch width and form. Full-size stainless steel wire is used.
One of the simplest applications would be a space maintainer where the anterior arc (the apex of the lingual arch)
touches the lingual surface of the incisors (Fig 12-13).
A lingual arch can also prevent side effects during space
closure. Figure 12-14 shows an extraction case treated
without a lingual arch. Only the force systems on the
mandibular posterior segments are depicted. The anterior
force (red arrows in Fig 12-14a) from the space closure
222
spring is buccal to the center of resistance (CR, purple
cirlces in Fig 12-14a); the replaced force system at each
CR is shown in yellow. As a result, the mandibular buccal
segments were not only displaced mesially but also
rotated mesial in after space closure (Fig 12-14b). In Fig
12-14c, the force system at the mandibular buccal
segment is once again replaced by lateral and anterior
components of the forces. If a rigid passive lingual arch
is placed (wire in Fig 12-14d), the major side effects of
displacement and rotation can be eliminated. Reciprocal
equal and opposite lateral forces and moments cancel
each other out and will be effectively avoided; however,
the anterior component of force (yellow arrows in Fig
Lingual Arch Configurations
FIG 12-15 A localized unilateral buccal
crossbite on the maxillary left second molar
was treated with a flexible Ni-Ti wire. (a and
b) Before treatment. (c) After leveling, good
tooth-to-tooth alignment is seen from the
occlusal view. (d) However, the lateral view
shows that alignment was produced in part
by buccal movement of the entire left posterior segment.
FIG 12-16 (a and b) A passive lingual arch
can serve as a rigid base structure for the
attachment of auxiliary springs or elastics.
Unique lines of force can be achieved working from the lingual.
a
b
c
d
a
b
12-14d) still exists. No matter how rigid the wire is, this
anterior component of force is not avoided. It cannot
overcome the laws of physics.
A localized unilateral crossbite on the maxillary left
second molar (Figs 12-15a and 12-15b) was corrected
with a flexible labial Ni-Ti wire. After leveling, good
tooth-to-tooth alignment is seen from the occlusal view
(Fig 12-15c); however, the lateral view (Fig 12-15d) shows
that alignment was produced in part by anchorage loss,
with buccal movement of the entire left posterior segment. Thus, a simple localized malocclusion became a
new generalized crossbite of many teeth that may be more
difficult to treat. A passive maxillary lingual arch (TPA
or horseshoe arch from first molar to first molar), if placed
before leveling, would have prevented this side effect.
A passive lingual arch can serve as a rigid base structure
for the attachment of auxiliary springs or elastics (Fig
12-16). Leveling and alignment performed with only a
labial archwire may produce unwanted side effects in
some patients. The lingual arch offers many creative
possibilities that can continually be modified during
treatment without depending on adjacent teeth for
anchorage. Hooks, lever arms, elastomers, and metal
springs are simple to design and fabricate.
In Fig 12-17, a finger spring with a helix soldered to a
mandibular lingual arch was used for incisor alignment.
Anchorage was provided by the bilaterally rigidly connected
first molars. A labial wire might have been considered, but
it uses adjacent teeth for anchorage, which could lead to
side effects. Sometimes the lingual arch can be placed
before incisor brackets are placed. A delay in bracket
placement due to occlusal interference, esthetic reasons,
or improved biomechanics can eliminate an unnecessary
side effect of facial bracket-wire alignment.
In a patient with a bilateral maxillary second molar
buccal crossbite (Fig 12-18a), the only forces needed for
correction are bilateral single forces on the second molars
(red arrows in Fig 12-18b). A single elastic or a coil spring
placed bilaterally on the second molars would be the
simplest and the best appliance mechanically; however,
it would be uncomfortable for the patient. Instead, elastics are placed on the right and left extensions soldered
223
12
a
Lingual Arches
b
c
FIG 12-17 (a) A finger spring with a helix soldered to a mandibular lingual arch was used for incisor alignment. Because the anchorage was
the bilaterally rigidly connected first molars, not the adjacent teeth, no side effects were seen during leveling. (b) Before. (c) After.
a
b
c
d
to a passive lingual arch, which delivers the same force
system as the single coil spring but is much more comfortable for the patient (Figs 12-18c and 12-18d). Without
the lingual arch, any labial alignment archwire would
have expanded the posterior end of the arch and produced
a tapered arch form (see Fig 12-15d).
The patient in Fig 12-19 required extraction of the
maxillary first molars instead of the premolars because
there was localized enamel hypoplasia on both first
molars (Fig 12-19a). Before extraction, provisional crowns
were placed on the first molars. Two passive TPAs were
placed, one anteriorly and one posteriorly. Hooks were
placed on the TPAs near the level of the center of resistance
(CR) (Figs 12-19b and 12-19c). The anterior segment was
a rigid unit including right and left premolars. Initially
the anterior teeth were not bracketed. The posterior
segment was also a rigid unit; it included only second
224
FIG 12-18 (a and b) In a patient with a
bilateral maxillary second molar buccal
crossbite, only bilateral single forces are
needed for correction (red arrows) of the
second molars. Therefore, a single elastic or
a coil spring placed bilaterally on the second
molars would be the simplest and the best
appliance mechanically; however, it would
be uncomfortable for the patient. (c) Instead,
elastics are placed on right and left extensions soldered to a passive lingual arch that
delivers (more comfortably) the same force
system in the occlusal view. (d) After bilateral
constriction.
molars. Note that the posterior TPA provided additional
space for the elastics (see Fig 12-19c). Because the elastic
forces were applied near the CR of both anterior and
posterior segments, space closure was primarily translation. Also note that the mesial movement of the second
molars provided the space for the third molars to erupt
(Fig 12-19d). After space closure, coordination of the
anterior segment width and second molar width is still
required. The biomechanics of this case are discussed in
more detail in chapter 13.
With two passive lingual arches, the line of force can
also be placed obliquely so that differential space closure
is achieved (Figs 12-20a to 12-20d). It is seen from the
lateral cephalometric radiograph of this patient that the
line of force (Ni-Ti coil spring) passes through the CR of
the posterior segment (Fig 12-20e). The line of force lies
occlusal to the CR of the canine, so the canine will tip
Lingual Arch Configurations
FIG 12-19 The patient required maxillary
arch extraction. (a) The maxillary first molars
were extracted instead of the premolars because there was localized enamel hypoplasia
on both first molars. (b) Two passive TPAs
were placed, one anteriorly and one posteriorly. Hooks were placed on the TPAs near
the level of the CRs. (c) Because the elastic
forces were applied near the CR of both anterior and posterior segments, space closure
was primarily translation. Note that the mesial
movement of the second molar provided the
space for the third molars to erupt (d).
a
FIG 12-20 Use of two maxillary lingual
arches for space closure. (a) Before treatment. (b) The line of force was placed
obliquely so that differential space closure (tipping versus translation) could be
achieved. (c) After space closure, the incisors
were aligned. (d) After debonding. (e) The
lateral cephalometric radiograph shows that
the line of force (Ni-Ti coil spring) passes
through the CR of the posterior segment and
lies occlusal to the CR of the canine. (f) Various lines of action from an elastic or spring
are possible between the anterior and posterior segments. It is even possible to place
a resultant force (yellow arrows) away from
the hooks by using two elastics on each side.
(g) Two TPAs with dual elastics on each side.
a
b
c
d
b
c
d
e
f
g
distally. Note the various lines of action that are possible
from an elastic or spring between the anterior and posterior segments for various types of tooth movements (Fig
12-20f ). Also note that it is possible to place a force off
the hooks (yellow arrows in Fig 12-20f ) by using two
elastics (Fig 12-20g). The yellow resultant force is equivalent to the red forces (bilaterally) of the two separate
elastics at the hooks.
225
12
Lingual Arches
a
b
c
d
e
f
g
h
FIG 12-21 The passive lingual arch can serve as effective rigid anchorage for asymmetric tooth movement. (a) This patient has a unilateral
buccal crossbite on the maxillary right second molar. (b) Before treatment. The deactivation force diagram shows a lingual force on the second molar and the reciprocal force system on the anchorage unit at its CR. (c) After treatment. The anchor teeth connected by a passive TPA
remained unchanged after treatment. (d) Before treatment. The same principle was applied in the mandibular arch using a buccal force from a
cantilever inserted in the first molar bracket on the buccal side of the tooth. (e) After treatment. Note that the mandibular left second premolar
was moved buccally with a loop anchored by the two molars connected by the passive lingual arch. (f to h) After debonding, the maxillary and
mandibular arch coordination was good, which suggests minimal side effects with this approach.
But can a passive lingual arch also be used in cases
requiring unilateral asymmetric tooth movement? We
have already seen how placement of a passive lingual arch
before leveling can prevent side effects. The passive application of a lingual arch can serve as effective rigid anchorage for asymmetric tooth movement. The patient in Fig
12-21 had a unilateral buccal crossbite on the maxillary
right second molar (Fig 12-21a). Before a leveling arch
was placed, a passive TPA with an auxiliary cantilever
spring welded to the maxillary right second molar was
placed. The rigidly connected first molars act as one unit,
with its CR midway between the CRs of the first molars.
The deactivation force diagram in Fig 12-21b shows a
lingual force on the second molar and the reciprocal force
system on the anchorage unit at its CR. Note that the
anchor teeth acted on by the reciprocal force system
remained unchanged (Fig 12-21c). The same principle
226
was applied in the mandibular arch using a buccal force
at the mandibular right second molar (Figs 12-21d and
12-21e) from a cantilever inserted in the first molar
bracket on the buccal side of the tooth. If the second
molar is also rotated mesial in, the hook should be placed
as far mesial as possible; a vertical loop at the distal of
the first molar tube can be used to lower the F/∆ rate.
Also note that the mandibular left second premolar was
moved buccally with a loop anchored by the two molars
connected by the passive lingual arch. After treatment,
the maxillary and mandibular arch coordination is good,
which also suggests minimal side effects with this approach
(Figs 12-21f to 12-21h).
In order to rotate the second premolar distal in, either
a distal force on the buccal or a mesial force on the lingual
can be used (Fig 12-22). If a mesial force is needed along
with the moment, the lingual arch is ideal for attaching
Lingual Arch Configurations
a
b
c
FIG 12-22 (a) In order to rotate the second premolar distal in and move it mesially, a mesial force (red arrow) from a lingual arch can be used.
The yellow arrows are the equivalent force system at the CR. A chain elastic on the buccal from the second premolar to the first molar would
have an inconsistent force system, with the force in the wrong direction. (b) During canine retraction, placing part or all of the distal force on
the canine lingual hook can solve or reduce unfavorable canine rotation. (c) The lingual arch prevents molar or posterior segment rotation.
FIG 12-23 (a and b) An elastic from a hook
on the lingual surface of the tooth attached
to a lingual arch can produce the desired
moment to rotate the molar, which is not
easy with buccal wire.
a
an elastic (Fig 12-22a) because bilateral molars are used
as anchorage, not the anterior teeth. The lingual elastic
along with a buccal elastic are needed to produce a pure
moment or a couple. During canine retraction, canines
tend to rotate distal in because the CR is lingual to the
bracket. Placing part or all of the distal force on a lingual
button on the canine can solve this problem (Figs 12-22b
and 12-22c). The labial archwire can also be used to
deliver an antirotation moment to the canine; the disadvantage is that this approach adds friction to the system.
Leveling of a buccally erupted second molar has an
inherent side effect of mesial-in rotation in continuous
arch alignment. This common problem is discussed in
detail in chapter 14. A buccal straight wire undesirably
expands the intermolar width at the first molars. Sometimes after treatment has been completed successfully in
the anterior region, second molars erupt rotated. In such
cases, an elastic from a hook on the lingual surface of the
tooth attached to a lingual arch can produce the desired
moment to rotate the molar (Fig 12-23). The lingual elastic also produces a mesial force at the CR (yellow equivalent force system in Fig 12-23b). If the second molar and
first molar contact areas collide, the force system on the
second molar approaches a couple.
b
The lingual elastic in Fig 12-24 is ideal to rotate the
second premolar mesial in and to move the premolar
distally to close a small space; the lingual arch offers good
anchorage to prevent the first molars from rotating mesial
out. Unlike a buccal wire that is depended on primarily
for alignment, here the friction is small. If the molar also
needs mesial-out rotation unilaterally, the lingual arch is
placed after the molar rotation is corrected. An equal and
opposite couple at each CR can be produced by a single
elastic only (Fig 12-25a). The maxillary left first molar is
rotated mesial in, and the second premolar is rotated
mesial out. A single force is applied at the lingual side by
an elastic without a buccal wire. A force at the lingual
(red arrows) is replaced at the CR with a force and a couple
(yellow arrows). These equal and opposite forces near the
contact area cancel each other out because the teeth are
already in contact. Only the couples that rotate the teeth
in the desired direction—molar mesial out and premolar
mesial in—remain (Fig 12-25b and 12-25c). After the molar
is sufficiently rotated, a symmetric passive lingual arch
can be inserted and the premolar rotation can continue
if needed. If only the second premolar is rotated, the
elastic can still be used—only now the passive lingual
arch must be in place.
227
12
Lingual Arches
FIG 12-24 The lingual elastic is ideal to rotate the second
premolar mesial in and move it distally to close a small
space. The lingual arch offers good anchorage to prevent
first molars from rotating mesial out.
a
b
c
FIG 12-25 (a) The molar is rotated mesial in, and the second premolar is rotated mesial out. A single force was applied at the lingual side by
an elastic without any archwire. (b) The single force at the lingual (red arrow) is replaced with a force and a couple at the CR (yellow arrows).
The couple rotates the teeth in the desired direction. (c) Final alignment. Adding an archwire on the buccal during rotation would only increase
the friction, and a passive lingual arch prevents desirable rotation of the first molar.
a
b
c
FIG 12-26 Auxiliary springs attached to a lingual arch. (a) Spring for palatal traction of the maxillary second premolar. (b) Springs for extruding
impacted canines. (c) Spring for moving a canine labially. A reverse articulation prevents bracket placement on the labial.
A spring is attached to the passive lingual arch for palatal traction of the maxillary second premolar (Fig 12-26a).
Other passive lingual arch applications include springs to
erupt impacted canines (Fig 12-26b) and teeth in reverse
articulation where occlusal interference would debond
brackets at the labial surface (Fig 12-26c).
Major tooth movement via finger springs from a lingual
arch can be considered an efficient adjunct to labial wires
for initial tooth alignment and leveling with or without
the labial brackets. Waiting to bond buccal brackets on
228
individual teeth until after initial major tooth movement
is completed can provide the most efficient mechanics
to avoid unnecessary tooth movement by inadvertent
leveling errors using a continuous archwire.
Active applications
The lingual arch passive shape can be modified so that
when it is inserted into the molar attachments, a force
system can be produced. This appliance can be very useful
Shape-Driven Method
a
b
FIG 12-27 Sequential changing of infinitely rigid ideal arches. (a) Initial position of the teeth. Wires are
made wider than the tooth position in many steps. (b) Forces are delivered to the molars to translate
them buccally. The amount of tooth displacement within the PDL space will be very small; therefore,
frequent adjustments must be made by small increments. (The amount of tooth displacement within
the PDL space is exaggerated in the figure.)
in delivering both symmetric and asymmetric force
systems. In addition, because only two attachments are
involved, a lingual arch provides a simple model with
which to understand force systems from an orthodontic
appliance. In chapter 14, two-bracket systems are studied
using straight wires. This chapter analyzes more complicated configurations in three dimensions.
Shape-Driven Method
An entire chapter could be written on the use of active
lingual arches without mentioning forces at all, which
would emphasize the “shape” of different lingual arch
applications. However, the major focus of this book is to
delineate both the correct and incorrect force systems
produced by any orthodontic appliance. The shape-driven
appliance usually uses an “ideal arch shape,” where the
brackets at the start of treatment are imagined to move
out to the passive shape or final shape of the arch.
E. H. Angle called this predetermined form the ideal
archwire. In some shapes today, it is referred to as straight
wire or, more specifically, preformed archwire. Using an
ideal archwire in a labial appliance is a typical shapedriven method where the wire is elastically bent and
placed into malaligned brackets. As the wire deactivates
to the original preformed and deactivated ideal shape, it
will hopefully bring the teeth into ideal positions. The
same approach is commonly used with lingual arches.
Suppose there is a patient who needs the intermolar
width increased, for example. The force system from the
ideal-shaped arch depends on the rigidity of the arch. For
better understanding, the wire is a rigid body, which is
depicted in gray (Fig 12-27a). There is no such wire, but
let us imagine one that is infinitely rigid (F/Δ = ∞).
Because the arch does not deflect during tying of the
molars, tooth movement is limited to deformation of the
periodontal ligament (PDL) support (ie, initial mechanical displacement and subsequent biologic response).
Only buccal forces (red arrows in Fig 12-27b) are applied
to the molars, which translate buccally from the occlusal
view. The amount of displacement of the teeth will be so
small within the PDL that it is exaggerated in the figure.
Perhaps the direction of the force system is correct;
however, high rigidity of the lingual arch will require
frequent adjustments and hence lacks efficiency. A jack
screw in the removable appliance produces this kind of
force system.
Let us now, by contrast, fabricate a low-rigidity arch
(Fig 12-28a), where the preformed ideal shape (green
shape) is identical. The lingual archwire is fabricated so
that it is passive to the final position of the teeth. In other
words, the interbracket distance of the final position of
the teeth (L) and the width of the ideal lingual arch (L)
is the same. To place the flexible arch into the two molar
attachments (orange shape) by keeping the free ends of
the lingual arch and the brackets parallel, not only lingual
constrictive forces but also moments are required. Why
are additional moments necessary? As the lingual arch
is constricted to fit into the brackets by lingual forces
only, the green U shape becomes an orange V shape. The
free ends cut across the brackets at an angle (dotted lines
in Fig 12-28b). This necessitates additional moments for
full insertion (Fig 12-28c). Unlike the rigid lingual arch
in Fig 12-27, the flexible lingual arch undergoes a complicated elastic deformation of its shape that introduces
moments on each molar. Each first molar initially receives
both a desired buccal force as well as unwanted moments
rotating the molars mesial out and distal in (Fig 12-28d).
The flexible lingual arch has the advantage of a greater
229
12
Lingual Arches
a
b
c
d
range of activation, requiring fewer adjustments. In fact,
its shape can be made wider than the desired width to
ensure more efficient force levels as the molars approach
their final widths. The shape can still be an exaggerated
ideal arch if the free ends are made parallel to the molar
brackets; however, undesirable side effects are produced
by the elastic archwire deformation. Thus, the ideal arch
shape works best for relatively rigid wires where the F/Δ
rate is high and the displacement of the tooth is confined
to PDL deformation at each activation.
Force system changes during
deactivation of the shape-driven
appliance
Suppose the intermolar width at the first molars (Fig
12-29a) is to be expanded (Fig 12-29b), and we select the
green ideal arch shape (Fig 12-29c). The shape of the arch
is determined without considering the force system. Clinically, the desired molar position is determined first (blue
teeth in Fig 12-29c), and then the wire is fabricated passive
to that position (green wire in Fig 12-29c). Then the lingual
arch is elastically deformed by applying the necessary
force system during activation to place it into the initial
bracket positions (orange wire in Fig 12-29c). The activated lingual arch exerts a force system on the molars,
and as the teeth move to the final desired positions, the
lingual arch eventually deactivates to its preformed ideal
shape. This common treatment procedure seems conceptually logical and very easy to understand and apply;
however, it has inherent limitations and some disadvan230
FIG 12-28 (a) A flexible ideal arch where
the preformed ideal shape (green) is identical to Fig 12-27. The lingual archwire is
fabricated so that it is passive to the final
position of the teeth. (b) As the lingual arch
is constricted to place it into the brackets
using a force only, its U shape becomes a V
shape. The free ends cut across the brackets
at an angle. (c) This necessitates additional
moments for full insertion. (d) Deactivation
force system on the teeth. Each first molar
initially receives a desired buccal force as
well as unwanted moments rotating the
molars mesial out and distal in.
tages as discussed previously. Unwanted forces or moments
can be produced, and the force system can also change
during deactivation of this so-called ideal shape approach.
First, let us consider the relative magnitude of the force
system only. The magnitude of the force system of the
activated lingual arch is initially 100% (orange wire in Fig
12-29c), whereas the final preformed and deactivated
ideal shape (green wire in Fig 12-29c) would exert no force
system at all (0%). If the magnitude of the initial force
system was set for an optimal range, the tooth will move
rapidly initially. After the tooth passes through the optimal zone and as force continues to reduce, reaching a
suboptimal force zone, the movement would slow down;
the molars may not reach their final target position
because force magnitude approaches zero at the final
stage of deactivation.
More importantly, the initial force system may not be
correct. The mandibular left first molar is shown in an
enlarged view in Fig 12-29d. When a constriction force
(blue arrow) is applied at the free end to insert the lingual
arch (activation force), the deactivated arch (green wire)
will elastically deform to the orange shape. When a lingual
force is applied for insertion, the free ends of the arch cut
across the brackets at an angle (see Fig 12-29d). To place
the orange wire into the molar bracket, not only is a clockwise moment needed but also a greater magnitude of
lingual force (Fig 12-29e). Note the difference in the
orange shape between Fig 12-29d and 12-29e. A greater
magnitude of lingual force is required because clockwise
moments tend to further expand lingual arches. In this
appliance configuration, the forces and moments are
associated and do not act independently. (The principles
Shape-Driven Method
a
b
c
d
e
f
FIG 12-29 The shape-driven method. (a) The narrow intermolar width needs expansion. (b) The target positions of the molars are shaded in
blue. (c) The selected ideal arch shape is shown in green. The shape is determined without considering the force system. The magnitude of
the force system is initially 100% and dissipates as it approaches the final position. (d) If a constriction force (blue arrow) is applied at the free
end to insert the lingual arch, the left free end of the lingual arch cuts across the bracket at an angle. (e) To place the orange archwire into the
molar bracket, not only is a clockwise moment needed but also a greater magnitude of lingual force. (f) The deactivation force system (red
arrows) acting on the molars is equal and opposite to the activation force system required for insertion.
of association and dissociation are discussed later in this
chapter.) The blue arrows comprise the activation force
system that the clinician applies to the lingual arch to
activate it during insertion. The deactivation force system
acting on the molars, depicted in Fig 12-29f, is equal and
opposite to the activation force system. In this example,
the mesial-out moments on the molars are unnecessary
and can cause undesirable side effects. This is similar to
the example shown in Fig 12-28. The changes in the force
system over time are shown in Fig 12-30.
There are disadvantages to this approach (Fig 12-30).
First, let’s consider the direction of the tooth movement.
Initially the large, unwanted mesial-out, distal-in
moments are associated with the expansion force, and
later the moment direction is reversed. Because of the
wire shape changes during deactivation and subsequent
changes in bracket geometry, the centers of rotation are
continually being modified. Second, the magnitude of
the force system is not desirable. Initial rapid tooth movement with unnecessary counterclockwise rotation in the
optimal force zone is followed by very slow tooth movement with clockwise rotation in the suboptimal force
zone. The optimal zone is very narrow. As the molar
approaches its final deactivated position (blue tooth) after
the optimal force zone, it may rotate in a reverse direction
to correct the rotation side effect; however, this may take
FIG 12-30 Force system changes in the shape-driven method.
During the course of required deactivation (from orange to green
wire), the molar rotates counterclockwise (left dotted arrow) and then
clockwise (right dotted arrow), which is unnecessary. Initial rapid tooth
movement occurs in the optimal force zone, which includes significant
side effects. Moreover, the tooth movement is very slow to reach the
target position in the suboptimal zone.
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Lingual Arches
a relatively long time because the moment and force
values are dissipated in the suboptimal zone. It is certainly
better to directly move teeth to their final positions without the side effects associated with this “round trip” ride.
Force-Driven Method
It is apparent that the ideal arch may be the desired shape
when teeth have moved to the predetermined fully deactivated shape; however, the force system to get there may
not always be ideal with this shape, as discussed previously. A better approach is to design the shape of a lingual
arch to produce the desired force system. This is called
a force-driven appliance. The tooth reacts to the applied
force system; it does not matter what kind of material,
cross section, or configuration is used. Therefore, in the
force-driven method, deciding the force system takes
priority over establishing the final tooth position. Considering any given lingual arch configuration (horseshoe or
TPA), the force system changes during deactivation. Both
force magnitude and M/F ratios can be changed. In other
words, the force system may not always be correct
throughout the full range of deactivation. Therefore,
certain principles must be applied. First, the initial force
system, which is set to an optimal force magnitude range
and a correct M/F ratio, must be correct. Second, because
the magnitudes of the forces and moments after appliance
insertion decrease as the teeth move, their optimal levels
need to be maintained as much as possible, particularly
during the terminal phase of molar movement. This is
accomplished by lowering the F/∆ rate and producing a
range of activation that is larger than the required tooth
movement. This allows for the delivery of more constant
and optimally lighter forces.
The accurate shape of a lingual arch to deliver a specific
force system can be obtained using computers applying
beam theory and iterative methods; however, the clinician
can easily apply these principles chairside to achieve a
close approximation of the appropriate shape. The clinical
procedure follows.
The first step is to determine the desired force system.
An equilibrium diagram is useful to assure that a valid
force system exists. See, for example, the mandibular first
molar expansion (translation from the occlusal view) with
a single buccal force on both molars in Fig 12-31a.
In the second step, a passive shape of the arch for the
original molar position is fabricated and contoured with
minimal clearance between the soft tissues for maximum
comfort (green wire in Fig 12-31b).
232
The third step is a simulation shaping of the lingual
arch by the deactivation force system. Here the clinician
performs a loading on the passive shape, applying the
predetermined deactivation force system (forces on the
teeth) from the first step. In our example, the loading is
two simple equal and opposite forces in the direction of
the desired molar movement (red arrows in Fig 12-31c).
Note that as bilateral expansive forces are applied, the
lingual arch gets wider at the free end and assumes a less
tapered V shape. This is the simulated shape (orange wire
in Fig 12-31c). During the simulation, sufficiently flexible
wire (eg, β-titanium wire) is selected so that the distance
between the free ends (L2 ) under given force must be
larger than the final interbracket distance (L1 ). By this
procedure, the force magnitude is always in the optimal
force zone from initial to final target position of the tooth.
The general shape needed in the lingual arch is shown
with the green wire in Fig 12-31c.
In the fourth step, the wire is permanently bent to the
deactivated shape until it becomes identical to the simulated shape. To determine the amount of activation for
the given shape, a force gauge can be used when single
forces alone are needed (Fig 12-31e). Grid paper could
also be helpful to record the amount of activation in any
given simulated shape (Fig 12-31f ). The simulated shape
is observed as the lingual arch is elastically deformed; it
is therefore necessary to permanently deform the lingual
arch to the simulated shape by increasing the load.
However, increasing the load only to bend the wire to the
correct deactivated shape is not enough. Arches are more
resistant to permanent deformation if they are bent
further than the simulation shape and then bent back to
the established shape (Bauschinger effect).
The fifth step is the trial activation, where the shape is
checked in the mouth for correctness before final insertion. Activation forces (blue arrow) are applied on the
deactivated lingual arch (Fig 12-31g), and if the lingual
arch is correctly fabricated, it will fit easily in the molar
attachments with a single force only. If any moment is
necessary to engage the lingual arch, the shape needs
further adjustment. Note the difference in shape of the
ideal arch shape (see Fig 12-29c, green wire) and the forcedriven shape (see Fig 12-31d, green wire), where the free
ends cross the molar brackets at an angle with a single
force. After placement (orange activated shape in Fig
12-31g), the activated shape is identical with the original
passive shape (green shape in Figs 12-31b and 12-31c) in
the force-driven method. Once it is placed and the clinician’s hand is released, it exerts the planned forces on the
teeth as it deactivates (Fig 12-31h). If the lingual arch is
Force-Driven Method
a
b
d
Deactivated shape
Passive shape
e
Deactivated shape
Activated shape
Activation
g force system
c
Passive shape
Simulated shape
f
Deactivation force system
h
FIG 12-31 The force-driven method. (a) The desired force system is first established. An equilibrium diagram is
useful to assure its validity. (b) A passive shape (green) to the original molar position is fabricated. (c) Simulation is
performed with the deactivation force system. (d) The deactivated shape, which is identical to the simulated shape,
is the final shape needed before placement. (e) To determine the amount of activation for the given shape, a force
gauge can be used. (f) Grid paper could also be helpful to record the amount of activation. (g) The activated shape is
identical to the passive shape. (h) Once the archwire is placed and the clinician’s hand is released, the initial forces
on the molar are correct.
carefully contoured for comfort in its passive shape, it
should maintain these patient-friendly contours after
insertion. Fabricating a comfortable passive lingual arch
that does not impinge on tissues is time well spent using
a force-driven lingual shape. Because it is force driven
and based on the original lingual shape, the deactivated
shape is usually more comfortable than the ideal arch
shape, where forces during insertion are ignored.
Computer iteration methods using beam theory follow
the same steps described above. If any deviation from
the activated shape and passive shape is detected, a little
modification to the deactivated shape is added, and the
cycle is repeated until the correct shape is obtained.
Force system changes during
deactivation of the force-driven
appliance
The force-driven lingual arch is more efficient than the
shape-driven one. It delivers the correct force system
initially, within the optimal force level zone, where the
most significant tooth movement occurs. The tooth will
move directly to the target position without any unnecessary wiggling or side effects. It is comfortable for the
patient because the activated shape bypasses problematic
anatomical structures since it is identical with the passive
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12
Lingual Arches
FIG 12-32 Force magnitude zones with the force-driven method.
Initially there is a single force, without any moments. The optimal
zone ensures that efficient force magnitude levels are maintained until
the molar approaches its target position. If the force-driven bilateral
expansion lingual arch remains in place for a sufficiently long time
after the target position is reached, it will enter the suboptimal force
zone (yellow area). Therefore, the lingual arch is removed when the
molar reaches the target position within the optimal zone.
shape. Let us track the displacement of the tooth in the
force-driven method. Initially the required tooth movement involves only a single force in the optimal force
zone. Therefore, displacement to the target position (blue
tooth in Fig 12-32) is quick, and there are no side effects
of associated rotations. This is a desirable feature because
tooth movement will not slow down as the final target
position is reached, and no side effect correction is
required.
If the force-driven bilateral expansion lingual arch
remains in place for a sufficiently long time after the target
position is reached, it will enter the suboptimal force zone
(yellow area in Fig 12-32). Even if the tooth movement in
the suboptimal zone is very slow, the molars may pass
beyond their target positions, and molar position may
follow the shape of the wire, which together would create
molar expansion and distal-out rotation (outlined tooth
and green wire in Fig 12-32). Therefore, when the tooth
reaches the planned desired position, the lingual arch is
removed and reformed to a passive shape. Unlike the
shape-driven ideal arch, the deactivated force-driven
shape has little clinical relevance; the lingual arch is
removed or made passive before the fully deactivated
shape is reached.
It is easy to determine the deactivated shape of the
lingual arch in the shape-driven method; however, the
force system may or may not be correct. The amount of
linear (parallel) and angular bends are determined by the
final position of the molar’s bracket, ignoring the force
system. In the force-driven method, on the other hand,
the deactivated shape is determined by the force system,
not the final lingual bracket position. Universally, the
correct shape is formed by applying the desired deactivation force system to any passive arch, called simulation.
This arch shape is first simulated and then deformed
permanently into that simulated shape. The amount of
234
activation, both linear and angular, may be dependent on
the overall configuration of the lingual arch, its dimensions, cross-sectional shape, and material. Therefore, we
will limit our suggestions of specified activations in millimeters or degrees and rather emphasize where and how
to make bends or twists in the wire to deliver relatively
correct force systems. Force system simulation provides
the correct shape; however, understanding where and how
the wire is bent or twisted is more important not only to
fabricate the correct shape but also to modify the force
system when necessary. Remember, even if the force
system and the shape are correct, the response of teeth
can vary, and the force system always needs monitoring
and modification. The principle of correct shape based
on the required forces is obtained by an understanding
of beam theory. Further discussion of beam theory is
beyond the scope of this chapter, but it is important for
the clinician to note that the amount of bending or twist
is proportional to the bending moment or torsional
moment (torque) at a given section of a wire. The sections
of a wire that have the highest stresses are called the
critical sections. At these sections, most of the bending
or torsion occurs during the simulation procedure. More
detailed analytical and computer evaluations can be
useful and require further discussion.3
Symmetric Applications
Bilateral expansion
Bilateral (symmetric) molar and posterior segment arch
width modification requires reciprocal anchorage and
hence is a logical application of the lingual arch. First a
passive lingual arch is fabricated (Fig 12-33). A flexible
typodont is used to demonstrate the movement of the
Symmetric Applications
FIG 12-33 A passive lingual arch is first
fabricated for every active application. Red
dots in alignment are marked on the mesial
and distal sides of the first molar to visualize
the movement of the first molar on the elastic
typodont.
a
b
c
FIG 12-34 Bilateral expansion by the force-driven method. (a) Simulated and deactivated shape. Note that the wire and brackets are not
parallel (dotted lines). (b) After an active arch is placed. The two red dots on the first molar moved equally to the buccal. (c) If controlled tipping
or translation should occur, a couple is added by torsion of the square wire. Double-headed arrows represent couples (buccal root torque)
using the right-hand rule of thumb (see Fig 3-8).
teeth using the lingual arch. The resin teeth are embedded
in an elastic material so that the forces displace the teeth,
demonstrating the resulting tooth movement. After the
passive lingual arch is placed in the typodont, red dots
are marked on the mesial and distal sides of the first molar
(see Fig 12-33). Along with the red dots on the adjacent
teeth that form a line, one can easily visualize the movement of the first molar after an active arch is placed.
Let us consider a clinical situation where mandibular
molars require expansion. Our thinking must be in three
dimensions. From the frontal view, tipping around the
root center can be allowed. From the occlusal view, translation is the defined movement. Bilateral expansion
requires two single forces acting buccally at the lingual
bracket. This required deactivation force system is applied
to the passive lingual arch, and the simulated correct
deactivation shape is formed. Figure 12-34a is the deactivated shape, which is identical to the simulated shape
for bilateral expansion. The amount of activation is determined during the simulation step by the deactivation
force system. Note that distal free ends are not parallel
with the brackets (dotted lines in Fig 12-34a). Next, before
final placement, constriction force is applied by squeezing both free ends with single forces as a trial activation
in the mouth, and the fit is checked. Modification can be
made to alter the magnitude or to ensure passivity. Figure
12-34b shows that the first molar moves to the buccal
without noticeable rotation after placement of an active
lingual arch. The red dots on the first molar indicate equal
mesial and distal contact area displacement. If a single
force is required, a round wire is preferable (0.032-inch
β-titanium) so that the clinician does not have to adjust
a rectangular wire for third-order passivity to eliminate
unnecessary moments (torque) from the frontal view.
Note that Fig 12-34b shows tooth movement within an
optimal force zone, as the lingual arch is not fully deactivated. The appliance is force driven with the initial force
system correct, with only a buccal force and no moment
from the occlusal view. The arch is made passive when
alignment is reached.
A rectangular wire can be useful if the center of rotation
is to be moved from a centered position on the root to
the root apex. In addition to the shape previously
described, a twist is placed along the posterior of the arch
to deliver equal and opposite couples in a buccal root
torque direction. This is not a localized twist but a shape
that simulates loading the lingual arch with couples
(double-headed red arrow in Fig 12-34c).
If an ideal arch was used with posterior free-end arms
parallel to the brackets for expansion (Fig 12-35a), an
incorrect force would be produced. Couples would rotate
the molars mesial out during the delivery of the initial
235
12
Lingual Arches
FIG 12-35 Bilateral expansion by the
shape-driven method (ideal shape). (a)
The free ends are fabricated parallel to
the brackets (dotted lines). (b) Moments
(mesial-out) are associated with buccal
force in the initial force system.
a
b
FIG 12-36 Bilateral constriction by the forcedriven method. (a) Simulated and deactivated
shape. Note that the wire and brackets are
not parallel (dotted lines). (b) After insertion,
only translation (no rotation) is observed in
the occlusal view.
a
b
force system (Fig 12-35b). Initially, the undesired rotation
would be seen more than the expected widening of the
intermolar width. Also note that if the wire deforms unexpectedly, it may impinge on the soft tissues.
Bilateral constriction
The same principles and sequence are used for bilateral
constriction using a force-driven appliance. The deactivated shape is determined by simulation (ie, applying the
deactivation constrictive force to the passive shape). Note
that the free ends of the lingual arch and the brackets are
not parallel as in the ideal shape (Fig 12-36a). After insertion, only forces (no moments) act on both first molars,
and red dots indicate that the molars do not rotate (Fig
12-36b).
Estimating bending and torsional
moments at arbitrary sections along
a wire
The simulation procedure will correctly determine the
deactivation shape of an arch, but fabrication of the
correct shape also requires technique; nevertheless,
understanding the principle of how a wire undergoes
deformation during simulation greatly helps during fabrication of the correct deactivation shape. Figure 12-37a
depicts a straight-wire cantilever with a deactivation force
(red arrow) applied at the free end. This could represent
236
an intrusion arch to the mandibular incisors with the
terminal tube at the mandibular first molar and only a
point contact at the incisor brackets. Note that the forces
that act on the wire to deform it are in the direction of
the forces that act on the teeth for intrusion. In response
to an intrusively directed force, the cantilever will bend
downward elastically, assuming the simulated shape. This
is the correct deactivated shape for intrusion, when the wire
is permanently deformed to that shape (Fig 12-37b, top).
During activation (see Fig 12-37b, bottom), the force
direction (blue arrows) and the bending of the cantilever
are in the opposite direction to the last bend we used to
permanently deform the wire. It is recommended to overbend past the simulation shape and then to bend back to
the final deactivated shape. If this is done, activation to
place the wire in the mouth is in the same direction as
the last bends used to permanently deform the wire. The
deactivated shape (A) in Fig 12-37b and the deactivated
shape (B) in Fig 12-37c are identical, but the deactivated
shape (B) in Fig 12-37c is more resistant to permanent
deformation because the wire is activated with the same
direction of the final bend to reach the shape of the wire
(Bauschinger effect).
Intuitively, we know that the general shape will be a
downward curvature from the terminal molar bracket
forward. The question is: Where should we bend the wire,
and how much curvature should we place?
Let us make an imaginary cut with a knife at point B
of the wire (Fig 12-38a). If the cut were real, the small
element in Fig 12-38a (bottom) would fall off; however,
Symmetric Applications
a
b
c
FIG 12-37 The procedure to obtain the correct shape for a downward single force at the free end of a cantilever spring.
(a) Passive shape and simulated shape. The red arrow at the free end is in the direction of the deactivation force system that
we want. The cantilever is in equilibrium. (b) The wire is permanently deformed to the deactivated shape (green), which is
identical to the simulated shape. The activation force system (blue arrows) is applied. The activated shape (orange) becomes
identical to the passive shape. (c) To increase the range of action, the wire is overbent first and then bent back to the deactivated shape. Note that both deactivated shapes (A, B) are identical, but during fabrication of B, the final bends are in the
direction of the activation force system (blue arrows in b).
a
b
FIG 12-38 A straight-wire cantilever with a single force applied at the free end. (a) An imaginary cut (section) at point B will make an element
that is also in equilibrium but with less bending moment because the distance between two forces is less in the element (L2 < L1). (b) The
magnitude of the bending moments at arbitrary sections along the wire is depicted by varying the color of the wire. The part with maximum
bending moment is depicted in orange, and the part with no bending moment at all is depicted in green. The curvature of the deactivated
shape of the wire at each section is proportional to the bending moment. More bending occurs near point A, where the bending moment is
the greatest, and there is no bending at point C.
it does not because stresses (forces) hold the wire together.
Therefore, the element is in a state of equilibrium. The
surface of the element where we imagined the cut is called
a section. When we shape a wire or an arch, permanent
deformation occurs at every imaginary section along the
wire. Let us look at the element (from point B) again.
Stress and strain will occur in three dimensions at each
section. The red vertical force perpendicular to the long
axis of the wire is the shear force. The moment acting at
the section, needed for equilibrium, is the bending
moment. Horizontal axial stresses (pure tension and
compression), which are not shown in this figure, act
parallel to the long axis of the wire (see chapter 13). If
the wire is twisted (not in this example), torsional
moments (torque) can operate around the long axis of
the wire. However, the torsional moment along the long
axis does not change the overall configuration of the wire.
The magnitude of the bending moments at arbitrary
sections along the wire is less than the bending moment
at the fixed end because the distance between two
forces is decreased (L2 < L1) (see Fig 12-38a). The bending moment gradually decreases from point A to point
C; therefore, the part with maximum bending moment
is depicted in orange, and the part with no bending
moment at all is depicted in green (see Fig 12-38b).
However, no matter how much bending moment is
produced, it is depicted in uniform orange color in this
book for simplicity.
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12
238
Lingual Arches
FIG 12-39 A straight-wire cantilever with a couple applied at the
free end. At all arbitrary sections, the bending moment is the same.
Therefore, the curvature is uniform, a segment of a circle.
FIG 12-40 An imaginary section is made at the apex of the
arch (dotted line), which is the critical section in bilateral constriction by a single force at the free end. The colored element
is in equilibrium by the activation force system (blue arrows).
It also informs us where and how much to bend the
wire. The magnitude of curvature at each section is
proportional to that of the bending moment. At point C,
no element or bending moment exists; hence, no bend
is placed there. Compare sections at points B and A. The
element at section A is the longest and, hence, the bending moment is the largest. The amount of the bend (or
curvature) must be proportionally increased at point A.
Where a section has the highest bending moment, it is
called a critical section. Critical refers to its sensitivity to
wire failure, such as permanent deformation or fracture.
So how is the straight-wire cantilever bent to produce
the force-driven shape for incisor intrusion? The wire is
curved downward gradually, increasing the bend magnitude as the plier is moved toward the fixed end (see Fig
12-38b). The simulation approach (see Fig 12-37a)
described previously gives the same result. Push down
on a cantilever anteriorly, and you will see it deform
increasingly more to the distal. Understanding simulation
and the principle of where to bend a wire greatly enhances
the proper use of a force-driven appliance.
Two basic loading conditions in bending a wire are
forces and couples. Let us now apply a couple to the free
end of a cantilever and see how the wire will bend (Fig
12-39). Note that at each section, no shear force is present,
and only an equal and opposite bending moment is
required to keep all elements in equilibrium. What is the
force-driven shape to deliver only a couple at the free end?
Unlike the single force example in Fig 12-37, all sections
require identical bending moments. Therefore, the varying color of the wire becomes uniform. This requires equal
bending all along the wire in a downward direction. In
other words, placing any straight wire in equilibrium with
only equal and opposite couples produces a wire curvature
that is a segment of a circle. The curved wires in Figs
12-38b and 12-39 may look very similar, but the curvatures
and their positions are very different; thus, the force
system is totally different in each.
Figure 12-40 shows a lingual arch used to narrow intermolar widths by a lingual force alone. An imaginary
section is made at the apex of the arch (dotted line). The
colored element is in equilibrium, and the activation force
system is depicted by blue arrows. Note the highest bending moment at this section (marked by the dotted line).
This is observed during the simulation procedure and is
also anticipated with the bending moment diagram.
During the simulation of bilateral constriction, single
forces at the free ends are applied, and because the largest perpendicular distance from these single forces is near
the apex, most bending occurs there. This region is a
high-stress critical section where overbending is advised
to minimize permanent deformation. If lingual arches
are overbent and then reformed, residual stresses are
operating in the correct direction to minimize permanent
deformation. Therefore, desirable residual stresses should
not be removed (Bauschinger effect). It is not strange to
find that a wishbone is always broken near the apex, which
is the critical section.
Symmetric Applications
FIG 12-41 Bilateral expansion in the shapedriven method. Parallel buccal wire (a) is accompanied by molar mesial-out moments (b).
a
b
a
b
FIG 12-42 Bilateral rotation in the shapedriven method. Mesial-out angular bends
at molars (a) are accompanied by buccal
forces (b).
Association and dissociation of moment
and force
The phenomenon of wire association and dissociation
can be demonstrated with an ideal arch. The lingual arch
in Fig 12-41 is shaped with arms parallel to the brackets
and wide. We have already learned that both buccal forces
and mesial-out moments are produced; furthermore, the
wider the parallel arms, the greater the increase in both
buccal forces and mesial-out moments. With this horseshoe configuration, an association exists between the
force and the moment (buccal force and mesial-out
moment). Now let us angle the free arms (Fig 12-42) so
that they cross the bracket in a direction of mesial out
with the width not changed. Not only mesial-out moments
but also expansion forces are produced. If width or angle
is modified, an association always exists. Narrowing the
width produces a moment directed mesial in; the association is the same but reversed.
Let us suppose that two identical bilateral maxillary
expansion horseshoe arches are placed for bilateral
expansion using an ideal shape, one inserted from the
anterior (Fig 12-43a) and the other from the posterior
(Fig 12-43b). Typical mesial insertion will produce expansion and a mesial-out rotation (see Fig 12-43a). Although
it would be practically impossible to place, the distally
inserted lingual arch with its apex directed posteriorly
will produce expansion and a mesial-in rotation, which
is opposite in direction from the mesial insertion (see Fig
12-43b). These differences in the force system despite the
same shape at the bracket are not accidental; they reflect
how wires bend in respect to any applied forces, and they
demonstrate the association of force and moment.
Because the arches in Figs 12-43a and 12-43b are mirror
images, the angles at which the archwires cross the brackets during activation are equal and opposite.
Let us now make a TPA, where the amount of wire
anterior and posterior to the attachment is about the
same (Fig 12-43c). It should not surprise us that the
correct expansive force only is produced by parallel
expansion, without any free-end angulation. In other
words, because of its unique wire configuration in 3D,
the TPA dissociates the force from the moment. Increasing the amount of expansion does not change or produce
a moment, and increasing a first-order angle does not
change the force. Interestingly, the force-driven shape
and the ideal arch shape are the same from the occlusal
view. During the force-driven simulation with forces only,
the free ends expand with a relatively constant angle
parallel to the passive shape. The linear parallel displacement in expansion takes place mostly by the bending of
the apex of the TPA. Dissociation is only present from
the occlusal view in this example; associations with a TPA
between expansion and x-axis moments (third-order rotation) from the rear view are discussed later (see Fig 12-51).
If a round-wire TPA is used, pure expansion or constriction (occlusal view translation) can be delivered with
single forces; here, the force-driven shape is the same as
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12
Lingual Arches
a
b
c
FIG 12-43 Bilateral expansion using an ideal shape inserted from the anterior (a) and from the posterior (b) will generate moments in opposite
directions in the occlusal view. With a vertically placed TPA, no moment will be produced with parallel expansion (c). (a and b) Associated
designs for expansion. (c) Dissociated design for expansion.
a
b
c
FIG 12-44 Dissociated type of application with a TPA. (a) Passive shape. (b) Simulated (deactivated) shape. Note that the wire and brackets
are parallel (dotted lines). (c) Once it is inserted, only force (no moment) is produced in the occlusal view.
the shape-driven shape, with the exception that it is exaggerated so that a larger range of activation maintains the
optimal force zone.
The dissociation phenomenon is demonstrated on the
elastic typodont in Fig 12-44. From the occlusal view, an
ideal arch shape is used. The free-end arms are parallel
to the brackets (Figs 12-44a and 12-44b). After insertion
(Fig 12-44c), only translation occurs because a force
acting at the CR and no moments are produced. The teeth
translate to the buccal as shown by the red dots (see Fig
12-44c). In this application, the ideal shape gives the
correct force system. Also, the M/F ratio of 0 mm is
constant throughout the entire range of action. It is desirable to use a round wire with small cross section and
lower modulus of elasticity such as 0.032-inch β-titanium
to lower the magnitude of forces and to deliver it more
constantly. If so, the deactivation shape should be wider
than the required amount of translation. When ideal
width is achieved, the TPA is made passive.
Bilateral rotation
Many patients require molar rotation (rotation around
the y-axis). Individual molars can be rotated, or the overall arch form can be modified by rotation of a segment.
240
The shapes of horseshoe and TPA lingual arches to
produce equal and opposite couples for rotation are very
different. First, let us consider a horseshoe lingual arch
that can be used in either the maxillary or mandibular
arch.
To rotate the molars bilaterally (mesial out in this case),
a simulation is performed on the passive shape using
equal and opposite couples (Fig 12-45). The deactivated
shape (or simulated shape) of the lingual arch is depicted
in Fig 12-45a. In the simulation process, every section of
the lingual arch feels the same bending moment; therefore, the entire lingual arch is bent evenly from one end
to the other. This bilateral rotation lingual arch looks
similar to the bilateral constriction arch (see Fig 12-36).
Both have a shape with a curvature that may suggest a
distal-in molar rotation effect; however, this shape with
its added uniform curvature only exerts bilateral mesialout moments (Fig 12-45b). This force-driven shape, which
looks nothing like an ideal shape, only rotated the molars
mesial out. In Fig 12-45b, a little width change is seen,
which suggests that it has passed the optimal zone. The
lingual arch should have been removed and reshaped for
passivity when the tooth target position was reached. A
round 0.032-inch β-titanium wire is a good choice for
this type of tooth movement. The deactivated shape
Symmetric Applications
FIG 12-45 Bilateral rotation with a mandibular lingual arch. (a) Simulated (deactivated)
shape. Every section of the wire needs an
even amount of bending. (b) Once it is inserted, only moments are generated initially.
c
a
b
a
b
d
e
FIG 12-46 Bilateral rotation with a maxillary TPA. (a) Passive shape. (b) Simulated (deactivated) shape. With this lingual arch (TPA) design, the
force-driven and shape-driven shapes look similar in the occlusal view. Horizontal arm–bracket crossover angles on the right and left should be
the same. (c and d) If the angles are too small to compare, only one side is placed, and displacement (∆) of the contralateral side is measured
and repeated on the other side sequentially with the same ∆. (e) The molars have rotated and maintained their original widths after placement.
should be narrower, produced by applying the simulation
couples. By using a less stiff wire, the moments can be
lower and act more constantly, and the range of the optimal force zone is increased. Orthodontists have learned
to use less stiff wires and lower forces on the buccal for
alignment; unfortunately, there is still a tradition of fabricating rigid high-force lingual arches.
Bilateral molar rotation with a TPA requires an entirely
different deactivated shape (Fig 12-46). The free ends of
the TPA cut across the centers of the lingual brackets,
forming equal angles bilaterally. Note that the deactivation force-driven shape is similar to the shape-driven
shape. When dissociation is present, a configuration like
an ideal arch should work. Wire-bracket angles produce
moments, and linear parallel displacements produce
forces. This is the special case where a straight wire or
an ideal arch works according to the older predictions.
First, the passive TPA is fabricated (Fig 12-46a), and
simulation of the force system is performed (Fig 12-46b).
The simulation-determined deactivated shape should be
overlaid on the molar brackets and the geometry checked.
Arch width should be maintained at the bracket center
(see Fig 12-46b). Equal angles of the TPA arms in respect
to the slot should be present on both sides. If the angle
is hard to evaluate with this method, only one side of the
TPA can be placed, and the distance (∆) of the other free
end should be measured to the contralateral molar
bracket (Fig 12-46c). This is repeated on the other side
(Fig 12-46d). The distances (∆) should be the same on
both sides. Note that the molars have rotated and maintained their original widths after placement (Fig 12-46e).
When a mesial-out rotation is simulated on the passive
TPA, most of the wire deformation occurs at the blue
parts of the TPA (Fig 12-47) by torsion. Torsion of the
241
12
Lingual Arches
a
FIG 12-47 Most of the elastic deformation
of a TPA during bilateral rotation occurs at the
marked blue regions of the arch by torsion.
b
FIG 12-48 (a) Both molars are severely rotated mesial in so that a buccal wire cannot be
placed initially. (b) Molars have been rotated independent of all other teeth by the TPA.
b
a
vertical arms of the TPA does not change the arch width.
By contrast, any bending in the horseshoe lingual arch
changes both width and rotation.
Both molars in Fig 12-48a are severely rotated mesial
in so that a buccal wire cannot be placed initially. Alignment using brackets on the facial surface can lead to side
effects, particularly arch expansion. Reciprocal mechanics across the arch solves the problem more efficiently;
later, brackets can be placed on the facial surfaces to finish
the maxillary alignment. Molars have been rotated independent of all other teeth (Fig 12-48b). This method has
the advantage of pure rotation around the CR of the
molars.
The mesial contact area of the molar moves distally (Δ
in Fig 12-49a) because the molar rotates around the CR—
which lies lingual to the central groove (Fig 12-49b)—even
though there is no distal movement of the CR. This distalization is helpful for Class II malocclusion correction. A
facial continuous archwire with molar rotation moments
when tied back will not allow the buccal tube to move
distally, and hence this movement is inhibited. The CR
on the maxillary first molar may lie lingually because of
the large lingual root, the lingual divergence of the lingual
root, and the buccal inclination of the molar (see Fig
12-49b).
242
FIG 12-49 (a) The mesial contact area of the molar moves distally
because the molar rotates around the CR—which lies lingual to the
central groove—even though there is no distal movement of the CR.
(b) The CR on the maxillary first molar may lie lingually because of
the large lingual root, the lingual divergence of the lingual root, and
the buccal inclination of the molar.
Associated versus dissociated type of
application
In the maxillary arch, a TPA has a clear advantage over
a horseshoe arch if pure rotation is required. Because
force and moment are dissociated, the selection of a
proper shape is simplified. This is also true for either
linear parallel expansion or constriction in the occlusal
view. Bend laterally in a parallel direction, and you get a
force only; bend at an angle, and you get a moment only.
This is not true of the horseshoe arch, where the association of force and moment can require complicated linear
displacements and curvatures for either pure translation
or pure rotation of a molar. Thus, for pure translation or
pure rotation, a dissociated mechanism is easier to use.
Many times, we may want to have both a couple to
rotate a molar mesial out and a lateral force to increase
arch width in the occlusal view. The horseshoe arch from
the mesial can give a favorable moment (mesial out) and
force direction if the wire is formed parallel to the bracket
(see Fig 12-43a). We call this a consistent configuration.
If the horseshoe had the arch apex placed posteriorly (see
Fig 12-43b), the moment to the molars would be mesial
in, the opposite direction with the same ideal shape.
Because the direction is wrong, this is an inconsistent
Symmetric Applications
FIG 12-50 If pure rotation or expansion is
required in the maxillary arch, a TPA (a) has
an advantage over a horseshoe arch (b) in
the occlusal view because the TPA exhibits
dissociation of force and moment.
a
configuration. In Fig 12-50, the required force system is
shown with red arrows. Two arches are shown: a TPA
(Fig 12-50a) and a horseshoe arch (Fig 12-50b). The TPA
is the better choice for this situation because dissociation
occurs in the occlusal view. Dissociated moment-force
appliances are the most efficient because the magnitude
of force and moments are controlled independently, and
the correct shape (correct force system) is the easiest to
determine and maintain throughout the full range of
tooth movement. The horseshoe arch, which has a consistent configuration to give both force and moment in the
correct direction with an ideal arch shape, gives at least
the correct direction of force and moment; however, it
may not give the correct M/F ratio to deliver the desired
balance of force and moment. Force and moment cannot
be controlled independently because they are associated.
Therefore, calibration becomes difficult. The greater the
distance to the apex from the bracket, the greater will be
the M/F ratio.
It may seem that the applications for a TPA with dissociation are simple and easy to apply—that is, until we
think in three dimensions. Let us consider two first molars
viewed from the posterior (Fig 12-51a). We prefer to tip
the molars to the buccal with a center of rotation at the
apex of the molar roots. This requires both a buccal force
and a couple in the direction of buccal root torque (red
arrows). If the buccal root torque is increased, translation
could occur.
Let us first look at an ideal arch configuration using a
TPA, where the green arch is fabricated so that its arms
are parallel and buccal to the molar brackets (Fig 12-51b).
Now we apply a blue activation force to constrict and
insert the TPA into the brackets; notice that not only a
force (Fig 12-51c) but also a moment with increased force
are required for engagement (Fig 12-51d). Equal and
opposite deactivation moments with forces that are
consistent are correctly produced (Fig 12-51e); although
we are not sure of the exact M/F ratio, only the direction
is required to identify this configuration as associated
b
using an ideal arch for analysis. The arm parallel to the
buccal expansion shape of the lingual arch increases
buccal force and buccal root torque. The M/F ratio
depends on many parameters such as the vertical height
of the arch; if controlled tipping with a center of rotation
at the apex should occur with this lingual arch, it would
be just sheer luck, although the force and couple directions are valid.
The association of force and moment is also shown in
Fig 12-52. Let us make a passive TPA arch and maintain
the arch width. The wire at its insertion area is twisted
(torsion) in the direction of buccal root torque (Fig
12-52a). When the wire is elastically twisted by trial activation (blue curved arrows in Fig 12-52b), the arch will
expand. The so-called twist at the bracket produces a
torque (couple) and an associated buccal force (Fig
12-52c). Thus, placing a twist (torsional angle) in an arch
again produces an association of a buccal force and buccal
root torque.
The lingual arch shape to produce a force and couple
is not trivial. We could simulate the shape by applying
the deactivation force system; however, we are applying
both a moment and a force, which is not easy to do or
accurate enough. If the curvature to produce buccal root
torque is placed in the wrong place, buccal forces or the
torques will be incorrectly altered. Theoretically, both
TPA and horseshoe-type lingual arches can deliver identical and correct force systems initially (Fig 12-53). Practically in this example, the better shape for applying a
buccal force and buccal root torque is the horseshoe arch
rather than the TPA. With the horseshoe arch, forces are
dissociated from moments; applying the couples does
not change the arch width. The TPA, on the other hand
(see Figs 12-51 and 12-52), behaves exactly the same as
a vertical U-loop retraction spring, which is considered
in detail in chapter 13. Because the height and interbracket distance (arch width) of a TPA may vary individually, it is difficult to standardize the shape, and hence
forces and moments are difficult to predict. Adding more
243
12
Lingual Arches
a
b
c
d
FIG 12-51 Force and moment association with a TPA from the rear view. (a) We want to tip
the molars to the buccal with a center of rotation at the apex of the molar roots. The required
deactivation force system is shown with red arrows. (b) The TPA was fabricated into an ideal
shape (green) with arms parallel and buccal to the molar brackets without considering the force
system. (c) We now apply an activation force (blue arrows) to constrict the TPA. Notice that the
wire arms develop an angle (third-order) with the bracket. (d) Not only a lingual force but also a
palatal root torque is required for placement. (e) Equal and opposite deactivation moments with
forces are a correctly produced association. It is a consistent configuration because the direction
of the force and moment is correct; however, we are not sure of the exact M/F ratio. If more twist
were added to increase the buccal root torque, the lingual arch width would increase; hence, the
buccal force would be increased. Exact control of the M/F ratio is difficult due to this association.
e
a
b
FIG 12-52 Effect of TPA twist on arch width. (a) The archwire before
insertion has the same width as the brackets. (b) When the wire is
elastically twisted during a trial activation (blue curved arrows), the
arch will expand. (c) Thus, placing a torsional angle in an arch produces an associated buccal force and buccal root torque.
c
a
b
FIG 12-53 Both TPA (a) and horseshoe (b) lingual arches can theoretically deliver identical
and correct force systems initially. Practically, the better shape for applying a buccal force
and buccal root torque is the horseshoe arch because its design dissociates the expansion
or constriction force from the torque. Fine-tuning can be accomplished because increasing
the torque does not change the arch width.
244
FIG 12-54 A horseshoe lingual arch has
been used to expand the maxillary posterior segments by tipping at the root apices.
Expansion forces and moments (buccal root
torque) were applied to the molars. In this
type of application, the force and moment
are dissociated; manipulation is more userfriendly because force and moment are controlled independently.
Asymmetric Applications
FIG 12-55 A common anchorage approach is to pit a larger segment
with more teeth against a segment with fewer teeth; however, this
rarely works.
root-moving moments by twisting the free end of the
arch induces unpredictable alteration of associated horizontal forces (Fig 12-53a). In the horseshoe type of application (Fig 12-53b), forces and moments are working
independently (dissociated), so it is easier to modify the
force system. If the molar tips when we need translation,
force can be decreased and the amount of moment maintained; alternatively, the moment can be increased by
twisting the free end without changing the force magnitude (arch width). Dissociated mechanisms, if available,
should be the appliance of choice because they reduce
the indeterminacy of the force system so that tooth movement becomes more predictable.
The horseshoe lingual arch (Fig 12-54) has been
selected to expand the maxillary posterior segments. The
goal is to tip the molars around a center of rotation at
their apices. Force application requires both a buccal
force and buccal root torque on both sides. The horseshoe
design should be kept simple; note that the entire arch is
in one plane and not stepped apically into the palate. If
off-plane, any moments can be associated with a width
change. An increase in the M/F ratio at the bracket can
produce translation; this type of activation should be
done carefully because the buccal plate of bone is thin.
In younger patients, it is possible that the midpalatal
suture will open during expansion if the force system
approaches translation in the frontal view and tipping is
minimized.
Asymmetric Applications
Unilateral expansion or constriction
We have learned from the laws of equilibrium that the
right and left horizontal forces acting on each molar
cannot have different force magnitudes when unilateral
expansion is required. A common asymmetric approach
is to pit a larger segment with more teeth against a
segment with fewer teeth (Fig 12-55). In practice, this
rarely works because it is difficult to control force magnitude so that the anchorage side is in a subthreshold range.
The approach described below uses the lingual arch to
produce differential moments between the right and left
sides to achieve unilateral expansion or constriction. Two
options for delivering differential moments are possible.
The first method is to apply bilateral expansive single
forces obliquely so that the line of action passes through
the CR on one side and is near the crown level on the
other side (Fig 12-56). The deactivation force diagram
(Fig 12-56a) shows the line of action of forces in respect
to both CRs. This is the most appropriate diagram to
understand the principles that underpin the mechanics.
The equivalent force system replaced at the left molar
bracket is given with yellow arrows in Fig 12-56b. The
anchor tooth on the left side translates laterally and
intrudes. The more uniform stress distribution in the
PDL prevents anchorage loss. The active molar on the
right has a high stress distribution because the single
force at the bracket produce a large moment, tipping the
molar rapidly to the buccal. The undesirable side effect
is the extrusion of the right molar; however, the vertical
component of force can be small, and occlusal forces can
minimize the molar eruption. The deactivation force
245
12
Lingual Arches
a
b
c
d
e
f
g
h
FIG 12-56 Asymmetric expansion by equal and opposite forces with a moment on the anchorage side (method 1). (a) The line of action passes
through the CR on the anchorage (left) side and is near the crown level on the mandibular lingual crossbite (right) side. (b) The equivalent
force system replaced at the left molar bracket (yellow arrows). The right side will tip, and the left side will translate. (c) The arbitrary 100-g
expansion force requires 800 gmm for translation on the left side. (d) Simulation is performed in two steps: horizontal activation for horizontal
force followed by vertical activation for moment. Therefore, the force is resolved into vertical and horizontal components (yellow arrows).
(e) The horizontal force component on the molars is used to simulate the shape using a force gauge, and the deactivated shape is produced.
(f) Trial activation. A force gauge is applied in the direction of the blue arrow for fine-tuning. (g) The vertical force component on the molars is
used to simulate the deactivated shape needed for the moment. (h) Trial activation. A force gauge is applied in the direction of the blue arrow.
A 34-g vertical force is needed to produce an 800-gmm counterclockwise moment.
diagrams are based on an equilibrium diagram of the
appliance; hence, all forces and moments must sum to
zero. For simplicity, the equilibrium diagram of the appliance has been omitted, but because the appliance is in
equilibrium, all forces at the bracket add to zero, and
hence the equivalent forces at the CR also add to zero.
The CR position of the molars can vary individually as
measured from the molar brackets due to variation of
the root length, shape, inclination, and occlusogingival
placement of the bracket; therefore, consideration must
246
be given to the replacement force system at the brackets,
where the lingual arch adjustment occurs. Fine-tuning a
lingual arch activation based on tooth inclination and
morphology (altered equivalent force system) is required
in achieving success. For that reason, it is necessary to
make a plan based on the CR position first and later figure
what is needed at the bracket. The following protocol
outlines how these principles can be transferred to daily
practice.
Asymmetric Applications
FIG 12-57 During simulation, torsion occurs at the blue regions (a and b), and bending is produced at the red region (c). (d) The
right-side free end of the lingual arch after
simulation has a second-order (tip) angulation to the bracket.
a
b
c
d
Unilateral expansion by force (method 1)
Step 3: Perform a vertical simulation
Step 1: Establish a valid force system
A vertical force simulation is performed (Fig 12-56g).
Because the passive lingual arch width is 23.5 mm, 34 g
of occlusal vertical force is needed. If the force is maintained at 90 degrees to the occlusal surface of the lingual
archwire at the free end of the right tooth, the left
restrained end will undergo torsion. The vertical force
for torque is measured using a plier to stabilize the lingual
arch on the left side and a force gauge on the right side.
The right free end is pushed occlusally with a force gauge.
The measured vertical force times the horizontal distance
to the opposing bracket is the moment (see Fig 13-56g).
A grid paper is helpful to measure and record the amount
of deflection (see Fig 12-31f ). Finally, shape is confirmed
by trial activation (Fig 12-56h). Force-driven arches can
be readily fabricated using the simulation principle. But
unlike with an ideal arch, where one learns to copy a
shape, an understanding of biomechanics is necessary.
The correct deactivation shape has been determined
by simulation in steps 2 and 3; however, more details of
where the bends and twists are placed are further
described here. The passive lingual arch undergoes both
torsion and bending in reaching its simulated shape in
three dimensions. Most of the torsion occurs between
the left lingual bracket and the apex of the lingual arch
(blue regions in Figs 12-57a and 12-57b), which is the
most distant region from the single force. Slight bending
occurs at the red region of the lingual arch (Fig 12-57c).
The deactivated shape is fabricated using torsion and
bending at these critical sections. The right-side free end
of the lingual arch after simulation has an angle with the
bracket (Fig 12-57d) that looks like it might tip the right
molar forward, but this does not happen.
A good diagram can be helpful where a valid force system
is designed (see Fig 12-56a). The red diagonal single force
acting at the CR of the left molar is replaced at the molar
bracket with yellow arrows (see Fig 12-56b). Let us arbitrarily use 100 g of expansion force. The measured interbracket distance is 23.5 mm, and the distance to the CR
is 8 mm; thus, a counterclockwise 800-gmm moment is
needed on the anchor molar (Fig 12-56c). Shape simulation is performed horizontally and vertically (independently of each other); therefore, the force is resolved
into vertical and horizontal components (Fig 12-56d).
Step 2: Perform a horizontal simulation
The horizontal force simulation is performed. The deactivated lingual arch shape is determined by simulation as
described previously. It is more complicated with both a
force and a moment delivered on one tooth, but vertical
and horizontal forces and moments are dissociated in
this type, so forming can be done in sequential stages.
The passive lingual arch is first horizontally expanded for
simulation (Fig 12-56e). A total of 94 g can be measured
with a force gauge.
An accurate horizontal deactivated shape that is the
same as the simulated shape is fabricated, and the final
deactivated shape is confirmed by a trial activation (Fig
12-56f ).
247
12
Lingual Arches
FIG 12-58 After insertion of the lingual arch
in the elastic typodont, the red dots show
little tooth movement on the left anchorage
side. The right side shows buccal movement
without any rotation.
Step 4: Perform a trial activation
After the deactivation shape is fully completed, a trial
activation is performed to check both the force system
and comfort (ie, the archwire is inserted into one or more
brackets as a check). Pushing the free end of the right
side apically to the level of the right lingual bracket with
a single force should show no first-, second-, or third-order
angle (potential moment) because the wire changes its
angle during activation; if the deactivated shape is correct,
it is ready for final insertion. If not, a little adjustment is
made until the right-side free end fits without applying
torque. Grinding off the rectangular edge of the right-side
free end could also ensure freedom from third-order
torque.
Figure 12-58 shows the effect on the elastic typodont
of the activated unilateral expansion lingual arch using
an oblique single force (a force and a couple on the anchor
bracket). The red dots show little tooth movement on the
left anchorage side. The right side shows buccal movement without any rotation. Vertical side effects are minimized by occlusal forces and the mechanical advantage
of a large transpalatal distance between the CRs of the
first molars.
Unilateral expansion by a couple
(method 2)
The second method of unilateral crossbite correction is
to directly apply a couple to the molar to be expanded.
The valid force system at the CR is presented in Fig
12-59a, and its equivalent force system at the bracket is
shown in Fig 12-59b with yellow arrows. In this second
method, the couple for tipping the problematic side is
produced by vertical (not horizontal) forces. The deactivation force diagram in Fig 12-59a has no horizontal
expansion forces and uses a couple instead on the right
side. The red arrows are the forces acting through the
molar CRs. The right molar will tip to the buccal around
248
an axis near the CR of the right root. Anchorage is
afforded by the intrusive-extrusive resistance of the
molars. As in the first method, the distance between the
CRs is large, and hence, the vertical forces are relatively
small.
A replaced equivalent force system on the lingual
brackets is shown in Fig 12-59b. It is somewhat complicated by the need to have two couples in opposite directions and with different magnitudes on the right and left
molars. Although not perfect, the force system in Fig
12-59c is simpler and gives a close approximation with
only a couple on the tooth in the problematic side.
Because the vertical forces do not act through the CR,
side effects are produced; however, moments produced
by these forces in respect to the CR are in a favorable
direction to help correct the asymmetry. They move the
molars to the right. For convenience, the force system of
Fig 12-59c, with one couple only, is usually selected. As
the right molar is tipped to the patient’s right side, the
left molar will be carried to the right. To hold its position,
a compensatory expansion may be placed in the lingual
arch.
Other than the compensatory few millimeters of expansion, the deactivation shape is simple to make because
only deformation from the couple on the active molar
(along with the vertical forces) is needed. The lingual arch
is held with a plier on the couple side, and a vertical force
is applied on the other side at 90 degrees to the occlusal
surface of the wire to simulate the shape for the desired
force system. During the simulation process, torsion and
bending occur. The anchorage-side free end will lie apical
to the bracket. A force gauge attached to the free end can
measure the force by pushing down on the archwire arm
until the required level of force is reached. After the
correct deactivated shape is fabricated, the required force
system should be obtained when the arm is pushed up
to the bracket level. In this example (Fig 12-60), a moment
of 1,000 gmm is wanted for correction; therefore, a 33-g
vertical force is needed to insert the wire into the left
Asymmetric Applications
a
b
c
FIG 12-59 Asymmetric expansion by a couple on the affected side (method 2). (a) The force system at the CR. The couple moves the right
molar to the buccal. Vertical forces act as anchorage. (b) Equivalent force system at the brackets (yellow arrows). (c) No couple at the bracket
on the anchorage side is a simpler force system that approximates the force system in b.
a
b
FIG 12-60 (a) The deactivation force system (red arrows) is used to simulate the correct shape. The vertical force is calculated to deliver
the required moment. (b) Trial activation. The activation force system is applied using a force gauge in the direction of the blue arrow for
fine-tuning of the shape.
a
b
FIG 12-61 In method 2, during simulation, torsion is produced at the blue region (a), and
bending occurs at the red region (b).
anchor bracket (33 g × 30 mm = ~1,000 gmm). Figure
12-60a shows all the forces on the teeth (deactivation
forces) after insertion. When all arrows are reversed, the
activation force diagram correctly shows the validity of
the force system (Fig 12-60b).
This simple simulation approach (applying the deactivation force system and observing the shape change)
determines the deactivated shape where both torsion
(blue region in Fig 12-61a) and some bending (red region
in Fig 12-61b) are needed. The effect of the vertical force
is negligible; therefore, little lingual tipping would occur
by the intrusive force acting on the left molar in Fig
FIG 12-62 The typodont result of applying
the unilateral couple of method 2. The right
side moved buccally, and the left side was
maintained. Note that there is no apparent
difference between method 1 (see Fig 12-58)
and method 2.
12-60a. Should lingual tipping of the left molar occur, it
could be compensated by adding proper moments to
counter the effect. Some prefer to round the archwire on
the noncouple side to assure that no moment is delivered
from the archwire on the “good” side.
Figure 12-62 shows the typodont result of applying the
unilateral couple of method 2. The right side moved
buccally, and the left side was maintained. Note that there
is no difference between method 1 (see Fig 12-58) and
method 2 (see Fig 12-62), and little difference would be
expected clinically. The choice between the first and
second method depends on clinical feasibility. Generally,
249
12
a
Lingual Arches
b
c
FIG 12-63 (a and b) A patient with a severely lingually tipped mandibular left second molar that needs unilateral expansion. The mandibular
left first molar is a pontic. Unilateral expansion by a couple (method 2) to the left second molar was the method of choice. (c) Note that the left
second molar uprighted and moved buccally (rectangle) while the anchorage of the right second molar has been preserved.
a
b
c
d
using a couple (second method) gives a more constant
force system in that the center of rotation does not change
much as the lingual arch deactivates. In other words, the
sensitivity of tooth movement is reduced in method 2.
Method 1, on the other hand, is clinically very difficult
because translation itself is very sensitive to force position. In method 2, the procedure of measuring the
moment requires only placing the archwire in one bracket
and having the other be displaced vertically, either gingivally or occlusally. If the archwire is displaced occlusally,
it is far easier to place. This is another factor to consider
when selecting between method 1 and method 2.
Figure 12-63 shows a patient with a severely lingually
tipped mandibular left second molar that needs unilateral
expansion. Note that the left second molar is so severely
tipped lingually that a pontic with an occlusal rest on the
first molar was used (Fig 12-63a). Unilateral expansion
was achieved by adding a couple to the problem tooth.
Note that the left second molar uprighted and moved
buccally (rectangle in Figs 12-63b and 12-63c). There was
no expansion force but only buccal crown torque for the
250
FIG 12-64 (a and b) A patient with a unilateral reverse articulation needed unilateral
constriction of the mandibular right first molar. A couple only on the affected side (method 2) was used for the unilateral constriction.
(c) A passive lingual arch fabricated with
comfort bends. (d) The deactivation force
system and shape produced by simulation.
(e) Unilateral constriction of the mandibular
right molar was achieved with no apparent
side effects.
e
asymmetric movement; the right second molar maintained its original position. If adjacent teeth had been
used for anchorage on the left side with a facial continuous arch, it would have been more difficult to achieve
success and could have potentially caused many problematic side effects.
The patient in Fig 12-64 with a unilateral reverse articulation needs unilateral constriction of the mandibular
right first molar. A couple was added only on the affected
side (method 2) for unilateral constriction. A passive
lingual arch was fabricated for comfort with minimal
clearance between all anatomical structures (see Fig
12-64c). The simulation or the correct deactivated shape
before placement into the right-side bracket is shown in
Fig 12-64d. The red force system in Fig 12-64d is the
deactivation force system that was used to simulate the
archwire to study its shape. Unilateral constriction of the
mandibular right molar was achieved with no apparent
side effects (see Fig 12-64e).
Figure 12-65 is a frontal view of a TPA for unilateral
constriction where the maxillary right side is to be
Asymmetric Applications
a
b
c
FIG 12-65 A TPA for unilateral constriction of the maxillary right side using method 2. (a) The deactivation force system and simulated shape.
(b) The deactivated shape is created by most bending occurring near the critical sections (red region). (c) The maxillary right side has narrowed,
and anchorage on the left side has been preserved.
FIG 12-66 Unilateral expansion by method
1. (a) The patient had a unilateral reverse articulation on the right side. (b) The maxillary
right side was expanded unilaterally. (c) The
lateral force simulation shape. (d) The vertical force and moment simulation shape.
a
c
narrowed. The red arrows are the force system during
simulation and are also the forces acting on the teeth
after full insertion (Fig 12-65a). The simulation from the
posterior view shows that the deactivated shape is created
by most bending occurring near the critical sections (red
region in Fig 12-65b) that lie close to the affected molar
bracket. Cyclic bending and sharp bends or twists should
be avoided in this high-stress region (especially near the
90-degree bend near the lingual bracket) because they
can lead to fatigue fracture. As before, the desired force
and moment were calculated, and the force was measured
with a force gauge. Trial activation was performed to
bring the free end to the level of the left molar bracket,
and the original width was maintained. Width should be
checked because there can be an association between the
force and the moment in a TPA. Note that there is no
torsion required in this simulation. In Fig 12-65c, the
response after insertion into the typodont can be seen.
b
d
The maxillary right side has narrowed, and anchorage on
the left side has been preserved.
The patient in Fig 12-66 had a unilateral reverse articulation without a mandibular shift. Correction involved
both maxillary and mandibular lingual arches (Figs 12-66a
and 12-66b). The force system of method 1 (adding a
couple to the anchorage side) was delivered by a maxillary
horseshoe arch. The maxillary force-driven shape was
accomplished in two separate steps. Step 1 was the expansion, where the typical free ends diverge (Fig 12-66c). A
unilateral bend-twist was placed on the anchorage left
side, and a small compensating second-order bend was
placed on the right side as step 2. A force gauge was used
to measure both the horizontal force and the left couple
magnitude (vertical forces). The original passive shape
and the simulated (deactivated) shape (Fig 12-66d) were
recorded on graph paper for reference.
251
12
Lingual Arches
a
a
a
b
b
b
For both method 1 and method 2 for unilateral reverse
articulation correction, there are a number of advantages
in selecting the horseshoe lingual arch over the TPA. The
force is dissociated from the moment, and the clinician
can use interactive mechanics and thus continually
modify the appliance to fine-tune the result. Horseshoe
arches are easier to make than TPAs, where contouring
for comfort is demanding. Nevertheless, the force systems
in methods 1 and 2 with the horseshoe arch or TPA
configuration are valid choices by the orthodontist.
Unilateral rotation
A lingual arch can be very useful if an asymmetric (unilateral) rotation needs correction. The mandibular right
molar shown in Fig 12-67 requires a mesial-out rotation.
The forces acting on the teeth (brackets) are shown in
Fig 12-67a. This force diagram is based on the equilibrium
252
FIG 12-67 Unilateral rotation with a mandibular lingual arch. The mandibular right
molar requires a mesial-out rotation. (a) Simulated shape. Most bending is done near the
right attachment (orange region). (b) After
insertion, the right molar has rotated mesial
out. No side effects are evident.
FIG 12-68 Unilateral rotation with a maxillary TPA. It is an efficient design because of
dissociation in the occlusal view. (a) Torsion
(blue region) and bending (red region) during
simulation. (b) The right molar is corrected.
Anchorage control is good because the distal force on the left side is small.
FIG 12-69 (a) The patient has a maxillary
right first molar rotated mesial in. (b) Unilateral rotation was accomplished with no
side effects.
diagram (not shown), which is identical except that the
force and moment directions are reversed; equilibrium
demonstrates that the solution is valid. The deactivation
force system was applied to deform the arch, giving the
simulated shape (see Fig 12-67a). Note that to create this
shape, most bending is done near the right attachment
(orange region), where the critical section is located. After
insertion (Fig 12-67b), the right molar has rotated mesial
out. No side effects are evident. According to the force
diagram, a distal force is present on the anchorage side
and a mesial force on the active right molar; however,
these mesiodistal forces are small because of the large
distance across the arch. This is another example of a
favorable mechanical advantage of a desired large moment
associated with unwanted small magnitude forces.
Similar mechanics are used in the maxillary arch, where
a TPA is an efficient design because of dissociation in the
occlusal view. The force system on the teeth (red arrows)
Asymmetric Applications
FIG 12-70 The maxillary right posterior segment requires rotation in a counterclockwise
direction. (a) Before treatment. A modified
active horseshoe-shaped maxillary lingual
arch was fabricated. The configuration has
a free end with a single-force contact at the
mesial of the left canine. The deactivated
shape is shown in green. (b) The unilateral
rotation was corrected without any noticeable side effects. (c and d) Before treatment.
This patient also needed reverse articulation
correction of the right canine with a midline
discrepancy. From the buccal tube of the
right molar, a labial cantilever was extended
to the maxillary right incisor segment, exerting a lateral force. The deactivated shape is
shown in green. The direction of the moment
at the molar (curved red arrow in d) is opposite to what is required; however, the
magnitude of the moment from the TPA
overwhelmed the moment from the labial
cantilever. (e and f ) After treatment. The
midline and reverse articulation were also
corrected.
a
b
c
e
and the deactivation shape are shown in Fig 12-68a. Most
twisting is done near the right attachment (blue region),
and supplementary bending (red region) is performed to
ensure passivity on the left side. Distributing torsion along
the vertical arms can minimize permanent deformation
of the TPA. Figure 12-68b shows the right molar corrected
with good anchorage control after the TPA was inserted.
As in Fig 12-67, the effect of the mesiodistal forces is
negligible.
The patient in Fig 12-69 has a maxillary right first molar
rotated mesial in. A straight wire on the buccal would
tend to expand the molar buccally; furthermore, if a facial
arch is tied back, the buccal tube on the molar will be
prevented from moving distally (Fig 12-69a). The efficient
rotation correction is shown in Fig 12-69b; side effects
are minimized. Along with the correction of the rotation,
no other side effects like distal movement are detected
on the left side.
The maxillary right posterior segment (Fig 12-70a)
requires rotation in a counterclockwise direction. A
modified active horseshoe-type maxillary lingual arch
was placed for unilateral rotation of the maxillary right
d
f
buccal segment. The configuration has a free end with a
single-force contact at the mesial of the left canine.
Because it is a cantilever, the force magnitude measured
by a force gauge is sufficient to make the force system
statically determinate. The deactivated shape is depicted
in green (see Fig 12-70a). Shape is not as critical with a
cantilever as long as magnitude, direction, and point of
force application are correct. The unilateral rotation was
corrected without noticeable side effects from the forces
(Fig 12-70b). In the frontal view, this patient also needs
correction of the reverse articulation of the right canine
and midline discrepancy. From the buccal tube of the
right molar, a labial cantilever was extended to the maxillary right incisor segment, exerting a lateral force (Figs
12-70c and 12-70d). The direction of the moment at the
molar is opposite to what we need; however, the magnitude of the moment from the TPA overwhelms the one
from the labial cantilever. The midline and reverse articulation were also corrected after the treatment (Figs
12-70e and 12-70f ). Here one of the largest interbracket
distances inside the oral cavity was used; therefore, the
force side effects were kept to a minimum.
253
12
a
d
a
Lingual Arches
b
e
b
Another example of a unilateral rotation with a cantilever of the maxillary second premolar is found in Fig
12-71. The premolar underwent a rotation of over 100
degrees (dotted lines from a to d); however, side effects
are hardly seen. When a single force is applied to an
anchor tooth, there are two choices; one is full bracket
engagement, and the other is a free end with a ligature
tie. Full engagement may be more secure in maintaining
the appliance, but the cantilever is the simpler option for
the delivery of the correct force system. The cantilever
does not require an equilibrium diagram for checking
the validity of the force system; the applied force diagram
(based on the force gauge reading) and its equivalent
force system through an appropriate CR (yellow arrows
in Fig 12-71e) should be sufficient. Also, the shaping is
not sensitive. As long as the force is correct, any shape
will work. Still, any fine-tuning of a cantilever involves
considering possible changes in force direction over time
and appliance comfort.
254
c
FIG 12-71 A cantilever from the maxillary
second premolar is used to correct its rotation. The premolar underwent a rotation of
over 100 degrees (dotted lines from a to d);
however, side effects are hardly seen. The
cantilever does not require an equilibrium
diagram for checking the validity of the force
system; the applied force and its equivalent
force system through an appropriate CR
should be sufficient (e).
FIG 12-72 Fallacy of moving right and left
molars distally in two steps. (a) First, the right
molar is moved distally with a single force by
shaping a TPA. (b) Then the force system is
reversed, and the left molar is moved distally.
This figure is actually the same as Fig 12-46
(bilateral rotation) except that the force is
applied sequentially in two steps.
It could be suggested that to accomplish bilateral distal
movement of the molars on a Class II patient, an asymmetric activation of the TPA could be used. The suggested
concept is first to move the right molar distally with a
single force by shaping a TPA as shown in Fig 12-72a.
The opposite anchorage molar feels a mesial force and a
moment that is mesial out on the molar. After the right
molar has moved distally, the force system is reversed,
and the left molar is moved distally (Fig 12-72b). Is it
possible to produce bilateral distal molar movement by
alternating sides with this force system? Unfortunately,
the anchorage molar feels very high shearing stress
because the mesial-out moment on the molar is so large.
The mechanical advantage is to rotate the anchorage
molar rather than move the opposite side distally. The
reason is the same as explained for the delivery of the
moment to accomplish unilateral rotation; the large interbracket distance favors the moment rather than the force.
Figure 12-72 is actually the same as the bilateral rotation
Asymmetric Applications
FIG 12-73 (a and b) A patient on whom the
asymmetric force system from Fig 12-72a
was used. Note that the left molar received
a very large moment while the right molar
received a small distal force, so little movement is observed on the right side.
a
in Fig 12-46, but the shape was applied sequentially in
two steps. Figure 12-73 shows a patient on whom the
force system from Fig 12-72a was used. Notice the effect.
The right molar did not appreciably move distally. The
left molar radically rotated because of the large mesial-out
moment. Alternating sides only will produce bilateral
molar rotation and no distalization of the CR. This could
help correct a Class II malocclusion for the reasons
explained in Fig 12-49; however, the same result can be
more simply achieved by applying equal and opposite
rotation couples to the first molars in one step.
Unilateral tip-back and tip-forward
mechanics
Typically, equal and opposite single forces at the facial
side of a continuous arch are considered when the distal
movement of a posterior segment or molar is indicated.
This is problematic unilaterally if the anterior teeth are
used as anchorage because incisors can flare and a midline
discrepancy can occur, along with other side effects. Can
a lingual arch offer other possibilities?
Let us first look from the occlusal view. It would be
easy to think of equal and opposite single forces (right
acting distally) on the lingual brackets to generate unilateral tip-back (right) and tip-forward (left); however, a
valid force diagram in equilibrium cannot be constructed
with these forces alone (Fig 12-74a), so this is impossible.
However, adding couples to the same plane of occlusion
creates a possible equilibrium force system. Figure 12-74b
is a valid force diagram that is in equilibrium after the
addition of equal-magnitude counterclockwise rotational
couples at both molars. If a lingual arch were inserted,
more buccal segment rotation (right side mesial in and
left side mesial out) than tip-forward and tip-back from
the mesiodistal forces would be observed. Both would
probably occur, so this lingual arch force system may be
b
valid for very specific malocclusions requiring such movements. Another possibility is to place the couple on one
side only (Fig 12-74c) if only that side (molar) requires
rotation.
Sometimes, rotating a molar or a posterior segment
can give a favorable distal force to distalize the affected
side; however, this is usually unlikely, as shown in Fig
12-73. A better approach is needed. An important and
unique force system that a lingual arch can provide is
equal and opposite couples that act in the sagittal plane
(Fig 12-74d). The effects from a single force at the crown
and those from couples are indistinguishable clinically.
Also, this force system is not available in a full facial
continuous arch where tip-back or tip-forward of posterior teeth is pitted against the anterior segment.
The starting shape is always passive, and then equal
and opposite couple loading is simulated. A mandibular
horseshoe lingual arch has been simulated by applying
equal and opposite couples: tip-back moments on the
right side and tip-forward moments on the left side (Fig
12-75a). The lingual arch is twisted at the apex (blue
region in Figs 12-75a and 12-75b) and bent with a gentle
curvature bilaterally (red regions). Note the smooth curvature of the arch, which is not surprising because couples
deform a straight wire into a segment of a circle. Once
the adjacent tooth is removed from the typodont, the
tooth displacement is more evident after insertion of the
activated lingual arch. Note that space has opened on the
right side mesial to the first molar as it has tipped back
and on the left side distal to the first molar as it has
tipped forward (Fig 12-75c). This action is independent
from the anterior segment because equilibrium is created
across the arch bilaterally. Basically, there is no difference
in deactivation shape of the TPA in the maxillary arch
(Fig 12-76), where equal and opposite tip-forward and
tip-back moments are generated simultaneously,
governed by the law of equilibrium.
255
12
Lingual Arches
a
b
c
d
a
b
FIG 12-74 Producing unilateral distal movement with a lingual arch. (a) An invalid deactivation force diagram. This is not possible because the archwire would not be in
equilibrium. (b) One possible valid deactivation force diagram with equal-magnitude
counterclockwise rotational couples at both
molars; however, this rotation is rarely indicated. (c) Another valid deactivation force
diagram with the moment on one side only;
this is also rarely indicated (ie, that a molar or
buccal segment needs rotation in this direction on one side only). In b and c, moments
are large, and the distal force is too small to
be effective. (d) A valid deactivation force
diagram with equal and opposite couples
acting in the sagittal plane; this is the most
practical force system and does not use a
force for distalization.
c
FIG 12-75 Unilateral tip-back and tip-forward with a mandibular lingual arch. (a) The simulated (deactivated) shape. Torsion (blue region)
and bending (red region) occur along the arch (a and b). (b and c) Unilateral tip-back and tip-forward occurs on the typodont without forces.
FIG 12-76 (a and b) Unilateral tip-back and
tip-forward with a maxillary TPA. The deactivated shape of the TPA is basically the
same as the horseshoe arch except that it
is turned at 90 degrees. The left side is the
tip-back side.
a
b
Let us see the force system developed in a unilateral
tip-back and tip-forward lingual arch if it were fabricated
in an ideal shape in three dimensions. Figure 12-77a
shows the passive shape of a TPA, assuming that the
molar lingual attachments are parallel with each other.
In Fig 12-77b, second-order bends were placed for unilateral tip-back and tip-forward based on the ideal shape
256
method. This ideal shape generates a very complicated
force system three dimensionally. When activated by
equal and opposite couples on both sides for the free ends
to be parallel to each other, the free ends displace anteroposteriorly (Figs 12-77c and 12-77d). This requires additional mesiodistal forces for engagement into the brackets.
This is not desirable because the force is acting in the
Asymmetric Applications
a
b
c
d
FIG 12-77 The reasons why the ideal arch shape for unilateral tipback and tip-forward mechanics is incorrect. (a) Passive shape. (b)
Second-order bends next to the brackets were placed for unilateral
tip-back and tip-forward based on the ideal shape concept. (c and
d) When equal and opposite couples were applied on both sides of
the free ends to make them parallel to each other, the tip-back arm
moved anteriorly (∆). Therefore, the tip-back side feels both a tip-back
moment and an anterior force. Instead of the molar crown moving
distally, the root could come forward. (e) The mesiodistal forces can
produce undesirable molar rotations from the occlusal view.
e
a
b
c
d
FIG 12-78 The force-driven shape (a to c) produces only equal and opposite couples (d) without any side effects.
opposite direction to what we want. The molar mesial
root movement can occur on the tip-back side (Fig 12-77e).
Furthermore, rotational side effects on the molars in the
occlusal view will be produced (see Fig 12-77e). This is
also the force system produced by a TPA with parallel
arms from the lateral view if right and left lingual molar
brackets are not parallel (one side molar bracket is
tipped back and the other side molar bracket is tipped
forward). In short, the ideal arch shape for this application should not be used because of very complicated
three-dimensional side effects. If the TPA in Fig 12-77 is
turned at 90 degrees, it becomes the horseshoe lingual
arch, and its improper shape gives the same side effects.
Therefore, the force-driven shape (Figs 12-78a to 12-78c)
is mandatory in unilateral tip-back and tip-forward applications to produce equal and opposite couples only (Fig
12-78d).
Another application of a lingual arch with equal and
opposite couples is to equalize right and left posterior
occlusal planes, acting on the entire buccal segment
rather than on the molar alone. What if we want to tip
back a molar on one side only? Would it be possible? The
anchorage side will tip forward from the lingual arch’s
equal and opposite couple; a method must be available
to prevent that from happening. In reality, there are two
possible approaches for handling the anchorage side
effect.
257
12
Lingual Arches
a
b
c
d
e
f
FIG 12-79 (a to c) The patient has an asymmetric molar relationship: Class I on the right side and Class II on the left side. (d) A unilateral
tip-back/tip-forward active TPA was placed. (e and f) On the labial, two symmetric tip-back cantilever springs were inserted bilaterally on the
anterior segment. Note the correction of the Class II molar relationship on the left side.
a
c
b
d
The first method is to counter the tip-forward moment
on the anchorage side by a labial appliance. The patient
in Fig 12-79 has an asymmetric molar relationship: Class
I on the right side and Class II on the left side. The treatment plan was to tip the left molar back using a tip-back/
tip-forward TPA. Figures 12-79c and 12-79d show the
maxillary arch before and after TPA treatment. It is seen
that only the left first molar has tipped back. On the
258
FIG 12-80 (a and b) An asymmetric tipback/tip-forward active TPA was placed in
a patient with a Class II malocclusion on the
right side. (c and d) Right and left tip-back
cantilevers were added on the labial side. No
buccal archwire but only a ligature wire was
placed in the premolars and canine to allow
them to drift distally individually on the right
side and to maintain the plane of occlusion
of posterior teeth.
labial, two symmetric tip-back cantilever springs were
inserted bilaterally (see Figs 12-79e and 12-79f ). The sum
of the force systems from both appliances are the following: The tip-forward lingual arch moment on the right
side and the cantilever spring sum to zero, whereas the
lingual arch tip-back moment is doubled on the left side
from the cantilever activation.
Summary
FIG 12-81 (a and b) A patient needed unilateral second molar tip-back on the maxillary
left side. (c) A tip-back moment was applied
to the left molar. (d) All other teeth were rigidly joined to comprise an anchorage unit.
Space opened with minimal side effects. A
couple, not a distal force, was used for distalization of the left second molar.
a
b
Tip-forward
Tip-back
c
An asymmetric tip-back/tip-forward arch was placed
in a similar patient with a Class II malocclusion on the
right side (Figs 12-80a and 12-80b). Right and left tip-back
cantilevers were added on the labial to enhance the rightside moment and to prevent the left posterior segment
from tipping forward. No buccal archwire but only a
ligature wire was placed in the premolars and canine to
allow them to drift distally individually on the right side
and to maintain the plane of occlusion of the posterior
teeth (Figs 12-80c and 12-80d). A segment of wire
connected at the posterior segment, including the first
molar, would undesirably cant the right-side plane of
occlusion as the first molar tipped back.
The second method to prevent the tip-forward side
from tipping forward is to connect the anchorage side to
an arch segment joining all the teeth on that side or, even
better, to connect more teeth around the arch. The patient
in Fig 12-81 requires unilateral tip-back on the maxillary
left side (Figs 12-81a and 12-81b). The required force
system is depicted in Fig 12-81c. The tip-forward moment
on the right side is prevented because all the teeth are
rigidly joined together as an anchorage unit (Fig 12-81d).
d
Summary
In this chapter, force systems from lingual arches of different configurations and applications have been described.
Because there are only two brackets involved, the lingual
arch offers an excellent opportunity to analyze simple
appliance equilibrium in three dimensions. Emphasis has
been placed not only on arch fabrication techniques but
also on developing the principles of correct force delivery.
Force-driven appliances have shapes that may look
unusual and nothing like an ideal arch. Absolute force
values in three dimensions were determined using beam
theory. The ideal arch shape is easy to visualize and simple
to make, but the result is unpredictable; the force system
is complicated, and many side effects may occur. The
starting point of all appliance design is to establish the
desired force system and to make sure it is valid; the
lingual arch must be in equilibrium. A simple equilibrium
force diagram does not waste time. Appliance selection,
specific design considerations, and fabrication follow in
later chapters.
259
12
Lingual Arches
References
Recommended Reading
1. Burstone CJ, Hanley KJ. Modern Edgewise Mechanics and the
Segmented Arch Technique. Glendora, CA: Ormco, 1986.
2. Burstone CJ. Precision lingual arches. Active applications. J Clin
Orthod 1989;23:101–109.
3. Burstone CJ, Koenig HA. Precision adjustment of the transpalatal
lingual arch: Computer arch form predetermination. Am J Orthod
1981;79:115–133.
Burstone CJ, Koenig HA. Force systems from an ideal arch. Am J
Orthod 1974;65:270–289.
Burstone CJ, Manhartsberger C. Precision lingual arches. Passive
applications. J Clin Orthod 1988;22:444–451.
DeFranco JC, Koenig HA, Burstone CJ. Three-dimensional large
displacement analysis of orthodontic appliances. J Biomech
1976;9:793–801.
Drenker EW. Forces and torques associated with second order bends.
Am J Orthod 1956;42:766–773.
Koenig HA, Burstone CJ. Analysis of generalized curved beams for
orthodontic applications. J Biomech 1974;7:429–435.
Koenig HA, Burstone CJ. Force systems from an ideal arch—Large
deflection considerations. Angle Orthod 1989;59:11–16.
260
Problems
The following problems may require the principle of equilibrium for their solution. The first step should be an
equilibrium diagram with forces and moments on the appliance. Solve for unknowns on the appliance. Then the
direction of the force system is reversed for the forces on the teeth.
1. The maxillary right second molar is erupted buccally with
mesial-in rotation. A passive lingual arch with a rigid
wire extension was used to pull the second molar palatally. Find the resultant force system at the CRs of the
second molar and the anchorage unit (two first molars).
2. The situation is the same as in problem 1, but a flexible wire was used for the extension instead of a rigid
one. How does it affect the force system?
3. The mandibular molars require bilateral expansion, and
the deactivation force on the left molar is given (200 g).
What would be the force magnitude required at the right
molar? It is still not in equilibrium, so find the moments
for the following conditions:
4. The maxillary right first molar requires unilateral
expansion. The TPA was fabricated with angulation
(buccal crown twist) of the right side arm only based
on an ideal shape-driven shape. Draw the directions
of the force system. What side effect is anticipated at
the anchorage tooth? Ignore the horizontal forces.
a. Add a moment on the right molar only for a valid deactivation force diagram.
b. Add a moment on the left molar only for a valid deactivation force diagram.
c. Equally distribute the moments at both molars for a
valid deactivation force diagram.
261
Problems
5. The lingual arch was designed to produce a single force
for bilateral constriction. Give the relative force system
at the bracket and molar CR (direction only is required,
not magnitude) acting at each molar. Assume that the
force at the bracket passes through the CR.
6. The situation is the same as in problem 5, but the
posterior segment is connected by rigid wire forming
right and left rigid units. Replace the force system at
the CR of each posterior segment. Describe the difference with problem 5.
7. A 1,000-gmm tip-back moment is delivered to the maxillary left first molar. What is the valid force system acting
on the maxillary right molar if no other forces exist?
8. The situation is the same as in problem 7 except that
the maxillary right molar is already located distally.
What is the valid force system acting on the maxillary
right molar if no other forces exist?
9. Bends were placed next to the molar brackets (tip-back
on the left and tip-forward on the right) based on an
ideal arch–shaped TPA. The ideal shape gives an
unwanted mesial force on the left molar. Calculate all
other forces and moments on the teeth. What are the
side effects if this TPA arch shape is used?
10. A patient has a unilateral buccal crossbite on the left
side with poor axial inclinations.
262
a. A lingual arch is used to correct the reverse articulation. Draw a deactivation force diagram (only direction
of force system is required, not magnitude).
b. What type of lingual arch is preferred? Why?
13
Extraction
Therapies and
Space Closure
“The Tao that can be described is not the true Tao.”
— Lao Tzu
Extraction cases require a solid understanding of biomechanics, whether or not
sliding or friction-free appliances are used. The ratio of anterior retraction to posterior protraction is primarily determined by the force system. Group A (anchorage
cases) mechanics may also use an increased number of teeth, rigid segments, and
undisturbed anchorage. Little evidence suggests that separate canine retraction
is more conservative of anchorage than en masse space closure. In both sliding mechanics and friction-free loops, the moment-to-force (M/F) ratios at the
bracket are continually changing. Distinct phases are tipping, translation, and
root movement. Cantilevers (gingival extensions) for space closure are statically
determinate and deliver more constant M/F ratios at the bracket, minimizing different stages of retraction. The scientific basis for spring design for a space closure
loop is discussed in this chapter. Variables include loop height, apical added wire
placement, and interbracket distance. The roles of the activation moment and the
residual moment are explained. Temporary anchorage devices also require sound
biomechanics because side effects can occur.
263
13
Extraction Therapies and Space Closure
a
b
c
FIG 13-1 Three categories of differential space closure. (a) Group A mechanics. Most of the extraction space is closed by retraction of anterior
teeth. (b) Group B mechanics. The extraction space is closed by equal attraction of both anterior and posterior teeth. (c) Group C mechanics.
Most of the extraction space is closed by protraction of posterior teeth.
M
any appliances and techniques have been introduced for closing space in extraction patients.
Emphasis has been placed on wires, brackets,
named techniques, and specially designed appliances. If
the clinician is to select the best methods and optimally
use any mechanism, less consideration should be given
to the hardware and more to sound biomechanical principles applied to space closure.
The movement of a tooth is solely a response to the
periodontal ligament (PDL) stress produced by the appliance. The PDL does not have a preference for the material, size, or shape of the wire or for the type of bracket
or kind of configuration used. This chapter discusses the
biomechanical principles governing both retraction
springs and sliding mechanics that are used for extraction
space closure. The force system developed during space
closure can be very sophisticated due to factors such as
friction, three-dimensional (3D) effects, cross-section
shape and dimension of the wire, bracket width, loop
shape, and modulus of elasticity, among others. However,
the discussion here is kept as simple as possible. The
objective of this chapter is not to advocate for any specific
appliance or technique but rather to introduce important
principles so that the clinician can creatively design,
select, and modify his or her appliance to fully control
the force system during space closure.
Differential Space Closure
Extraction therapy is frequently necessary in orthodontic treatment of patients with severe crowding or protruding anterior teeth (with or without crowding). Once
extraction is decided upon, the anteroposterior position
of the incisors must be established, and only then can the
optimal force system be determined. For example, if the
treatment goal is to retract a canine and maintain the
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anteroposterior position of the molar, differential space
closure is required.
Differential space closure is divided into three categories, depending on how proportionally anterior and
posterior segments contribute to the space closure. In
group A mechanics, most of the extraction space is closed
by retraction of anterior teeth (Fig 13-1a). In group B
mechanics, the extraction space is closed by somewhat
equal attraction of both anterior and posterior teeth (Fig
13-1b). In group C mechanics, most of the extraction
space is closed by protraction of posterior teeth (Fig
13-1c). Obviously, differential space closure with group
A and group C mechanics is more challenging than with
group B mechanics.
Strategies for Maintaining Posterior
Molar Position: Group A Mechanics
The stress developed in the PDL by the orthodontic appliance is the initiator of tooth movement. Even the insertion
of the separation elastic ring for banding the molar could
create a large stress on the molar, causing mesial movement by initiating an orthodontic tooth movement
cascade. Therefore, the strategies for maintaining posterior anchorage are concentrated on keeping the stress in
the PDL of the anchorage unit as low as possible.
The simplest method of reinforcing the anchorage is to
increase the number of teeth in the anchorage unit.
However, this method is very limited unless stress is
evenly distributed among the roots. In addition to the
number of teeth included in the anchorage unit, the
posterior teeth can be connected rigidly so that individual movements are not allowed. Passive lingual arches
provide cross-arch stabilization, and the insertion of fullsize wires may prevent play between the wire and the
brackets so that the stress is more evenly distributed to
Strategies for Maintaining Posterior Molar Position: Group A Mechanics
FIG 13-2 The posterior anchorage unit. (a)
Rigid wires are inserted into the posterior
segments. A transpalatal arch is used for
cross-arch stabilization. (b) Rigidly connected posterior teeth with minimum play form
the posterior anchorage unit.
a
b
a
b
FIG 13-3 Bonded FRC segment. FRC provides better passivity and stability of the
segment than a wire does.
FIG 13-4 A patient with incisor protrusion.
(a) The maxillary anterior segment and the
mandibular buccal segments are connected
with an FRC. (b) Space is closed. Note that
the mandibular anterior teeth have been
retracted with minimal anchorage loss and
that the good intercuspation was not disturbed.
the whole unit (Fig 13-2). Rigidly joined posterior teeth
are called a posterior anchorage unit.
Starting space closure with undisturbed posterior
anchorage may be an important factor in group A
mechanics. Technically, insertion of “straight” full-size
wires into the posterior brackets may require a leveling
process because most sliding techniques require rigid
archwires for space closure. The leveling process may be
avoided by using completely passive wires, but this is very
difficult to achieve because both wire bending and torsion
may be required to ensure passivity. Even if the wire may
look passive, very small wire deflections could create
instantaneous heavy forces, leading to high stress on the
anchorage unit. These stresses added to the stress from
the space closure mechanism can lead to anchorage loss.
A bonded fiber-reinforced composite (FRC) segment
is a good alternative to a passive heavy wire for an anchorage unit because passivity is ensured (Fig 13-3). Figure
13-4 shows the mandibular buccal segment connected
by an FRC so that the stress is evenly distributed and
initial extraneous heavy leveling forces between posterior
teeth are avoided. Note that the mandibular anterior teeth
have been retracted with minimal anchorage loss. Also,
the good intercuspation is not disturbed, which is another
contributing factor to reinforcing the anchorage. Temporary anchorage devices (TADs) are used as anchorage for
space closure in the maxilla (see Fig 13-4a). Supplemental forces from headgears, maxillomandibular elastics
(previously referred to as intermaxillary elastics), and
TADs are additional possibilities for consideration in
group A mechanics.
Perhaps the most important factor in differential space
closure is the application of a force system that provides
differential stress on each unit so that the anchorage unit
(posterior teeth) translates and the active unit (anterior
teeth or canine) tips. Figure 13-5 shows a physical model
of tooth displacement occurring within the PDL. The red
line is drawn on the background, and the green line is
drawn on the transparent teeth that are suspended by a
series of elastics around the root, simulating the principal
fibers of the PDL. In Fig 13-5a, an elastic is placed at an
angle from the left tooth (β, posterior or anchorage tooth)
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Extraction Therapies and Space Closure
a
b
FIG 13-5 A physical model of teeth showing differential space closure. (a) An elastic is placed at
an angle from the left tooth to the right tooth. (b) The T-loop acts identically to the angled elastic.
a
b
c
d
FIG 13-6 A physical model of teeth showing the effect of location of the force application point and
magnitude of force. (a) A light force is applied near the alveolar crest. (b) A heavier force is applied
at the CR. Note that the crown movement at the bracket is about the same. (c and d) The same
amounts of force are applied at different locations. Note that less crown tooth movement is observed
from the force near/at the CR.
to the right tooth (α, anterior or canine tooth). Compare
the red and green lines. The red force on the anchorage
tooth acting through its center of resistance (CR) translates the tooth. By contrast, the right anterior tooth with
the red force applied at the alveolar crest tips around a
point at its apex. Translation produces a more uniform
stress distribution in the PDL and, hence, preserves
anchorage. One should also notice with the angled
266
elastic an intrusive force on the anterior unit. Is it possible to run an elastic in this manner clinically? The answer
is no, of course, because of anatomical limitations. But
we can place an equivalent force system at the brackets
with the T-loop shown in Fig 13-5b that acts identically
to the angled elastic (see discussion of equivalence in
chapter 3).
En Masse Versus Separate Canine Retraction
a
b
FIG 13-7 Two basic types of space-closing mechanics. (a) Sliding (or friction) mechanics. (b) Loop (or frictionless)
mechanics. In sliding mechanics, the tooth feels less force than that applied as a result of friction. In loop mechanics,
there is no loss of applied force due to friction.
Let us compare the effect of force placement level on
the amount of crown tooth movement using the same
model. A light force is applied near the alveolar crest (Fig
13-6a), and a heavier force is applied at the CR (Fig 13-6b).
Note that the elastic chain is stretched more in Fig 13-6b.
Crown movement at the bracket is about the same. Why?
Because forces away from the CR produce higher stresses
and strain. The sum of all strain, clinically, we call tooth
movement. If we use the same model again and keep the
forces identical, more crown tooth movement will be
observed from the force occlusal to the CR (Figs 13-6c
and 13-6d). This is to be expected because higher PDL
strains are present. Thus, differential tooth movement
can be produced by changing the point of force application.
This model represents the initial stage of tooth movement (mechanical displacement) and relates only to the
stress-strain pattern. The biologic displacement occurs
later. But is it possible to achieve differential tooth movement with the same force? Yes, because tooth morphology and tooth number can influence the stress. As
demonstrated by the above model, even if the resultant
forces on the anterior and posterior segments are equal
and opposite, differential tooth movement can be
produced if the force is angled relative to the occlusal
plane. Other equilibrium situations are possible where
the resultant force system has equal and opposite forces
and unequal moments. (The forces do not have to be on
the same line of action. The elastic used in this model is
a special case.)
Sometimes the phrase differential force is used to
describe the differences between anterior and posterior
tooth movement. Without explanation, this concept can
be confusing. If only one appliance is used, the force
magnitudes to the anterior and posterior teeth must be
the same (equilibrium principle). However, because we
normally apply our forces at the brackets, a differential
M/F ratio system can exist. Forces may be equal and
opposite, but moments can be different at the anterior
and posterior brackets; hence, differential M/F ratios can
be expected.
En Masse Versus Separate Canine
Retraction
In cases of severe crowding, separate canine retraction
is necessary for gaining space for incisor alignment. Classically, anterior teeth were retracted in two stages. It was
believed that separate canine retraction followed by fourincisor retraction would preserve the posterior anchorage
because lighter forces could be used at each stage. Perhaps
this could work if sufficiently low magnitudes of force
were used; however, most clinicians use about the same
forces for separate canine retraction as for en masse space
closure. Clinical studies show that there is no difference
in anchorage loss between en masse and two-stage retraction.1 Therefore, there is no reason to retract the anterior
segment in two stages unless it is indicated for special
situations such as anterior crowding, flared incisors,
extruded or high canines, and midline discrepancies.
Two-stage retraction is more complicated and less
esthetic, due to transient spacing between the canine and
lateral incisor; it also has a longer treatment time and is
more likely to lead to iatrogenic side effects like incisor
extrusion, especially during sliding mechanics.
Intra-arch space-closing mechanics
There are two basic types of space-closing mechanics:
sliding (or friction) mechanics (Fig 13-7a) and loop (or
frictionless) mechanics (Fig 13-7b). In sliding mechanics,
the force is applied by an elastic or spring, and the bracket
slides along the guiding archwire. There is always friction
between the bracket and the archwire, so the tooth feels
less force than the force applied by the elastic or coil
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13
a
Extraction Therapies and Space Closure
b
spring. The guiding wire provides moments required for
prevention of tipping and rotation (see Fig 13-7a). In loop
mechanics, there is no guiding wire, and the spring
provides both force and moment so that there is no loss
of applied force due to friction; because friction is usually
unknown, results may be more predictable with loop
mechanics (see Fig 13-7b) than with sliding mechanics.
On the other hand, sliding mechanics may provide better
control over all tooth movement because the archwire
serves as a definitive guide. Placing differential moments
is easier with frictionless loop design and allows for better
delivery of group A and group C specialized mechanics.
Continuous Versus Segmented
Arches
When we apply sliding space closure mechanics, the
archwire can be continuous from the posterior tooth
(second or first molar) on one side to the posterior tooth
on the opposite side; the shape of the wire cross section
and the material used are usually unchanged (Fig 13-8a).
A loop can be incorporated in a continuous archwire for
frictionless mechanics (Fig 13-8b). However, with a
continuous archwire many possible interactions can
occur between each tooth (bracket); therefore, activations
can become complicated and indeterminate. The small
interbracket distances limit the amount of activation, and
furthermore, the accuracy of both linear and angular
activations is limited.
Figure 13-2 shows a possible segmented archwire
where the maxillary arch is divided into an anterior
segment and two posterior segments. If the right and left
posterior segments are connected by a passive lingual
arch (transpalatal arch [TPA]) and rigid stabilizing archwires are inserted, one could consider the maxillary arch
as composed of only two teeth, a multirooted anterior
tooth and a multirooted posterior tooth. These two
segments can be connected by two springs of varying
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FIG 13-8 Continuous archwire. (a)
The archwire is relatively straight for
sliding mechanics. (b) A loop can be
incorporated for loop mechanics.
cross section and material. By reducing an arch into two
segments, the orthodontist needs only place a single activation per side between the auxiliary tubes on the canine
and the molar. The use of only these two attachments per
side also increases the intertube distance (distance
between auxiliary tubes), allowing for larger activations
and greater accuracy of activation.
Friction (Sliding) Mechanics
Tooth movement from the facial view typically follows
four phases during sliding mechanics (Fig 13-9). In three
dimensions, it is more complicated because from the
occlusal view, rotational moments must be considered.
This is described in more detail in chapter 16 (on friction).
In this chapter, only the force system in the lateral view
is considered. During phase I, after the tooth is leveled,
a distal force is applied. Uncontrolled tipping can occur
because of the play between the bracket and the wire. In
phase I there is no friction. With wide brackets, little
phase I tipping occurs because play is small. In comparison, narrow brackets used in the Begg technique allow
for considerable tipping without friction. In phase II,
more tipping is observed; friction is now increasingly
present as the tipped bracket deflects the wire more and
more. As a reaction from the deflected wire, two vertical
forces (couple) act on the bracket slot, fighting the tipping
tendency. Forces from these moments produce normal
forces, which cause the friction. When the tooth tips
sufficiently to deflect the wire enough to create higher
moments, the tooth translates (phase III). Distal tooth
movement or space closure can stop as the root-uprighting moments increase or the distal force lessens. Root
correction now occurs (phase IV).
One problem with sliding mechanics is that friction is
unpredictable, and hence, the delivered force system is
unpredictable. The usual treatment goal is to translate
adjacent teeth at the extraction site during space closure.
Friction (Sliding) Mechanics
FIG 13-9 The facial view of four phases in sliding mechanics (right
to left). A force is applied distally from right to left (red arrows).
Phase I: Uncontrolled tipping with play. The tooth freely tips without
friction until the bracket slot and wire touch each other. Phase II:
Controlled tipping. More tipping is observed, and the wire starts to
deflect. The deflected wire exerts two equal and opposite normal
forces (couple), and friction increases. Phase III: Translation. If the
moment from the deflected wire is high enough, translation occurs.
Phase IV: Root movement. The tooth movement can stop as the rootuprighting moments increase or the distal force lessens.
a
b
FIG 13-10 The side effects of sliding mechanics with an inadequately
stiff guide wire. (a) Before retraction of the canine. (b) After retraction
of the canine. (c) Iatrogenic curve of Spee. Note that the curve of Spee
was developed in the mandibular arch. As the canine tipped distally,
the wire deflected occlusally in the incisor region and gingivally in
the premolar region (phase II tipping). The anterior vertical overlap
was increased (dotted lines show the axis of the canine before and
after retraction).
c
Unfortunately, the highest level of friction will occur
during translatory tooth movement. In fact, frictional
forces acting in the opposite direction to the applied force
may be large enough to stop the tooth movement
completely. This phenomenon is referred to as appliance
ankylosis.
In phase IV, the tooth uprights without space closure;
the orthodontist then reactivates the spring or chain
elastic, and the tipping phases can start again. With each
repeating phase, the tooth will wiggle back and forth until
space is closed. In other words, the center of rotation
keeps changing during space closure, which biologically
is not the most direct way to stimulate the PDL. Along
with a varying M/F ratio at the bracket, the absolute
magnitude of the force is fluctuating due to the variable
amount of friction in accordance with each phase. Therefore, even if the magnitude of applied force is constant
and not excessive, the stress in the PDL might be too high
in the uncontrolled tipping phase or too low in the translation phase.
Space closure such as canine retraction with sliding
mechanics may appear to be pure translation from the
start to the end; however, a complicated series of phases
normally occurs. Much friction can be eliminated by
avoiding translatory movement and allowing primarily
tipping, but this has the disadvantages of requiring later
root movement (axial inclination correction) and potentially leading to undesirable side effects.
If the guiding continuous wire is not stiff enough, a
canine might tip, resulting in increased vertical overlap
(also known as overbite). Figure 13-10 shows this effect.
The canine was retracted by an elastic force along a
0.016-inch stainless steel wire. As the tooth tipped
distally, the wire deflected occlusally in the incisor region.
Note that a curve of Spee was developed in the mandibular arch with an increase in vertical overlap as the
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13
a
Extraction Therapies and Space Closure
b
c
FIG 13-11 Cantilever intrusion spring in a continuous archwire. (a) Before retraction. (b) During retraction. (c) After retraction. The incisor eruption side effect is prevented by an additional cantilever intrusion force during canine retraction. Also, it provides additional tip-back moment
at the molar for better anchorage. Note that the amount of anterior vertical overlap is not changed.
FIG 13-12 A second overlay continuous archwire with an intrusion
force anterior to the canine prevents incisor eruption during canine
retraction.
mandibular incisors erupted and premolars intruded. To
compensate for these side effects, a so-called compensating reverse curve may be incorporated into the archwire; however, it will make the force system more unpredictable and many times incorrect. Some might place a
V-bend (or better, a curvature) between the canine and
the second premolar. This might prevent the canine from
tipping, but it increases the sliding friction. The high
friction can impede retraction. Larger–cross section
wires can also minimize this side effect, but again the
frictional forces are increased.
Another approach to prevent incisor eruption during
sliding canine retraction mechanics is the use of a separate cantilever that bypasses the canine with an intrusive
force anterior to the canine (Fig 13-11), or a second
continuous intrusion arch can be inserted into an auxiliary tube on the molar and tied to the anterior segment
(Fig 13-12).
If the anterior segment is retracted en masse, the sliding occurs within the posterior segment; therefore, posterior teeth should be leveled prior to sliding (see Fig 13-7a).
As mentioned previously, this leveling of posterior
segments may initiate anchorage loss during tooth movement and may not be desirable in group A mechanics.
Because the applied force system is unknown to the
operator with friction, and because the interbracket
270
distance between the canine and second premolar bracket
is small, differential space closure via the application of
differential M/F ratios is very limited. Hence, most orthodontists can only accomplish group B mechanics when
using sliding mechanics. Differential space closure (group
A and group C mechanics) may require additional appliances including headgears, maxillomandibular elastics,
and TADs, among others.
Frictionless (Loop) Mechanics
For simplicity, let us consider the force system using sliding mechanics only from the facial view (Fig 13-13a).
What is the role of the guide wire? The archwire provides
the required amount of moment to control the tooth
movement. If the canine is translating (phase III movement), the correct moment for the needed M/F ratio at
the bracket comes from the wire. If wire extensions (Fig
13-13b) are attached to the auxiliary tubes on the molar
and canine so that the line of action of the force passes
through each CR, the canine and molar also translate.
Here no guide wire is necessary.
In frictionless mechanics, there is no guide wire, and
a specially designed spring may be used. The spring
provides the required M/F ratios in three dimensions.
Frictionless (Loop) Mechanics
a
b
FIG 13-13 Separate canine retraction. (a) Facial view of the force system in phase III (translation) of sliding mechanics. The wire provides
required moments. (b) Sliding mechanics with an extension hook. The line of action of the force passes through each CR. There is no moment
from the guide wire; therefore, a guide wire may not be necessary.
FIG 13-14 Separate canine retraction by frictionless mechanics. The
spring delivers required forces and moments for controlled movement
of the canine without any guide wire (facial view).
FIG 13-15 Separate canine retraction using frictionless mechanics.
(a) Force system in the facial view
of the spring. (b) Because the force
is applied buccal to the CR, an occlusal antirotation moment is also
required in the occlusal view. Note
that all 3D forces are incorporated
in the spring.
a
There is no loss of applied force due to friction, which
means greater predictability and versatility. Typically,
with a properly shaped loop or frictionless spring, space
closure will go through three of the same phases seen
with sliding mechanics: tipping followed by translation
and then root movement. The differences in a properly
designed appliance are a greater activation range and a
more constant force level and M/F ratio, leading to a
more constant center of rotation (less wiggling). Differential moments between anterior and posterior segments
become easier to achieve and more predictable.
b
During separate canine retraction with a T-loop spring,
the facial view force system (Fig 13-14) delivers forces
and moments for controlled movement of the canine
without any guiding wire. Because the force is applied
buccal to the CR, an occlusal antirotation moment is also
required (Fig 13-15). These 3D effects are discussed in
detail later in this chapter. The appliances used in frictionless mechanics can be either statically determinate
or statically indeterminate (Fig 13-16).
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Extraction Therapies and Space Closure
FIG 13-16 Two types of frictionless
mechanics. (a) Statically indeterminate system. The point of force application is at the bracket. (b) Statically determinate system. The point
of force application is on the root of
the tooth. The red force on the left is
acting at the CR of the tooth. Note
that the indeterminate and determinate systems deliver equivalent force
systems.
a
b
Statically Determinate Space
Closure Appliances
A statically determinate appliance or spring means that
the law of statics (equilibrium) is sufficient to solve for
all unknown forces and moments acting on the wire and
the teeth. Unless we know all the forces and moments,
we cannot predict the result. A common statically determinate appliance is the cantilever intrusion spring, where
the force is measured at the incisors with a force gauge.
All of the appliances previously discussed for space
closure are indeterminate, even if the forces are measured
or known. If sliding mechanics are used, the many
unknowns include friction and the moments acting along
the arch. A frictionless spring (loop) requires the determination of both forces and moments at their anterior
and posterior ends. Thus, even if forces are known, the
system is indeterminate unless the moments are also
determined. We can make these springs determinate by
calibrating springs in which the material and dimensions
are kept constant either experimentally or theoretically.
Another approach is to design a space closure device
that is so simple that it delivers only a force and no
moments at either end. Such an appliance is statically
determinate if the force is measured at one end. A simple
rubber band, coil spring, or elastic fits that description.
Of course, a spring placed parallel to the occlusal plane
at the level of the brackets would tip teeth during space
closure. If the point of force application is moved from
the bracket to a point apically in line with the CR, translation will occur. This system has both control and determinacy using equal and opposite forces without requiring moments. From the occlusal view, a rigid anterior
segment and posterior segments connected with a TPA
272
allow rotational control without the use of an archwire.
Figure 13-16b shows differential space closure using an
oblique single force that is applied at the root, below the
clinical crown of the tooth on both sides. Applying the
force near the CR of the tooth is limited due to anatomical considerations, such as a shallow vestibule or a buccal
frenum. The point of force application may freely slide
along the line of action of the force by the law of transmissibility. If we move the elastic forces closer to the right
tooth along the line of action of the forces, the forces have
the same relationship to the CR (same effect), but the
more occlusal position of the force will impinge less on
the mucobuccal fold.
Figure 13-17 shows a cantilever spring made of rectangular 0.017 × 0.025–inch β-titanium wire with a helix
used for retracting the anterior segment. Note that the
point of force application of the activated spring is located
anteriorly as much as possible so that the spring is not
extended too far apically, yet the line of force (black dotted
line) passes near or apical to the CR. Predicted types of
tooth movement by this spring are controlled tipping of
the anterior segment and translation or slight mesial root
movement of the posterior segment based on the imaginary visualized line of action of the force. The dotted
curved arrow is the estimated deactivation path of the
hook of the spring. The deactivation path of the spring
produces a relatively constant line of action during space
closure. Figure 13-18 shows an extraction case treated
with the cantilever spring (statically determinate retraction system) described above. Controlled tipping of the
anterior segment followed by root movement of the anterior segment occurred as predicted. There was no translation phase, so greater magnitude of force was not
required. Note that two elastics were used to redirect the
line of force in Fig 13-18c. This is discussed in detail later
Statically Determinate Space Closure Appliances
FIG 13-17 Statically determinate retraction system made
of 0.017 × 0.025–inch rectangular β-titanium wire. The
orange wire shows the activated shape of the spring, and
the green wire shows the fully deactivated shape of the
spring. The dotted line is the initial line of action. The dotted
curved arrow is the imaginary deactivation path of the hook
of the spring, which shows that the line of action remains
relatively unchanged within the initial effective range of
tooth movement.
a
b
c
e
f
d
FIG 13-18 A uniarch extraction case treated with the statically determinate retraction
system. (a) Before treatment. (b) Controlled
tipping of the anterior segment. (c) Root
movement of the anterior segment. (d) After
treatment. Note that there is no translation
phase. (e and f) Cephalometric radiographs
before and after treatment. (g) Superimposition of cephalometric tracings.
g
in the chapter. The cephalometric radiographs and superimposition show that the treatment goals were achieved
(see Figs 13-18e to 13-18g).
There are anatomical limitations to placing a force far
enough apically on the facial side to produce translation.
Better possibilities are available on the lingual. If the
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Extraction Therapies and Space Closure
palatal vault is sufficiently high, the line of action can be
placed near or apical to the CR.
The patient shown in Fig 12-19 (see chapter 12) had
localized enamel hypoplasia on the first molars only.
Therefore, the left and right first molars were extracted,
and two passive TPAs were placed anteriorly and posteriorly with hooks near the estimated occlusogingival
location of the CR. Anterior and posterior segments were
translated into the extraction space, and mesial movement of the second molars allowed the space for the third
molars to erupt. The patient shown in Fig 12-20 shows
differential space closure using two TPAs.
It is theoretically possible but practically difficult for
translation to deliver a line of action of force at the CR.
The location of the CR is compounded by many factors,
such as root length and shape, alveolar bone level, and
the position of the tooth itself. Furthermore, even if the
exact location of the CR is identified, placing the force
slightly occlusal or apical to the CR can produce significant second-order rotation (tipping or root movement).
Therefore, it is recommended that the lingual hook be
placed sufficiently apical to the estimated CR and that
additional force be applied at the brackets of the crown
so that the resultant force can be easily modified. If the
tooth tips too much, rather than relocating the hook
further apically by fabricating a new TPA, the magnitude
of force of the elastic at the crown is reduced, or the apical
elastic force is increased. If the tooth undergoes root
movement, the magnitude of force of the elastic at the
crown is increased or the apical elastic force is reduced.
This is another example of equivalence—using many
forces rather than a single force for better clinical ease
and control.
The statically determinate spring, which delivers force
at the desired point of force application so that no
moments are required, has another advantage over frictionless springs occlusally positioned at brackets and over
sliding mechanics. In occlusally positioned space closure
mechanisms with well-designed springs, the change in
M/F ratios is not abrupt but very gradual; hence, the
center of rotation is relatively constant. With an elastic
or spring without moments, the M/F ratio at the CR is a
constant that is independent of the force magnitude (see
Fig 13-6). Therefore, the “wiggling” side effect is reduced
to a minimum.
Statically determinate space closure utilizes a single
force without moments. Statically indeterminate mechanisms ideally achieve the same result with an equivalent
force system using moments and forces at the level of the
brackets. The result can be the same, but it is simpler for
the orthodontist to visualize a single force and an
274
imaginary line of action. Force determination or calibration is usually simple, requiring only a force gauge. Clinically or experimentally statically indeterminate mechanisms involve sophisticated theory or equipment.
Statically Indeterminate Spring
Design
If both ends of a spring are inserted in the bracket or tube
by activations involving forces and moments, it becomes
statically indeterminate. Because the number of
unknowns increases, the laws governing static equilibrium are not sufficient to determine all unknown forces
and moments. A simple linear force gauge is not adequate
to measure the force from the appliance because all
moments should be measured at the same time. One
solution is to measure the force system in the laboratory
with sophisticated sensors; test results may be presented
by graphs or data tables (Fig 13-19).
When buying a car, we do not need to test every aspect
of its function ourselves. Its technical performance has
already been tested in the factory, and the results are
presented in the user’s manual. The orthodontic loop or
spring with test results is called a calibrated spring and
is similar to an automobile with specifications. The calibrated test results provide the amount of force and
moment at any given activation, the moment differential
between the anterior and posterior ends, and the maximum amount of activation allowed within the elastic
limit to avoid permanent deformation.
Many loops and springs have been suggested for space
closure characteristics, making it hard to choose between
them without a scientific basis for evaluation (Fig 13-20).
With the exception of a few examples, most loops have
not been fully or even partially tested in the laboratory
or in the clinic; furthermore, the shape of the loop usually
lacks a sound biomechanical basis. The important variables that should be considered in designing a space
closure spring are discussed here. The advantage of a
statically indeterminate spring is that the point of force
application lies at the occlusal plane level of the bracket,
yet the replaced equivalent single force is placed far apical
to the bracket on the root.
No matter what type of spring is used, some properties
are required to deliver the optimal force system and
ensure operator convenience. The force system that can
move the teeth as quickly as possible to the target position
without any adverse side effects to the teeth or periodontium is considered the optimal orthodontic force system.
The exact value of the optimal level of stress in the
Statically Indeterminate Spring Design
FIG 13-19 Spring tester. Sophisticated sensors are used to measure the forces and moments at the same time. The test results may
be presented by graphs or data tables.
FIG 13-20 Various shapes of loop springs
(upper rows, deactivated shape; lower rows,
activated shape). Most of them are not precalibrated in the laboratory and thereby produce
unknown force systems and unpredictable
results.
periodontium is not known; however, continuous light
force is thought to be the most efficient for producing
reasonable rates of tooth movement, minimizing anchorage loss, and reducing possible discomfort and tissue
damage. More importantly, an optimal force system
should control the center of rotation to move the tooth
to its correct position. Therefore, the force magnitude,
force/deflection (F/∆) rate, and M/F ratio of the spring
are very important characteristics or specifications for
space closure springs.
Physical properties such as cross section, shape, and
length of the wire, along with material properties such
as modulus of elasticity and yield strength, determine the
force system of the spring. In this chapter, emphasis is
placed on the shape properties of the spring and how
they relate to the F/∆ rate and M/F ratio. Shape properties, unlike material properties, are fully under the clinician’s control in designing a biomechanically efficient
loop or spring. Knowing where and how to bend and
place additional wire is very important, but first we must
investigate what happens inside a wire during activation
of an orthodontic appliance.
275
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Extraction Therapies and Space Closure
a
b
FIG 13-21 The role of a loop in a wire. (a) A straight wire has a tremendously high F/∆ rate in tension. (b) Even a simple vertical loop dramatically reduces the F/∆ rate (∆L1 << ∆L2 ).
a
c
b
d
e
f
Stress Distribution in a Spring Wire
Figure 13-21a shows a segment of straight wire. If forces
are applied at each end, the wire will elongate elastically;
however, the F/∆ rate is tremendously high so that the
amount of elongation (∆L1 ) is not detectable with a naked
276
FIG 13-22 Bending of a wire. (a) Before force application. (b) After
force application. Note that the length of the upper surface (A) decreases and that of the lower surface (C) increases during bending.
The length of the neutral axis (B) remains the same. (c and d) The
stress-distribution pattern at an imaginary cut (section) of the wire.
The stress is linearly increasing from the neutral axis to the outer
surfaces with upper compression and lower tension. (e and f) The
imaginary wire element produced by the arbitrary cut is in equilibrium
by the applied force (FA ) and shear vertical force (FS ), and equal and
opposite compressive and tensile stresses along the long axis of
the wire are replaced with a bending moment depicted as a curved
arrow (MB ).
eye. Every clinician knows that a wire is very stiff in axial
tension and would make a useless spring. Adding just a
few bends to the wire by making a loop can dramatically
reduce the F/∆ rate so that it elongates much more (∆L2 )
under the same amount of applied force (Fig 13-21b).
The F/∆ rate is reduced because the wire bends and
Stress Distribution in a Spring Wire
FIG 13-23 Bending moment analysis of
the vertical loop spring. (a) Before activation. (b) Activation by single force only.
(c) Activation by force and moment while
keeping the legs parallel. The magnitude of
the bending moment is depicted in various
color schemes. (Based on Halazonetis.2)
0.017 × 0.025–inch β-titanium
a
b
c
undergoes a nonuniform stress distribution instead of a
uniform stress distribution with a pure axial load.
When a wire bends, the length of the upper longitudinal surface of the wire (A in Figs 13-22a and 13-22b) is
decreased by compression, while the length of the lower
surface (C) is increased by tension. Somewhere near the
center of the wire, where the length is unchanged and
the stress is zero, is the neutral axis (dotted line at B in
Figs 13-22a and 13-22b). Let us define the direction of
this bending (upper compression, lower tension) as positive and the other direction (upper tension, lower
compression) as negative.
The stress-distribution pattern at an arbitrary section
(at the middle of the wire) is shown in Fig 13-22c. Horizontal stress is linearly increasing from the neutral axis
and reaches the maximum value (σmax ) at the outer
surfaces of the wire (Fig 13-22d). These equal and opposite stresses are called bending moments and act at any
arbitrary cut (section) of the wire (Figs 13-22e and
13-22f ), as described briefly in chapter 12 (see Fig 12-37).
For example, a 0.017 × 0.025–inch rectangular β-titanium
wire (with a typical yield strength) reaches its maximum
yield stress at its outermost surface if loaded with a secondorder bending moment (approximately 3,000 gmm); greater
bending moments at any section along the wire will result
in permanent deformation.
When a vertical loop is activated by a pulling action
from single forces at each end, the horizontal legs are not
parallel any longer and become angled so that they cut
across the bracket with an angle (Figs 13-23a and 13-23b).
This is because the apex of the spring is under the highest stress and bends most. To engage an activated spring
into parallel brackets, not only a force but also additional
moments are necessary to keep the legs parallel (Fig
13-23c). The stress distribution in the vertical loop was
determined by numerical analysis, and the bending
moments at each cross section along the length of the
spring are given in the diagram. The highest bending
stress occurs at the apex of the loop (red), and the
second-highest stress occurs near the bends at both legs
(yellow). The region with the highest stress is called the
critical section.
Three bends (apex and both legs) are bent in an
unwinding direction (from the direction of bending
during fabrication) as the loop is activated. Also note the
direction of the bending. The bends at the apex are in
one direction, whereas the bends in the horizontal legs
are in the opposite direction. There is a stress-free region
(blue) in between where the curvature of the spring
changes from convex to concave (see Fig 13-23c).
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Extraction Therapies and Space Closure
FIG 13-24 The Bauschinger effect. (a) Loop 1.
All bends (A, B, C) are bent in the unwinding
direction during activation. (b) Loop 2. All
bends (D, E, F) are activated in the winding
direction. Winding-direction bending is more
resistant to permanent deformation during
activation.
a
a
b
b
The Bauschinger effect
Two loops are shown in Fig 13-24. When loop 1 (Fig
13-24a) is activated, all bends (A, B, C) are bent in the
opposite direction (unwinding) from the bends made
during its fabrication. Whereas in loop 2 (Fig 13-24b), all
bends including the helices (D, E, F) are activated in the
same direction (winding) as during its fabrication. Loop
2 is more resistant to permanent deformation and
provides more elastic range of action not only because
more wire is used in the loop design but also because all
bends during activation are bent in the same direction
as during the initial forming. This directional phenomenon is called the Bauschinger effect and is explained by
the creation of favorable residual stresses in the wire
during fabrication. Vertical loops like that in Fig 13-24a
frequently show permanent deformation (opening of the
loop) after an archwire is removed from the mouth.
Therefore, wire bends should be in the same direction
during both forming and activation. Some designs require
the bend to be overbent, followed by a reversal in the
direction of the bending, in order to reach the final shape.
By first overbending and then reversing the direction, the
direction of the last bend is correct to give a favorable
residual stress pattern during activation. This overbending provides resistance to permanent deformation and
278
FIG 13-25 Sharp bends create stress points.
(a) A loop with a squashed apex. (b) The
close-up view shows extreme deformation
of the wire. The sharp bend at the apex can
easily lead to permanent deformation or fracture during activation.
increases the range of activation. Some orthodontists
may heat-treat a stainless steel archwire. This process is
better described as a stress relief. If the residual stress is
favorable after loop forming as described in this section,
stress relief by heating is not recommended because
possible permanent deformation will be increased.
Stress raisers
A sudden change of wire cross section, nicks or other
defects on the surface of the wire, or sharp bends can
lead to unpredicted high stress concentrations during
activation, followed by permanent deformation or fracture. Therefore, sharp bends should be avoided in loop
design. It is not a good idea to squash the apex of the
vertical loop (Fig 13-25). The sharp bend at the apex (a
high-stress section of the loop) can lead to permanent
deformation or fracture.
Effect of Shape and Dimension on
Spring Properties
What is the effect of the loop on the force system if the
vertical height (V), number of helices (N), location of
helices (K), horizontal width (T), or interbracket distances
Effect of Shape and Dimension on Spring Properties
a
b
c
d
FIG 13-26 (a) Configuration of the standard reference loop. The wire is 0.017 × 0.025–inch β-titanium. (b) Vertical loop height (V) was varied
from 2 to 10 mm. (c) F/∆ rate plotted against vertical height. (d) M/F ratio plotted against vertical height. In c and d, the value of the standard
loop is given in red.
(L) are changed? Let us use a 0.017 × 0.025–inch rectangular β-titanium wire with a maximum bending
moment of 3,000 gmm. First we will consider the standard
form as a reference. The height of the loop (V), apical
width of the loop (T), and the interbracket distance (L)
were arbitrarily set to 6, 2, and 7 mm, respectively, and
this standard loop is shown in Fig 13-26a.
Vertical height of the loop
The apical width (T) of the loop and the interbracket
distance (L) were not changed, and the vertical height (V)
was varied from 2 to 10 mm (Fig 13-26b).
Figure 13-26c plots the F/∆ rate as a function of the
vertical height (V) of the loop. The F/∆ rate dramatically
decreases as the vertical height (V) increases. A
2-mm-high vertical loop is more of an “omega stop” than
a space closure loop because its stiffness is very high
(9,370 g/mm). The values of the standard loop (V = 6
mm) are given in red. Figure 13-26d shows the M/F ratio
as a function of the vertical height. The M/F ratio linearly
increases with vertical height. If the line is extrapolated,
approximately 24 mm of loop height is required to deliver
a 10-mm M/F ratio, which might be required for some
tooth translation. However, 24 mm of loop height is not
practical considering anatomical factors. Although
increasing the vertical height of the loop is desirable
because it both decreases the F/∆ rate and increases the
M/F ratios, practically it has its limitations.
An edgewise vertical loop with a height of 6 mm is a
typical clinical application. It delivers 437 g/mm and has
an M/F ratio of 2.2 mm. The high F/∆ rate makes it too
sensitive for accurate activations; an error in activation
of ±1 mm can cause an error in force of ±437 g. The M/F
ratio of 2.2 mm is far too low and initially produces
uncontrolled tipping. Moreover, if this loop were made
with stainless steel, the modulus of elasticity would be
approximately double that of the β-titanium loop, and
hence, the force value would be twice as high.
279
13
a
Extraction Therapies and Space Closure
b
FIG 13-27 The effect of number of helices at the apex. (a) A helix was
added at the apex of the standard loop, and the number of helices
(N) was varied. (b) F/∆ rate plotted against the number of helices. (c)
M/F ratio plotted against the number of helices.
c
a
b
c
d
FIG 13-28 The effect of location of the helix. (a) The location of the helices (K) was varied vertically. (b) F/∆ rate plotted against the location
of the helix. (c) It is useless to add the wire near the inflection section (near K = 2 mm). Both loops show the same amount of deflection (∆).
(d) M/F ratio plotted against the location of the helix.
280
Effect of Shape and Dimension on Spring Properties
FIG 13-29 The effect of width of the loop apex in the T-loop spring.
(a) Wire was added at the apex, producing a T shape. (b) F/∆ rate
plotted against the width of the loop apex. (c) M/F ratio plotted against
the width of the loop apex.
b
Number of helices
Because increasing the loop height is anatomically
limited, wire can be added at the critical section (apex of
the loop) in the form of a helix. A helix was added at the
apex of the standard loop, and the number (N) of helices
was varied (Fig 13-27a). The F/∆ rate decreases as the
number of helices increases; however, more than three
helical turns leads to minimal effect (Fig 13-27b). The
M/F ratio also increases as the number of helices
increases; however, again it is not useful to have more
than three helical turns (Fig 13-27c). No matter how many
helices are incorporated into the apex of the loop, the
M/F ratio can hardly reach the effect produced by increasing the height of the loop. Multiple helical turns are not
only inefficient biomechanically but also uncomfortable
and unhygienic because of buccolingual bulk.
Location of helices
The placement of the helices (K) was varied vertically to
establish an optimal position (Fig 13-28a). The F/Δ rate
is the smallest with the helix at the apex (K = 5 mm) and
increases as the helix moves occlusally up to K = 2 mm
(Fig 13-28b). From here, if the helix is placed further
occlusal to the brackets, the F/∆ rate will decrease. The
highest F/∆ rate is at a location near the occlusal one-third
of the loop (K = 2 mm). It is useless to add the wire near
a
c
any inflection point (blue region of the loop in Fig 13-23),
where the bending direction changes during activation.
The loop designs in Fig 13-28c show the same F/∆ rates
even if wire is added to the vertical loop. Thus, it is not
always true that adding more wire in a loop will lower
the force. The M/F ratio increases linearly as the helix is
moved apically. The M/F ratio is smaller (1.7 mm) than
that of the standard vertical loop (2.2 mm) without a helix
if the helix is placed far occlusally, although more wire is
incorporated in the loop (Fig 13-28d). Even if the helix
is placed as far apical as possible, unfortunately, the M/F
ratios are only modestly increased.
Horizontal width (T-loop spring)
It can be seen from the previous data that the most efficient place for adding additional wire is at the apex. Therefore, wire was added at the apex, producing a T shape.
The T-loop is more comfortable and hygienic than the
loop with the helix because it has less bulk. The horizontal width at the apex (T) was varied (Fig 13-29a). The F/∆
rate decreases dramatically as T increases (Fig 13-29b).
This design is more efficient for lowering the F/∆ rate
than adding a helix at the apex (assuming the same
amount of wire is used). The M/F ratio also increases as
the horizontal dimension (T) increases. The T-loop is
thus more effective than the use of a helix for preventing
tipping (Fig 13-29c).
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Extraction Therapies and Space Closure
a
b
FIG 13-30 The effect of interbracket distance. (a) As the interbracket
distance increases, the length of the horizontal leg (L) also increases.
(b) F/∆ rate plotted against interbracket distance. (c) M/F ratio plotted
against interbracket distance. Note that in b and c, the slope of each
curve is relatively flat considering the amount of wire added.
c
FIG 13-31 Composite T-loop spring. A loop made of 0.018-inch round
β-titanium wire is welded to a straight 0.017 × 0.025–inch rectangular
β-titanium wire. The stiffer wire in the leg prevents loss of the M/F
ratio by an increased length of the horizontal leg.
Interbracket distance
As the interbracket distance increases, the length of the
horizontal leg (L) also increases (Fig 13-30a). The F/∆
rate and M/F ratio decrease only slightly with greater
interbracket distance, and more wire is added below the
inflection point as L increases (Fig 13-30b). The slope of
both curves is relatively flat, considering the amount of
horizontal wire added. Decreasing the M/F ratio is a
disadvantage because translation of the teeth becomes
more difficult (Fig 13-30c), but this is less significant clinically with the T-loop. The main advantage of an increased
interbracket distance (intertube for segments) is that it
increases the horizontal space for activation and makes
more accurate appliance placement possible. It also allows
for fewer reactivations during space closure.
The reduction in M/F ratio with larger interbracket
distances can be prevented by making the horizontal leg
282
stiffer. An example is shown in Fig 13-31. The horizontal
leg of a composite T-loop is made with stiffer wire (larger
cross section), preventing the reduction of the M/F ratio
with an increased interbracket distance.
T-Loop Moments
Activation moments
Figure 13-32a shows the actual shape of a clinically applicable passive T-loop spring made from 0.017 × 0.025–
inch β-titanium wire. The vertical height of 8 mm and
the horizontal apical width of 10 mm are a compromise
between efficacy and comfort. As the spring activates,
the force and the moment at both ends increase linearly
(Figs 13-32b and 13-32c). There is no moment differential
between the anterior (α) and posterior (β) ends because
T-Loop Moments
a
b
c
d
FIG 13-32 The T-loop. (a) The shape and dimension of a clinically applicable T-loop made from 0.017 × 0.025–inch β-titanium wire. (b) Force
plotted against activation. (c) Anterior (α) and posterior (β) moments plotted against activation. The slope represents the activation moment
of the spring. (d) M/F ratio plotted against activation. Note that both force and moment linearly increase with activation, and therefore the M/F
ratio is kept relatively constant during activation. The residual moment bend angle is 0 degrees anteriorly and posteriorly.
the loop is symmetric in shape. The M/F ratios (which
are approximately 3 to 4 mm) remain relatively constant
throughout the whole activation range (Fig 13-32d). These
moments are produced when the horizontal loop legs
are activated and pulled apart, keeping the legs parallel
for bracket insertion. These moments, which appear only
when the spring is longitudinally activated, depend on
the design factors previously discussed. Therefore, this
type of moment is called an activation moment. It is represented by the slope of the moment plotted against activation. Unfortunately, the activation moment is not sufficient for translation of the teeth. The M/F ratio of 3 to 4
mm may tip the adjacent teeth to an extraction site, where
ratios of 10:1 or so may be required for translation. More
moment is necessary, but where do you get this additional
moment?
Adding a residual moment
The shape of the T-loop is now slightly changed so that
the horizontal legs are no longer parallel but angled at 40
degrees at each end (Fig 13-33a). The moments produced
after insertion from these bends are independent of the
activation and are called residual moments.
Let us now look at the force and moment graphs when
this T-loop with angled bends is activated (Figs 13-33b
and 13-33c.) The force graph has a similar pattern and
slope to that of the original shape without angled legs
(see Fig 13-32b). The moment curve in Fig 13-33c also
shows a similar slope to that of the original shape (see
Fig 13-32c), so the activation moment is the same;
however, because of the angled bends, there is now residual moment of 900 gmm. The residual moment is represented by the y-intercept of the moment plotted on the
activation curve. The total moment of the angled spring
is the sum of the residual moment and the activation
moment. Figure 13-33d shows the M/F ratio at the
bracket of the spring with added residual moment bends.
The curve is not linear anymore. A typical T-loop with
full activation of 6 mm generates an M/F ratio of 6 mm,
which is suitable for controlled tipping. As the tooth
moves (spring deactivates), the M/F ratio increases. At
2.3 mm of activation, the M/F ratio reaches approximately
10 mm for translation. Less than 2 mm of activation
produces M/F ratios large enough for root movement
(axis of rotation at the bracket). Theoretically, 4 to 5 mm
of space closure can be accomplished with a single activation. The center of rotation of the anterior teeth may
283
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Extraction Therapies and Space Closure
a
b
c
d
FIG 13-33 Residual moment bend in a T-loop. (a) Bends were added to the T-loop shown in Fig 13-32. (b) Force plotted against activation.
(c) Moment plotted against activation. Note that the slope of the curve is unchanged, but the curve starts at 900 gmm at 0 mm of activation
(y-intercept). This moment is independent from activation. The y-intercept represents the residual moment of the spring. (d) The M/F ratio
curve is not linear; its shape is a hyperbola.
not be constant. Space closure will occur in phases:
controlled tipping followed by translation and finally root
movement. However, as seen in Fig 13-33b, absolute force
magnitude decreases as the spring deactivates, and the
tooth movement becomes very slow during translation
and root movement because force levels are suboptimal.
Clinically, specially designed root springs are used for
efficient root movement after rapid space closure by
controlled tipping.
The neutral position and angled bends
in a spring
Figure 13-34a shows the amount of deflection (∆) required
during activation with a passive T-loop spring. Once the
spring is activated and engaged, the amount of activation
is easily measured by the distance between the vertical
arms measured at their junction with the horizontal legs
(Fig 13-34b). When treating a patient, the orthodontist
need not disengage the spring to measure the amount of
activation; however, measuring between the vertical legs
284
after activation can be erroneous if angled bends are
incorrectly employed (Fig 13-34c). Let us consider why.
Empirically, orthodontists have learned to prevent tipping
during space closure by adding a V-bend or gable bend
to their loops. Figure 13-35a shows a passive T-loop before
bends were placed. If the gable bends producing spring
angulation are placed only at the occlusal portion of the
spring (Fig 13-35b), the vertical arms will cross each other
as a result of the moment needed to keep the horizontal
legs parallel during insertion into the brackets (Fig 13-35c).
This will cause the loop to shorten horizontally (Δ). Thus,
more activation will be present than that anticipated by
measuring between the vertical arms (see Fig 13-34c).
Let us introduce another important concept of a
specific shape. The shape of the spring with a 0-g horizontal force when placed into the brackets is called the
neutral position. In the neutral position, the spring has
no horizontal force. However, there may be other vertical
forces due to moment differentials at the neutral position,
depending on the shape. The passive spring without angulation in Fig 13-35a is shown in its neutral position. If we
T-Loop Moments
a
b
FIG 13-34 Measuring the amount of activation of the spring. (a) The
passive shape of the spring is ready for activation (∆1 ). (b) The amount
of activation measured between the vertical legs of the spring (∆2) is
the correct amount of activation (∆1 ). It is not necessary to disengage
the spring to measure the amount of activation in a passive T-loop
because ∆1 = ∆2. (c) In a spring with residual moment bends, the
measurement of the amount of activation must consider the neutral
position because ∆1 ≠ ∆2.
c
a
b
c
FIG 13-35 Neutral position of the spring. (a) Passive shape. (b)
V-bends or gable bends are placed for residual moments. (c) Trial
activation of the spring by moments only. The spring is crossed
at the neutral position, and the length of the spring shortens (∆).
Therefore, more activation will be present than anticipated by measuring between the vertical arms.
have calibrated data from an F/∆ curve of this spring and
we measure the distance between the vertical arms, we
can correctly estimate the force by measuring the horizontal activation, because the vertical arms touch in the
neutral position.
The occlusally angled spring in Fig 13-35b has a different “neutral position” when activated with equal and
opposite moments to make the horizontal legs parallel
(see Fig 13-35c). The starting position (neutral position)
for zero horizontal force is with the vertical arms crossed.
Therefore, one cannot assume that 0 g of force is present
FIG 13-36 The angled bends are distributed between
the occlusal and apical portions of a spring. In this case,
after trial activation by moments alone, the vertical legs
touch each other (similar to a passive spring), and the
amount of activation measured between the vertical
legs is correct.
if the vertical arms are just touching. Because the arms
cross in the neutral position, more force would be present, leading to either permanent deformation or abusively
high forces if the clinician was not aware that angulation
can change the neutral position. Activation of a spring
or loop outside of the mouth with moments and/or forces
is called a trial activation.
Any neutral position is acceptable provided that we
know where it is. Sometimes the angled bends are distributed between the occlusal and apical portions of a spring,
as shown in Fig 13-36. In this case, after trial activation
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Extraction Therapies and Space Closure
FIG 13-37 Residual moment bends with a
curvature. (a) Before insertion. (b) After insertion and activation. There are many advantages of a gentle curvature over sharp bends.
a
b
a
b
c
d
e
f
FIG 13-38 A patient treated with group B mechanics. (a to c) Mild crowding is
present. The patient does not require lip retraction. (d to f) Group B mechanics with
a symmetric T-loop were used. The extraction space was closed by attraction of the
posterior and anterior segments. (g) The facial profile was maintained after treatment.
g
by moments alone, the neutral positions of the spring
and passive T-loop are similar. It may be more comfortable for the patient, or the forces may be more predictable
because we do not have to disengage the spring to check
the neutral position and measure the amount of activation. The distance measured between the vertical arms
is the correct amount of activation if the neutral positions
are identical in angled and passive shapes. In other situations, having the vertical arms cross in the neutral position can increase the available distance for reactivation
when interbracket distance is limited.
286
Before the horizontal force is applied for activation on
the patient, the horizontal legs need to be parallel to each
other to be engaged into the brackets. The clinician should
simulate this action outside of the mouth (trial activation).
This is done by applying moments only to each end of
the spring. In the neutral position, the spring has no
horizontal force but a residual moment. During a trial
activation, the horizontal legs of a spring with a gable
bend may or may not cross each other, depending on
where the angulation is placed. The orthodontist can
accept the neutral position or make a correction.
Differential Space Closure with a T-Loop Spring
Correction could be done by changing the position or
amount of the angulation. Other trial activations in which
moments and forces are applied to permanently deform
a spring or wire to take advantage of the Bauschinger
effect are discussed later in the chapter.
Sharp bends versus curvature in the
spring
Figure 13-37a shows a spring with angulation bends
before insertion into the canine bracket. Note that there
is a gentle curvature rather than sharp, well-defined
bends. There are several advantages to placing a distributed curvature rather than sharper localized bends. Along
the horizontal leg region is the second-highest stress
section of the spring; therefore, it can be easily deformed
accidentally during activation or by masticatory forces.
A smooth curvature will more effectively resist higher
stresses than a sharp bend. As noted earlier, with an
increased interbracket distance, a greater angle for residual moment bends is necessary if a sharp bend is used.
However, with a curvature, the same radius of curvature
will give the same moment independent of interbracket
distance; hence, activation templates are simplified. A
sharp bend near the attachments may interfere with the
reactivation of the spring. In addition, a smooth curvature
after activation in the mouth will have more comfortable
contours (Fig 13-37b).
The patient shown in Fig 13-38 had mild crowding with
no facial problems. Because she did not require retraction
of the lips, the posterior teeth could come forward during
anterior retraction; group B mechanics with a symmetric
T-loop were used (see Fig 13-38c). After the symmetric
T-loop was inserted, the center of rotation varied during
space closure from controlled tipping to translation and
finally to root movement of both the anterior and posterior segments. It is important to keep the spring shape
symmetric during the space closure to optimize reciprocal space closure. Because the posterior teeth were
allowed to protract, the good facial profile was maintained
(see Fig 13-38g).
Differential Space Closure with a
T-Loop Spring
In Group A mechanics, we construct a force system to
produce differential space closure by maintaining posterior anchorage and only allowing anterior retraction.
With a single spring, the equilibrium principle tells us
that the forces must be equal and opposite; hence, differ-
ential force is impossible. Perhaps greater or smaller
forces could make a difference. Some say that light forces
maintain posterior anchorage better than heavy forces.
This is unproven in the range of forces most clinicians
use. Therefore, differential space closure is best accomplished by applying differential M/F ratios at each end of
the spring (if indicated by the treatment plan). Other
additional factors such as headgears, TADs, or increasing
the number or size of the teeth in the anchorage unit are
not considered here.
Controlling the residual moment
There are two moments under the control of the operator using a loop or spring for space closure: the residual
moment and the activation moment. The angles of the
residual moment bends can be made differently on each
side of the loop (α is the anterior leg and β is the posterior leg) so that the shape is asymmetric. In Fig 13-39a,
the anterior (α) angle was formed at 40 degrees, and the
posterior (β) angle was increased to 60 degrees. The
moments at each side of the spring for every millimeter
of activation are plotted in Fig 13-39b. The slopes of the
activation moments (α and β) are the same; however, the
starting points determined by the residual moments are
different. Therefore, the M/F ratio at each end of the
spring is different (Fig 13-39c). Suppose that this spring
was activated at 3 mm; the M/F ratio at the anterior end
(α) is approximately 7 mm (typical for controlled tipping),
and the M/F ratio at the posterior end (β) is 10 mm (typical for translation). The difference is smallest at full activation and increases as the spring deactivates. The more
differential residual moment angle we place, the greater
will be the M/F ratio differential (Fig 13-39d).
Figure 13-40 shows the final shape of a composite
T-loop spring used for group A mechanics that is designed
based on a number of principles. The position of the
flexible loop is fixed more anteriorly (α position) to a
more rigid rectangular wire. The moment from the bend
at the posterior leg (Mβ ) is greater because a stiffer edgewise base wire is used. The clinician can bend the wire
according to this template shape and will get a known
force system in tabular form based on experimental data.
Note that the angle of the β bends must increase as the
interbracket distance increases. The M/F ratio during
deactivation is more constant at the anterior end than at
the posterior ends when the T-loop is off-center to the
anterior. Therefore, the center of rotation of the active
unit (anterior segment) is kept relatively constant regardless of the amount of activation.
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a
b
c
d
FIG 13-39 Differential residual moment bends (asymmetric shape). (a) The posterior residual moment bend is increased to 60 degrees. (b)
Anterior (α) and posterior (β) moments plotted against activation. Note that the slope (activation moment) is the same, but the y-intercepts
(residual moments) are different. (c) M/F ratio at each end of the spring. Note that the difference is smallest at full activation and increases as
the spring deactivates. (d) Eighty-degree β angle. The moment differential at any given activation increased.
FIG 13-40 Template and force system of a composite T-loop spring for group A mechanics. The position of the more flexible wire loop is fixed
anteriorly (α position), and the greater posterior leg (β position) moment is provided from the stiffer wire. Note that shapes (residual moment
bend angles) vary with interbracket distance.
288
Differential Space Closure with a T-Loop Spring
a
b
d
e
f
g
i
j
c
h
k
l
FIG 13-41 A patient treated with an asymmetric T-loop spring. (a to d) Before treatment. There is not much crowding, but the patient shows
lip protrusion. (e) Treatment objectives. Group A mechanics were planned. (f) The asymmetric spring was placed. (g) Controlled tipping of the
anterior segment and translation or slight mesial root movement of the posterior segment are shown. (h) Separate canine root spring with
canine bypass arch. (i) After treatment. (j) Facial photograph after treatment. (k and l) Cephalometric radiographs before and after treatment.
Maximum retraction of the lip using group A mechanics was achieved as planned.
Figure 13-41 shows a patient with lip protrusion and
minimal crowding treated by an asymmetric T-loop
spring. The treatment objective was to maintain the
posterior anchorage and to maximally retract the anterior
segment using group A mechanics (see Fig 13-41e). The
space was closed by controlled tipping of the anterior
segment and translation or slight mesial root movement
of the posterior segment (see Fig 13-41g). Note that the
loop of the spring was placed anteriorly with asymmetric
angular bends. After the extraction space was closed,
separate canine root springs (cantilevers) were used to
correct the axial inclination (see Fig 13-41h). The treatment goals were accomplished as planned (see Figs 13-41j
to 13-41l).
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FIG 13-42 Controlling the activation moment (eccentric placement
of the spring). (a to c) The activated passive T-loop by single force
only acts as a single V-bend (dotted line) on a straight wire. The side
closest to an eccentrically placed T-loop receives a larger moment.
a
b
c
a
b
FIG 13-43 (a) A residual moment is added to the entire T-loop evenly with a curvature of radius 25 mm. (b) Off-center T-loop. Differential
activation moments can be anticipated when activated.
Controlling the activation moment
The position of a single V-bend on a straight wire between
two brackets significantly alters the force system at each
side of the wire. The side closest to an eccentrically placed
V-bend receives a larger moment. When a T-loop with
parallel legs (no residual moment angulation) is activated
by a single force only, the parallel legs now become angled
with each other as the apex opens. Additional force and
moments are necessary to engage it into the brackets (Fig
13-42a). This angle of the horizontal legs works exactly
the same as a V-bend on a straight wire, which is discussed
in chapter 14. Note that in Figs 13-42b and 13-42c, the
activation of a T-loop (or other shaped loop) produces
an off-center V similar to a wire with a single V-bend
(dotted line), and hence, differential moments are created
290
at each end. During activation, the off-center T-loop
produces activation moments that are much less than
required. Therefore, a residual moment can be added to
the entire T-loop evenly by incorporating a smooth curvature with a 25-mm radius (Fig 13-43a). If the spring has
a location that is properly off-center and a standard residual moment from a uniform curvature, desirable differential moments can be anticipated (Fig 13-43b). Figure
13-44 shows how the eccentricity of the loop is measured
by a B/L ratio that defines the force system in the passive
shape before the curvature is placed, where B stands for
the spring length of the anterior horizontal leg and L
stands for the total spring length. The loop with curvature
in Fig 13-45a has a B/L ratio of 0.61 and is off-center with
the loop placed posteriorly.
Differential Space Closure with a T-Loop Spring
FIG 13-44 Loop placement eccentricity. The B/L ratio defines the
eccentricity of the T-loop.
a
b
c
d
FIG 13-45 Off-center loop. (a) B/L ratio = 0.61, off-center to the posterior. (b) Moment plotted against activation curve. The residual moment
(y-intercept) is the same, but the activation moment (slope) is different. (c) M/F ratio at each end of the spring. Note that the M/F ratio differential at any given activation is relatively constant (∆). (d) In the spring with more eccentricity (B/L = 0.72), the M/F ratio differential increased.
The α and β moments (moments at each end of the
spring) are plotted in Fig 13-45b. In the neutral position
(zero horizontal force), the moments are the same: 1,200
gmm (only residual moments are present). However, the
slope of the curves (caused by the activation moments)
are different. The posterior moment is the larger, and the
differential between Mα and Mβ increases with activation.
Of course, we are most interested in the differential
between the M/F ratios on each end of the spring.
Suppose this off-center spring is activated at 4 mm; the
posterior M/F ratio is 10 mm, and the anterior M/F ratio
is approximately 7 mm, a desirable difference for a group
A mechanics case (Fig 13-45c). In other words, translation
of the posterior teeth is pitted against controlled canine
or anterior tipping. This difference of M/F ratios between
the anterior and posterior ends is relatively constant
throughout the entire range of activation (∆ in Fig 13-45c).
Placing the spring further off-center (B/L = 0.72) will
deliver a greater differential between the M/F ratios at
the α and β positions (Fig 13-45d).
Figure 13-46 shows a template with a T-loop spring
superimposed, employing the concept of a constant
curvature for the residual moment. In group A mechanics, the spring is placed off-center toward the posterior;
in group B mechanics, the spring is centered; and in group
C mechanics, the spring is placed off-center anteriorly
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Extraction Therapies and Space Closure
FIG 13-46 A template with a universal
T-loop superimposed. One template shape is
used for groups A, B, and C mechanics and
is used regardless of interbracket distance.
a
b
c
FIG 13-47 A patient treated with the group A
mechanics universal T-loop spring. (a to e)
The loop is placed off-center posteriorly.
d
e
FIG 13-48 Typical placement of a vertical
loop in a continuous archwire. Theoretically,
the anteriorly placed loop can induce posterior protraction, which is not desirable in
group A mechanics.
(see Fig 13-43b). One of the advantages of using this
spring design is that a universally shaped T-loop spring
is used for groups A, B, and C mechanics; also, one
template shape (curvature) can accurately estimate the
residual moment with different interbracket distances.
Figure 13-47 shows a case treated with the universal
T-loop spring. Note that the loop is placed off-center
posteriorly (for group A mechanics). The space was closed
by controlled tipping of the anterior segment while the
posterior segment minimally translated forward. A root
spring was used for intersegmental leveling at the final
stage of treatment.
Figure 13-48 shows a typically placed vertical loop in
a continuous arch. The vertical loop is placed anteriorly
292
as far as possible near the canine bracket so that the wire
posterior to the loop has sufficient room for reactivation
by a cinch-back on the terminal molar. Theoretically, this
design is not desirable if group A mechanics are used
because the activation moment is greater in the anterior
segment than the posterior segment, so the anterior
segment may act as anchorage for posterior protraction.
In addition, an extrusive force on the anterior teeth can
increase the vertical overlap, which is not indicated if
excessive vertical overlap is present.
In the patient shown in Fig 13-49, group A mechanics
were needed in both the maxillary and mandibular
arches. Similar results were achieved in each arch with
different appliances because both delivered a similar force
Differential Space Closure with a T-Loop Spring
b
a
c
d
e
g
f
h
j
k
i
l
FIG 13-49 A patient showing upper and lower lip protrusion. (a to d) Before treatment. There is not much crowding. (e) Group A mechanics
were planned on both arches. (f to i) A statically determinate spring was placed in the maxillary arch, while a statically indeterminate spring
(T-loop) was placed in the mandibular arch. (j) Facial photograph after treatment. The lip protrusion was relieved. (k and l) Cephalometric
radiographs before and after treatment. Note the controlled tipping of both the maxillary and mandibular anterior segments.
system. Initially, incisor protrusion and little crowding
was present (Figs 13-49a to 13-49d). The treatment objective was to maintain the posterior anchorage and retract
the anterior segment as much as possible (Fig 13-49e).
Leveling was performed on the anterior segment only,
and heavy wire was inserted passively into the posterior
segment to preserve anchorage and to keep the maximal
intercuspation of the posterior occlusion. A cantilever
statically determinate spring was used in the maxillary
arch, and a T-loop spring with an asymmetric residual
angulation bend was placed in the mandibular arch. The
extraction space was closed mostly by controlled tipping
of the anterior segment. A root spring was used for a
small amount of root movement (Figs 13-49f to 13-49i).
Although the springs used in each arch have differences
in appearance, they delivered similar group A mechanics
force systems. The results confirm that most of the
extraction space was closed by controlled tipping of the
anterior segment with little anchorage loss (Figs 13-49j
to 13-49l).
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a
c
b
d
e
FIG 13-50 A patient treated with a statically determinate spring for group C mechanics. (a) The maxillary right side needs further space
closure. (b) Maxillary and mandibular midline discrepancy. Continued space closure may result in further midline deviation. (c) A statically
determinate cantilever spring was placed. The force direction is oblique, where an imaginary line of action passes above the CR of the anterior
segment. (d) The posterior segment tipped into the extraction space with minimal movement of the anterior segment. The force was released,
and spontaneous eruption was expected. (e) After treatment. The maxillary midline did not deviate to the right more than its original position.
Group C mechanics: Posterior
protraction
Group C mechanics during space closure are most challenging because the anterior segment, with its small roots,
is not the best anchorage for protracting posterior teeth.
Posterior protraction can be indicated in Class II mandibular arches where growth is expected to mainly correct
the Class II malocclusion. Some Class III patients can
utilize maxillary posterior segment protraction. On the
other hand, if an extraction space the full width of a
premolar needs to be closed by protraction of posterior
segments, there should be no reason to extract teeth. Yet
some patients may have missing teeth, so protraction
may be desired.
The strategies for handling group C mechanics are
similar to those for group A mechanics in applying differential space closure, but group C mechanics use a reversed
force system. In Fig 13-50, a patient needed further space
closure by protraction on the maxillary right side after
the extraction space on the left side closed. However, the
maxillary midline could deviate more to the right with
continued space closure (Fig 13-50b). A statically determinate cantilever spring (Fig 13-50c) was placed so that
the force direction was oblique, where an imaginary line
of action passed above the CR of the anterior segment.
As predicted, the posterior segment tipped into the
294
extraction space while translation or slight root movement occurred at the anterior segment. Instead of root
movement being performed on the controlled tipped
posterior segment, the force was released and spontaneous eruption occurred, because root movement
mechanics of the posterior segment can significantly alter
the position of the anterior segment (Fig 13-50d). At the
end of treatment, the maxillary midline had not deviated
to the right more than its original position (Fig 13-50e).
Another possibility is to use maxillomandibular elastics
to aid in achieving posterior protraction. The patient in
Fig 13-51a had an end-to-end Class II molar relationship
on the right side. The maxillary first premolar was
extracted, which required significant protraction of the
maxillary right segment. The following group C mechanics were employed: A symmetric T-loop spring was
inserted and activated only 2 to 3 mm. The M/F ratios
on the anterior and posterior segments from the spring
were both around 10 mm; therefore, these segments
would translate. The force magnitude was about 100 to
150 g. A Class III maxillomandibular elastic was then
added to alter the force system on the posterior segment
(Fig 13-51b). The horizontal component of force from
the elastic added to the posterior segment increased the
total force acting on the posterior segment, and the
moment still remained the same, so the final M/F ratio
was decreased, resulting in controlled tipping. There was
Separate Canine Retraction
FIG 13-51 A patient treated with a symmetric and centered T-loop and a maxillomandibular elastic for group C mechanics.
(a) Before treatment. The patient has an
end-to-end Class II molar relationship on
the right side. (b) A symmetric T-loop spring
was inserted and activated only 2 to 3 mm. In
addition, a Class III maxillomandibular elastic was added to alter the force system on
the posterior segment. (c) A root spring was
used for intersegmental leveling. (d) After
treatment. The maxillary posterior segment
was protracted for a full Class II molar relationship.
FIG 13-52 (a and b) Molar protraction
using a TAD. A TAD is recommended if a
significant amount of molar protraction is
indicated.
a
b
c
d
a
no change in the force system on the anterior segment.
The 100 to 150 g of force magnitude was not so effective
for translation of a segment, so anchorage of the anterior
segment was preserved. Later, a root spring was used for
intersegmental leveling (Fig 13-51c). The final result
shows that the maxillary right posterior segment moved
forward with good axial inclinations and intercuspation
(Fig 13-51d).
If a significant amount of protraction of a molar is
indicated, protraction headgear or TADs can be recommended. Note the protraction of the mandibular left
second molar using TADs for anchorage in Fig 13-52.
Separate Canine Retraction
Separate canine retraction may be indicated in a patient
showing severe crowding or a midline discrepancy. The
b
spring used in separate canine retraction can be basically
the same as the one used for en masse retraction. Figure
13-53a shows a simple coil spring that is engaged between
the molar and canine brackets. Because the forces are
acting away from the CR, the teeth will be tipped into
the extraction space, and an additional moment should
be provided either by a guiding wire when using sliding
mechanics or by building it into the spring itself.
For separate canine retraction, first-order rotation must
also be considered because the force is applied buccal
(red arrow) to the estimated location of the CR in the
occlusal view (Fig 13-53b). The replaced equivalent force
system (yellow arrows) at the CR shows that the canine
not only moves to the distal but also rotates mesial out
and distal in. The amount of moment may vary in accordance with the location of the CR. Buccolingual angulation of the canine significantly affects the occlusal location of the CR.
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Extraction Therapies and Space Closure
FIG 13-53 3D force system from
separate canine retraction. (a) A coil
spring is engaged between the molar and canine brackets, and a force
(red arrows) is applied. The force is
away from the CR so that the teeth
will tip. (b) The force is applied buccal to the CR (red arrow) in the occlusal view. The replaced equivalent
force system (yellow arrows) at the
occlusal CR shows that the canine
will rotate.
a
a
296
b
b
a
b
FIG 13-54 Using an additional lingual attachment for separate canine retraction. (a) By adding additional force on the lingual side of
the canine, the resultant force (yellow arrow) passes through the CR.
(b) A patient with a separate canine retraction spring with additional
force at the lingual side.
FIG 13-55 Antirotation bends in the spring. (a) A typical “toe in”
antirotation bend is placed anteriorly. Not only is an antirotation
moment produced at the canine, but also an unwanted buccal force
is generated. (b) The additional posterior bend eliminates the labial
force to the canine. If properly balanced, equal and opposite couples
can be produced. Mesiodistal force was not depicted for the sake
of simplicity.
A lingual attachment can be used with additional force
at the lingual side of the canine so that the resultant of
the labial and lingual forces passes through the occlusal
CR of the canine (Fig 13-54). This method uses part of
the force on the buccal and part on the lingual. More
force is usually applied on the lingual side because the
canine CR is usually found nearer to the lingual crown
surface due to canine inclination.
Antirotation bends in the spring can be placed to
prevent rotation during canine retraction. Figure 13-55a
shows a typical “toe in” antirotation bend placed anteriorly so the anterior leg of the spring cuts across the canine
bracket with an angle. The occlusal view of the labiolingual component of the force system of the spring is
similar to an off-center V-bend placed toward the canine
bracket. A simplified equilibrium force system diagram
is shown without mesiodistal retraction forces. As a
result, not only is an antirotation moment applied to the
canine, but also a buccal force is generated. It is better to
modify the spring by adding another bend posteriorly to
eliminate a labial force to the canine; if properly balanced,
an equal and opposite couple can be produced (Fig
13-55b). However, if the intercanine width must be
narrowed during separate canine retraction, difficulties
arise. Lingual forces and moments to rotate a canine distal
out are inconsistent with a wire originating at the distal
of the canine. This concept is discussed in detail in chapter 14. Even if proper antirotation bends were made, it is
difficult to place them comfortably because the wire must
be bent and twisted in three dimensions (Fig 13-56).
A passive canine-to-canine incisor bypass arch can be
used to prevent rotation of the canine during retraction
(Fig 13-57a). In addition, if the canine is initially rotated
distal out, the incisor bypass archwire alone is capable of
distal movement of the canine without any additional distal
horizontal force being applied. Suppose a canine is bilaterally rotated mesial in and the occlusal CR is found
lingually (a typical canine inclination places the CR
Canine Bypass Archwire and Canine Root Spring
b
a
FIG 13-56 (a) T-loop spring with antirotation bends and curvature. (b) A patient with a separate canine retraction spring. It is difficult to place
the spring comfortably because the spring is bent and twisted in three dimensions.
a
FIG 13-57 Canine-to-canine incisor bypass arch for prevention of
canine rotation. (a) Passive application. (b) Active application.
a
b
b
c
FIG 13-58 A patient treated with a canine-to-canine incisor bypass arch. (a) Lateral view. The labial spring does not need complicated antirotation bends. (b) Before canine retraction. (c) After canine retraction.
lingually); the equal and opposite couples from the incisor bypass archwire derotates (turns back) the canine, and
the distal contact area moves to the distal (Fig 13-57b).
Although the canine crown moves distally, the CR of the
canine does not move distally because there is no genuine distal force and only a moment acts. Clinically, we
can take advantage of this “free” distal crown movement.
The patient in Fig 13-58 underwent separate canine
retraction with a canine-to-canine active incisor bypass
archwire and a frictionless spring. The incisor bypass
archwire effectively derotates the canine (crown distal
in) during separate canine retraction. A protective sheath
was used in the incisor region for comfort.
Canine Bypass Archwire and
Canine Root Spring
Once a canine is retracted sufficiently to provide adequate
space, the anterior segment can be leveled. If the canine
is tipped distally after retraction, it is better not to level
the canine but to keep its angulation until the extraction
space is completely closed. Moving the canine roots
distally by applying a couple from a continuous leveling
archwire or even with an archwire segment will develop
many undesirable side effects. If a leveling continuous
archwire is inserted for canine root movement, not only
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13
a
Extraction Therapies and Space Closure
b
FIG 13-59 (a) Leveling a canine with a
continuous archwire. (b) If the archwire is
engaged at the canine only, vertical forces
(blue arrows) are necessary to engage the
rest of the arch. Therefore, the premolars will
intrude and the incisors will extrude. (c and
d) When a couple is applied at the canine
bracket, the crown will move mesially and
pull the entire dentition forward (∆). (e) The
rowboat effect. When an oar is pushed back,
the water does not go back but rather the
boat goes forward.
c
d
e
is a couple produced at the canine bracket, but also vertical forces and moments will be developed on adjacent
teeth. In Fig 13-59, a leveling archwire is engaged into a
full arch. The actual force system developed at each
bracket by the continuous archwire is more complicated;
however, we can roughly visualize its effects by a simple
thought experiment. Suppose the archwire is only
engaged at the canine (see Fig 13-59b); to engage the wire
into the other brackets, the wire is lifted upward anteriorly and downward posteriorly by activation forces. The
adjacent lateral incisors and premolars are poor anchorage, so the premolars will intrude and the incisors will
extrude from these vertical forces. Moreover, root movement of the canine (rotation around the canine bracket)
can tax anchorage because a couple rotates a tooth
around its CR. When a couple is applied at the bracket
(see Fig 13-59c) to move the roots to the distal, the crown
will move mesially (see Fig 13-59d). To prevent this, we
must tie back the arch or tie the canine to the posterior
segment; hence, a distal force to the canine is required,
and an equal and opposite force wants to bring the posterior segments forward (see Fig 13-59d). A rough analogy
is an oar on a rowboat. The end of the oar in the water is
the root apex. The oar is pushed back, but the ocean does
298
not go back; rather, the boat shoots forward. We therefore
sometimes refer to this type of anchorage loss (Δ in Fig
13-59d) as the rowboat effect (Fig 13-59e). A more scientific explanation is given in chapter 14. For instance, the
leveling arch in Fig 13-59 produces a Class III geometry
between the canine and the incisors. Not only are
moments present, but a large incisor extrusive force is
expected.
Therefore, after leveling of the canine by a continuous
archwire, side effects such as iatrogenic reverse curves
of Spee, open bite at the premolar area, reopened space,
and deep bite can be generated (Fig 13-60). Similar effects
result in sliding mechanics if the incisors are included in
the continuous archwire (see Fig 13-10).
The patient in Fig 13-61a shows severe linguoversion
of the maxillary incisors after retraction and lacks overjet. A continuous round wire was inserted, and a rectangular nickel-titanium (Ni-Ti) wire was attached over the
incisors to produce counterclockwise moment. The round
wire allows free third-order rotation. The vertical forces
from the rectangular Ni-Ti wire could be kept to a minimum because the interbracket distance is large (Fig
13-61b). After 3 months of treatment, the maxillary incisors rotated and pulled posterior teeth forward (Fig
Canine Bypass Archwire and Canine Root Spring
FIG 13-60 Side effects of leveling the canine by a continuous archwire. An iatrogenic reverse curve of Spee, an open bite at the premolar
area, reopened space, and a deep bite can be observed after leveling.
a
b
c
d
e
a
b
FIG 13-61 (a) A patient shows severe linguoversion of the maxillary incisors after
retraction and also lacks overjet. (b) The
force system from a rectangular Ni-Ti wire.
(c) Occlusion 3 months after treatment. (d)
Posttreatment cephalometric radiograph. (e)
Cephalometric superposition showing the
change in angulation.
FIG 13-62 Canine bypass archwire. (a) Before canine root movement. (b) After treatment. Undesirable effects are eliminated on
the adjacent teeth.
13-61c). The lateral cephalometric radiograph after treatment (Fig 13-61d) and superimposition (Fig 13-61e) show
the rowboat effect.
In Fig 13-62, a full archwire was stepped around the
canine (canine bypass arch), and a separate canine root
spring was placed. Forces are distributed to the rest of
the arch, not to the adjacent teeth, so that the undesirable
effects are kept to a minimum.
A cantilever canine root spring was used after space
closure in partial lingual treatment (Fig 13-63). Note that
the canine is tied back using elastics on the lingual side
to prevent mesial movement of the canine crown. Care
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13
a
Extraction Therapies and Space Closure
b
c
FIG 13-63 A cantilever canine root spring during partial lingual treatment. (a) Before canine root movement. (b) After root movement. (c)
Occlusal view. The canine is tied back using elastics on the lingual side to prevent mesial movement. The force system of the root spring
produced a desirable moment, and the extrusive force helped to erupt the canine that was in infra-occlusion (red arrows in a).
FIG 13-64 3D applications with a rectangular loop. A rectangular loop was welded
on a continuous β-titanium wire. This loop
is very versatile in that it controls the tooth
movement in three dimensions. Note that a
small vertical loop welded on both sides was
used for regaining a little space.
a
b
c
FIG 13-65 (a to e) Leveling a canine with
a Ni-Ti overlay wire. The rest of the arch is
stabilized with a posterior FRC segment and
a rigid stainless steel rectangular archwire
to prevent side effects.
d
e
must be taken if elastics are used as a tie-back because
space could open if they stretch. Note that no canine
bypass was used, so the extrusive force on the root spring
slightly extruded the canine, which was indicated.
Another method of canine root movement is a rectangular loop welded on a β-titanium wire, as shown in
Fig 13-64. (A small vertical loop welded on both sides
was also used for regaining a little space.) The
300
rectangular loop is very versatile in that it controls the
tooth movement in three dimensions (see chapter 14 for
more information). Similar side effects can be generated
in leveling a high canine even though it is not tipped;
eruptive forces alone on the canine will induce intrusion
force and moments at adjacent teeth if a full continuous
archwire is used for leveling. Figure 13-65 shows such a
severely blocked-out canine. The canine was leveled with
Incisor Root Movement
a
b
c
d
FIG 13-66 (a to d) A patient showing spontaneous eruption of a high canine. If sufficient space is regained, spontaneous
eruption may be successful even though the apex of the canine root has closed.
a Ni-Ti overlay wire. The rest of the arch was stabilized
with a posterior FRC segment and a rigid stainless steel
rectangular archwire; in this way, the stress was more
evenly distributed to the full arch as a unit. Side effects
such as skewing of the arch form or vertical side effects
at the adjacent teeth were minimized.
The treatment of choice for a high canine is to provide
sufficient space and just let the canine erupt spontaneously. Sometimes, but not always, this may be successful (Fig 13-66). Of course, nonorthodontic eruption eliminates any potential for side effects. Here, even though
the apex of the canine root had closed, it still erupted
spontaneously.
Incisor Root Movement
Incisors can be tipped too much lingually after space
closure. This is likely to occur when a round wire or
undersized wire is used for the anterior segment during
retraction. Moving the incisor roots lingually can be one
of the most challenging procedures for the orthodontist.
Many methods are suggested, such as twisting the wire
or using specially designed springs; however, some of the
methods make no sense scientifically (see Fig 1-3), are
difficult to do, or produce many side effects. The force
system required for root movement is a single force (red
arrow) applied slightly apical to the CR (Fig 13-67). An
equivalent force system is replaced at the bracket (yellow
arrows), and it is obvious that not only torque (moment)
at the bracket but also a lingual force is needed. (The M/F
ratio of 12 mm is somewhat typical for a maxillary incisor.) If only torque is applied, the tooth will rotate around
the CR, and the crown will move forward.
When torque is applied at the incisor bracket by localized twisting on a full-sized, rectangular continuous
archwire, the torque/twist rate is very high and ineffective
for several reasons. Interbracket distance is limited no
matter how narrow the bracket width is. Selection of wire
cross section is limited—full-size wire should be used. If
a smaller–cross section wire is used for a lower torque/
twist rate, there would be significant play between the
bracket and the wire. A high-springback, low-stiffness
material like Ni-Ti is difficult to permanently form to
increase or decrease the amount of torque. Straight-wire
applications that produce small angular activations are
usually inadequate.
A high–torque/twist rate wire requires a very small
amount of activation; therefore, too frequently wire
adjustment is required. Small errors in the twist angle
could lead to very high stress at the apex, which could
cause undesirable side effects such as discomfort, root
resorption (Fig 13-68), and anchorage loss. Even if a desirable moment from the twist in an edgewise wire was
301
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Extraction Therapies and Space Closure
FIG 13-67 Force system of incisor root movement. A single force
slightly apical to the CR is required for incisor root movement (red
arrow). The replaced equivalent force system at the bracket is shown
in yellow. Note that not only torque (moment) at the bracket but also
a lingual force is needed.
a
FIG 13-68 Root resorption probably produced by the high torque/
twist rate of the wire.
b
FIG 13-69 The activation force system for a 0.017 × 0.025–inch β-titanium incisor root spring. (a) The downward force on the hook (blue arrow)
is necessary to activate the spring. (b) The deactivation force system acting on all the brackets (red arrows). Note that the spring delivers a
continuous large incisor moment, unlike a localized twist of a continuous wire, because the moment arm is large.
a
b
c
d
e
f
FIG 13-70 A patient treated with an undersized wire during anterior retraction. (a) Before treatment. (b) Uncontrolled lingual crown tipping.
The apex penetrated the labial cortical plate. (c) The roots were moved lingually into the bone. New bone formation is seen at the labial side
of the apex. (d to f) The root spring was inserted into four incisors. The roots of the four incisors were moved lingually as a unit, and the spring
was modified to be inserted into the central incisors for further root movement.
302
Incisor Root Movement
a
b
FIG 13-71 3D effects of four-incisor root movement. (a) En masse movement of an incisor segment can produce canting of the occlusal plane
(from A to B). (b) An extrusive force is acting on the incisor segment (yellow arrow), and the CR moves occlusally (C); however, the incisors
may look intruded (D) because the incisal edges of the central incisors move apically.
initially delivered, the moment would rapidly drop to
suboptimal levels over a few degrees. Even more problematic may be adjacent bracket side effects as adjacent
teeth or segments receive equal and opposite moments.
In short, the desired twist in a continuous edgewise wire
to produce effective torque is not obvious or easily
measured by the orthodontist. One possible solution is
the use of more than one wire, a base arch, and an auxiliary root spring.
The force system from a 0.017 × 0.025–inch β-titanium
incisor root spring produces sufficient incisor moment
by a small force because the moment arm is large (Fig
13-69a). The moment is delivered continuously, unlike a
localized twist of the wire between two brackets in a
continuous archwire. The downward force on the hook
(blue arrow) is necessary to activate the spring (activation
force system). The deactivation force system (red arrows)
acting on all the teeth of the activated incisor root spring
looks like a horizontally flipped image of an incisor intrusion spring (Fig 13-69b). It is a molar intrusion spring
anchored by incisors, with the active region being the
incisors and not the posterior teeth. The incisors feel the
desired counterclockwise moment (lingual root torque)
but also a small amount of unwanted extrusive force. As
the incisor inclinations correct from the action of the
moment, the crown moves labially and the root apex
lingually. (A couple spins a tooth around its CR.) To
prevent this eruption and labial movement of the crown,
a stabilizing archwire is placed, bypassing the incisor
brackets. The anterior part of the archwire is stepped so
that the wire contacts the occlusal surface of the incisor
bracket and is tightly cinched back to prevent labial movement of the incisors. This stabilizing archwire prevents
side effects and forces the incisors to rotate around the
bracket.
A patient treated with an undersized wire during anterior retraction demonstrated uncontrolled lingual crown
tipping (Figs 13-70a and 13-70b). As a result, the apex
penetrated the labial cortical plate (see Fig 13-70b). An
incisor root spring was inserted, and the roots were
moved lingually into the bone of the maxilla; new bone
formation is seen at the labial side of the apex (Fig 13-70c).
Many times, the lingual root movement of a four-incisor
segment requires two stages of treatment, such as fourincisor root movement followed by central incisor root
movement (Figs 13-70d to 13-70f ). The reason is that the
four-incisor segment undergoes an en masse movement
in three dimensions that results in a cant of the anterior
occlusal plane as seen from the lateral view (Fig 13-71a).
Note the change of cant from A to B; therefore, two stages
of root movement may be necessary to keep the plane of
occlusion level in the lateral view. On the other hand, if
the four incisors were tipped lingually en masse, only one
stage should be required. Root correction will correct the
cant of the anterior occlusal plane simultaneously.
Even though an extrusive force is acting on the incisor
segment and the CR moves occlusally (C in Fig 13-71b),
it is also frequently observed in the final stages of incisor
root movement that the vertical overlap is reduced, and
the incisors may look intruded because the incisor brackets creep apically away from the anterior base arch (D in
Fig 13-71b).
303
13
a
d
Extraction Therapies and Space Closure
b
c
e
f
FIG 13-72 (a) Blocked-out maxillary right lateral incisor. The lateral incisor was aligned to the labial side by uncontrolled
tipping; therefore, it needed labial root movement. (b) Deactivated shape before labial root movement. (c) Activated shape
and the force system. (d) The distance from the fixed end (maxillary right lateral incisor) to the free end is the largest distance
we can obtain in the oral cavity. (e and f) Before and after root movement.
For a single-tooth root movement, the same principle
is used. Figure 13-72 shows a patient with a blocked-out
maxillary right lateral incisor (Fig 13-72a). The premolars
were extracted, and the canines were retracted to regain
space for the lateral incisor. The lateral incisor was aligned
to the labial side by uncontrolled tipping; therefore, it
needed labial root movement. A cantilever root spring
was bonded on the maxillary lateral incisor and extended
to the contralateral side of the arch. The deactivated
spring shows torsion and bending in three dimensions
(Fig 13-72b). For a detailed procedure of fabricating the
root spring, scan the QR code and refer to the video clip.
The free end should be located far and perpendicular to
the labial surface of the lateral incisor as much as possible, which would be in the molar region on the contralateral side. This is the largest distance we can obtain in
the oral cavity so that the vertical forces can be kept to a
minimum (Figs 13-72c and 13-72d). The torque/twist
rate is very low; torque was applied on the lateral incisor
continuously so that only one or two reactivations were
performed during the 3 months of root movement (Figs
13-72e and 13-72f ). There were no side effects on adjacent
teeth (maxillary right canine and maxillary right central
304
incisor), and the round wire prevented intrusive and
lingual movement of the crown of the maxillary lateral
incisor during application of torque by this cantilever
root spring.
TADs can be used indirectly to reinforce the anchorage
to prevent side effects from a root spring or directly to
apply a single force without the root spring. Figure 13-73
shows a case of linguoversion of the maxillary incisors
that required root correction after space closure using
TADs. An archwire with a poorly designed spring (Fig
13-73a) was removed, and bilateral TADs were used to
reinforce the anchorage indirectly to prevent the rowboat
effect as described above. Elastics from the TAD attached
to a small piece of β-titanium wire welded on the root
spring were used to prevent labial movement of the
crown, and two stages of treatment were performed (Figs
13-73b to 13-73d). The horizontal side effect (rowboat
effect) was prevented by TADs, and extrusion of incisors
was prevented by an incisor bypass stabilizing archwire
(see Figs 13-73b to 13-73d). The cephalometric radiographs
(Figs 13-73e and 13-73f) show that the incisor was rotated
around the incisal tip or the bracket.
Two-Phase Space Closure
a
b
c
e
f
d
FIG 13-73 (a) The maxillary incisors required root correction after space closure.
(b) A TAD was placed and tied to the spring
to prevent labial movement of the four incisors. (c) Root movement of two incisors
followed. (d) After treatment. (e and f) The
incisor was rotated around the incisal tip
without anchorage loss.
A TAD can be used for the direct application of a single
force required for incisor root movement rather than for
reinforcing the anchorage. A patient was treated with an
FRC anterior segment and a statically determinate retraction system (Figs 13-74a to 13-74e). The anterior segment
looks excessively tipped to the lingual because all of the
extraction space was closed by controlled tipping of the
anterior teeth (Figs 13-74f to 13-74h). However, the
center of rotation was kept near the apex. Successful
space closure with minimal anchorage loss accounted for
the lingual incisor inclination. The TAD that was used
for retraction was utilized again for root movement to
correct the incisor axial inclination (Fig 13-74i). The force
system was accurately designed so that the resultant force
(FR, yellow arrow) passed slightly above the estimated CR
using two forces from elastics (red arrows). The patient
was carefully monitored. If the crown of the incisor moves
labially in the course of root movement, the force from
one elastic (Fa) is increased and/or the force from the
other elastic (Fb) is reduced (see Fig 13-74i). The final
incisor inclination is good following incisor root movement (Figs 13-74j to 13-74p). This is more difficult to
accomplish with a single elastic because modifying the
line of action of force would be very difficult and perhaps
anatomically impossible.
Two-Phase Space Closure
Traditionally, when incisor crowding is presented, separate canine retraction followed by incisor retraction or
en masse retraction is performed after leveling of the
anterior teeth. Three phases of treatment are the usual
sequence—leveling, major tooth movement, and finishing. By using an FRC, the major tooth movement can be
performed from the beginning without leveling. The
leveling and finishing can be done simultaneously at a
later stage of treatment; thus, extraction therapy is done
in only two phases.
The patient shown in Fig 13-75 complained of lip
protrusion and minor crowding (Figs 13-75a and
13-75b). An FRC was used to stabilize the anterior teeth
as a rigid unit. Force was applied to a gingival extension
on the anterior teeth (Figs 13-75c to 13-75e). A TAD was
used for group A mechanics (Figs 13-75f to 13-75h).
After the maxillary and mandibular spaces were closed,
the maxillary and mandibular FRCs were replaced with
brackets for simultaneous final leveling and finishing (Figs
13-75i to 13-75k). The patient’s chief complaint was
resolved initially without a leveling stage, and the duration
of bracketing was minimized because the traditional three
stages of treatment were reduced to two stages. Satisfactory results after treatment are shown (Figs 13-75l to
13-75o).
305
13
Extraction Therapies and Space Closure
a
b
d
e
f
g
j
306
h
FIG 13-74 A direct application of a TAD for incisor root movement without
an incisor root spring. Before treatment, the patient showed good buccal
intercuspation. (a to e) The anterior segment was stabilized using an FRC.
(f to h) A wire extension was used for controlled tipping of the anterior
segment. Note that there are no attachments on the posterior segment.
The force system was adequate, but too much linguoversion of the incisors
occurred because a large extraction space was closed by controlled tipping of the anterior segment only. (i) The resultant (FR ) of two forces from
elastics (Fa , Fb ) passed slightly above the CR of the anterior segment. (j to
n) After treatment, a favorable incisor inclination was obtained. (o and p)
Profile views before and after treatment.
i
m
c
k
n
l
o
p
Two-Phase Space Closure
a
b
c
d
e
f
g
h
i
j
k
l
m
n
FIG 13-75 Two-phase space closure. (a and b) A patient complained of lip protrusion and
minor crowding. (c to e) An FRC was used to stabilize the anterior teeth as a rigid unit. Force
was applied to a gingival extension on the anterior teeth. (f to h) A TAD was used for group
A mechanics. (i to k) After the maxillary and mandibular spaces were closed, the maxillary
and mandibular FRCs were replaced with brackets, and leveling and finishing were done
simultaneously. (l to o) The patient’s chief complaint was resolved initially, and the duration
of bracketing was minimized. Satisfactory results after treatment are shown.
o
307
13
Extraction Therapies and Space Closure
Summary
This chapter has discussed several methods of space
closure for extraction patients. There is no best one. Some
methods may be better in special situations, or the clinician may have his or her own preferences. No matter
what method is used, a sound understanding of the
biomechanical principles is necessary. Many designs of
retraction springs have been suggested; clinicians can
even design their own. The spring should be selected or
designed based on sound biomechanical principles, not
by intuition. There may be some attractive-looking shapes
that are biomechanically inefficient. No matter how accurately we make a spring and know the exact force system
acting on the tooth, the tooth may not move as we
predicted for many reasons. M/F ratios may have to be
modified because of the periodontal and root support.
Force magnitude may have to be altered based on individual considerations. Accurate fabrication of a given
loop involves only technique; design and proper adjustment relies on principles and requires thinking.
References
1. Heo W, Nahm DS, Baek SH. En masse retraction and two-step
retraction of maxillary anterior teeth in adult class I women. A
comparison of anchorage loss. Angle Orthod 2007;77:973–978.
2. Halazonetis DJ. Design and test orthodontic loops using your
computer. Am J Orthod Dentofacial Orthop 1997;111:346–348.
Recommended Reading
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orthodontic appliances. IEEE Trans Biomed Eng 1977;24:538–539.
Burstone CJ. Rationale of the segmented arch. Am J Orthod 1962;
48:805–822.
Burstone CJ. The biophysics of bone remodeling during orthodontics:
Optimal force considerations. In: Biology of Tooth Movement. Boca
Raton, FL: CRC, 1989;321–333.
Burstone CJ. The mechanics of the segmented arch techniques. Angle
Orthod 1966;36:99–120.
Burstone CJ. The segmented arch approach to space closure. Am J
Orthod 1982;82:361–378.
Burstone CJ, Baldwin JJ, Lawless DT. The application of continuous
forces to orthodontics. Angle Orthod 1961;31:1–14.
Burstone CJ, Hanley KJ. Modern Edgewise Mechanics Segmented
Arch Technique. Glendora, CA: Ormco, 1995.
Burstone CJ, Koenig HA. Creative wire bending—The force system from
Step and V-bends. Am J Orthod Dentofacial Orthop 1988;93:59–67.
Burstone CJ, Koenig HA. Optimizing anterior and canine retraction.
Am J Orthod 1976;70:1–19.
308
Burstone CJ, Pryputniewicz RJ. Holographic determination of center
of rotation produced by orthodontic forces. Am J Orthod 1980;77:396–
409.
Caldas SGFR, Martins RP, Viecilli RF, Galvãoa MR, Martins LP. Effects
of stress relaxation in beta-titanium orthodontic loops. Am J Orthod
Dentofacial Orthop 2011;140:e85–e92.
Choy K, Kim K, Burstone CJ. Initial changes of centres of rotation of
the anterior segment in response to horizontal forces. Eur J Orthod
2006;28:471–474.
Choy K, Kim K, Park Y. Factors affecting force system of orthodontic
loop spring. Korean J Orthod 1999;29:511–519.
Choy K, Pae E, Kim K, Park Y, Burstone CJ. Controlled space closure
with a statically determinate retraction system. Angle Orthod
2002;72:191–198.
Choy K, Pae E, Park Y, Kim K, Burstone CJ. Effect of root and bone
morphology on the stress distribution in the periodontal ligament. Am
J Orthod Dentofacial Orthop 2000;117:98–105.
Faulkner MG, Fuchshuber P, Haberstock D, Mioduchowski A. A parametric study of the force/moment systems produced by “T”-loop
retraction springs. J Biomech 1989;22:637–647.
Faulkner MG, Lipsett AW, El-Rayes K, Haberstock DL. On the use of
vertical loops in retraction systems. Am J Orthod Dentofacial Orthop
1991;99:328–336.
Gjessing P. Biomechanical design and clinical evaluation of a new
canine retraction spring. Am J Orthod 1985;87:353–362.
Kojima Y, Fukui H. Numerical simulation of canine retraction by sliding
mechanics. Am J Orthod Dentofacial Orthop 2005;127:542–551.
Kuhlberg AJ, Burstone CJ. T-loop position and anchorage control. Am
J Orthod Dentofacial Orthop 1997;112:12–18.
Manhartsberger C, Morton JY, Burstone CJ. Space closure in adult
patients using the segmented arch technique. Angle Orthod 1989;
59:205–210
Martins RP, Buschang PH, Gandini LG Jr. Group A “T” loop for differential moment mechanics: An implant study. Am J Orthod Dentofacial
Orthop 2009;135:182–189.
Martins RP, Buschang PH, Martins LP, Gandini LG Jr. Optimizing the
design of preactivated titanium T-loop springs with Loop software.
Am J Orthod Dentofacial Orthop 2008;134:161–166.
Mulligan TF. Common sense mechanics. J Clin Orthod 1980;14:
546–553.
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with transverse forces: An experimental study. Am J Orthod Dentofacial Orthop 1991;99:337–345.
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Siatkowski RE. Continuous arch wire closing loop design, optimization,
and verification. Part I. Am J Orthod Dentofacial Orthop 1997;112:393–
402.
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Weinstein S. Minimal forces in tooth movement. Am J Orthod
1967;53:881–903.
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Problems
1. The loop height of a standard vertical loop is increased
from 6 mm to 8 mm. How would it affect the F/∆ rate
of the spring? (Refer to Fig 13-26c.)
2. The standard vertical loop made with β-titanium wire
shows an F/∆ rate of 437 g/mm and an M/F ratio of
2.2 mm. If it were made with stainless steel, how
would that affect the F/∆ rate and M/F ratio (provided
that the modulus of elasticity of stainless steel is twice
that of β-titanium)?
3. Two different vertical loops with the same height are
shown: A, teardrop shape; B, keyhole shape. Compare
the M/F ratio between the two shapes.
4. Two types of vertical loop with the same height are
shown: A, open type; B, closed type. Compare the
range of action between the two.
5. Two types of loop with the same height are shown: A,
with four helices; B, with three helices. Compare the
M/F ratio, F/∆ rate, and range of action between the
two.
6. An L-loop is used for space closure. There are no residual moment bends in the spring, and the vertical part
of the loop is at the center of the two brackets. Would
a moment differential between the anterior and posterior ends exist? Explain why.
309
Problems
310
7. The T-loop shown in Fig 13-33 is activated at 2 mm to
be cautious on first use. Space did not close and even
increased. Explain why.
8. An eccentric universal T-loop spring (B/L = 0.61) is
used for group A mechanics. If this spring is fully
activated (∆ = 6 mm), what type of tooth movement
is expected in the posterior and anterior segments?
(Refer to Fig 13-45.) After space has closed to 4 mm,
how is the tooth movement different?
9. The spring in problem 8 (B/L = 0.61) was kept at full
activation (6 mm) during the entire space closure by
constant reactivation. How will the space closure be
different?
10. No guide wire is present, and the arch is segmented
with separate stabilizing wires and a passive TPA. In
the anterior segment, the extension arm was used to
prevent lingual tipping of the incisors. Would it be
suitable for group A mechanics?
14
Forces from
Wires and
Brackets
“If you can't explain it simply, you don't understand it well enough.”
— Albert Einstein
Straight wires are commonly placed in malaligned brackets that have been
precisely oriented on the tooth crown, and it is very important to understand the
force system produced. This chapter begins with a discussion of the force system
on two brackets and then builds to multibracket alignment. This is a study in equilibrium because the archwire is in equilibrium. The force system is a continuum
in which six geometries exist, from Class I to Class VI. The vertical forces are the
greatest in Class I, and there are no forces but only equal and opposite couples in
Class VI. The same ratio of moments for the same malalignment geometry exists
between brackets as interbracket distance is altered. Force systems produced
by V-bends and Z-bends are explained. This chapter discusses how forces with
multiple brackets are determined. In some malocclusion bracket geometries, a
straight wire works efficiently, whereas in other geometries adverse side effects
develop. This chapter explains how to eliminate these side effects.
311
14
Forces from Wires and Brackets
FIG 14-1 A button attachment with a single force. The measurement
of this one force is enough to predict the response of the posterior
segment.
O
rthodontic appliances at the most sophisticated
level usually involve bracket placement on the teeth
with an archwire as the force-delivering mechanism. The prediction of forces can be very complicated
when a wire is inserted in the malaligned brackets associated with a malocclusion. Forces and moments can
operate in three-dimensional space. Small differences in
bracket position can significantly alter the force system.
As teeth move, the bracket geometries change, resulting
in a continually changing force system. Friction and
bracket-wire play are important variables in the ultimate
force achieved. Beam theory, large deflection considerations, and mechanics of materials may be referenced.
To facilitate a more approachable understanding of this
material to the reader, most major concepts in this chapter are developed considering only two-dimensional
space and small deflections of orthodontic wires.
Forces from a Straight Wire in
Malaligned Brackets
Brackets and tubes can be accurately placed on individual teeth so that if aligned along a straight archwire, an
ideal occlusion will result. This can only occur at the end
of treatment because archwires placed earlier in treatment can unleash many unpredictable forces, preventing
good bracket alignment from proceeding in a straight
line. This chapter discusses what happens when a relatively straight wire is inserted into crooked brackets.
Because this clinical scenario is complicated by many
factors, we will start by analyzing the force system
produced by a wire inserted into just two brackets. With
this knowledge, we can then move forward to an understanding of a multibracketed appliance such as the continuous edgewise arch.
312
Let us consider the effect on a posterior segment of an
intrusive force (red arrow) on the canine (Fig 14-1). If a
button rather than a bracket is placed on the canine, a
single force measurement is adequate to predict the
response of the posterior segment. The occlusal force on
the canine (blue arrow) is the activation force on the wire
at the canine, and it is equivalent to the deactivation force
on the posterior segment (left red arrows). Because it is
the only force acting on the teeth, the posterior segment
will extrude, and its occlusal plane will steepen (as shown
in Fig 14-1 by the equivalent force system at the center
of resistance [CR] of the segment). This first system is
statically determinate with the measurement of one force
alone.
In Fig 14-2, the maxillary right canine is highly positioned, and the maxillary right lateral incisor is tipped
counterclockwise. A button is placed on the canine
instead of a bracket so that the single force is applied at
the canine (Fig 14-2a). The force system is statically determinate between the canine and the lateral incisor and is
highly predictable (Fig 14-2b). However, the force system
from the archwire between the lateral and central incisors
is indeterminate; this relationship is discussed in detail
later in this chapter. The panoramic radiograph shows
that the canine erupted parallel to the other teeth and
that the lateral incisor intruded and rotated clockwise
(Figs 14-2c and 14-2d). This indicates that an edgewise
bracket with six degrees of freedom is not always the best
appliance. Let us see why.
Let us replace the button on the canine in Fig 14-1 with
a bracket (Fig 14-3). If an occlusal activation force is
placed at the canine bracket, the wire will curve so that
the wire cuts across the bracket with an angle (orange
wire), and full insertion will also require a counterclockwise moment and greater force during activation (see Fig
14-3b). Thus, both a force and a moment should be
Forces from a Straight Wire in Malaligned Brackets
FIG 14-2 A button instead of a bracket
was placed at the canine so that the single
force is applied at the canine. (a) The force
system is statically determinate, and the
tooth movement is highly predictable. (b)
The canine erupted parallel, and the lateral incisor intruded and rotated clockwise.
(c and d) Radiographs before and after
treatment.
a
b
c
d
a
b
FIG 14-3 The force system when a button is replaced by an edgewise bracket. (The activation force system is depicted at the canine only for
the sake of simplicity.) (a) If an occlusal activation force is placed at the canine bracket, the wire will curve and cut across the bracket so that
full insertion will also require a counterclockwise moment and increased force during activation. (b) Thus, both a force and a moment should
be simultaneously measured in order to predict the effect.
simultaneously measured, if we want to predict the effect.
This is different than the situations shown in Figs 14-1
and 14-2, where only a single force measurement is
required at a button; in Fig 14-3, moments are generated
at both anterior and posterior ends. Thus, if only the force
or the moment is measured at one bracket, there is not
enough information to determine the complete force
system.
This chapter shows that a single measurement of the
force or moment at the canine becomes adequate if the
geometric configuration of the brackets is defined. In Fig
14-4, the canine and first premolar brackets are parallel
but not in alignment. This spatial relationship of the
bracket slot (geometry) helps to determine the force
system. A straight wire placed between the two brackets
will need equal moments during activation along with
equal and opposite forces (Fig 14-4a). Therefore, with
this added information derived from initial bracket
geometry, the system becomes statically determinate
even if only one force or moment is measured. However,
measuring this force or moment is not easy because they
are associated with each other (see chapter 13). In Fig
14-4b, the force directions are reversed (red arrows),
showing the deactivation force system acting on the
premolar and canine. The force and moment acting at
the CR of the posterior segment (yellow arrows in Fig
14-4c) are equivalent to the force and moment acting at
the first premolar bracket; the posterior segment will
extrude, and its occlusal plane will steepen.
In the next section, various bracket geometric configurations are defined, and the associated relative force
systems produced are given. This background analysis
allows the force system to be determinate with just a few
measurements and allows us to better understand what
a straight wire will do during an alignment phase even
without any measurement of the forces.
313
14
Forces from Wires and Brackets
a
b
FIG 14-4 (a) Moments are required at both the anterior and the
posterior ends to place a wire. The ratio of moments is determined
by observing the geometry of two brackets, and it provides an additional boundary condition so that the force system becomes statically
determinate. (b) Deactivation force system. (c) Replaced force system
at the CR of the posterior segment. The posterior segment (blue teeth)
will extrude, and the occlusal plane will steepen.
c
If a straight wire is placed into malaligned brackets,
the force system is determined by many factors. The
behavior of the wire depends on cross section, material,
configuration, friction, bracket play, large or small deflections, and many other factors. It is further complicated
by the possibilities of many teeth (up to 14 per arch) being
actively involved. Initially, it is important not to get lost
in all the details so as to better understand the fundamental relationships between the insertion of active wires
into brackets and the forces produced. The following
section begins with the simplest unit: two brackets.
Different geometric relationships are explored as straight
or bent wires are inserted. Deflections are purposely kept
small, friction and play are ignored, and mesiodistal forces
are not considered. Three-dimensional (3D) analyses are
considered later in this chapter.
Ideal Modeling of Two-Bracket
(Tooth) Segments
The spatial relationship of one bracket to another can be
classified as one of six geometries, with an expected force
system for every geometric configuration. A simplified
model is given in Fig 14-5. Commonly, wide brackets can
be used; these brackets can have wire-bracket interactions
between the mesial and distal sides of the bracket.
However, the bracket widths (and any intrabracket width
314
effects) are purposely not modeled. In Fig 14-5a, an ideal
edgewise bracket is modeled where the bracket width
becomes a zero horizontal dimension; nevertheless, whatever size of wire is inserted, it has no play at the bracket
slot. Remaining diagrams in this chapter used to develop
the concepts and present data show more typical (wider)
bracket widths (Fig 14-5b). The intent is to show how the
force system changes with different bracket-wire geometries and not to discuss the effect of bracket width. For
example, actual wire stiffness varies as 1/I 3 of the actual
interbracket distance (I) and not the distance between
the bracket centers (see chapter 18); therefore, if stiffness
is to be calculated, the distance between the proximal
surfaces of the brackets must be used. From this point
forward, the brackets are depicted with wide shape;
however, for simplicity, the interbracket distance is
measured between each bracket slot center (Fig 14-5c).
The purpose of the illustrated wide bracket slot in orange
is for demonstration of the angulation of the slot, because
the angulation of the ideal bracket may not be visible (see
Fig 14-5a). Remember that we are using an ideal bracket
and not studying the effect of different bracket mesiodistal
widths, where the distance between proximal surfaces is
required. Also, as we are considering small deflections,
the length of the wire (L) between the two brackets is
assumed to be the same as the interbracket distance (see
Fig 14-5c).
Geometry Classification and Determination
a
b
c
FIG 14-5 (a) An ideal edgewise bracket is modeled where bracket width becomes a zero horizontal dimension. (b) For more clinical relevance,
wide bracket widths can be equivalent with two ideal brackets per tooth. (c) In the clinic, the distance between the centers of the bracket slots is
measured for interbracket distance. The length of the wire (L) between two brackets is assumed to be the same as the interbracket distance (I).
a
b
c
FIG 14-6 (a) When the two bracket centers are connected, the interbracket axis (black dotted line between two brackets) is formed. The
black dotted line through each slot is the slot axis. At each bracket (A, B), the angle (θA, θB) between the interbracket axis and the slot axis is
measured. By convention, the absolute value of θB is always equal to or larger than the absolute value of θA. (b) The deflected wire placed in
the bracket is in equilibrium by the activation force system (blue arrows). (c) The deactivation force system (red arrows) is produced at each
bracket by deflected wire.
Geometry Classification and
Determination
If the two bracket centers are connected, the interbracket
axis is formed (Fig 14-6a). A line through each slot is the
slot axis. At each bracket (A, B), the angle (θA , θB ) between
the interbracket axis and the slot axis is measured. By
convention, θB is always equal to or larger than θA. The
ratio of θA/ θB defines the classification. When a wire
segment is placed into the malaligned bracket, it produces
fixed support and three unknowns: M A (moment), FA
(vertical force), and FH (horizontal force) (see Fig 8-7).
However, FH is not depicted (see Fig 14-6c) because we
ignored the friction, which produces horizontal forces.
The wire is in equilibrium, and hence the correct relative
force system for each geometry is the activation force
(Fig 14-6b). Then the forces and moments are reversed
to give the deactivation force system on each bracket for
the various geometries (Fig 14-6c). Because these forces
are equal and opposite (Newton’s Third Law), the typical
force diagrams in this section only show the deactivation
forces acting on the teeth. It should be remembered that
this correct force system was determined from the activation force diagram (see Fig 14-6b), which is the equilibrium diagram. The diagram in Fig 14-6c is not an equilibrium diagram (although all forces and moments sum
to zero) but a correct force diagram. For better understanding, the reader should initially make an equilibrium
diagram (forces on the wire) and then reverse the forces
for the deactivation force diagram (forces on the teeth).
The direction of each slot-interbracket angle is determined as depicted in Fig 14-6a. An example geometry in
Fig 14-6c has equal moments in the same direction on
each tooth, which is a Class I geometry. In a Class I geometry, θA/θB = 1 and M A/MB = 1. With these additional
boundary conditions (ratio of the moments) determined
by geometry, the force system becomes statically
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Forces from Wires and Brackets
FIG 14-7 Determination of each of the six geometries. The θA/θB ratio
determines the geometry. As bracket A rotates clockwise between
the black dotted slot axes, geometries from Class I to Class VI are
produced sequentially.
determinate. The maxillary arches in Figs 14-3 and 14-4
are also Class I geometries. Now let us describe each of
the six geometries in greater detail.
The two mandibular teeth in Fig 14-7 have a starting
reference position marked with parallel Class I geometry
black dotted lines. The bracket slots are parallel, not in a
straight line, and θA /θB = 1. This is a Class I geometry. If
the angle of bracket B in Fig 14-7 is kept constant and
tooth A is rotated around the bracket, the various geometries will be produced. The final rotation of the slot axis
as shown is a Class VI geometry. In a Class VI geometry,
as in a Class I geometry, the angles of the bracket are
equal but opposite in direction. The following description
for each bracket geometry first gives the ratio of θA /θB
and then the associated ratio of M A/MB. The forces are
then described. All forces for each geometry described
in the force diagrams are deactivation force systems acting
on the brackets from the wire. All diagrams show the
facial view of the mandibular arch; however, force and
moment relationships hold for all planes of space and for
both maxillary and mandibular arches. The force system
of each geometry is derived from small deflection beam
theory.
Class I geometry
In a Class I geometry, the bracket slots are parallel but
not in a straight line (Fig 14-8). The first step to identify
a specific geometry is to select the bracket with the greatest angle between the interbracket axis (black dotted line)
and the slot axis (black dotted line); in this case, either
angle (θA or θB) can be selected because they are equal.
Let us arbitrarily select the bracket A angle (θA ). The
direction of the premolar slot axis angle is positive (see
Fig 14-6a). The ratio of θA /θB in a Class I geometry is +1.0,
with both angles positive in this example. If we can identify the angle relationship between two brackets as a Class
316
I geometry, what is the force system that is produced?
The red arrows in Fig 14-8a give the forces acting on the
bracket. Equal moments (couples) in the same direction
act on bracket A and bracket B. The moments are positive
and are in a direction to rotate the teeth clockwise. In
addition, there is an intrusive force on bracket A and an
extrusive force on bracket B. The forces and moments in
this deactivation force diagram (see Fig 14-8a) are in
equilibrium. The ratio of the moments (MA /MB ) for a
Class I geometry is +1.0. This is an additional boundary
condition of the relative force system that allows us to
determine the forces and moments on both brackets if
only one force or one moment is measured and if the
distance between the bracket centers is also known.
Let us now apply this additional information derived
from the classification of bracket geometries. Even if
somehow we could measure the intrusive force on bracket
A as 100 g, for example, the force system acting on the
two brackets is still indeterminate. However, once we
have classified the bracket arrangement as a Class I geometry, the new boundary condition allows us to determine
the forces and moments on both brackets. The force
diagram in Fig 14-8a is the reverse of the equilibrium
diagram, where the forces and moments act on the archwire. Based on the equilibrium diagram (not shown),
note the following: If there is a 100-g intrusive force on
bracket A (Fig 14-8b), there must be a 100-g extrusive
force on bracket B (Fig 14-8c). Calculating the moments
from the vertical forces around any point gives us a
moment of –1,000 gmm. To meet the condition of equilibrium, each bracket will feel the same moment magnitude of 500 gmm because (MA /MB ) = 1 (Fig 14-8d).
Clinically, even measurement by a simple force gauge
of a single force acting on a tooth is also difficult because
the associated moment due to the slot angle may increase
or decrease the force. Therefore, as clinicians we are most
interested in the relative force system, and it is recognized
Geometry Classification and Determination
a
b
c
d
FIG 14-8 Class I geometry. (a) The bracket slots are parallel but not in a straight line. θA/θB = +1.0. MA/MB = +1.0. Also, equal and opposite
vertical forces (FA = –FB ) are produced. (b) The measured vertical force at bracket A is 100 g (intrusive), and the interbracket distance is 10
mm. (c) Bracket B feels an equal and opposite (extrusive) force of 100 g. (d) The total clockwise moment of 1,000 gmm to cancel the moment
from the vertical forces should be equally distributed to each bracket (500 gmm at each) because MA/MB = +1.0.
that because of the limits of our analysis (boundary conditions), our results are only approximations. Thus, bracket
A will have both an intrusive force and a clockwise
moment. To better understand the tooth movement, the
force system at the bracket should be replaced at the CR
for each tooth or segment, as we have done before in
chapter 3; however, this chapter only considers the force
system at the bracket from straight or bent wires. How
teeth move in respect to a force system is an entirely
different and complicated question (see chapter 9).
Another important consideration is that as the tooth
moves to a new position, the forces reduce, and the CRs
change. Note in our analysis that we have determined
only the initial or instantaneous force system; these are
the forces present as the patient leaves our office. Force
levels are initially the highest and will decrease during
tooth movement; also, geometries and moment-to-force
(M/F) ratios change and must be monitored during treatment. The partially or fully deactivated shape of the wire
may not be relevant because it is in the suboptimal stress
range and is usually replaced before full deactivation has
occurred. The initial force system is important, but so
are the proper bracket geometries or wire shapes as tooth
movement progresses. The reader should always be aware
that the description of the six geometries refers only to
the initial force system. The force system that is present
at a given bracket over time is very complicated, involving changing bracket geometries that depend on both
physics and biology.
Class II geometry
Let us now rotate bracket A slightly clockwise so that
θA /θB = +0.5, producing a Class II Geometry (Fig 14-9).
The force system is very similar to the Class I geometry.
Although bracket A has half (0.5) the θ angle of bracket
B, M A only drops to 0.8 times MB. For all practical
purposes, the force systems of Class I and Class II geometries can be considered identical. At this beginning of
the angle spectrum, a large change in bracket angulation
produces only a little change in force system.
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14
Forces from Wires and Brackets
FIG 14-9 Class II geometry. θA/θB = +0.5. The moment is not proportional to the amount of angulation. MA/MB = +0.8. This force system
is very similar to the Class I geometry.
a
b
FIG 14-10 Class III geometry. (a) θA/θB = 0. Remarkably, the force
system does not change radically. MA/MB = +0.5. Note that bracket
A feels moment even though the slot axis is not angulated. A little
experiment will show why a moment is present. (b) Place a wire in
bracket B (green wire). Lift it up with a single force to the level of the
center of bracket A. It will be seen that the wire cuts across bracket
A at an angle. (c) Therefore, it requires a counterclockwise moment
to insert the wire into bracket A.
c
Class III geometry
Now let us rotate bracket A slightly more so that the slot
axis lines up with the interbracket axis (Fig 14-10a). What
will bracket A and bracket B feel? Remarkably, the force
system does not change radically. As in a Class I geometry, moments are in the same direction except that M A
is half that of MB. Vertical forces may be reduced, but
they are still intrusive on bracket A and extrusive on
bracket B. We know that bracket B has the larger moment
because the greater slot angle is found there.
318
It is incorrect to assume that because the slot axis and
interbracket axis of bracket A coincide that bracket A
would feel no moment. A little experiment will show why
a moment is still present. Place a wire in bracket B; it now
lies apical to bracket A (green wire in Fig 14-10b). Lift it
up with a single force to the level of bracket A. It will be
seen that the wire cuts across bracket A with an angle
and requires a counterclockwise moment to insert the
wire into the bracket (Fig 14-10c). Note the different
shapes of the orange wire between Figs 14-10b and 14-10c.
Geometry Classification and Determination
FIG 14-11 Class IV geometry. θA/θB = –0.5. In comparison to Class
I and Class II geometries, the slot angle is in the opposite direction
(–). MA/MB = 0. This is a special equilibrium situation where only one
bracket feels a moment. It is the same force system as a free-end
cantilever where a force alone operates at the free end.
FIG 14-12 Class V geometry. θA/θB = –0.75. MA/MB = –0.4. The vertical
forces are very small because the sum of the two bracket couples
is very small.
FIG 14-13 Class VI geometry. θA/θB = –1.0. MA/MB = –1.0. Equal and
opposite couples are produced. No vertical forces are required or
present.
Class IV geometry
In Fig 14-11, we will rotate bracket A even more so that
θA /θB = –0.5. In comparison to Class I and Class II geometries, the slot angle is in the opposite direction (–). With
this configuration, no moment is present on bracket A.
This is a special equilibrium situation where only one
tooth feels a moment. Equal and opposite intrusive and
extrusive forces are evident on brackets A and B. It is the
same force system as a free-end cantilever where a force
alone operates at the free end. Note that the shape of the
wire in Fig 14-11 is identical to that in Fig 14-10b.
on bracket A is 0.4 times that on bracket B and in the
opposite direction. Equilibrium dictates small vertical
forces because the sum of the two bracket couples is very
small.
Class VI geometry
Finally, if bracket A is rotated so that θA is equal and
opposite to θB, it becomes a Class VI geometry (Fig 14-13).
Equal and opposite couples are produced. Because equal
and opposite moments place a wire in equilibrium, no
additional vertical forces are required or present.
Class V geometry
If bracket A is rotated slightly more (Fig 14-12) so that
θA /θB = –0.75, a Class V geometry is created. The moment
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14
Forces from Wires and Brackets
FIG 14-14 The bracket geometries as a continuum. Moving from left to right, the couple on the
left bracket becomes smaller until it disappears
in a Class IV geometry and reverses its direction
in Classes V and VI. Vertical forces are gradually
reduced and finally disappear in Class VI. Proportionately changing the bracket angulation
(θA/θB ratio) does not produce an equal change
in the moments between the two brackets. The
numbers are relative magnitudes of moments
and vertical forces.
Bracket Geometries as a
Continuum
The categorization of six geometries as discrete classes
is arbitrary. Figure 14-14 shows a continuum from Class
I, where equal unidirectional couples (clockwise) at each
bracket are placed in equilibrium by the vertical forces.
Moving to the right, the couple on the left bracket
becomes smaller until it disappears in a Class IV geometry. Vertical forces are gradually reduced. The couple on
the left bracket reverses in a Class V geometry, and finally
in the Class VI geometry right and left bracket couples
are equal with the vertical forces dropping to zero.
Proportionately changing the bracket angulation (θA /θB )
does not produce a proportional change in the moments
between the two brackets (MA /MB ).
The actual force systems from the wires are given as
the following:
FA = –FB = 6EI
(θ + θB )
L2 A
M A = 2EI (2θA + θB )
L
MB = 2EI (θA + 2θB )
L
where F is the vertical force, M is the moment, E is the
modulus of elasticity, I is the moment of inertia, L is the
length of the wire (interbracket distance), and θ is the
slot angle in radians. The vertical force linearly decreases
from 100% at Class I geometry and finally reaches 0% at
Class VI geometry (Fig 14-15a). Both moments decrease
320
linearly from 100% at Class I geometry and reach ±33%
at Class VI geometry; therefore, 3× larger activation (slot
angle) is allowed in Class VI geometry than in Class I
geometry (Fig 14-15b).
Suppose θA /θB = k; according to the formulas above, the
following ratio is derived:
(2k + 1)
MA
=
MB
(k + 2)
Figure 14-15c shows the plots of MA /MB versus k. Note
that the ratio of the moments (MA /MB ) between Class I
(k = 1) and Class II (k = 0.5) is only a small change. An
equal bracket geometry change at the right end of the
continuum, between Class V and Class VI, has a much
larger effect. This sensitivity to any change in bracket
angulation or bend in the wire is commonly observed
clinically. In particular, a Class V geometry with small
differences between opposite-directed couples on adjacent brackets is not only difficult to achieve with a single
wire but may also quickly change the couple ratios as the
teeth move. Therefore, clinically, a Class V geometry force
system is not usually applied with a single wire between
two brackets. Rather, a cantilever spring on an extension
wire may be preferred because it can deliver the force
system of a Class V geometry continuously (Fig 14-16).
Note that the intrusive force at the hook on the extension
wire of bracket A is replaced by an equivalent force system
(yellow arrows). This gives us the Class V geometry force
system. It is obvious now that more than one wire is
sometimes needed to optimize special force systems
involving two brackets. However, this chapter discusses
mainly two-bracket (teeth) relationships and the forces
produced by a single wire.
Bracket Geometries as a Continuum
a
b
FIG 14-15 (a) Vertical force linearly decreases with k. (b) Both moments decrease linearly with k, becoming ±33% at Class VI. (c) The
MA/MB versus k curve shows less change of moment differential
between Class I and Class II and radical change between Class V
and Class VI.
c
FIG 14-16 Class V geometry is very difficult to produce due to inherent sensitivity of the force system. A cantilever spring on an extension
wire may be preferred for a Class V force system because it can deliver
the force system of a Class V geometry continuously.
Increasing the interbracket distance
The basic relationships presented are independent of the
interbracket distance. Figure 14-17a shows a second
molar tipped mesially. The bracket geometry between
the second premolar and second molar is a Class III. The
force system on the teeth includes two moments in the
same direction: crown tip-back. We select the largest slot
axis as bracket B (ie, the second molar in this case). This
is the mirror image of the Class III geometry described
above. Therefore, the second molar has twice the moment
of the premolar. Vertical forces are needed to place the
wire in equilibrium. In Fig 14-17a, a –2,000-gmm moment
is applied to the second molar. This is fine; if space opening is not indicated, it should be tied to the anterior
segment. The anterior segment may have sufficient
anchorage potential to support the counterclockwise
moment from the arch at the premolar bracket and the
moment from the downward force at the premolar acting
at the CR of the anterior segment. However, the anterior
segment will tend to tip toward the second molar.
321
14
Forces from Wires and Brackets
a
b
FIG 14-17 The effect of interbracket distance. (a) A second molar is
tipped mesially. The bracket geometry between the second premolar
and second molar is a Class III geometry. (b) An increased interbracket distance produces the same geometry; however, the M/F ratio at
each bracket will be increased, which is beneficial if vertical force is
not wanted. (c) The replaced force system at the CR (yellow arrows)
is needed to predict the tooth movement.
c
The other major problem is that there is an extrusive
force on the second molar. This second molar eruption
could be an undesirable side effect; if a –2,000-gmm
moment is applied to the second molar, a 300-g extrusive
force acts to extrude the molar. By contrast, let us bypass
the premolars and place brackets on the canines. Let us
keep the geometry the same (Fig 14-17b). We have to
increase the wire cross section to achieve identical initial
magnitude of moments, which does not change the geometry. With the larger interbracket distance, the force drops
to 120 g. How do we know this? The reader should make
an equilibrium diagram and solve for the vertical force
magnitude. Compare the M/F ratio of the second molars
in Figs 14-17a and 14-17b (2,000 gmm / 300 g = 6.7 mm,
and 2,000 gmm / 120 g = 16.7 mm). The larger interbracket distance with its higher molar M/F ratio will
produce a smaller extrusive side effect on the molar.
Furthermore, the intrusive force, if delivered to a fully
bracketed anterior segment, will create no tipping or less
tipping toward the posterior.
In short, the ratios of bracket angles to moment ratios
that comprise the six geometries are independent of interbracket distance. However, M/F ratios at each bracket
will change as the interbracket distance increases if the
interbracket geometry is kept constant.
322
Force system equivalence at the CR
If we want to predict how a tooth will move, two steps
are considered. First, the force system at the bracket must
be determined by identifying the geometry. The vertical
force may not have a line of action passing through the
CR of a tooth. Note in Fig 14-17c that the vertical force
is 4 mm anterior to the molar CR. Second, the force
system at the bracket must be replaced with an equivalent
force system at the CR. This gives an extrusive force of
120 g and a moment of –2,480 gmm at the CR of the
second molar. Considering prediction of tooth movement
as a single step is a common mistake in orthodontic
research; when considering the effect of a given appliance,
two steps are required as described above. The accurate
force system must first be obtained; then the effect of this
force system on stress and strain in the periodontal ligament and bone must be studied. This second step involves
physics and biology.
Boundary conditions and limitations
It is important to point out the limitations of the classification and application of bracket geometries (ie, the
boundary conditions). The relationships are only valid
for relatively small deflections or activations. It is further
assumed that stress versus strain is linear in all other
alloys. The formulations describe the insertion of straight
Additional Methods for Visualization of Geometries
FIG 14-18 Additional methods for
visualization of geometries. The
θA/θB ratio defines the accurate
geometry; however, with a small θ,
it is difficult to identify the angles.
Instead, DA/DB or KA/KB is useful
and may be easier to visualize.
These ratios are identical if θ is
small enough. A Class IV geometry is depicted in the figure as an
example.
wires in brackets in any given plane. No play between the
wire and bracket is considered, and mesiodistal forces
are ignored. Nevertheless, if appropriately used, identification of bracket geometries can aid the clinician in
predicting the force system when a straight wire is
inserted and thus can allow for good estimates of the
clinical response. Calculations are based on beam theory
of traditional alloys, which follows Hooke’s law. With
newer alloys such as superelastic nickel-titanium (Ni-Ti)
wires, very large activations without permanent deformation are possible. In this situation, large deflection
beam theory should be applied, which is different from
small deflection beam theory.
Because of the small bracket-wire angles and small
interbracket distances, the practicality or accuracy of this
type of “reading” of the appliance could be questioned.
This is more a limitation of a continuous archwire than
the method of evaluation, because small angular changes
can produce large changes in the force system. The goal
of this analysis is usually to establish the relative force
system in clinical practice (and not to find the absolute
numbers) so that the orthodontic appliance can be better
designed and any undesirable side effects minimized. It
should also be noted that predictability can be improved
when there are fewer brackets and increased interbracket
distances (see Fig 14-17b).
Additional Methods for
Visualization of Geometries
Figure 14-18 is a Class IV geometry. How do we know it
is Class IV? By definition, the θA /θB ratio defines the accu-
rate geometry. As described earlier, the force system is
valid only when the deflection angles are small; however,
with very small angles, it is difficult to identify the ratio
of the angles between bracket A and bracket B. If θ is
small enough, θ = sin θ = tan θ. Therefore, instead of
θA /θB , the measurement of vertical distances (DA/DB) or
the location of the intersection (red dot, KA/KB or KA/L)
is useful and may be easier to visualize in some malocclusions. In particular, the location (red dot) of the intersection of the bracket slot axes can be a valuable aid. In
Fig 14-18, the bracket arrangement is a Class IV geometry because the slot axes are drawn so that they intersect
at one-third of the interbracket distance (L) from bracket
B (which has the larger angle):
KA/KB = 0.5 or KA/L = 0.33
The six classes are depicted in Fig 14-19. Note how the
intersection point (red dot) changes from one class to
another. In a Class I configuration, the slot axes are parallel and intersect at infinity (Fig 14-19a). Note that in a
Class II bracket arrangement, the intersection changes
from infinity to a point on the outside part of bracket B
(Fig 14-19b). A Class III geometry has the intersection
at the center of bracket B (Fig 14-19c). Intersections ranging from infinity (Class I) to the center of bracket B in
Class III make very little difference in the force system,
even though the red point travels a very long distance.
An intersection at one-third the distance from bracket
B produces a moment only on bracket B (Class IV,
Fig 14-19d). The high sensitivity of producing opposite
directions to the couples along with forces is present in
Class V geometry (Fig 14-19e), where K A/L = 0.43
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Forces from Wires and Brackets
a
d
b
e
c
f
FIG 14-19 A second method to determine the geometry. (a) Class I geometry. The slot axes (black dotted lines) are parallel. They intersect at
infinity (red dot). (b) Class II geometry. The extension of the slot axis intersects at a point outside the two brackets. (c) Class III geometry. The
slot axis intersects at bracket B. The change in intersection (red dot) from infinity (Class I) to the center of bracket B in a Class III geometry
makes very little difference in the force system, even though the red dot travels a very long distance. (d) Class IV geometry. An intersection
at KA/L = 0.33 or KA/KB = –0.5. There is a moment on bracket B only. (e) Class V geometry. KA/L = 0.43 or KA/KB = –0.75. The high sensitivity
of identifying the intersection is evident. Producing opposite-direction couples along with forces is difficult to achieve. (f) Class VI geometry.
The intersection meets at the center of the two brackets. KA/L = 0.5 or KA/KB = –1.
(or KA/KB = –0.75). With an intersection at the center
between the brackets, the equal and opposite couples of
the Class VI geometry are seen (Fig 14-19f ). With small
interbracket distances in a continuous archwire, a difference of intersection between 0.5 and 0.43 would be difficult to distinguish if required. Compare the intersection
points for Figs 14-19e and 14-19f.
324
The relationship between the patient’s occlusal plane
and the interbracket axis has no bearing on the force
system. Compare the three scenarios of different levels
and orientations of the interbracket axis in Fig 14-20. In
Fig 14-20a, the wire is inserted into bracket A first. Will
bracket B erupt to the level of bracket A? In Fig 14-20b,
the wire is inserted into bracket B first. Will bracket A
Additional Methods for Visualization of Geometries
a
b
c
d
FIG 14-20 (a to d) The effect of level and angulation of the interbracket axis. Parts a to c produce the same force system, as shown in part d.
The force system is independent of the level and angulation of the interbracket axis.
FIG 14-21 (a and b) Will the ideal archwire level the teeth from a to b? Initially,
teeth may not follow the shortest path
to their final positions, and they may assume a new malocclusion (geometry).
Later, even if teeth eventually align in a
straight line, the overall orientation (occlusal plane) may be changed.
a
b
intrude to the same level as bracket B? In Fig 14-20c, the
wire is placed into brackets A and B simultaneously. Will
the teeth rotate clockwise simultaneously around the
bracket? The force system is the same (Class I geometry)
in all three cases. Equal couples and vertical forces are
produced (Fig 14-20d); thus, the tooth movement is the
same.
If a straight wire is placed in malaligned brackets
(Fig 14-21a), will the teeth align in a straight line after
initial leveling (Fig 14-21b)? The answer is no. Initially
teeth may not follow the shortest path to their final positions, and they may assume a new malocclusion (geometry) quickly in the highest force range of the wire. Later,
even if teeth may eventually align in a straight line, the
overall orientation (occlusal plane) may be changed. Also,
final alignment may be very slow because the wire is
acting in the suboptimal force range. Refer to the shapedriven methods described for the lingual arch in chapter
12. Brackets cannot think and therefore do not understand that the straight wire is their goal.
A third method to help in identifying bracket geometry
is to use the bracket slot axes deviation (see D in Fig 14-18).
Instead of looking at the point of intersection, the vertical distance (D) is measured from the brackets to the
green wire. The ratio of the vertical distances (DA/DB ) is
the same as the ratio of θA /θB . Figure 14-22 shows Class
IV geometry because DA/DB = 0.5.
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14
Forces from Wires and Brackets
FIG 14-22 A third method to determine the geometry. The vertical
distance (D) is measured from the brackets to the slot axis of the
adjacent bracket. The ratio of the vertical distances (DA/DB) is the
same as θA/θB.
a
a
c
b
b
d
A deep bite case (Fig 14-23a) was leveled in the maxillary arch with the goal of intruding the central incisors
and maintaining the cant of the occlusal plane. The major
discrepancy was between the lateral incisor and the
central incisor. This is a 3D problem because the archwire
curves from canine to central incisor. Let us simplify our
analysis by considering only forces parallel to the midsagittal plane, since we are interested in the anchorage of
the posterior teeth. The 2D projection of the right central
and lateral incisor brackets (to the sagittal plane) is
326
FIG 14-23 A deep bite case was leveled in
the maxillary arch with the goal of intruding
the central incisors and maintaining the cant
of the occlusal plane. (a) The slot axes of the
right central incisor and lateral incisor brackets show a Class I geometry. (b) The maxillary arch has been leveled, but the deep bite
has not improved. As predicted, the occlusal
plane of the posterior teeth has steepened.
FIG 14-24 A buccal crossbite of the maxillary left second molar. (a and b) The brackets
of the first and second molars form a Class
I geometry (white dotted slot axes) before
leveling. (c and d) The first and second molars have been leveled, but the maxillary and
mandibular arches show a generalized arch
form discrepancy that is harder to treat than
the original malocclusion.
basically a Class I geometry. The lateral incisor in this
projection feels an extrusive force and a moment. Let us
anticipate what will happen with this wire. There is an
intrusive force on the central incisor, and intrusion would
definitely be favorable to achieve the goal. Now let us
check the posterior segment. From lateral incisor to first
molar, the wire is relatively passive so that the entire
posterior segment will move as a unit. The forces on the
lateral incisor are replaced by an equivalent force system
at the CR of the buccal segment (yellow arrows). It is
Bent Wires and the Six Geometries
a
b
FIG 14-25 A Z-bend with parallel arms produces a Class I geometry with both vertical forces and equal, unidirectional moments. (a) If the wire
is placed in one bracket (A), the origin of the vertical force is easily visualized. (b) When the wire is angled, it is clear that moments are at work.
obvious that a large moment steeping the plane of occlusion will be produced. Compare the before and after
intraoral photographs in Figs 14-23a and 14-23b. The
maxillary arch has been leveled, but the deep bite has not
improved. As predicted, the occlusal plane of the posterior teeth has steepened. Actually, the leveled arch is
harder to treat than the original malocclusion (see chapters 6 and 7). The same phenomena can happen in the
occlusal view. Figures 14-24a and 14-24b show a buccal
crossbite of the maxillary left second molar. The brackets
of the first and second molars form a Class I geometry
(white dotted slot axes before leveling). In Figs 14-24c
and 14-24d, the first and second molars have been leveled,
but the maxillary and mandibular arches show a generalized arch form discrepancy that is harder to treat than
the original malocclusion (see Fig 14-24d). Even without
calculations giving exact force quantities, the recognition
of different bracket geometries can help the clinician to
avoid undesirable side effects.
So far we have discussed placing a straight wire into
malaligned brackets and have recognized the pattern of
resulting forces. In this straight-wire application, the force
system is not under the operator’s control but rather is
geometry dependent. Also, the magnitude of the force
system decreases as the teeth move to their final positions, and new geometries may present themselves, leading to side effects. Because teeth are not directly moving
to their terminal desired positions, it may take a long
time to achieve alignment. In some cases, the correct
final positions are never reached. Therefore, we may need
to increase or modify the force system during the leveling
process by bending the wire. Let us now consider the
effects of placing strategic bends in a wire to control the
force system if brackets are well aligned. With this knowledge, we will be able to modify the force system creatively
during the alignment of malaligned brackets.
Bent Wires and the Six Geometries
Let us now place some basic shapes in a wire such as
Z-bends (step bends) and V-bends and insert the wire
into brackets that are fully aligned.
Force systems from Z-bends
A Z-bend (step bend) with parallel arms produces a Class
I geometry with both vertical forces and equal unidirectional moments (Fig 14-25). If the wire is placed in one
bracket, the origin of the vertical force is easily visualized
(Fig 14-25a). If the wires are angled as in Fig 14-25b so
that the center of each bracket touches the wire, it is
apparent that moments are at work. The wire can touch
the mesial and distal edges of the bracket; however, the
interbracket width in the description to follow is measured
from the center of each bracket in our ideal modeled
bracket.
The horizontal position (Z/L) of the Z-bend (Fig
14-26a) hardly changes the magnitude of the ratios of
force (Fig 14-26b) or moment (Fig 14-26c) because the
angle between the wire and the brackets (ie, the geometry) remains the same. A slightly decreased force near
Z/L = 0 is observed because a very small amount of wire
(0.35 mm of step) is added at the critical section, which
is at each bracket; this effect is clinically irrelevant. A
physical model with varying positions of Z-bends also
shows little difference in tooth movement (Fig 14-27). Of
course, increasing the vertical height of the step will
proportionately increase both force and moment. Because
couples are in the same direction, it is not surprising that
equilibrium dictates that very large vertical forces must
be evident. Figure 14-28 shows the force system from a
0.016-inch stainless steel wire (relatively flexible) with an
interbracket distance or wire length of 7 mm. With only
327
14
a
Forces from Wires and Brackets
b
FIG 14-26 (a) The horizontal position (Z/L) of the Z-bend barely
changes the force system. (b) The calculated magnitude of the force
remains almost unchanged. (c) The calculated magnitudes of MA
and MB remain almost unchanged. A slight decrease near Z/L = 0
is observed because very small amounts of wire (step) are added
at the critical section, which is next to each bracket. This effect is
clinically irrelevant.
c
a
b
FIG 14-27 (a to c) A physical model with varying positions of a
Z-bend shows little difference in tooth movement.
c
328
Bent Wires and the Six Geometries
FIG 14-28 (a) The calculated activation force system from a 0.016-inch
stainless steel wire with an interbracket distance of 7 mm. (b) The equal and
opposite deactivation force system
acting on the teeth. With only a very
small step (0.35 mm), 347 g of vertical
forces are delivered to the teeth.
b
a
a
FIG 14-29 A Class II geometry is produced by a
slight modification of the Z-bend. For all practical
purposes, Class II and Class I geometries can be
considered identical.
b
FIG 14-30 (a and b) A V-bend placed next to bracket A produces a Class III geometry.
Vertical displacement of the wire (D/L) also shows consistency of the bend angle or
bend location.
the very small amount of step (activation) of 0.35 mm,
347 g of vertical forces are delivered to the teeth.
Figure 14-28a is the activation force diagram, showing
that the wire segment is in equilibrium. Figure 14-28b is
a deactivation force diagram showing the forces on the
brackets (teeth). While it is based on Fig 14-28a where
all forces and moments are in equilibrium for the wire,
conceptually it is not an equilibrium diagram because
unbalanced forces are acting on the teeth.
As seen in Fig 14-26b and 14-28, placing bends in a
continuous archwire is very limited because the forcedeflection (F/Δ) rate is very high even with a relatively
flexible wire (0.016-inch stainless steel). Therefore, bends
are usually placed in appliances with large interbracket
distances.
Force systems from V-bends
A single V-bend can be placed in a wire segment, and its
mesiodistal location of placement can produce an array
of different forces and moments. The Z-bend described
in the previous section can also be modified so that one
arm is angled, and thus the lines between both arms
intersect to the right of bracket B, outside of the two
brackets (Fig 14-29). In this way, force systems approaching a Class II geometry can be obtained. For all practical
purposes, however, Class II and Class I geometries can
be considered identical because the force systems are so
similar.
A V-bend is made right next to bracket A (Fig 14-30a).
The wire, when angled, contacts the mesial and distal
edges of the bracket. Note that a bend at a bracket is
actually mesial to the bracket and not at its center.
However, the ideal bracket we modeled does not have
bracket width; therefore, bracket width is ignored, and it
is assumed that the bend is made at the center of bracket
A. Remember that the depicted width of the bracket slot
is only for demonstration purposes of the slot axis angle.
Vertical displacement of the wire (D/L) also shows the
consistency of the bend location (Fig 14-30b). Now let
us change the position of the V-bend (apex); the VB/L
ratio defines the bend position eccentricity (Fig 14-31a).
329
14
Forces from Wires and Brackets
a
b
FIG 14-31 (a) The location of the V-bend and the VB/L ratio define
the bend position eccentricity and the various geometries. (b) Force
versus VB/L. The vertical force decreases as the V-bend approaches
the center of the interbracket distance. (c) Moment versus VB/L. By
changing the horizontal position of the V-bend, the moment differential from Class III to Class VI geometry can be obtained.
c
Table 14-1 Force systems from Z-bends and
V-bends
Bend location
Z-bends (Z/L)
All
V-bends (VB /L)
0.00
0.10
0.20
0.33
0.43
0.50
MB/MA
Class (geometry)
1.00
I
0.50
0.41
0.29
0.00
–0.40
–1.00
III
III
III
IV
V
VI
The further from the center between the two brackets
that the bend is placed, the greater the vertical force
(Fig 14-31b). The plot in Fig 14-31c shows that by changing the horizontal position of the V-bend, the entire
moment differential from Class III to Class VI geometry
can be obtained. This is summarized in Table 14-1 for
both Z-bends and V-bends. Figure 14-32 shows the position for the V-bend placement, the VB/L ratio, and its
class geometry. A Class III geometry is produced by a
bend at the left tooth (actual bends at attachments are
made proximal to the bracket) (Figs 14-32a and 14-32b).
330
A Class IV geometry, where one tooth has no moment,
is found at a VB/L ratio of 0.33 (Figs 14-32c and 14-32d).
Equal and opposite couples (Class VI geometry) are found
at a VB/L ratio of 0.5 (Figs 14-32e and 14-32f ). A Class
IV geometry is again found at aVB/L ratio of 0.66 (Figs
14-32g and 14-32h), with opposite direction to the
moment on the other side (see Figs 14-32c and 14-32d).
A Class V geometry bend is found between a Class IV
position and a Class VI position. For small interbracket
distances, this is obviously very difficult to place (Figs
14-32i and 14-32j).
Bent Wires and the Six Geometries
a
b
c
d
e
f
g
h
i
j
FIG 14-32 (a and b) Class III geometry is produced by a bend at the left tooth. VB/L = 0. Half the amount of moment acting on bracket A
acts on bracket B, with the same direction. (c and d) Class IV geometry, where one tooth has no moment, is found at VB/L = 0.33. Bracket B
feels no moment but a single force. (e and f) Class VI geometry is found at VB/L = 0.5. Equal and opposite couples are produced without any
vertical forces. (g and h) Class IV geometry found again at VB/L = 0.66, with opposite direction to the moments in c and d. (i and j) A Class V
geometry bend is found between a Class IV position and a Class VI position. Note that this position requires great accuracy to achieve, and
hence, clinically this force system is difficult to obtain.
331
14
Forces from Wires and Brackets
FIG 14-33 Force systems from three VB/L ratios. The interbracket distance is 7 mm, and the
height of the V-bend is 0.35 mm. (a) Centered
V-bend, with VB/L = 0.5; equal and opposite
couples. (b) VB/L = 0.29; a larger moment exists at MA. (c) VB/L = 0.14; forces and moments
are much greater than in parts a and b. In parts
b and c, the wire would permanently deform.
a
b
c
a
b
c
d
Three examples are given in Fig 14-33 of the actual
magnitude of V-bends from a 0.016-inch stainless steel
wire: centered (Fig 14-33a), VB /L = 0.29 (Fig 14-33b), and
VB /L = 0.14 (Fig 14-33c). Note that small changes in bend
position can give large changes in moments and forces.
Placement is also a continuum and is not limited to
discrete classes. The wires with the highest forces and
moments are at the Class III part of the spectrum, assuming that the bend is made within the elastic range (Δ =
0.35 mm). If superelastic Ni-Ti wires are used, moments
may be too low in a Class VI geometry to produce adequate
equal and opposite couples for pure rotation of each
tooth. Also, the amount of activation may be too small
332
FIG 14-34 Variation of the V-bend. (a) The
V apex would extend occlusally and may interfere with the occlusion from the opposing
tooth. (b) A truncated V-bend also produces
equal and opposite couples. (c) If bending of
a wire is not possible, an angulated bracket
produces the same effect. (d) A curvature with
a segment of a circle also produces couples
alone. One advantage of a smooth curvature
over a V-bend is that it is not position sensitive.
to reach the superelastic zone. Ni-Ti wires can differ from
traditional linear materials in each of the geometries.
A V-bend is a simple appliance adjustment and an
important tool for the clinician. However, it is not the
only simple bend available and may not always be practical. For example, a V-bend can be placed in a patient
requiring equal and opposite couples. The V apex would
extend occlusally and may interfere with the occlusion
from the opposing tooth (Fig 14-34a). An alternate solution is a truncated V that also produces equal and opposite couples (Fig 14-34b). The truncated V-bends are
composed of two V-bends at each side; however, they
show the same force system provided that the angulations
Consistency and Inconsistency
FIG 14-35 Consistent force system. The maxillary second molar is
rotated mesial out and buccally displaced. The arrows in the green
circle are the desired force system. The geometry is Class II, and the
force system produced is depicted with red arrows, which are in the
desired direction.
FIG 14-36 Inconsistent force system. The arrows in the green circle
are the desired force system. The force system from the given Class
III geometry is depicted with red arrows. The moment is correct, but
the force is acting in the opposite direction. As shown, a straight wire
does not always produce the desired force system.
formed by the bracket slot and the wire are the same.
This works exactly the same as rotating the bracket using
a straight wire without any bends (Fig 14-34c), which is
discussed later in this chapter in the section on virtual
bracket repositioning.
Another solution is a curvature with a segment of a
circle (Fig 14-34d). If a straight wire is loaded with equal
and opposite couples, a segment of a circle is formed
(refer to the simulation procedure in chapter 12). If a
curvature (constant radius) is placed into an archwire, it
is possible to produce couples alone. One advantage of
a smooth curvature over a V-bend is that it is not position
sensitive. The wire can be moved mesiodistally without
changing the force system.
Theoretically, Figs 14-34a to 14-34d all produce the
same force system; however, the effects are different from
each other clinically. The interbracket distance in Figs
14-34b and 14-34c will be increased, but it will be
decreased in Figs 14-34a and 14-34d due to friction
between the bracket slot and the wire. This concept is
further discussed in chapter 16.
It should be remembered that Z-bends and V-bends
only describe the initial force system if the brackets are
in good alignment. As teeth move, new geometries are
created; therefore, the force system can be continually
changing with the same bend.
the moment at each bracket to be in the correct direction.
If every direction is compatible with the desired force
system, the force system is called consistent. If only some
forces or moments but not all are in the desired direction,
the force system is considered inconsistent. Consistency
usually refers to a desired force system where both forces
and couples are required. The M/F ratio may be adequate
or not. Sometimes, however, only a force or a couple is
required. If a side effect force or couple is present, the
force system can also be considered inconsistent. Many
times a poor response to straight-wire alignment can be
explained by inconsistency. Sometimes inconsistency is
inevitable, but most of the time it can be avoided by “reading” a wire. Consider the maxillary second molar in
Fig 14-35 that is rotated mesial out and buccally displaced.
The first molar position is correct and is our reference
(the rest of the arch is the anchorage). We want the
moment to be mesial in (counterclockwise) and the force
to be in a lingual direction; the goal force system is found
in the green circle (see Fig 14-35). The intersection of the
slot axes (red dot) demonstrates a Class II geometry. The
second molar tube will feel a lingual force and a moment
to rotate the second molar mesial to the lingual (red
arrows). This is what is desired and what a straight wire
will reliably produce because there is consistency. Unfortunately, however, a straight wire does not always produce
the force system that we want.
In the patient in Fig 14-36, the first molar is in its
correct position, but the second molar is rotated mesial
in. The arrows in the green circle are the force system
required to move the second molar into its desired position. The geometry between the two molars is a Class III.
The second molar will feel a favorable moment that will
Consistency and Inconsistency
An archwire inserted into a series of malaligned brackets
will produce both forces and couples at the brackets. To
reach our treatment goals, we want both the force and
333
14
Forces from Wires and Brackets
a
b
FIG 14-37 (a) The maxillary second molar is tipped buccally, and a straight wire is placed. Only a lingual force (green circle) is required. The
geometry is Class I. The lingual force on the second molar is correct; however, the moment produced (curved red arrow) is unnecessary.
Therefore, it is an inconsistent force system. (b) Simply removing the second molar tube will make this force system consistent.
FIG 14-38 An inherently difficult clinical situation with a Class V
geometry. The force system acting on the second molar is consistent;
however, Class V geometry is difficult to achieve because of the
sensitivity of V-bend placement.
rotate it mesial out, but there is also a wrong direction
of buccal force that could produce a buccal crossbite;
hence, the force system is inconsistent because only the
moment is in the correct direction. A straight wire would
improve the rotation but may create a buccal crossbite.
The clinician could then step the wire lingually, but that
would only make the rotation worse. A better solution
to rotate the second molar would be an elastic on the
lingual (see chapter 12).
Is the two-tooth geometry in Fig 14-37a consistent or
inconsistent for the second molar? We need a lingual
force only (green circle). The reference first molar position
is correct. The geometry is Class I. The lingual force on
the second molar is correct; however, the moment
produced (curved red arrow) is unnecessary because it
would rotate the second molar mesial to the lingual. Diagnosis of class geometry can be helpful in alerting the
orthodontist to a potential side effect associated with an
inconsistent force system. Simply removing the second
molar tube will convert the force system from inconsistent to consistent. A lingual force from a cantilever to
move the molar lingually without using a tube on the
second molar is a better solution (Fig 14-37b). This solution is discussed in greater detail later in this chapter.
334
An inherently difficult clinical situation is a Class V
geometry. If a lingual force with a clockwise moment is
required (green circle in Fig 14-38), it is problematic for
several reasons, even though it is theoretically possible.
We have seen that a Class V geometry is difficult to
achieve because of the sensitivity of V-bend placement.
Figure 14-38 shows a second molar positioned to the
buccal and rotated mesial in. The first molar position is
correct. The moment on the second molar will rotate the
molar mesial out, and the force will move the molar to
the lingual. Both moment and force are consistent;
however, this Class V geometry quickly disappears as the
tooth moves, and then the geometry becomes inconsistent. Moreover, the magnitude of force is very low, and
the magnitude of moment acting on the active tooth
(second molar) is much lower than that acting on the
reciprocal unit (first molar), which is already in the
correct position. Therefore, a Class V geometry is often
theoretically consistent but practically inefficient.
It is not surprising that a straight wire in malaligned
brackets does not always give a force system that coincides with our goals. If we are lucky, both force and
moment will be in the correct direction and display consistency. If not, the problem is more serious than what
Consistency and Inconsistency
FIG 14-39 (a) Cantilever test to diagnose consistency. At the mandibular right canine, the desired force system is shown in the green
circle. Place an imaginary distal bar or wire in the bracket slot and
direct the force (red arrow) as desired. The replaced equivalent force
system at the CR (yellow arrows) shows both force and moment
acting in the correct direction; therefore, the wire to the canine from
the posterior teeth will be consistent on the canine. (b) If we wanted
to extrude the canine while keeping the same direction of moment
(green circle), an inconsistency would arise because the force at the
end of the bar (red arrow) produces the correct direction of force but
a moment in the wrong direction.
a
can be solved by just a straight-wire application. It means
that a straight wire has an unavoidable, fundamental
problem and that creative bending or loop design of the
wire based on sound principles is needed.
Is there a fast and easy way to diagnose inconsistency?
Yes. Use the cantilever diagnostic test. Figure 14-39a
shows a mandibular right canine that requires an intrusive force and a tip moment to move the crown distally
from the adjacent tooth. Our goal force system is shown
in the green circle. Place an imaginary distal bar or wire
in the bracket slot. Direct the force (red arrow) in the
direction that is needed, and then check the direction of
the moment at the bracket or at the CR (yellow arrows).
In Fig 14-39a, both force and moment are acting in the
correct direction (arrows in the green circle and yellow
arrows coincide). The wire to the canine from the posterior teeth will be consistent on the canine. But what will
happen if we want to extrude the canine while keeping
the same direction of moment (arrows in the green circle
in Fig 14-39b)? The red force at the end of the bar is now
directed occlusally. The force direction is correct, but the
moment is incorrect and would tip the canine to the
mesial; therefore, an inconsistency exists.
Let us consider the best approach to leveling a high
central incisor that is tipped mesially (Fig 14-40a).
Suppose a continuous archwire is used. The force system
at the left central incisor is affected by the adjacent teeth;
therefore, it is necessary to compare the force system as
a series of two teeth: the right central incisor with the left
central incisor (Fig 14-40b), and the left central incisor
with the left lateral incisor (Fig 14-40c). Now we have
moved from a two-tooth to a three-bracket analysis.
Enlarged views of the brackets are shown in Figs 14-40d
and 14-40e, and the arrows in the green circle show the
force system needed. The right side of the misplaced
central incisor displays a Class II geometry (see Fig
b
14-40d), and the left side of the central incisor has a
Class III geometry (see Fig 14-40e). Both force and
moment at the left central incisor are consistent from the
right central incisor (see Fig 14-40d). However, the force
system from the left lateral incisor is inconsistent. The
force is correct, but the moment is in the wrong direction.
In other words, the wire to the left of the offending incisor is being less helpful. One solution to improve the
force system is to cut the wire (Figs 14-40f and 14-40g).
Figure 14-40f is consistent in that both force and moment
are in the proper direction, but Fig 14-40g is inconsistent.
Therefore, the correct cut of the wire is shown in Fig
14-40f.
A similar geometry in the occlusal view is demonstrated in Fig 14-41. A second premolar has erupted to
the buccal and is rotated mesial out. It is more efficient
to cut a continuous archwire distal to the second premolar to produce a Class II geometry force system (Fig 14-41a).
A wire coming from the distal of the premolar produces
a lingual force and a clockwise moment, which is inconsistent (Fig 14-41b).
It is common for a second molar to erupt rotated mesial
in and moved out to the buccal. A Class II geometry
between the first and second molars (first molar position
is correct) is evident in Fig 14-42. The force system needed
is shown in the green circle (Fig 14-42a). The force system
is inconsistent; the force direction is correct, but the
moment is incorrect, rotating the second molar mesial
in. One solution is to apply only a lingual force from the
facial wire, leaving the bracket off the second molar
(Fig 14-42b). The needed moment (green circle) for rotation is picked up by an intra-arch elastic (Fig 14-42c);
hence, two mechanisms are employed to avoid inconsistency. The first molar can be stabilized by a passive lingual
arch, which effectively prevents side effects from the
reciprocal force system.
335
14
Forces from Wires and Brackets
a
b
c
d
e
f
g
FIG 14-40 (a) A high central incisor is tipped mesially. It is necessary to compare the force system as a series of two teeth: the right central
incisor with the left central incisor (b), and the left central incisor with the left lateral incisor (c). (d and e) In the enlarged views, each green
circle shows the force system needed, and the red arrows show the force system from the straight wire. The right side of the misplaced central
incisor displays a Class II geometry with a consistent force system (d), but the left side of the central incisor has a Class III geometry with an
inconsistent force system (e). Cutting the wire distal to the left central incisor (f) makes the force system consistent, while cutting the wire
mesial to the left central incisor (g) makes the force system inconsistent.
a
b
FIG 14-41 (a) A second premolar has erupted to the buccal and is rotated mesial out. It is more efficient to cut a continuous archwire distal to
the second premolar to produce a Class II geometry force system. (b) A wire coming from the distal of the premolar produces a lingual force
and a clockwise moment, which is an inconsistent Class III geometry.
336
Consistency and Inconsistency
a
b
FIG 14-42 A Class II geometry between the first and second molars.
(a) The force system needed is shown in the green circle; however,
the force system is inconsistent. (b) One solution is to remove the
bracket from the second molar. (c) The needed moment (red curved
arrow) for rotation is produced by an intra-arch elastic.
c
a
b
FIG 14-43 (a) The tip-back spring produces the desired tip-back moment, but the vertical extrusive force is not indicated. (b) The distal end
of the continuous archwire is placed on top of the molar tube so that an intrusive force can be delivered.
Another example of an inconsistency problem solved
by two wires instead of one is the tip-back spring in
Fig 14-43. The tip-back spring produces the desired
tip-back moment, but the vertical extrusive force is not
indicated (Fig 14-43a). How can that inconsistency be
removed? One way is to lay the distal end of the continuous archwire on top of the molar tube so that an intrusive force can be delivered (Fig 14-43b). The unwanted
vertical forces can be canceled by two wires.
In Fig 14-44, the second premolar requires an intrusive
force and a moment in a clockwise direction (arrows in
the green circle). A straight wire is inconsistent because
a counterclockwise moment would be produced, associated with a Class II geometry (Fig 14-44b). Usually loops
are placed in arches to lower the F/Δ rate and to increase
the range of action. A more important reason to design
a specific loop is to change the force system (ie, to make
consistency possible). Figure 14-45 shows a rectangular
loop. By placing sufficient wire mesial to the premolar
bracket (Fig 14-45a), a consistent force system is produced
so that the tooth intrudes and the root moves distally. In
other words, it can maintain a Class V geometry force
337
14
Forces from Wires and Brackets
FIG 14-44 (a) The second premolar requires
an intrusive force and a moment (green circle). (b) A straight wire is inconsistent because
a counterclockwise moment would be produced, associated with a Class II geometry.
a
b
a
c
b
d
e
f
g
system continuously, which is practically impossible with
a straight wire connecting adjacent brackets as described
earlier. The rules of the six geometries do not apply where
loops are incorporated because the force systems from
six geometries are based on the assumption that the
straight wire is deflected very little.
338
FIG 14-45 (a) Loops can be placed to make consistency possible. (b) Proper distribution of the
wire mesial and distal to the premolar gives only
a force without a moment if the anterior arm is
kept parallel to the bracket. (c) This unique design
of the rectangular loop has a point of dissociation where only a single force is produced (red
dot). (d and e) A Class IV geometry force system
is delivered continuously during deactivation. (f
and g) The point of dissociation exists in three
dimensions.
Wire curvature can make a difference. Some literature
has applied the six geometries to explain the force system
from various loop designs or lingual arches; however, this
is not always a correct application of the six geometries.
With a rectangular loop, for example, proper distribution
of the wire mesial and distal to the premolar gives only
Three-Bracket Segments
a
b
FIG 14-46 (a) A patient presented with a canine with poor axial
inclination, with the root forward, and a continuous full archwire was
placed. (b) The geometry between the lateral incisor and canine was
Class III, and the canine-premolar geometry was also Class III. (c)
The resultant force system.
c
a force without a moment if the anterior arm is kept
parallel to the bracket (Fig 14-45b). This unique design
of the rectangular loop has a point of dissociation where
only a single force is produced (red dot in Fig 14-45c). Its
exact location depends on the configuration of the loop,
so it is not presented here. Instead, it is recommended
that the reader fabricate a loop and find a point where
parallel activation is found using a simple experimental
trial (Fig 14-45c). A Class IV geometry force system is
delivered continuously during deactivation (Figs 14-45d
and 14-45e). Interestingly, this point of dissociation exists
not only during occlusogingival activation but also during
labiolingual activation—thus, in three dimensions
(Figs 14-45f and 14-45g).
Three-Bracket Segments
Let us now extend our analysis from two brackets to three
brackets. A patient presents with a canine with poor axial
inclination, with the root forward (Fig 14-46a). This can
be observed in either nonextraction cases or commonly
in extraction therapy where the canine has tipped during
retraction. We will analyze the force system by selecting
two teeth at a time: the lateral incisor and the canine, and
the canine and the first premolar. To simplify our discussion, the canine and lateral incisor will be considered as
acting in one plane. The geometry between the lateral
incisor and the canine is Class III (Fig 14-46b). The
canine-premolar geometry is also Class III (see Fig 14-46b).
No horizontal interaction between brackets is assumed.
A number of simplifying assumptions are made to facilitate our discussion, because only the general force system
is wanted. More accurate specifications (boundary conditions) can give us a more accurate answer, if required.
Now we add the forces and couples from each bracket;
the summation for each tooth is shown in Fig 14-46c.
The main side effect on the incisors is extrusion, leading
to an increase in deep bite. Note that the dotted black
canine bracket slot axis lies occlusal to the incisors. The
canine is fine receiving a large moment (along with a
distal tie force) needed to move its root posteriorly. The
vertical forces on the canine tend to cancel each other
out.
What will happen to the posterior segment? The first
premolar bracket feels an intrusive force and a counterclockwise moment. Therefore, the posterior segment
tends to tip mesially with an overall development of a
reverse curve of Spee in the maxillary arch. The analysis
of the three-bracket geometries tells us that a straight
wire will produce many unacceptable side effects (also
known as the rowboat effect; see chapter 13). Can we bend
the continuous archwire so that no side effects are
produced? Unfortunately, the answer is no. The geometry
may be changed, however. Side effects are concentrated
on adjacent teeth such as the first premolar and lateral
339
14
Forces from Wires and Brackets
a
b
c
d
a
b
incisor because a flexible continuous archwire is not
capable of properly distributing the forces to eliminate
the side effects.
One solution for correcting the canine axial inclination
is to fabricate a continuous archwire with a canine bypass
(Fig 14-47a). This archwire is relatively rigid, which allows
the remaining teeth to move as a unit and hence offers
good anchorage. The canine requires a distal force above
the CR to rotate around the bracket or cusp tip. The
replaced equivalent force system at the bracket is depicted
in yellow with a distal force and a counterclockwise
moment (Fig 14-47b). The moment to the canine comes
from a cantilever root spring (Fig 14-47c). A downward
force produces the correct moment direction for canine
root movement. In addition, the distal force is provided
by the full bypass arch and a canine tieback to complete
the force system. The treatment result is shown in
Fig 14-47d. The downward force may be beneficial in this
case, and because the cantilever is long, the force is small.
340
FIG 14-47 A solution for correcting the
canine axial inclination. (a) A continuous
archwire with a canine bypass is placed,
which offers good anchorage. (b) The canine requires a distal force above the CR
to rotate it around the bracket or cusp tip.
The replaced equivalent force system at the
bracket is depicted with yellow arrows. (c)
The moment to the canine comes from a
cantilever root spring. The eruptive force is
prevented by the continuous canine bypass
archwire, and the distal force is provided by
a canine tieback. (d) Treatment result.
FIG 14-48 (a and b) If an increased downward extrusive force is required, the length
of the spring can be shortened.
If an increased downward extrusive force is required, the
length of the spring can be shortened (Fig 14-48).
A β-titanium spring is shown in Fig 14-49. Because
this type of spring is a cantilever with a free end, it is easy
to calibrate. Typically, the bypass arch should touch the
occlusal edge of the canine bracket to prevent any extrusion of the canine. As depicted, with a large occlusal
bypass step away from the bracket, the canine would
extrude, which is a desirable effect for a canine that has
not fully erupted.
The patient in Fig 14-50a seems to require very simple
treatment, perhaps a few alignment arches. Unfortunately, the root of the maxillary left canine is too far mesial
and needs correction. A straight continuous archwire
could lead to a deep bite malocclusion (Fig 14-50b). In
this case, the canine inclination discrepancy was only on
the left side; the leveling sequelae would also include a
midline deviation and an occlusal plane cant from the
frontal view. In short, leveling performed without consideration of forces would make a simple case difficult and
Three-Bracket Segments
a
b
FIG 14-49 (a and b) A β-titanium cantilever spring with a free end. Typically, the bypass arch should touch the occlusal edge of the canine bracket to prevent any extrusion of the canine; however, a step away from the bracket may be
indicated if the canine needs extrusion.
FIG 14-50 (a) The root of the maxillary left
canine is too far mesial. (b) A straight continuous arch could lead to the formation of
deep bite. (c) The maxillary canine bypass
arch was placed, and a cantilever spring was
inserted. (d) The final finishing arch was fabricated after the canine root was corrected.
a
b
c
d
extend the treatment time. The initial archwire was a
leveling archwire that did not include the left canine, on
which no bracket was bonded. The second maxillary
archwire was a bypass arch designed to correct the canine
inclination (Fig 14-50c). Then a final finishing archwire
was placed (Fig 14-50d). In this way, a major adverse side
effect was avoided.
Using a flexible wire to treat a high canine requiring
extrusion to the level of the arch can create adverse side
effects, particularly in the adjacent teeth (Fig 14-51a). Let
us carry out a three-bracket analysis of a straight wire
inserted in the malocclusion. The net force system
(Fig 14-51b) on the canine is consistent, but on both sides
of the canine a Class I geometry is observed. An intrusive
force is exerted on the incisor and the first premolar.
Unless there is an open bite, the intrusive force on the
incisor should not be a problem (see Fig 14-51b). The
moment at the CR of the incisors should be small, and
little axial inclination change should be expected because
the incisor CR is located distal to the upward vertical
force, which may cancel the clockwise moment acting
on the incisor (Fig 14-51c). On the other hand, the intrusive force and moment at the first premolar produce a
large counterclockwise, tip-forward moment, and as the
first premolar rotates and intrudes, the rotation of the
whole posterior segment will follow (Fig 14-51d).
Figure 14-52 shows a patient with a high canine. A
straight Ni-Ti continuous archwire was placed; note,
however, that as the canine extruded favorably, the posterior teeth predictably tipped mesially (Figs 14-52a and
14-52b). Figure 14-52c shows the estimated replaced
force and moment directions calculated at the CR of all
of the posterior teeth. Note that the moment is very large.
In contrast, the bypass arch technique can be very useful
in the treatment of the high canine. Figure 14-53 shows
a bypass arch for stability along with a second continuous
Ni-Ti active archwire from molar auxiliary tube to molar
tube. The Ni-Ti archwire is initially tied just at the midline;
its force can be increased by tying further distally on the
anterior segment.
341
14
Forces from Wires and Brackets
a
b
c
d
FIG 14-51 A high canine requiring extrusion. (a) A Class I geometry is observed on both sides of the canine. (b) The resultant force system
(red arrow) on the canine is consistent, but a flexible wire can create an adverse side effect, particularly on adjacent teeth. (c) The moment at
the CR of the incisors should be small, and little axial inclination change should be expected. (d) On the other hand, the intrusive force and
moment at the first premolar (yellow arrows) produce a large counterclockwise rotation, tipping the posterior teeth mesially.
FIG 14-52 (a) In a patient with a high canine, a straight Ni-Ti continuous archwire
was placed. (b) The canine extruded favorably; however, the premolar and lateral
incisor predictably tipped toward the canine.
(c) The enlarged view shows the estimated
replaced forces and moments calculated at
the CR of all of the posterior teeth. The moment is very large.
a
b
c
342
Making Bends Creatively in Continuous Archwires
a
b
FIG 14-53 (a and b) A bypass arch for stability along with a second continuous Ni-Ti active arch from molar
auxiliary tube to molar tube.
FIG 14-54 A combination of a bypass arch and a
cantilever spring. A cantilever with a helix and hook
at the canine (green ligature or O-ring) delivers only
a force to the canine.
Another approach for aligning the canine is the combination of a bypass arch and a cantilever spring from the
molar auxiliary tube (Fig 14-54). A cantilever with a helix
and hook at the canine (green ligature or O-ring) delivers
only a force. It is usually preferable not to place the wire
in the canine bracket slot, where a moment may be inadvertently present. Although anchorage is good with a stiff
bypass arch, the extrusion force on the canine should be
maintained at a low level. Heavier forces may elastically
deform the base arch (bypass arch) and allow adjacent
tooth tipping, or the entire group of anchor teeth may
rotate counterclockwise even with a very rigid bypass
archwire.
Look again at the patient in Fig 14-2. What is the
bracket geometry between the lateral incisor and the
central incisor? Does the force system between those two
brackets help correct the lateral incisor inclination?
Making Bends Creatively in
Continuous Archwires
We have discussed how a straight wire works in malaligned
brackets and how bends alter the force system in well-
aligned brackets. Now we will learn how to bend the wire
creatively in malaligned brackets to generate the specific
force system we want.
In Fig 14-55a, the mandibular second molar is tipped
forward. Two possible goals—to open space or to maintain or close space—both require a counterclockwise,
tip-back moment. The bracket geometry between the
second premolar and second molar brackets is a Class
III. The force system produced by a straight wire gives a
moment in the correct direction on the second molar;
however, unwanted moment acting on the anterior
segment will tend to tip into the edentulous site. Also,
the unwanted vertical forces produce undesirable side
effects. The second molar will extrude (Fig 14-55b).
A better force system is to deliver equal and opposite
couples by a Class VI geometry force system (Fig 14-56a).
We could reposition the bracket on the second premolar
and angle it so that a Class VI geometry exists (Fig 14-56b).
We can make a Class VI geometry in two ways: (1)
Make the wire passive and then place a V-bend in the
center between the brackets, or (2) follow the virtual
bracket repositioning approach.
343
14
Forces from Wires and Brackets
a
b
FIG 14-55 (a) The mandibular second molar is tipped forward. The bracket geometry between the second premolar and the second molar
is a Class III. (b) The force system produced by a straight wire gives a moment in the correct direction on the second molar; however, an
unwanted moment acts on the anterior segment, and the second molar will extrude.
a
b
FIG 14-56 (a) A better force system is to deliver equal and opposite couples by a Class VI geometry force system. (b) We could reposition
the bracket on the second premolar and angle it so that a Class VI geometry exists.
1
2
a
1
2
b
FIG 14-57 The first method to make a Class VI geometry from an undesirable existing geometry is by placing a single V-bend. (a) First make
the wire passive. (b) Then place a V-bend in the center between the brackets.
Placement of a single V-bend in the
center
The wire is contoured so that it can be passively inserted
into the existing malocclusion (Fig 14-57a). Then a single
V-bend is placed at the middle of the span between the
second molar and second premolar brackets (red circle
in Fig 14-57b). If the span is large enough, a curvature
(segment of a circle) could be formed instead of a V-bend
or truncated V-bend. Unlike a V-bend, a curvature does
not depend on accurate mesiodistal positioning.
344
Virtual bracket repositioning
This method is equivalent to repositioning the bracket
with a changed angulation to create a Class VI geometry.
Let us call this a “virtual” bracket repositioning. Here it
is usually not necessary to make the wire passive in the
second-order view. One or two V-bends are placed immediately next to the brackets as if the brackets were repositioned so that if the wire is separately placed in the
molar and premolar brackets, the wire would cross at the
center between the two brackets (Fig 14-58). The clinical
procedure follows.
Making Bends Creatively in Continuous Archwires
FIG 14-58 The second method is by virtual bracket repositioning.
The V-bends are placed immediately next to the brackets as if the
brackets were repositioned so that if the wire is separately placed in
the molar and premolar brackets, the wire would cross at the center
between the two brackets.
1
2
1
2
1
2
a
FIG 14-59 The clinical procedure for virtual bracket repositioning. (a) A short wire
segment is placed into the molar tube, and
the vertical displacement (DA ) is measured.
(b) A bend distal to the premolar bracket
is formed so that this presents the same
amount of vertical displacement (DA = DB)
from the second molar bracket. (c) The arms
will now cross at the center of the two brackets, giving us a Class VI geometry. (d) As
the molar changes inclination, the geometry
is changed to Class IV. (e) As the second
molar needs more rotation, it becomes necessary to place a bend mesial to the molar
to increase its angle to the interbracket axis
(black circle).
b
1
2
c
2
3
1
2
1
3
d
In Fig 14-59a, a short wire segment (green) is placed
into the molar tube. The second molar bracket is already
angled in respect to the interbracket axis; therefore, it
may not be necessary to place a bend mesial to the tube
if the angle is sufficient. The vertical displacement (DA )
from the second premolar bracket is measured. In
Fig 14-59b, a bend distal to the premolar bracket has now
been formed so that this extension of the anterior wire
presents the same amount of vertical displacement (DB )
as the second premolar bracket (DA ). The arms will now
cross at the center of the two brackets, giving us a Class
VI geometry (Fig 14-59c). We can elect to place only one
bend next to the premolar because the tipped molar may
1
2
e
give sufficient moment. As the molar changes inclination,
the wire projections no longer cross in the center, and
the geometry is changed to a Class IV (Fig 14-59d). As
the second molar needs more rotation, it becomes necessary to place a bend mesial to the molar to increase its
angle to the interbracket axis (black circle in Fig 14-59e).
Fine-tuning involves placing either one or two adjacent
bracket bends. It would be better to place the bend at the
center of the bracket, but this is not always possible.
Sequential virtual bracket repositioning replaces repeated
debonding and bonding of brackets and is a close approximation.
345
14
Forces from Wires and Brackets
FIG 14-60 A molar is both tipped and in infraocclusion. A good
solution is to apply a Class IV force system with tip-back moment
and extrusive force. The posterior extension of the wire is bent at the
distal of the premolar bracket so that it crosses the molar slot axis
at one-third of the distance from the centers of the brackets or with
half the amount of vertical displacement (½ D).
1
3
2
3
a
b
1
2
c
1
3
1
2
2
3
The virtual bracket repositioning approach is applicable to other geometries. In Fig 14-60, a molar is both
tipped and in infraocclusion. A good solution is to apply
a Class IV force system. This allows for both molar
uprighting and extrusion. The posterior extension of the
wire is bent at the distal of the premolar bracket so that
it crosses the molar slot axis at one-third of the distance
from the centers of the brackets.
The canine in Fig 14-61a requires intrusion. A continuous archwire with a canine bypass is fabricated (gray
wire); a long-span wire from the auxiliary tube of the
first molar to the canine is used for canine intrusion.
The canine could be moved vertically by a cantilever
with a free end at the canine, as described in Fig 14-54;
this might be the best solution if a highly predictable
single force is wanted. If we elect to have full bracket
346
FIG 14-61 The canine requires intrusion. (a) The straight wire–bracket
classification between the molar and canine brackets is Class I. (b)
First, the wire segment is made passive with small bends for comfort,
usually next to each bracket. (c) Second, an appropriate V-bend can
be placed in the wire segment in accordance with the force system
required.
engagement on the canine, perhaps to rotate the canine
and change its width with full control of six degrees of
freedom, then we will not use a straight wire but will
rather change the canine-molar geometry into a Class
IV, where only an intrusive force is present at the canine.
This is easily done by first making the wire segment
passive with small bends, usually next to each bracket
(Fig 14-61b). Now an appropriate V-bend can be placed
in the wire segment; a bend placed one-third of the
interbracket distance from the molar tube will approach
a Class IV geometry (the correct force system for an
intrusive force alone). A bend placed one-half of the
interbracket distance from the molar tube will approach
Class VI (Fig 14-61c). Other positions for the V-bend
can be selected to alter the force system for any desired
geometry.
Clinical Applications of Two-Bracket Geometries
a
b
c
d
e
f
g
h
i
j
FIG 14-62 Clinical case with Class I geometry. (a and b) The mandibular left second molar is mesially inclined and partially
impacted, and a short clinical crown due to
partial eruption makes it difficult to bond an
attachment; therefore, a labial approach was
impossible. (c) Equal and opposite forces at
the crown will tip the anterior anchor unit
and the second molar in opposite directions.
(d) A bonded button was used for a single
force at the crown. (e and f) After single force
application, the bracket slot relationship became a Class I geometry. (g and h) A Z-bend
was added to increase the magnitude of
force and moment. (i and j) After treatment.
The anterior segment also rotated clockwise
(tip-back).
Clinical Applications of
Two-Bracket Geometries
A straight wire can be the correct
geometry
In Fig 14-62, the mandibular left second molar was mesially inclined and partially impacted. A short clinical
crown due to partial eruption made it difficult to bond
an attachment; therefore, a labial approach was impossible (Figs 14-62a and 14-62b). The equal and opposite
forces at the crown level will tip the anterior anchor unit
and the second molar in opposite directions (Fig 14-62c).
A bonded button at the occlusal surface of the crown
ensured that only a single force acted at the second molar
crown (Fig 14-62d). Note that to reduce the F/∆ rate, the
spring to the second molar originated from the mesial of
the first molar. After the single force application, the
bracket slot relationship became a Class I geometry
(Figs 14-62e and 14-62f ).
In the new geometry, both the second molar and also
the anterior anchor unit could utilize a clockwise rotation
moment (Fig 14-62g). A Class I geometry force system
347
14
Forces from Wires and Brackets
a
b
c
FIG 14-63 Clinical case with Class II geometry. (a) In a patient showing protrusion, controlled tipping of the anterior segment and slight root
movement of the posterior segment occurred after space closure. (b) The required force system is depicted with red arrows, which is a Class II
geometry force system. (c) Two bends were made at each side of the bracket, with the posterior angulation being half that of the anterior
angulation, and each segment rotated as predicted.
a
b
c
d
e
f
FIG 14-64 Clinical case with Class IV geometry. (a) A surgery patient with a prognathic mandible is ready for retraction of the maxillary anterior
segment for decompensation. (b) Controlled tipping of the anterior segment and translation of the posterior segment occurred by group A
mechanics. (c) Two bends were made at each side of the bracket. Note that the posterior angle is opposite. (d to f) The anterior segment
primarily rotated, and a single occlusal plane was produced.
is ideal here. An extrusive force and tip-back moment
are delivered to the second molar. The clockwise moment
to the anchor unit is also helpful. A straight wire alone
between the molars will generate the desired force system;
however, a Z-bend was added to increase the magnitude
of force and moment while maintaining the same
geometry (Fig 14-62h). Figure 14-62i shows the occlusal
view after treatment, and the lateral view after Z-bend
application is shown in Fig 14-62j. The tipped-forward
anterior unit also rotated clockwise (tip-back), resulting
in good occlusion (compare Figs 14-62f and 14-62j).
348
Increasing the interbracket distance
In a patient showing anterior protrusion (Fig 14-63a), the
first premolar was extracted, and the anterior segment
was retracted using group A mechanics with a universal
T-loop spring. Controlled tipping of the anterior segment
and slight mesial root movement of the posterior segment
occurred after space closure. The required force system
(a Class II geometry) is depicted in red arrows (Fig 14-63b).
The T-loop spring was removed, and a root spring was
placed (Fig 14-63c). Two bends were made at each side
Clinical Applications of Two-Bracket Geometries
FIG 14-65 Clinical case with Class V geometry. (a) After space closure, the mandibular arch required a Class V geometry force
system. (b) A cantilever was used to deliver
the Class V force system. The free end was
placed as far distal to the CR of the posterior segment as possible. (c) A single force
placed posteriorly provided a Class V force
system continuously during the treatment.
(d) After treatment.
a
b
c
d
of the bracket; the angulation at the posterior was half
that at the anterior based on the arms of the spring starting at a passive angle. Each segment rotated as predicted,
resulting in favorable maxillary arch leveling (see
Fig 14-63c). The long span between the canine and the
first molar has many advantages. It lowers the F/∆ rate,
allows for greater accuracy in bend placement or evaluation, gives a more constant M/F ratio during deactivation, and also delivers larger moment with less vertical
force.
Using a Class IV geometry force system
to align segments in an extraction arch
A surgery patient with a prognathic mandible required
retraction of the maxillary anterior segment for decompensation (Fig 14-64a). Controlled tipping of the anterior
segment and translation of the posterior segment
occurred by group A mechanics (Fig 14-64b). Two bends
were made in the straight wire at each bracket (green lines
in Fig 14-64c). The anterior segment rotated to correct
the incisor axial inclinations, and a single maxillary arch
occlusal plane was produced (Figs 14-64d to 14-64f ).
Note that the posterior angle is opposite to that in
Fig 14-64c.
Using a cantilever to produce a Class V
geometry force system
After space closure, the mandibular arch of the patient
in Fig 14-65 required a Class V geometry force system
(Fig 14-65a). Class V geometry is very difficult to achieve
clinically with a single V-bend. Even if the bend is accurately placed initially, the geometry will change quickly
with any tooth movement of small magnitude. Therefore,
a cantilever was used to deliver a Class V force system.
The free end was placed as far distally as possible to the
CR of the posterior segment, between the first and second
molar (Fig 14-65b). A single force placed posteriorly to
the bracket provided a Class V force system continuously.
Figures 14-65c and 14-65d show the occlusion during
and after treatment.
Two-tooth root movement independent
of adjacent teeth
In Fig 14-66, the roots of the mandibular right canine
and the first premolar converge toward each other
(Fig 14-66a); all other roots are normal. A small segment
of a 0.017 × 0.025–inch β-titanium wire (green) was
formed as a segment of a circle and placed into the canine
349
14
Forces from Wires and Brackets
a
c
b
d
FIG 14-66 Clinical case with Class VI geometry. (a) The roots of the mandibular right canine and first premolar
converge toward each other to develop a Class VI geometry. The roots of the rest of the arch are normal. (b) A small
segment of 0.017 × 0.025–inch β-titanium wire is formed into a segment of a circle (green) and placed into the canine
and first premolar brackets only. A Ni-Ti wire is used for minor leveling of the incisors and overlaid on the two-tooth
segment for stabilization purposes only. (c) After treatment. (d) Panoramic radiograph after treatment.
a
b
c
d
e
350
FIG 14-67 Clinical case with Class VI geometry. (a) The
maxillary left second premolar is rotated. (b) During the
7-month observation period, it did not show any spontaneous eruption; the roots of adjacent teeth probably
interfered with its eruption. The force system needed is
depicted with red arrows. (c) The needed moment is produced by the brackets that are intentionally angulated
to produce a Class VI geometry. The horizontal force is
produced by an open coil spring. (d and e) Divergence
of the roots provided spontaneous eruption of the second premolar.
Clinical Applications of Two-Bracket Geometries
a
b
c
d
e
f
FIG 14-68 Clinical case with Class VI geometry. (a and b) The maxillary
left lateral incisor is missing. (c) A cantilever spring was inserted to regain
the space between the maxillary central incisor and canine. (d) The roots
are converged by a single force. (e) Activation force system of a small,
straight piece of rectangular Ni-Ti wire. (f) Deactivation after 4 months.
(g) Due to the large friction between the wire and resin, the interbracket
distance was maintained and the roots became divergent.
g
and first premolar brackets only (Fig 14-66b). Considering the initial angulations and the wire curvature, the
force system approached a Class VI geometry. The
secondary Ni-Ti wire was used for minor leveling of the
incisors and overlaid on the two-tooth segment for stabilization purposes only (see Fig 14-66b). Figures 14-66c
and 14-66d show posttreatment views of the occlusion
and the root position.
Increasing space between apices with
a Class VI geometry force system
A maxillary left second premolar was rotated (Fig 14-67a).
During a 7-month observation period, it did not show
any spontaneous eruption; the roots of the adjacent teeth
probably interfered with its eruption. The force system
needed to rotate the teeth around their brackets is shown
with red arrows (Fig 14-67b). The required moments are
produced by the brackets that are intentionally angulated
to produce a Class VI geometry. The horizontal force was
produced by open coil springs. Virtual repositioning by
wire bending is not applicable to Ni-Ti wire because
superelasticity is lost by plastic bending. Divergence of
the roots provided spontaneous eruption of the second
premolar (Figs 14-67c to 14-67e).
Another patient presented with a missing maxillary
lateral incisor (Figs 14-68a and 14-68b). A cantilever
spring was inserted to regain the space between the
maxillary central incisor and canine, but the roots were
converged by a single force (Figs 14-68c and 14-68d). An
equal and opposite couple was applied to a small, straight
piece of rectangular Ni-Ti wire so that the activated shape
was a segment of a circle (Fig 14-68e). The elastically bent
Ni-Ti wire continuously applied an equal and opposite
couple and was deactivated in 4 months (Fig 14-68f ).
Because of the large amount of friction between the wire
and the resin, the interbracket distance was maintained
and the roots became sufficiently divergent for the placement of an implant (Fig 14-68g).
351
14
Forces from Wires and Brackets
a
b
FIG 14-69 Clinical case with Class VI geometry. (a) The
maxillary left second premolar needs space for eruption.
(b) The brackets were intentionally angulated at the first
premolar and first molar to produce a Class VI geometry.
An open coil spring was inserted to produce the needed
force. (c) The second premolar erupted spontaneously
after translation of adjacent teeth.
c
a
b
c
d
FIG 14-70 Clinical case of modifying geometry from
Class III to Class IV. (a) Severe crowding was present
that required first premolar extraction. (b) The canine
was retracted separately by controlled tipping. (c) The
relationship shows a Class III geometry. A straight wire
will produce unnecessary moment at the posterior segment. (d) A single V-bend was placed so that slot angles
produced a Class IV geometry. (e) After treatment. Note
that a minor bend was placed mesial to the first molar tube
for fine-tuning (circle).
e
352
Recommended Reading
Enhancing eruption with a Class VI
geometry force system
Similar to Fig 14-67, in Fig 14-69 a maxillary left second
premolar required space for eruption (Fig 14-69a). The
brackets were intentionally angulated at the first premolar and first molar to produce a Class VI geometry force
system (Fig 14-69b), and an open coil was inserted for
the horizontal force needed. The second premolar
erupted spontaneously after translation of adjacent teeth
(Fig 14-69c).
Modifying the geometry from Class III to
Class IV
A severe crowding case required first premolar extraction
(Fig 14-70a). The canine was retracted separately by
controlled tipping (Fig 14-70b). The relationship showed
a Class III geometry (Fig 14-70c). A straight wire producing the force system of Class III geometry (green wire)
includes an unnecessary moment at the posterior
segment, so a single V-bend was placed so that the slot
angle changed to a Class IV geometry (Fig 14-70d). After
this treatment, a second minor bend was placed mesial
to the first molar tube for fine-tuning (circle in Fig 14-70e).
Summary
This chapter has discussed the relationship between wires
and brackets with either straight-wire or bent-wire applications. It is often easy to get lost in the details; therefore,
it was not the primary goal of this chapter to develop the
exact forces in any given instance. Clinicians want and
need the important basic and general principles to guide
them throughout treatment, so emphasis has been placed
on the relative force systems and the initial forces at work
immediately after archwire placement; wherever possible,
examples have been presented without any specific
values.
Recommended Reading
Burstone CJ. Application of bioengineering to clinical orthodontics.
In: Graber LW, Vanarsdall RL Jr, Vig KWL (eds). Orthodontics: Current
Principles and Techniques, ed 5. Philadelphia: Elsevier Mosby,
2012:345–380.
Burstone CJ. The biomechanical rationale of orthodontic therapy. In:
Melsen B (ed). Current Controversies in Orthodontics. Chicago: Quintessence, 1991:131–146.
Burstone CJ. Variable modulus orthodontics. Am J Orthod Dentofacial
Orthop 1981;80:1–16.
Burstone CJ, Goldberg AJ. Maximum forces and deflections from
orthodontic appliances. Am J Orthod Dentofacial Orthop 1983;84:
95–103.
Burstone CJ, Koenig HA. Creative wire bending—The force system
from step and V bends. Am J Orthod Dentofacial Orthop 1988;93:
59–67.
Burstone CJ, Koenig HA. Force systems from an ideal arch. Am J
Orthod Dentofacial Orthop 1974;65:270–289.
Burstone CJ, Qin B, Morton JY. Chinese NiTi wire—A new orthodontic
alloy. Am J Orthod Dentofacial Orthop 1985;87:445–452.
Choy KC, Sohn BH. Analysis of force system developed by continuous
straight archwire. Korean J Orthod 1996;26:281–290.
Drake SR, Wayne DM, Powers JM, Asgar K. Mechanical properties of
orthodontic wires in tension, bending, and torsion. Am J Orthod Dentofacial Orthop 1982;82:206–210.
Drenker E. Calculating continuous archwire forces. Angle Orthod
1988;58:59–70.
Drescher D, Bourauel C, Thier M. Application of the orthodontic
measurement and simulation system (OMSS) in orthodontics. Eur J
Orthod 1991;13:169–178.
Goldberg AJ, Burstone CJ. An evaluation of beta titanium alloys for
use in orthodontic appliances. J Dent Res 1979;58:593–600.
Goldberg AJ, Burstone CJ, Koenig HA. Plastic deformation of orthodontic wire. J Dent Res 1983;62:1016–1020.
Koenig HA, Burstone CJ. Force systems from an ideal arch—Large
deflection considerations. Angle Orthod 1989;59:11–16.
Kusy RP, Greenberg AR. Effects of composition and cross section on
the elastic properties of orthodontic wire. Angle Orthod 1981;51:325–
341.
Popov EP. Mechanics of Materials, ed 2. Englewood Cliff, NJ: PrenticeHall, 1978.
Rock WP, Wilson HJ. Forces exerted by orthodontic aligning archwires.
Br J Orthod 1988;15:255–259.
Ronay F, Melsen B, Burstone CJ. Force system developed by V bends
in an elastic orthodontic wire. Am J Orthod Dentofacial Orthop 1989;96:
295–301.
Schaus JG, Nikolai RJ. Localized, transverse, flexural stiffness of continuous arch wire. Am J Orthod Dentofacial Orthop 1986;89:407–414.
353
Problems
In the following problems, a straight wire with malaligned brackets was used, unless otherwise stated. Consider only
the force system in the plane shown, and ignore mesiodistal forces. Known or unknown forces and moments are given,
but the reader must determine their direction. Both the activation force system (blue forces) for an equilibrium diagram
and the deactivation force system (red forces) are required. This two-step analysis is critical for understanding the
correct equilibrium and avoiding confusion with multiple wires or appliances. With experience, clinicians can work
from the deactivation force diagram (red force system) only.
354
1. The maxillary left central incisor needs intrusion. A
button is bonded, and a two-tooth leveling wire is
placed. What will happen to the right incisor?
2. This is the same as problem 1 but with two more boundary conditions. The interbracket distance is 7 mm, and
a 100-g force is measured by a force gauge when the
wire is activated. Solve for all forces and moments on
both teeth.
3. When a bracket instead of a button is bonded on the
maxillary left central incisor, the number of unknowns
increases. Depict every unknown. Can FB be measured
by a force gauge? Discuss any problems with using a
gauge.
4. This is the same as problem 3 but with one more
boundary condition. FB = 150 g is given. Is it possible
to solve for all other forces and moments on the two
teeth?
Problems
Problems
5. The green dot is the intersection of the slot axes. What
is the geometry? What is the relative force system in
this geometry?
6. Using the identified geometry in problem 5, solve for
the complete force system.
7. The geometry between the brackets is Class I. Find all
other forces and moments acting on the teeth.
8. This is the same as problem 7; however, the left tooth
is a dental implant and fully ankylosed. Find all other
forces and moments acting on the teeth.
9. A Z-bend is made at the center of the brackets. Find
all other forces and moments acting on the teeth.
10. Two V-bends are made (one at each bracket) with the
same angle. Find all other forces and moments acting
on the teeth.
355
Problems
11. A V-bend was placed at one-third of the interbracket
distance measured between the auxiliary tubes of the
anterior and posterior segments. Each segment is
stabilized by rigid wire. Identify the geometry, and
predict the movement of each segment.
12. (a) Two teeth are tipped toward each other. Identify
the geometry and the force system of the brackets if
straight wire is placed. (Note that the brackets are
improperly placed.) Draw the deactivation force
system for this geometry. (b) Equal and opposite
couples, as depicted with green arrows, are required.
Design the correct shape of the wire to produce the
desired force system. (Keep the wrong position of the
brackets.)
a
b
13. The brackets of three teeth (first premolar, canine, and
lateral incisor) are shown. The canine bracket is angled
because the root is positioned forward. What happens
to the incisor, canine, and first premolar?
356
15
Principles of
Statically Determinate
Appliances and
Creative Mechanics
Giorgio Fiorelli / Paola Merlo
Design is not just what it looks like and feels like.
Design is how it works.”
— Steve Jobs
A statically determinate orthodontic appliance is defined as an appliance by which a
single force measurement allows for the computation of the full force system on both
the active and reactive teeth. Examples are cantilevers and coil springs. Cantilevers
can be simple straight wires or a wire with a special shape. Unlike appliances that
deliver both forces and moments that change their ratios during deactivation, cantilevers can be designed to maintain a virtually constant direction and distance to the
center of resistance. In this chapter, clinical examples are presented to demonstrate
how cantilevers can upright teeth, correct rotations, and resolve more complicated
situations. A sequence of mechanical treatment planning is presented in which
treatment goals are achieved by the determination of the three-dimensional (3D)
location of cantilever forces or their reacting moments. In special geometries, more
than one cantilever may be required (especially in Class V geometry force systems).
Special contouring of cantilevers can produce needed mesiodistal forces. Although
finishing detail is usually efficiently accomplished with a shape-driven appliance,
the clinician can take advantage of force-driven cantilevers, with their simplicity and
ease of force determination, to achieve major tooth movements.
357
15
Principles of Statically Determinate Appliances and Creative Mechanics
FIG 15-1 Different examples of statically determinate appliances. The colored
arrows represent force vectors applied
to the unit of the same color. In each
system, arrows lie on a line of action
where all features of the force system
can be summarized. The represented
vectors can be measured clinically
with a force gauge, and the relevant
distances of the equivalent force systems for each unit can be calculated at
the bracket or directly at the estimated
position of the CR.
Principles of Statically Determinate
Appliances
According to Newton’s First Law, the sum of all forces
and all moments acting on an object at rest or with uniform
velocity must equal zero.
∑F = 0
∑M = 0
In a force diagram with two dental units, these formulas can be used to compute the total force system on
both units if the force system is completely known in
one of them in any place. In a clinical environment, where
only forces but no moments can be measured, the total
force system can still be statically determinate. It is possible to classify different statically determinate mechanics
where only forces are known (and moments are
unknown), depending on the location and orientation
of the line of action (Fig 15-1).
Statically determinate biomechanical systems therefore allow for exact measurement and computation of
the whole force system on all units to which they are
applied. In order to estimate the appliance effect, it is
necessary to transfer these forces as equivalent force
systems to the estimated center of resistance (CR) position of each unit, where a force and possibly a moment
will be acting. By following these steps, the orthodontist
can easily and accurately predict tooth movement.
358
Cantilevers
In the construction industry, a cantilever is a beam
anchored at only one end.1 The beam carries the load to
the support, where moment and shear stresses are forced
against it. In orthodontics, a cantilever is any piece of
wire with one end inserted into a bracket or a tube (or
included in the acrylic of a removable appliance) and the
other end tied or hooked to another unit; all cantilevers
have, at one end or the other, only one point of contact.
As for any statically determinate appliance, when using
a cantilever, the orthodontist can easily estimate the force
system by measuring the activation force, identifying the
single point of contact, and determining the distance
between the site of full engagement of the wire and the
point of ligature, measured perpendicularly to the line of
action of the force.
As can be seen in Fig 15-2, the biomechanical force
system generated by a cantilever is characterized by a
combination of a moment and a force at the unit where
the cantilever is inserted, whereas only a single force is
present at the other end. This force system must be
expressed with respect to the estimated CR of each dental
unit in order to predict the tooth movement. The two
forces are equal in magnitude but opposite in direction,
according to Newton’s First Law, and the activation force
can be measured by a force gauge. The value of the
moment (M) is equal to the length of the cantilever—or
more significantly, the distance between the site of full
engagement and the site where a single point of contact
exists—(d), measured perpendicularly to the line of action
of the produced force, multiplied by the force (F):
Cantilevers
a
b
FIG 15-2 Deactivation force system of a cantilever. Equivalent force
systems at the estimated CR (blue dot) of the posterior segment
should be computed in order to assess the clinical effects.
M=F×d
In orthodontic therapy, cantilevers can be utilized in
all planes of space, and they can be applied both buccally
and lingually. Indications for cantilever use include
control of the labiolingual position of incisors and
canines, the buccolingual position of molars and premolars, rotations, vertical positions, extrusion or intrusion
of lateral and anterior teeth, third-order movements,
and molar uprighting. Transpalatal and lingual arches
can be used as cantilevers as well, if fully engaged on one
side and with a single point of contact on the other side.
Although cantilever design is based on a rational and
systematic procedure, in the end, it is the imagination of
the clinician that sets the scope of their use.
One important feature of cantilever mechanics is the
generation of a force system with a high degree of
constancy over time from activation to deactivation. The
forces and moments at its two ends maintain their
direction and decrease proportionally in magnitude with
the cantilever deactivation. In addition, there is also a
high degree of constancy of the moment-to-force (M/F)
ratio (with respect to the bracket). This also means a
homogenous dental movement with a relatively stable
center of rotation (CRot).
FIG 15-3 (a) The molar requires uprighting with minimal extrusion
of the CR (blue dot). The CRot is located very close to the CR.
Therefore, a very large M/F ratio (32 mm in this example) must be
generated to obtain the displacement. The equivalent force system
including a single force is a force vector passing 35 mm mesial to
the CR. The cantilever needs to be ligated or hooked along this
line of action. (b) More extrusion of the molar is needed with the
CRot located about 6 mm from the CR. Here the needed M/F ratio
at the CR is 18 mm, so the cantilever will be shorter in order to be
ligated along the line of action of the equivalent single vector.
Cantilever design
When the orthodontist decides to implement a cantilever,
several factors must be considered when designing it.
The most important of these is the movement that should
be generated by the cantilever, which in turn determines
the force system that needs to be delivered to the active
dental unit. Several experimental studies have described
the relationships between the applied force system and
the dental movement.2–6 Based on these data, the force
system can be estimated by considering where the CRot
of the needed movement should be located and the single
force needed to get it there. Calculations that yield these
estimates can be performed using a software program
called Dental Movement Analysis (DMA) (IOSS).7
Once the needed single force for a specific movement
has been established, a general idea of the cantilever shape
can easily be clinically visualized by referencing the point
where the single force should be generated by the cantilever, which is located where the cantilever is ligated or
hooked at a single point of contact. This point must lie
along the line of action of the needed single force. An
example of this principle is given in Fig 15-3, where two
different movements are described. The needed force
systems are expressed at the CR and as the equivalent
359
15
a
Principles of Statically Determinate Appliances and Creative Mechanics
b
c
FIG 15-4 (a) A cantilever is used for the correction of the heavily rotated maxillary left canine. The bracket is placed on the lingual surface
of the tooth. The cantilever is made of 0.021 × 0.025–inch stainless steel wire; when inserted into the canine lingual bracket passively (before
its activation), its other extreme end is situated approximately 10 mm linguodistally from the molar tube. When ligated, a force of 30 g and a
moment of 1,050 gmm, rotating the canine distally, are produced. (b) After 2 months. (c) After 5 months. (From Biomechanics in Orthodontics,
www.ortho-biomechanics.com.)
FIG 15-5 A short cantilever is engaged in the canine bracket slot
and ligated mesial to the molar tube. Helices are used to reduce the
load-deflection ratio. The activation force produces an extrusive
force and crown mesial moment; both vectors are consistent with
the needed displacement of the canine. The M/F ratio of 14 mm
determines a movement where both the rotary and translatory
components take place. (From Biomechanics in Orthodontics,
www.ortho-biomechanics.com.)
a
b
FIG 15-6 An extension connecting the two maxillary canines is used to displace forward the point of application of
the intrusive force. In this way, a mesial moment is coupled
with an intrusive force at the CR of these teeth.
c
FIG 15-7 (a and b) A rigid stainless steel wire extends mesially from the canine bracket, and at its end a cantilever delivers a palatal force. The
line of action of the applied force passes about 5 mm anterior to the estimated position of the canine CR, thus creating there a clockwise moment
with an M/F ratio of 5 mm. (c) Clinical effect 5 weeks later. Both a mesial-in rotation and a palatal displacement of the canine were achieved.
single force away from the CR. At the CR, we have a
moment and a force and an M/F ratio that is related to
the distance between the CRot and CR; the position of
360
the single force vector indicates the position of the single
point of contact of the cantilever.
Cantilevers
FIG 15-8 (a to d) In this sequence, two
cantilevers are guiding the canines from the
palate to their correct positions. The needed
movement on the occlusal plane is a buccal
translation and a distal rotation. To increase
the amount of moment created at the canine
CRs, the cantilevers are ligated buccally, using some composite to stabilize the metal
ligatures. Note in b that the cantilevers are
in their passive state (before activation). Also
note the progressive distal rotation of the
canines.
FIG 15-9 (a) A buccal displacement of the
mandibular left canine and first premolar is
achieved using a force delivered by a cantilever inserted in the molar tube and ligated
to the canine and premolar, which are splinted together. The molar is the reactive unit
and is connected with a temporary anchorage device (TAD). (b) Occlusal view when
the cantilever was applied. (c) After 45 days.
a
b
c
d
a
The principles discussed so far yield simple rules based
on the needed M/F ratio at the bracket of the active unit:
• If a pure rotation or inclination is desired (M/F ratio
at the CR should be infinite), or if the translation
component of the desired movement is clinically insignificant, use a cantilever as long as possible, insert it in
the slot of the active unit, and tie it (with a single point
of contact) to the reactive unit. See the clinical example
in Fig 15-4.
• If a combination of rotation or inclination with translation is desired and the needed M/F ratio is > 10 mm,
use a shorter cantilever (its length should be approximately equal to the desired M/F ratio at the bracket)
inserted into the active unit and ligated to the reactive
one (Fig 15-5). If the needed M/F ratio is < 10 mm,
insert the cantilever into the reactive unit and ligate it
to a rigid wire extending from the active unit bracket
(Figs 15-6 to 15-8). Otherwise, special configurations
b
c
of the cantilever must be adopted in order to reduce
the load-deflection ratio (see later section) and to
engage the cantilever into the active unit.
• If an M/F ratio of 0 is needed at the bracket, insert the
cantilever into the reactive unit and ligate it to the
bracket of the active unit (Fig 15-9).
This classification does not include the cases where
there is no reactive unit and the force system is desirable
in both units. We call these systems absolutely consistent,
and they represent an excellent chance to expedite the
treatment, solving two problems at one time without any
anchorage need. Examples of such systems are shown in
Figs 15-10 to 15-12.
Once the orthodontist has created the design of the
cantilever and determined the site of engagement into
the slot as well as the location of the single point of ligature, other characteristics of the cantilever should be
defined.
361
15
Principles of Statically Determinate Appliances and Creative Mechanics
a
b
c
d
FIG 15-10 In this sequence, a correction of
the cant of the anterior teeth can be seen.
(a) The maxillary anterior teeth show a cant
in the frontal view. (b) A long cantilever
that delivers an intrusive force passes to
the right of the anterior group's CR, delivering a clockwise moment. The cantilever
is inserted in the left molar tube, which was
mesially inclined and had a reverse articulation (also known as a crossbite) tendency.
These mechanics are consistent both for the
maxillary anterior group and for the maxillary
left molar (see Fig 15-11). (c) Movement of the
anterior teeth, while correcting the anterior
cant, moves the midline to the right side.
Therefore, in a second step, two-vector mechanics were used to translate the incisors
to the left and extrude them (d).
FIG 15-11 This force diagram, corresponding to Fig 15-10b, explains how the cantilever
can simultaneously correct the anterior cant
and the molar crossbite tendency.
a
b
Wire selection
There are several issues to be considered when selecting
a wire for a cantilever. First of all, the wire should be easily
formed, which in most cases excludes nickel-titanium
(Ni-Ti) wires because of their very low formability.
The load-deflection ratio of a cantilever, as for all the
active elements of the appliance, should be as low as
possible, leaving the force system with a high degree of
constancy. The load deflection of the cantilever depends
on the following factors:
• Cantilever length
• Wire stiffness
• Configuration of the wire
362
FIG 15-12 (a and b) The cantilever generates a tip-back of the molar, and the metal
ligature limits its distal drift while improving
the canine and premolar occlusal intercuspation. Once occlusal stability is achieved,
the posterior segment will be splinted and
used as anchorage.
Because the cantilever length is dependent on and
largely decided by the needed force system of the active
unit, the wire should be selected based on the following
two key factors:
1. The stiffness of the wire should allow an acceptable
load-deflection ratio for the given cantilever length.
2. The wire should have a yield moment (the moment
causing 1% permanent deformation of the cantilever)
high enough to allow the generation of needed forces
and moments for the given cantilever length.8,9
With these concepts in mind, as well as an eye toward
cantilever length and needed force magnitude, the
following wires are recommended.
Cantilevers
FIG 15-13 (a) A cantilever ligated to a TAD is
used to complete space closure by molar mesial displacement. A loop is included in the wire
to reduce the load-deflection rate. Note that the
loop lays on a sagittal plane, the same plane as
the activation force. In this case, the rotation was
controlled by an activation of the transpalatal bar.
(b) The space is closed, and some molar uprighting has been achieved. The extension attached
to the TAD is used to deliver a transverse force to
the posterior teeth.
a
Short cantilever (10–15 mm)
A wire with low stiffness such as a round 0.018-inch
β-titanium wire would be appropriate to obtain a good
load-deflection ratio. However, unless its activation is
parallel to the bracket or tube orientation (mesiodistal
activation for a horizontal tube), this type of wire will roll
in the bracket or tube. In this case, in order to prevent
the wire from rolling, this wire should be welded to a
short piece of rectangular wire (typically 0.017 × 0.025–
inch β-titanium) that will be inserted into the bracket or
tube. This kind of cantilever is called a composite cantilever, and due to its short span length, it usually has sufficient yield force for most dental movement needs.
Rectangular 0.016 × 0.022–inch β-titanium wires can
also be used for cantilevers that are not extremely short
if the clinician accepts a higher load-deflection ratio. In
such cases, it should be remembered that rectangular
wires have double the rigidity depending on their
cross-section geometry. For this reason, when the cantilever requires buccolingual activation, a 90-degree twist
of the wire immediately close to the bracket of insertion
decreases its rigidity (ribbon-wise orientation of the
wire).
Another possibility to avoid the use of composite cantilevers with short-length welded wires is to add wire to
the cantilever using an appropriate configuration as
described later in the chapter.
Medium-length cantilever (16–24 mm)
Most cantilevers fall into this length range. For these
lengths, a 0.017 × 0.025–inch β-titanium wire is recommended. Such a wire cross section has the great advantage
of having very limited freedom to roll in 0.018 × 0.025–
inch slots and tubes. A typical 20-mm cantilever, corresponding to the common distance between the molar
tube and the canine bracket, can deliver about 150 g of
buccolingual activation before permanent deformation
occurs. If a load-deflection reduction is desired when a
b
buccolingual activation is needed, a 90-degree twist of
the wire will reduce the cantilever rigidity by 50% because
rectangular wires have different wire stiffnesses depending on the plane of activation.
Long cantilever (25 mm or more)
An increase in cantilever length dramatically reduces the
load-deflection ratio; however, for such long cantilevers,
the moments generated at one end of the wire are very
high, and permanent deformation of the wire can be a
problem. For this reason, if small forces are needed (50 g
and below), 0.017 × 0.025–inch β-titanium wires are
recommended. If larges forces are needed, stainless steel
wires with the same or larger cross section are required.
Configuration
The cantilever configuration can be altered for two main
purposes: (1) to reduce the load-deflection ratio, or (2)
to modify the line of action of the force by changing its
angulation.
Load-deflection reduction
Load-deflection reduction, when required, can be achieved
by adding wire to the cantilever in one of two configurations: either looped or shaped into a zigzag pattern.
(These configurations can also change the force vector
direction, which will be discussed later in this chapter.)
Loops are effective in reducing the load-deflection ratio
when ideally oriented on the same plane of activation as
the cantilever. While they can be applied in many clinical
scenarios, they are not conducive to buccal cantilevers
used in buccolingual activations; in this case, a mechanically correct orientation of the loop would cause patient
discomfort (Fig 15-13).
Zigzag shapes can be even more effective in reducing
the load-deflection ratio of cantilevers because they can
add a large extension of wire even if the distance between
363
15
Principles of Statically Determinate Appliances and Creative Mechanics
FIG 15-14 A zigzag-configured cantilever is used to apply an extrusive and
slightly buccal force to the anterior teeth. The cantilever is activated by ligating
it to a miniscrew. The limited horizontal distance between the distal end of the
canine bracket and the miniscrew would determine a high load-deflection
ratio for a vertical activation. The blue line shows its passive shape. Blue arrows
show the activation force and the force system at the CR of the anterior teeth.
the point of engagement and the single point of ligature
is very limited. They can easily be used for activations in
all planes of space (Figs 15-14 and 15-15).
Cantilever and force vector angulation
Force vectors produced by a linear (or almost linear)
cantilever are in a roughly perpendicular orientation to
the line connecting the point of engagement to the single
point of contact of the cantilever (structural axis). This
orientation, along with the concomitant movement of
the active unit, can be changed by modifying the shape
of the cantilever itself. These modifications can be classified as either minor or major configurations (major
configurations are characterized by loops).
Minor configurations. The orientation of the force
vector generated by the cantilever activation can be
changed by modification of the shape of the cantilever.
Cantilevers with different curves will change shape upon
activation. Such shape changes will impact behavior of
the cantilever, causing it to expand or to contract during
the activation, thus skewing the force vector that it
produces away from its otherwise perpendicular orientation.
In a study by Dalstra and Melsen,10 six different cantilever configurations were analyzed in a finite element
model. Among these were some configurations that can
be considered “minor”; three of these are represented in
Fig 15-16. It can be seen that the configuration in Fig
15-16b delivers a force vector perpendicular to the structural axis of the cantilever, while the configurations in
Figs 15-16a and 15-16c demonstrate expansion or
364
FIG 15-15 A composite cantilever with a
zigzag configuration is used to intrude the
premolar and move it lingually.
contraction that cause the vector to deviate from the
perpendicular axis. Each modified cantilever will apply
a unique intrusive force to the anterior teeth, which will
in turn produce a unique sagittal force to be added to the
vertical force. Thus, the choice of a uniquely shaped cantilever will determine whether the anterior teeth undergo
a purely vertical movement or are also displaced buccally
or lingually as they move along the vertical line.
Cantilevers with major curves. These cantilevers are
usually used between the molars and near the midline
area; they have a curvature in the canine area and a structural axis that is oblique in the occlusal plane. If activated
on the occlusal plane, this type of cantilever automatically
delivers a force that has both sagittal and transverse
components, as illustrated in Fig 15-17. It is often used
for midline correction and proclination of the anterior
teeth. In Fig 15-18, this type of cantilever created a large
amount of space in the maxillary right canine area.
Lingual and palatal arches can also be used as cantilevers with major curves. In Figs 15-19 and 15-20, a lingual
arch is used in a statically determinate way to move a
molar mesially.
Major configurations. If a loop is added to the cantilever as shown in Figs 15-21 and 15-22, it is possible to
have separate and independent activations of its two
components. When these activations are balanced, any
force vector angulation can be achieved. These cantilevers
can be activated by bending the wire close to its full
engagement, where a vertical or transverse component
is usually produced, and at the loop, where a mesiodistal
component of the force is generated.
Cantilevers
a
b
c
FIG 15-16 Three cantilever configurations depicted in their deactivated and vertically activated shapes. (a) A V-bend is placed close
to the engagement. The activated shape becomes shorter, and with
a concavity facing downward, the deactivation force (red vector)
will have an expansion component besides the vertical one. (b) The
cantilever is bent with a homogeneous curvature. The activated shape
almost keeps the same length and has a very limited curvature. In
this case, the deactivation force is almost purely vertical. (c) The
cantilever is bent with a logarithmic shape. Its activation determines
an elongation of the beam, and its activated shape has a concavity
facing upward. The deactivation force will have a contraction component besides the vertical one.
FIG 15-17 A curved cantilever is inserted in a molar
tube; if activated, as shown by the dashed line, it will
generate both transverse and sagittal components. If
this force is applied to the anterior teeth, it will generate a moment at the CR that will determine a great
space opening in the canine area contralateral to the
side of insertion of the cantilever.
a
b
c
d
FIG 15-18 (a and b) A curved cantilever is applied to the anterior segment, including the four incisors, to create the
space for a buccally erupted maxillary right canine. (c and d) After 3 months. Note how the correct line of action of the
force vector has contributed to the alignment of the maxillary left lateral incisor and canine, in addition to creating space
for the maxillary right canine, which is extruding spontaneously.
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Principles of Statically Determinate Appliances and Creative Mechanics
FIG 15-19 Force diagram for the mechanics applied in Fig 15-18.
Two units are considered: the active anterior (red) and the reactive
posterior (blue), which included six teeth connected by a transpalatal
arch. Assuming that the cantilever produces a force of 50 g applied
mesial to the bracket of the right central incisor, force systems are
represented for both units at the estimated CR position. Considering
the heavy occlusal contacts distributed to all posterior teeth, it was
assumed that the displacement of the posterior unit would have
been negligible. The estimated position of the CRot for the anterior
teeth implies a larger movement of the left central incisor than the
left lateral incisor due to the rotation of the group.
a
b
c
d
a
b
FIG 15-20 (a) A lingual arch is used in a
statically determinate manner to apply a mesially directed force on the lingual side of the
mandibular first molar to close the space for
a missing second premolar. The force is applied through a 7-mm apical extension of the
molar lingual sheath. On the buccal side, another cantilever is delivering a second mesial
force. In order to increase anchorage, a TAD
is connected to the reactive unit (the whole
arch excluding the left molars), and Triad gel
(Dentsply) is bonded in both the maxillary
and mandibular arches to maximize occlusal
contacts. (b) The deactivation force diagram
(red, forces from the lingual arch; blue, forces from the buccal mechanics). (c) After 3
months. (d) After 9 months.
FIG 15-21 Same patient as in Fig 15-20. (a)
The buccal cantilever, like the lingual arch, is
activated to generate a slightly extrusive force
to the molar. In this way, the line of action of
the force should pass close or maybe even
apical to the CR position. (b) Lateral view 9
months later. At this time, a T-loop was used
to perform final space closure. Note that the
anterior anchorage was maintained and that
the molar has both moved mesially and been
uprighted.
FIG 15-22 Cantilever with a major configuration corresponding to the shape in Fig 15-23a.
The loop insertion in the anterior part of the
cantilever makes the independent activation
of vertical and sagittal components possible.
366
Cantilevers
a
b
c
d
FIG 15-23 Different cantilever configurations. All cantilevers can produce vertical components when the wire
is bent mesial to the molar tube. A sagittal force component can be added by activating the anterior loop. Cantilevers a and b are used to produce retraction forces, while cantilevers c and d can produce protraction forces.
Note that configurations a and c have a loop placed at bracket level and produce a single force 6 to 8 mm apical
to this level. Conversely, configurations b and d have a loop located apically and produce a single force at bracket
level. Obviously cantilevers a and c produce smaller moments at the CR level than cantilevers b and d for their
sagittal activations. (From Biomechanics in Orthodontics 4.0.)
a
b
FIG 15-24 (a) A cantilever with a configuration corresponding to Fig 15-23b is used to produce a retraction
and intrusion vector applied to a stainless steel extension from the maxillary central incisors (blue arrow). The
photograph depicts its passive state, while the blue drawing represents its activated shape. (b) The two central
incisors are intruded and palatally inclined, while the horizontal overlap (also known as overjet) and vertical
overlap (also known as overbite) are reduced.
Melsen et al11 have published a finite element analysis
of the mechanics of cantilevers with helices in anterior
retraction and intrusion. These cantilevers, however, can
also be used for extrusion and protraction, and when
oriented on a different plane, for production of transverse
forces. Fig 15-23 presents different examples of cantilever
configurations with different positions and orientations
of the loop. It can be seen in this lateral view that, if the
loop has an occlusal position, an apical extension from
the bracket level (6 to 8 mm) will be required to apply
the single force. This displacement of the force vector
reduces the distance of its sagittal component from the
CR, thus reducing the rotational component of the movement. As shown in Figs 15-24 to 15-27, there are a great
number of clinical uses for these cantilevers.
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15
Principles of Statically Determinate Appliances and Creative Mechanics
a
b
FIG 15-25 (a) A cantilever with configuration is ligated distal to the mandibular first premolar bracket and inserted into a
tube distal to the canine. The anterior unit includes the six anterior teeth, while the posterior unit includes premolars and the
first molar connected by a lingual arch to a contralateral segment. The cantilever has a configuration corresponding to
Fig 15-23d and is used to produce an oblique force vector. (b) The force diagram of the appliance. The blue drawing shows
the passive shape of the cantilever. The blue arrow represents the force applied to the anterior unit (at activation). Note that
the line of action of the force intersects the estimated position of the anterior unit CR. Red arrows represent the force system
(at the point of force application and at the CR) delivered to the posterior unit.
FIG 15-26 Same patient as in Fig 15-25 but 3 months later.
Note how a large space has opened mesial to the mandibular left first premolar, where an implant will be placed.
The mandibular anterior teeth were intruded and moved
buccally, reducing both horizontal and vertical overlaps
and achieving a Class I canine relationship. The posterior
teeth kept their position thanks to the occlusal forces.
a
b
c
d
e
FIG 15-27 A cantilever with major configuration is attached anteriorly to a TAD and used to correct a transposition of the maxillary left
canine. (a) The cantilever is applied at the start of treatment. (b) In the first stage, the movement was mostly a root mesial displacement.
(c) A late stage in the correction of the transposition. (d) The finished case. (e) The cantilever activation (same stage as a) showing how
the line of action of the force is passing above the estimated position of the CR, thus producing the initial movement of the canine root.
Composite cantilevers
Composite cantilevers are composed of two different
wires that are joined together by spot welding. These
368
cantilevers can be very useful because they offer components with different rigidities. They are usually made of
β-titanium wire, which has the best joining properties.12,13
Most of the time, composite cantilevers include a stiffer
Cantilevers
a
b
FIG 15-28 (a and b) A 0.018-inch β-titanium cantilever has been welded to a β-titanium base archwire. Being welded at
canine level, this cantilever will produce a more sagittal force vector than a cantilever inserted directly into the molar tube.
(From Biomechanics in Orthodontics, www.ortho-biomechanics.com.)
FIG 15-29 (a) The maxillary left
second molar is erupting buccally
and mesially, partly overlapping
the first molar. Only its mesiobuccal cusp can be seen. The needed
force vector is distal and palatal
(blue arrow). The production of
such a vector would be of utmost
difficulty from the buccal side, and
for this reason a composite cantilever (0.016 × 0.22–inch β-titanium
welded to a transpalatal arch)
has been designed and activated (dashed line shows passive
shape). (b) After 3 months, the
tooth is now remarkably more
distal and palatal and can complete its eruption normally. (From
Biomechanics in Orthodontics,
www.ortho-biomechanics.com.)
a
FIG 15-30 (a and b) A 0.017 × 0.025–inch
cantilever is welded to a 0.036-inch transpalatal bar and is activated to deliver a buccal force in the area of the maxillary right
canine (blue arrow). This force vector will
create asymmetric expansion of the anterior teeth while a 0.014-inch Ni-Ti wire aligns
them. (c and d) After 10 weeks, the maxillary
right lateral incisor and canine show buccal
inclination that can be appreciated from both
anterior and occlusal views. This movement
created sufficient space for rotation of the
maxillary right central incisor and therefore
the possibility of aligning the maxillary anterior teeth. (From Biomechanics in Orthodontics, www.ortho-biomechanics.com.)
b
a
b
c
component made out of 0.017 × 0.025–inch β-titanium
wire and a more elastic component made out of round
0.018-inch β-titanium wire (Fig 15-28).
Composite cantilevers can also be created by joining
0.018-inch, 0.016 × 0.022–inch, or 0.017 × 0.025–inch
wires to a β-titanium lingual or transpalatal arch.14
d
Examples are given in Figs 15-29 and 15-30. These cantilevers can easily produce force vectors that may be
extremely difficult to generate by other means. In fact,
joining different wires allows significant change to the
structural axis of the cantilever.
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15
Principles of Statically Determinate Appliances and Creative Mechanics
FIG 15-31 The right cantilever will be activated both vertically and transversely to
obtain an oblique force vector applied to the
anterior teeth. This force is part of more complex two-vector mechanics used to translate
the maxillary anterior teeth to the right.
a
b
c
d
FIG 15-32 (a and b) A canine needs both distal crown tipping and distal rotation. A cantilever is activated in two
planes of space to obtain both rotations. In blue, see the passive shape of the cantilever. (c and d) Displacement
after 2 months. The canine has been uprighted and rotated at the same time. (From Biomechanics in Orthodontics,
www.ortho-biomechanics.com.)
Combining activations on different
planes
Cantilevers can be activated in two different planes when
displacement in different directions is needed (eg, vertical and buccolingual) (Fig 15-31) as well as when a tooth
must be rotated in two planes of space (ie, a tipped and
rotated tooth that requires 3D rotation in an oblique
plane) (Fig 15-32).
Closed Coil Springs and Elastics
Coil springs and elastics are used to produce single forces
between dental units. It is simple to make clinical eval370
uations of the magnitude (with some limitations) and
line of action of the produced vectors. The force magnitude can be measured with a force gauge or is given by
the manufacturer (in the case of Ni-Ti material), while
the line of action corresponds to the orientation of the
elastic or coil springs.
Closed coil springs
Closed coil springs, when activated, produce a force
between the two points of attachment. Closed coil springs
can be used in space closure with sliding mechanics, but
they can also be used in frictionless, statically determinate
mechanics, as shown in Figs 15-33 and 15-34. These
springs can be made out of superelastic Ni-Ti wire, which
Closed Coil Springs and Elastics
FIG 15-33 A coil spring is activated between two extensions to close a posterior space. The line of action of the
force is determined by the position of the two points of
attachment. In this case, the line should pass a few millimeters away from the CR of the anterior and posterior
units, thus limiting crown tipping of the teeth. The use
of the same mechanics on the left side, together with a
transpalatal arch, will prevent the mesial rotation of the
molars during space closure.
a
b
FIG 15-34 In this patient, both maxillary central incisors were to be extracted, due to previous dental trauma. (a) The treatment goal was planned
using the T3D Occlusogram software, and the needed force system to close
the extraction space was estimated by moving the dental segment from the
lateral incisor to the first premolar on both sides. The needed force systems
are represented by their equivalent single forces. (b) The appliance at the start
of treatment. The two units are solidarized by a cast bonded metal structure,
from which two extensions lead to the points of attachment of the coil springs,
which are the points of force application. Two Ni-Ti coil springs are attached
between these extensions and the transpalatal bar, thus creating the two
needed forces. (c) Clinical results of these mechanics 5 months later, with
space closure between the two lateral incisors of about 12 mm.
has the unique ability to generate force plateaus, which
allow for constancy in applied force. However, plateau
force levels vary widely between different manufacturers,
and they are influenced by both temperature and mechanical loading cycles. Furthermore, they cannot be evaluated
with a simple force gauge, and product information from
manufacturers is not always reliable.15,16 Therefore, if a
specific magnitude of force is categorically needed, such
as in two-vector mechanics, the orthodontist should
probably use a cantilever system to produce that specific
force.
c
Elastics and elastic chains
Elastics are widely used to produce a needed force, both
in single-arch mechanics and maxillomandibular
mechanics. Their advantages include ease of use, low
price, and a reduced possibility of intraoral impingement.
Disadvantages include the impairment of oral hygiene
and decay of the applied force over time. Rapid force
decline has been reported within the first 24 hours, with
loss in the range of 40% to 70% of the initial value. Thereafter, a more stable phase has been observed, with only
minor changes for the succeeding 4 weeks.17–20
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Principles of Statically Determinate Appliances and Creative Mechanics
a
b
c
FIG 15-35 An elastic chain is used to close a unilateral space and correct an arch asymmetry in a patient seeking an esthetic appliance. In
this case, the amount of force exerted by the elastic chain is not critical. (a) The line of action of the force passes close to the CR of the anterior
group, thus generating translation of the group on the occlusal plane. (b) The power arm was bent to change the line of action of the force,
creating a small counterclockwise moment to the anterior group's CR and limiting the lingual displacement of the maxillary left lateral incisor,
which is now aligned to the canine. (c) Space closure almost complete.
Also, because the degree of stretch in elastics is usually
determined by the convenience of the attachment rather
than the needed force, clinical measurement of the
applied force can be difficult. For these reasons, elastic
chains are only recommended when the magnitude of
force is not a critical factor, affording the orthodontist
latitude to manage a wide range of force values (Fig
15-35). Elastics can be used quite safely if they can be
changed daily by the patient.
Two-Vector Mechanics
In some clinical situations, it is not possible to obtain the
desired force vector by means of a single cantilever, coil
spring, or elastic. This happens when the line of action
of the desired force cannot be reached within the orthodontic direct working area, as shown in Fig 15-36.
When a dental movement is generated by one of the
numbered red vectors shown in Fig 15-36, a special and
more complex design of the statically determinate
mechanics is required: two-vector mechanics. In this
example, vector 1 would produce a retraction of the
maxillary anterior roots with minimal movement of the
crowns. Vector 2 would produce retraction, intrusion,
and palatal root torque to the maxillary anterior teeth.
Vector 3 would generate a translatory distal displacement
of the mandibular anterior teeth. Vector 4 would cause
uprighting and intrusion of a mandibular molar. If any
of these red vectors is needed in a statically determinate
system, two-vector mechanics should be designed. Note
that the lines of action of these vectors (dashed lines)
never cross the green or blue areas. It is interesting to
note that these dental movements are universally considered difficult movements, regardless of which mechanical orthodontic approach is applied.
372
Mathematic procedure
The procedure used to design a proper two-vector
mechanics system is based on vector calculation and has
been discussed by Fiorelli et al.21 The following section
is reprinted from their article with permission.
Once the necessary force has been defined (FR), it is
necessary to find two forces (F1 and F2) whose combination will generate the resultant FR. An infinite number
of force combinations will have FR as the resultant. If the
points of application of F1 and F2 are established in
advance, the number of possible solutions will be limited.
The mathematic procedure to resolve a given resultant
force into two vectors applied at two chosen points (P1
and P2) can be divided into three steps:
1. The equivalent force system (F1P1 and MP1) of the
given vector is calculated at one of the two chosen
points, which will be called point P1 (Fig 15-37a).
2. A couple (F2P1 and FP2) with its two vectors applied
at points P1 and P2, equivalent to MP1 that has already
been calculated at point P1, must be computed. This
couple will substitute MP1 (Fig 15-37b). Note that there
are many different couples of vectors applied at points
P1 and P2 that can be equivalent to MP1. As illustrated
in Fig 15-37c, there is a wide range of directions that
can be chosen for the two vectors, with the limit determined by the line connecting points P1 and P2. Thus,
the problem of resolving one vector into two vectors
applied to two chosen points has a unique solution
only if the line of action of the force vector applied to
one of the two points is also chosen.
3. At point P1, two force vectors must be added to find
a single resultant vector FP1. FP1 and FP2 are the two
needed vectors to replace vector R (Fig 15-37d).
Two-Vector Mechanics
FIG 15-36 Orthodontic working areas in sagittal view. The green area
represents the working area on the buccal side, while the blue area
represents the working area in the palate. Note that this area can be
more or less extended, depending on the palatal vault anatomy. Green
vectors can be produced easily with a single cantilever, coil spring,
or elastic on the buccal side. If a blue vector is needed (only for the
maxillary arch), palatal single-vector mechanics can be designed.
a
b
c
d
FIG 15-37 (a) R is the needed vector. The equivalent force system at P1 includes the force F1P1 and the moment MP1. (b) The MP1 moment
can be replaced by two vectors applied at P1 and P2, which together make a couple equivalent to MP1. (c) The couple described in b can also
be replaced by other couples where the single vectors have different angulations. Of course, in these cases there will be different distances
between the two vectors, so their magnitude should be adjusted to produce the same moment. (d) The sum of vectors F1P1 and F2P1 will
result in the vector FP1. Vectors FP1 and FP2 applied simultaneously will generate the same mechanical effects as vector R. Because P1 and
P2 are located inside the working area (see Fig 15-36), it will be possible to design a statically determinate appliance including two different
parts, one of them producing FP1 and the other producing FP2. (Reprinted from Fiorelli et al21 with permission.)
Clinical applications
The clinical application of this computation procedure
can take place only if the desired dental movement has
been defined and if the force system leading to this movement has been calculated. These two steps can be
performed by using two specific software programs: the
T3D Occlusogram22 and DMA.7
T3D Occlusogram allows the computerized execution
of an occlusogram in conjunction with the lateral cephalometric radiograph. This program provides a visual reproduction of the treatment goal. DMA then calculates the
force system needed to generate the dental movement(s)
defined by the T3D Occlusogram software using formulas
that are derived from experimental data.2–5 Both steps can
also be executed manually, if necessary.23
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15
Principles of Statically Determinate Appliances and Creative Mechanics
a
b
FIG 15-38 (a) In this force diagram, the two blue vectors are combined to obtain the same effect given by the vector represented in
red, which cannot be produced directly because of its position. This
vector would generate intrusion, retraction, and palatal root torque if
applied to the anterior teeth. Clinically, the two vectors are generated
by a torque composite arch (80 g) and a cantilever with configuration
(120 g). (b) The appliance, activated bilaterally, has been applied for
5 months. (c) Note the change in inclination of the maxillary anterior
teeth and the bite opening, while no space was opened between the
lateral incisor and the canine.
a
Mandibular midline
c
b
FIG 15-39 (a) The maxillary anterior teeth require lateral displacement with moderate intrusion, while the mandibular anterior teeth are mostly
intruded and proclined by a cantilever inserted into the left molar tube. Two-vector mechanics have been designed after vector calculation
by DMA software. The red vector should produce the desired movement of the teeth. Two cantilevers are applied to deliver the two vectors
represented in blue. (b) After 5 months, the dental midlines are completely aligned. The deep bite is also improved with a major contribution of mandibular tooth displacement. Note how the mesial aspect of the maxillary left canine is totally visible, while the same surface of
the contralateral canine is mostly hidden due to the transverse movement of the incisor group. (From Biomechanics in Orthodontics, www.
ortho-biomechanics.com.)
In the case presented in Fig 15-38, movement including
intrusion, retraction, and palatal root torque is achieved
via two-vector mechanics. The two needed force vectors
were generated by two cantilevers applied bilaterally,
which allowed for excellent precision both for angulation
and magnitude of the two forces. It is important to stress
that in two-vector mechanics the amount of force delivered by the appliance is critical. An insufficient or excessive magnitude of one of the two vectors will skew the
direction and position of the resultant vector, producing
clinical results that may differ significantly from the
desired outcome.
374
In the case illustrated in Fig 15-39, two cantilevers were
used to translate the incisor group in the frontal plane,
with a transverse and slightly intrusive movement. It is
interesting to visualize the potential for generating different vectors by modifying the amount of activation on the
two sides; results could include a larger midline discrepancy or deep bite correction.
The case depicted in Fig 15-40 shows how it is possible
to produce a buccal translation of the mandibular incisor,
dramatically reducing the patient’s horizontal overlap
(also known as overjet) without decreasing the interincisal
angle.
Two-Vector Mechanics
a
b
c
d
e
f
g
h
i
FIG 15-40 (a to d) This patient shows a large horizontal overlap and an open bite due to her thumb-sucking habit that was maintained until
the age of 14 years. Her skeletal pattern is a Class I (A-NPg = 0) with a normal vertical pattern, while her profile is agreeable. For this reason, it
was decided to avoid any functional or surgical advancement of the mandible and to try translatory advancement of the mandibular anterior
teeth and extrusion of the maxillary teeth. (e) Vectorial calculations executed with the DMA software. The two blue vectors added together
will produce the same effect as a sagittal horizontal force passing close to the CR level (red vector). These two vectors can easily be obtained
using cantilevers with major configurations inserted in the anterior segment and ligated to the posterior teeth and by anterior vertical elastics.
Such mechanics can be used bilaterally with similar forces to obtain symmetric displacement. (f) The appliance producing the two vectors
includes a cantilever with configuration and vertical elastics, delivering the forces described in e to the six anterior teeth. After about 1 year of
treatment with the appliance, a large space has opened in the first premolar area, the horizontal overlap has normalized, and the molar relation
is now a full Class II. Some more extrusion of the anterior teeth is still needed to close the horizontal overlap. Note how the mandibular anterior
teeth seem to have good inclination. (g) The cephalometric radiograph after 1 year of treatment shows the reduced horizontal overlap and
still some open bite. The angle of the mandibular anterior teeth to the mandible has increased minimally from the start of treatment. (h and
i) The orthodontic treatment was completed in 28 months, and two implants were later inserted between the mandibular canines and first
premolars. Note how the mandibular incisors do not seem proclined. (From Biomechanics in Orthodontics, www.ortho-biomechanics.com.)
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15
Principles of Statically Determinate Appliances and Creative Mechanics
a
b
FIG 15-41 (a) Two-vector mechanics applied to obtain a forward movement of the roots of the maxillary incisors, which are
flared as a compensation for a skeletal Class III condition. The two vectors applied to the maxillary incisors are produced by a
cantilever with major configuration attached at premolar level and by vertical elastics. The resultant force vector is a horizontal
force passing several millimeters above the CR of the anterior teeth. (b) Clinical results after 5 months; the maxillary incisor
inclination appears significantly less flared.
a
b
c
d
FIG 15-42 (a) A second molar is mesially tipped. (b) A cantilever is used to upright the tooth, and the force system is appropriate
to obtain the needed movement with a limited extrusive effect, due to the large M/F ratio. (c) After 4 months, the molar is totally
uprighted, but a reverse articulation has resulted. (d) Frontal view of the force diagram. The drawing represents the molar shape
seen from the frontal view. Note how a buccal crown moment is generated by the cantilever activation, thus explaining the reverse
articulation side effect. This movement created sufficient space for the rotation of the maxillary right central incisor and therefore
the possibility for aligning the maxillary anterior teeth. (From Biomechanics in Orthodontics, www.ortho-biomechanics.com.)
In Fig 15-41, two-vector mechanics are used to produce
a buccal root movement of the maxillary incisors in a
borderline skeletal Class III case.
376
3D Problems
When planning a dental movement, it is important to
consider the needed force system in three planes of space.
In fact, a simple two-dimensional analysis sometimes
does not allow expression of the correct force system.
3D Problems
Represented forces are those delivered
to the anterior unit
Cantilever SS 0.017 × 0.025–inch
Activation FX 67g, F Y 45g, FZ 80g
Composite torque arch
β-titanium 0.018–inch welded to
0.017 × 0.025–inch
Resultant force vector
FIG 15-43 In this image, a 3D vector calculation is represented. Three vectors are used (two red vectors delivered by a composite torque arch,
and the blue one by a cantilever). The three vectors are shown in the three planes of space, unless they are normal to them. For example, the
two red vectors visible in the frontal view cannot be represented in the occlusal view. The same vectors share the line of action of the force
in a sagittal view so they can be represented by a single vector. The goal is to move the anterior teeth forward and to the left, controlling the
root position on the sagittal and frontal plane; some rotation of the anterior teeth on the occlusal plane is expected. Note how an asymmetric
activation of the torque arch is needed. In the clinical application, of course, forces will be applied with some approximation.
Figure 15-42a shows a mandibular second molar that
is mesially tipped. In Fig 15-42b, simple cantilever
mechanics for molar uprighting are illustrated together
with a force diagram in the sagittal view. The force system
is appropriate to upright the molar, as can be seen in Fig
15-42c; however, a reverse articulation (also known as
crossbite) has developed. The reason for this side effect
can be understood if an analysis of the frontal view is
performed. In fact, as shown in Fig 15-42d, a buccal crown
moment is created by the cantilever activation because
the cantilever is ligated closer to the mandibular midline
with respect to the CR of the mandibular molar. Usually,
this side effect is desirable because mesially tipped molars
often are also lingually tipped; however, when a reverse
articulation tendency is present, the buccal inclination
should be controlled.
3D mechanics planning
For more complex clinical conditions, a true 3D mechanics design can offer better solutions. The theoretical basis
for a 3D design is not much different from that used in
a 2D environment. The first step is always to define a
force vector corresponding to the desired movement. A
single force vector can always be determined, but identifying and generating it in a 3D space may be much more
complex. Figure 15-43 provides an example of this calculation. In this case, the treatment goal was to correct a
negative overjet through translation of anterior teeth
along an oblique line while simultaneously correcting the
midline position with translation in the frontal plane.
377
15
Principles of Statically Determinate Appliances and Creative Mechanics
a
b
c
d
FIG 15-44 Phases of treatment for patient in Fig 15-43. (a) Before treatment. (b) Start of treatment; the appliance is activated as described
in Fig 15-43. Overlays are bonded to the mandibular posterior teeth to allow the free movement of the maxillary anterior teeth. Unfortunately,
these were often broken by the patient in the interval between visits. (c) After 4 months, some advancement of the anterior teeth and some
space distal to the maxillary right canine can be appreciated; however, the anterior teeth appear too vertical, probably due to the occlusal
interferences at the incisal edge that modified the force system. The mechanics were adjusted by reducing the amount of activation of the
torque arch and by moving the point of ligature of the cantilever about 5 mm to the left. This adjustment generated some crown proclination
in the sagittal plane, while keeping a translatory movement in the frontal one. (d) After 8 months of treatment, the anterior teeth are in their
desired position, and composite restorations have been performed on them.
In the occlusal plane, a group rotation of these teeth
contributes to the opening of space in the right premolar
area. The vectors that are shown in the diagram were
used as guidance for the design and activation of statically
determinate mechanics.
378
Figure 15-44 shows some stages of the treatment,
demonstrating the clinical effects of the mechanics. In
Fig 15-45, the superimposition of the cephalometric tracings confirms the translatory movement of the anterior
teeth in a sagittal view.
References
a
b
c
FIG 15-45 Cephalometric radiographs of the patient in Figs 15-43 and Fig 15-44. (a) Before treatment. (b) After movement of the maxillary
anterior teeth, corresponding to Fig 15-44d. (c) The incisor positions were superimposed using the posterior contour of the alveolar process
as a reference. The dental movement appears to be a translation with an oblique direction (buccal-extrusive). The anterior contour of the
alveolar bone seems slightly displaced anteriorly.
References
1. Hool GA, Johnson NC. Elements of structural theory. In: Handbook of Building Construction. Vol 1: Data for Architects, Designing and Constructing Engineers, and Contractors. New York:
McGraw-Hill, 1920.
2. Burstone CJ, Pryputniewicz RJ. Holographic determination of
centers of rotation produced by orthodontic forces. Am J Orthod
1980;77:396–409.
3. Vanden Bulcke MM, Burstone CJ, Sachdeva RCL, Dermaut LR.
Location of the center of resistance for anterior teeth during retraction using laser reflection technique and holographic interferometry. Am J Orthod Dentofacial Orthop 1987;91:375–384.
4 Nägerl H, Burstone CJ, Becker B, Kubein-Messenburg D. Center
of rotation with transverse forces. Am J Orthod Dentofacial Orthop 1991;99:337–345.
5. Meyer BN, Chen J, Katona TR. Does the center of resistance depend on the direction of tooth movement? Am J Orthod Dentofacial Orthop 2010;137:354–361.
6. Cattaneo PM, Dalstra M, Melsen B. Moment-to-force ratio, center of rotation, and force level: A finite element study predicting
their interdependency for simulated orthodontic loading regimens. Am J Orthod Dentofacial Orthop 2008;133:681–689.
7. Fiorelli G, Melsen B, Modica C. The design of custom orthodontic mechanics. Clin Orthod 2000;3:210–219.
8. Burstone CJ. Variable modulus orthodontics. Am J Orthod 1981;
80:1–16.
379
15
Principles of Statically Determinate Appliances and Creative Mechanics
9. Burstone CJ. Maximum forces and deflections from orthodontics
appliances. Am J Orthod 1983;84:95–103.
10. Dalstra M, Melsen B. Force systems developed by six different
cantilever configurations. Clin Orthod Res 1999;2:3–9.
11. Melsen B, Konstantellos V, Lagoudakis M, Planert J. Combined
intrusion and retraction generated by cantilevers with helical
coils. J Orofac Orthop 1997;58:232–241.
12. Nelson KR, Burstone CJ, Goldberg AJ. Optimal welding of betatitanium archwires. Am J Orthod Dentofacial Orthop 1987;92:213–
219.
13. Krishnan V, Kumar KJ. Weld characteristics of orthodontic archwire materials. Angle Orthod 2004;74:533–538.
14. Burstone CJ. Welding of TMA wire. Clinical applications. J Clin
Orthod 1987;21:609–615.
15. Wichelhaus A, Brauchli L, Ball J, Mertmann M. Mechanical behavior and clinical application of nickel-titanium closed-coil
springs under different stress levels and mechanical loading cycles. Am J Orthod Dentofacial Orthop 2010;137:671–678.
16. Melsen B, Terp S. Force systems developed from closed coil
springs. Eur J Orthod 1994;16:531–539.
17. Ash J, Nikolai R. Relaxation of orthodontic elastic chains and
modules in vitro and in vivo. J Dent Res 1978;57:685–690.
18. Andreasen GF, Bishara SE. Relaxation of orthodontic elastomeric chains and modules in vitro and in vivo. Angle Orthod
1970;40:319–328.
19. Lu TC, Wang WN. Force decay of elastomeric chain. China Dent
J 1988;7:74–79.
20. Buchmann N, Senn C, Ball S, Brauchli L. Influence of initial strain
on the force decay of currently available elastic chains over time.
Angle Orthod 2012;82:529–535.
380
21. Fiorelli G, Melsen B, Modica C. Two-vector mechanics. Prog Orthod 2003;4:62–73.
22. Fiorelli G, Melsen B. The “3-D occlusogram” software. Am J Orthod Dentofacial Orthop 1999;116:363–368.
23. Fiorelli G, De Oliveira W, Merlo P, Vasudavan S. Exercises in Orthodontic Biomechanics. Radolfzell am Bodensee, Germany:
IOSS, 2019.
Recommended Reading
Alexander RG. The vari-simplex discipline. Part 1. Concept and appliance design. J Clin Orthod 1983;17:380–392.
Andrews LF. The straight-wire appliance. Explained and compared. J
Clin Orthod 1976;10:174–195.
Andrews LF. The straight-wire appliance, origin, controversy, commentary. J Clin Orthod 1976;10:99–114.
Damon DH. The rationale, evolution and clinical application of the
self-ligating bracket. Clin Orthod Res 1998;1:52–61.
McLaughlin RP, Bennett JC. The transition from standard edgewise
to preadjusted appliance systems. J Clin Orthod 1989;23:142–153.
Roth RH. The straight-wire appliance 17 years later. J Clin Orthod
1987;21:632–642.
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16
The Role of Friction
in Orthodontic
Appliances
“The gem cannot be polished without friction,
nor man perfected without trials.”
— Confucius
Because frictional forces are part of any orthodontic force system, friction
must be both studied and understood. In a continuous arch, wires continually slide through bracket slots, which produces frictional forces. Frictional
forces act parallel to the long axis of a wire and are produced by normal
forces at 90 degrees to the wire. Some frictional force is produced by the
tying mechanism, but most is associated with normal forces that are needed
for tooth movement: labial or lingual, occlusal or apical, or various moments
and torques. Fundamental classic equations govern the determination of
frictional force. Frictional force equals the coefficient of friction times the
normal force. Classic theory, although limited in the real world, is a good
introduction for the clinician. Frictional force varies during canine retraction. Four stages can be described, from tipping to root movement. The
frictional force is different in all stages, with the smallest during tipping.
For forces alone, friction is independent of bracket width. For moments
used to prevent canine tipping and rotation, the wider the bracket width,
the lower the friction. Many biologic considerations will influence friction,
including root length. The role of vibratory motion to reduce friction is also
discussed in this chapter. Friction can be both good and bad clinically,
but the biggest problem with friction is that it is unknown. If known, many
times it can be overridden.
383
16
The Role of Friction in Orthodontic Appliances
FIG 16-1 If a force is applied to a tooth, the tooth will not feel the
applied force (FA ) if there is friction (FF ) in the appliance. What the
tooth feels is the effective force (FE ).
a
b
FIG 16-2 (a) In classic friction theory, the friction is calculated from the coefficient of friction (µ) and the normal force (FN ) perpendicular to
the archwire: frictional force (FF ) = coefficient of friction (µ) × normal force (FN ). (b) The bracket is pushing on the wire in an occlusal direction
with an equal and opposite magnitude according to Newton’s Third Law.
T
here has been an increasing interest in understanding and controlling friction during the use of
orthodontic appliances. Frictional forces can operate along with active tooth-moving forces or can be from
the restraint of the tying mechanism. Friction can be both
good and bad. It is the intent of this chapter to delineate
the role of friction on a scientific basis so that the clinician
can optimize treatment and better evaluate the utility of
so-called low-friction brackets and wires. Understanding
friction can help us select a new appliance or improve our
use of an existing appliance system. It determines, in part,
the efficiency of tooth movement and anchorage, and it is
a factor in eliminating undesirable side effects. Moreover,
understanding friction can help to reduce commercialism
in the marketing of appliances and techniques.
Frictional Forces, Their Origin,
and Classic Formulas
If a force is applied to a canine from a chain elastic or a
coil spring (Fig 16-1), the tooth will not feel the full force
384
if there is friction in the appliance. What the tooth feels
is the effective force (FE ), not the applied force (FA ):
FE = FA – Frictional force (FF )
When the frictional force is the same as the applied force,
the tooth will feel no force from the spring. As long as
there is frictional force, effective force is always less than
the applied force. Of course, it is the effective force that
is relevant for the clinician.
Tooth movement is an intermittent start-and-stop
phenomenon; hence, we are interested in static friction.
Measurement of friction along moving surfaces (dynamic
or kinetic friction) will give somewhat smaller values.
Rolling friction involves wheels and is not relevant to
our appliances.
Where do frictional forces come from? The nature of
friction is still being debated between adhesion and interlocking theory, even among modern physicists; however,
classic friction theory tells us that forces perpendicular
to the archwire are responsible for friction. Figure 16-2a
shows a canine sliding along an archwire. For simplicity,
Frictional Forces, Their Origin, and Classic Formulas
FIG 16-3 Applied force (FA ) plotted against frictional force (FF ).
The frictional force and applied force are proportional up to a
certain level of force called the maximum static friction force.
Frictional force increases the same amount as the applied force
but in the opposite direction. If the applied force increases
above the maximum static friction force, the bracket will start
to move with slightly decreased frictional force (kinetic friction).
all moments are ignored. The applied force (FA) is 100 g.
FN is a normal force perpendicular to the wire; the bracket
is pushing on the wire in an occlusal direction, and
according to Newton’s Third Law, an equal and opposite
force is pushing on the bracket (Fig 16-2b). The classic
law of friction is also known as Amontons-Coulomb Law
and is very simple:
FF = Coefficient of friction (µ) × Normal force (FN )
There are no terms in the formula regarding the amount
of contact area, duration of contact, temperature, or sliding speed. The coefficient of friction is not an inherent
property of a material, such as modulus of elasticity. It
is a dimensionless property that represents the amount
of friction between two materials and is determined by
experiment only and not by theory. If a material is used at
the interface of two other materials to reduce the coefficient of friction, it is called a lubricant. If it increases the
coefficient of friction, it is called an adhesive. For a stainless steel wire and stainless steel bracket in the mouth, an
average value for the coefficient of friction (µ) is 0.16. The
magnitude of normal force can be unpredictable because
of the many variables, including three material interfaces
that can be present: wire, bracket, and polymeric O-ring.
Suppose a 50-g normal force is applied to the bracket in
Fig 16-2a. The frictional force can be calculated, and the
effective distal force is 92 g.
FF = 0.16 × 50 g = 8 g
FE = 100 g – 8 g = 92 g
Figure 16-3 shows a bracket with forces acting on it.
Note that as the applied force is increased, the frictional
force increases at a constant rate up to a certain level of
force called the maximum static friction force. Before this
point, frictional force increases proportionally with the
applied force but in the opposite direction. If the applied
force increases above the maximum static friction force,
the bracket will start to move. The applied force must
overcome this maximum frictional force for any tooth
to move. The maximum static friction force is observed
just before movement starts. If movement continues,
frictional force values decrease slightly, and so-called
kinetic friction is observed. Either static or kinetic friction
values are applicable as long as we understand that wirebracket interfaces are responding to static friction. Tooth
movement is not continuous; rather, it is an intermittent
start-and-stop phenomenon. Hence, static friction is the
most relevant for us. For the remainder of this chapter,
frictional force and maximum static friction force are
used interchangeably.
Plots of frictional force versus displacement are more
irregular than Fig 16-3. The plot (green) in Fig 16-4a
demonstrates fluctuating frictional force. This is explained
by the stick-slip phenomenon, where the surfaces adhere
together and then separate (break) apart. This behavior
can be explained at the microscopic level, where jagged
surfaces induce up and down motion during sliding
(Fig 16-4b).
The coefficient of friction is the lowest with stainless
steel wires and the highest with β-titanium wires. Ceramic
brackets have higher coefficients than stainless steel, and
the high variation is related to design and manufacturing
385
The Role of Friction in Orthodontic Appliances
Frictional force
16
Stick-slip
phenomenon
a
Displacement
b
FIG 16-4 (a) The actual plot from experimental data (green) demonstrates fluctuating frictional force. The average of the data is the red curve.
(b) The fluctuating curve in part a is better understood with interlocking theory at the microscopic level, where jagged surfaces must slide
past each other.
FIG 16-5 The schematic graph shows that at the extremes, both
rough and highly smooth materials have high coefficients of friction.
This phenomenon is well understood by adhesion theory at the ultramicroscopic level. At intermediate roughness, a good correlation
exists between roughness and coefficient of friction.
FIG 16-6 If the forces are high, destructive changes can
occur in either the bracket or the wire, and the subsequent
behavior will not follow classic friction theory.
methods. It is often assumed that the smoother the material, the lower the coefficient of friction; however, the
relationship is not so simple. The schematic graph in Fig
16-5 shows that at the extremes, high- and low-roughness
materials have high coefficients of friction. It is well
386
FIG 16-7 Ion impregnation by nitrogen bombarding on
β-titanium wire increases the hardness and reduces the
coefficient of friction of the wire.
known that highly polished surfaces show very high
coefficients of friction, which is well understood by adhesion theory at the ultramicroscopic level. Only at
intermediate roughness does a good correlation exist
between roughness and coefficient of friction. Hardness
Source of Normal Forces
FIG 16-8 The normal force can come
from a number of sources and in any
direction: buccal, lingual, occlusal, or
apical.
FIG 16-9 In the passive wire, the O-ring
produces a lingual force that can lead
to a frictional force. Thus, the ligation
method is only one source of friction.
is typically related to a low coefficient; nevertheless, Teflon
is a soft material, and yet its coefficient of friction is low.
The classic formulas presented in this chapter only
operate within reasonable ranges of perpendicular forces.
If the forces are high, destructive changes can occur
in either the bracket or the wire, changing the subsequent behavior. One example of this is wire notching, as
depicted in Fig 16-6. A tipped tooth can notch a wire,
producing effects not easily predicted.
Some surface treatments, such as ion impregnation by
nitrogen bombarding, increase the hardness and reduce
the coefficient of friction of a wire. Figure 16-7 shows a
group of β-titanium archwires; the various colors are
produced after titanium nitride particles are distributed
in the wire’s surface by ion impregnation.
Source of Normal Forces
Frictional forces are evident at all stages of orthodontic
treatment. They are involved in any mesiodistal sliding
between wire and bracket. This occurs not only with
purposeful sliding mechanics such as canine retraction
but also in alignment arches where, if the wire cannot
slide, buccal or lingual forces can be attenuated. Friction
can also be employed to open space in cases of arch length
discrepancies. Not all friction is bad.
Forces perpendicular to the wire can come from a
number of sources and in any direction: buccal, lingual,
occlusal, or apical (Fig 16-8). The O-ring produces
a lingual force in Fig 16-9 that can lead to a frictional
force. Thus, the ligation method is only one source of
friction. Any other forces required for tooth movement,
FIG 16-10 Of particular importance are forces originating from pure moments or couples. Normal forces
exist on the wire in three dimensions.
if perpendicular to the archwire, can also lead to friction
and in many situations can produce much more friction
than the ligature tie.
Of particular importance are forces originating from
pure moments or couples. By definition, couples are equal
and opposite forces not in the same line of action. Normal
forces exist on the wire, although the sum of the forces
is zero (Fig 16-10). Moments are used in a first-order
direction to rotate teeth, in a second-order direction to
change axial mesiodistal inclinations, and in a third-order
direction to change buccolingual axial inclinations. A
moment (couple) at the bracket is required to give an
equivalent force system for full control of a tooth. This
moment is one major source of friction with the edgewise
appliance. Some brackets are designed to allow a tooth
to tip or rotate. With this type of bracket, this source of
friction can be eliminated, but control of tooth movement
is lost as a result.
Sometimes we hear the phrase “friction-free brackets.”
These brackets use a locking cap mechanism instead of
a ligature tie. But is “friction free” possible under typical
clinical conditions? These brackets, when placed on a
wire, slide easily because no normal forces from ligation
or moments from the bracket are present. In Fig 16-11, a
low-friction self-ligating bracket has been used to rotate
a second premolar. The ideal-shaped frictionless archwire
will produce approximately a couple that should rotate
the premolar around its center of resistance (CR), near
the center of the crown. However, two normal forces
from the deflected wire produce very large frictional
forces so that the bracket can hardly slide (Fig 16-11a).
Therefore, the tooth rotates around the bracket according to the replaced equivalent force system (in yellow).
387
16
The Role of Friction in Orthodontic Appliances
FIG 16-11 (a) Even in a low-friction self-ligating bracket, frictional
force operates at the distal of the bracket in a mesial direction due to
the normal forces from the deflected wire. (b) The replaced equivalent
force system (yellow arrows) shows that the frictional force produced
a side effect that opened up space, and the crown moved mesially.
In clinical situations, forces on the wire, not just the ligature tie, are
a major source of friction.
a
b
Notice that the frictional force produced a side effect
that opened up space and that the crown moved mesially
(Fig 16-11b). This demonstrates once again that other
frictional forces operate beyond the ligation mechanism.
In other words, we should be careful in using the terms
friction-free or frictionless brackets to describe actual
clinical situations. In most clinical situations, the archwire
will deliver normal forces and moments to the teeth to
produce tooth movement; the reactive perpendicular
forces on the wire, not just the ligature tie, are a major
source of friction.
Some orthodontists differentiate between simple
normal forces and normal forces from couples. Classical friction theory allows couples to be handled like any
other forces. Terms such as binding should not be used to
suggest a different theoretical mechanism at work in the
situation where a tipping moment or torque is applied.
Canine Retraction
An in-depth consideration of canine retraction using sliding mechanics provides the opportunity to develop how
friction works with a major treatment phase. Without a
wire for control, a distal force on a canine produces wellknown side effects. The canine rotates distal in, and the
crown tips distally. To prevent the unwanted effects, an
archwire can be placed; the archwire elastically deforms
and, during recovery, prevents or minimizes the rotation
and tipping by exerting couples on the teeth (Figs 16-12a
and 16-12b). Figures 16-12c and 16-12d show the same
diagram with the couples (curved arrows) replaced by two
normal forces to further show the origin of the frictional
force. In Fig 16-13, as the tooth tips distally, the wire
388
curves, and energy is stored in the wire. As the curved
wire straightens out, normal forces control tooth movement and prevent tipping. The same forces and moments
that give control also initiate frictional forces as these
normal forces act on the wire. In short, no friction during
sliding mechanics means no control.
The “control” couples will vary depending on what is
required. The yellow arrow in Fig 16-14a is the location of
the force if translation is the objective. An equivalent force
system at the bracket requires large equal and opposite
vertical forces (a couple). This is contrasted in Fig 16-14b
with a tipping movement around the apex (as the center
of rotation), where the needed moment is much smaller.
Because the control moment is smaller, there will be less
friction during tipping than during translation.
To figure out how much frictional force occurs during
canine retraction, we must consider the phase of canine
retraction as evaluated from both the facial and occlusal views. Four phases can be recognized (Fig 16-15).
After a distal force is placed, the canine may have play
between the wire and the bracket, and initially the tooth
will display uncontrolled tipping. This is phase I. No
moments or normal forces operate in this plane. For now,
ligation forces are ignored. The tooth continues to tip
more, and the play is eliminated. Increasing moments are
created by the elastically deformed wire, and a controlled
tipping phase occurs (phase II). Perhaps we have a tipping
center of rotation at the apex. Note that normal forces are
produced in phase II as the tipping is being minimized,
but only low levels of friction are produced. When the
tooth tips some more and a sufficiently high moment is
delivered by the wire, translation occurs (phase III). The
greatest frictional forces are produced during translation.
During phase IV, as the force is reduced, no more distal
Canine Retraction
a
b
c
d
FIG 16-12 During canine retraction, the canine rotates distal in, and the crown tips distally. The archwire elastically deforms and, during recovery, prevents or minimizes the rotation (a) and tipping (b) by exerting couples on the teeth. (c and d) The same diagram with the couples
(curved arrows in a and b) replaced by two normal forces (arrows) to further show the origin of the frictional force.
FIG 16-13 As the canine tips distally, the wire curves, and energy is
stored in the wire. As the curved wire straightens out, normal forces
control tooth movement and prevent tipping (red arrows).
a
b
FIG 16-14 (a) The yellow arrow is the location of the force if translation is the objective. An equivalent force system at the bracket requires
vertical forces (a couple). (b) A tipping movement around the apex (as the center of rotation), where the needed moment is smaller.
389
16
The Role of Friction in Orthodontic Appliances
FIG 16-15 Four phases can be recognized during canine retraction
(facial view). Phase I: The canine may have play between the wire and
the bracket, and initially the tooth will display uncontrolled tipping.
Phase II: Increasing moments are created by the elastically deformed
wire, and controlled tipping occurs. Phase III: When the tooth tips
some more and a sufficiently high moment is delivered by the wire,
translation occurs. Phase IV: The force is reduced, no more distal
sliding occurs, and the axial inclination is corrected. Note that the
largest moment occurs during phase III, translation.
390
FIG 16-16 In the occlusal view, the same four phases can be recognized during canine retraction.
FIG 16-17 The amount of frictional force from the occlusal view
depends on the perpendicular distance of the bracket to the CR.
sliding occurs, and the axial inclination is corrected. Here,
of course, the high frictional force is acceptable because
sliding is not desired at this stage (see also Fig 13-9).
In short, frictional force varies depending on the stage
of canine retraction: none initially with play and the
highest levels during translation. Even with rigid edgewise arches, a retracted tooth will go through these four
phases; however, the angle of tip will be smaller. The
angle of tip during translation is mainly a function of
wire stiffness and the applied distal force. Clinically, it
may appear that the tooth has translated in one phase.
In reality, however, it has first tipped, then translated,
and then finally uprighted. Ligation forces and forces in
other planes are considered separately in this chapter.
As the bracket width decreases, the friction will increase
because the normal force must increase to provide the
same amount of moment. However, the mechanism of
narrow brackets (eg, Begg brackets) is different. They
produce only a single force and negligible frictional forces
because they do not prevent tooth tipping (no control
moments) and do not demonstrate phases II, III, and IV
of space closure. In Begg treatment, a separate individual
root spring is used for tooth uprighting during phase IV.
From the facial view, frictional forces are developed
because the CR is apical to the bracket. In a similar evaluation from the occlusal view, the bracket is labial to the
CR, and hence, a distal force will rotate the canine distal
in. The archwire prevents or minimizes canine rotation
in four phases (Fig 16-16). During phase I, if play exists
between the wire and the bracket, the canine is free to
rotate. No wire restraining of the rotation occurs; therefore, there is no friction in this phase in the occlusal view.
During phase II, the tooth continues to rotate; however,
the archwire is minimizing the rotation by elastic deformation. Because of the restraining archwire moments,
friction increases and finally reaches its maximum magnitude during phase III, translation. No sliding occurs in
phase IV when the rotation is being corrected.
The amount of frictional force from the occlusal view
depends on the perpendicular distance of the bracket to
the CR. The greater this distance, the larger is the moment
rotating the canine and the greater is the moment needed
from the archwire to prevent this rotation (Fig 16-17).
The patient in Fig 16-18a had blocked-out canines; if
canine retraction were started with their initial labial
positions, high friction levels would be anticipated for
Torque and Friction
a
b
c
FIG 16-18 If canine retraction were started with these initial labial positions, high frictional levels would be anticipated for three reasons: lingual and occlusal forces (a), a couple in the facial view, and a couple preventing distal-in rotation on the canine in the occlusal view (b and c).
a
b
FIG 16-19 (a) The moments associated with the prevention of tipping and rotation of a canine can lead to high frictional forces. (b) Thirdorder moments (ie, torque) can lead to particularly high frictional forces. Note that the same moment magnitude of 1,000 gmm requires very
high normal forces for torsion.
three reasons: (1) lingual and downward normal forces,
(2) normal forces from the couple preventing tipping
in the facial view, and (3) a moment preventing and
correcting distal-in rotation of the canine. Note that the
large distance from the applied force to the canine CR, as
observed from the occlusal view (Figs 16-18b and 16-18c),
leads to an unusually large moment. Because the frictional
forces are additive from the facial and occlusal views,
a canine abnormally flared to the buccal creates larger
frictional forces than average. It is usually wise to reduce
intercanine width as soon as possible before full canine
retraction.
Torque and Friction
It has been seen that moments associated with the
prevention of tipping and rotation of a canine can lead
to high frictional forces. In addition, third-order moments
(ie, torque) can lead to particularly high frictional forces.
Figure 16-19 compares two activations on a canine; both
have the same moment magnitude of 1,000 gmm, but one
is in the bending mode (Fig 16-19a), and the other is in
the torsion mode (Fig 16-19b).
The torque produces the largest vertical force of
2,000 g because the distance is small across the wire
cross section. Because the normal forces from torque are
greater than those from the second-order couple, the
friction will be eight times higher in torque than in tipping
for the same moment. (In this example, the ratio of the
moment arms is 4 mm/0.5 mm = 8; hence, the normal
force is eight times greater.) For this reason, it is not
recommended to use edgewise wires that fully engage
the brackets (with possible unwanted torque) for canine
retraction. The high friction can potentially make for
inefficient or unpredictable retraction. Round or undersized wires are preferable to eliminate possible unwanted
torque problems.
391
16
a
The Role of Friction in Orthodontic Appliances
b
c
FIG 16-20 Methods of ligation of the wire. (a) A wire can be placed
passively into a bracket by a locking mechanism. No force is exerted
on the tooth, and the tie function is purely restraint. (b) The tie mechanism activates the wire, producing an active force for desired tooth
movement. (c) After the wire is fully seated, a greater ligature tie force
does not increase the force to move the tooth. This wedging causes
high friction, and sometimes it can be used to keep teeth from sliding.
FIG 16-21 Leonardo da Vinci’s illustration showing that the size of
the contact area does not change the frictional force because the
normal force (weight) does not change.
Bracket Design and Friction
can deliver low tie forces; also, some clinicians are very
adept at forming light metal ties. If the frictional forces are
known, they can be overridden. It should be remembered
that, during treatment, the orthodontist applies forces
perpendicular to the arch during wire placement and
that it is these forces that can produce the most friction
during sliding mechanics; self-ligating brackets are not
an exception. The same forces are required for delivering
the correct force system with self-ligating brackets as
with more traditional brackets; hence, friction is similar.
Which bracket produces the most friction during
retraction: a wide bracket or a narrow bracket? It depends
on the phase of space closure. Many clinicians believe
that the size of the contact area between the bracket
and the wire affects the friction. Yet in the 15th century,
Leonardo da Vinci correctly observed that the frictional
force is proportional to the contact load and independent
of the contact area. The classic friction formula states
that for the same normal force, the size of the contact
area does not make any difference. Therefore, the orientation of the wire or replacement of rectangular wire
with round wire does not reduce the friction, provided
that the restraining or active normal forces are the same.
Note that in Leonardo da Vinci’s illustration (Fig 16-21),
the amount of contact area does not change the frictional
force because the normal force (weight) does not change.
The selection of a ribbon or edgewise wire orientation
must have another rationale for correct usage. Wire shape
and dimension can affect friction only if it alters wire
stiffness.
Let us consider two bracket design parameters: (1) method
of ligation and (2) bracket width. A wire can be placed
passively into a bracket, and a ligature or locking mechanism holds it in place. No force is exerted on the tooth,
and the tie function is purely restraint (Fig 16-20a). In Fig
16-20b, the tie mechanism activates the wire, producing
an active force for desired tooth movement. Displacing
the ligature tie with more force will cause the wire to more
fully seat in the bracket. After the wire is fully seated, a
greater ligature tie force does not increase the force to
move the tooth (Fig 16-20c). The added perpendicular
force will only produce a frictional force that most likely
is not required or wanted. This friction from tight ties
is sometimes used to keep teeth from sliding. Normal
force from metal ligature ties are difficult to control if
predictable ligating forces are to be achieved. Elastomeric
O-rings can deliver initially higher forces than a lightly
tied metal ligature wire. However, elastomers will undergo
degradation (or relaxation) over time, making the ligation
force unpredictable; after degradation, their normal forces
may be as low as some self-ligating brackets. If one only
considers friction from ligation, so-called self-ligating
brackets do have the advantage of more predictably delivering lighter restraining forces (forces at 90 degrees to
the archwire) and, hence, lower friction. Both active and
passive self-ligating systems can produce lower normal
forces by ligation alone than elastomeric rings or metal
ties. On the other hand, after degradation, elastomers
392
Is Friction Always Bad?
a
a
b
b
FIG 16-22 Narrow brackets may show initial faster tooth movement
than wide brackets, but that does not mean they have less friction. The
quicker tooth movement is due to the play between the bracket slot and
the wire during phase I of sliding mechanics. The narrow bracket (a)
tips more than the wide bracket (b) because of the increased play.
FIG 16-23 A narrow 2-mm bracket (a) and a wide 4-mm bracket (b)
are compared. Both teeth need a counterclockwise moment of
1,000 gmm for translation. The narrow bracket requires equal and
opposite 500-g forces (500 g × 2 mm = 1,000 gmm), while the wide
bracket requires 250-g forces (250 g × 4 mm = 1,000 gmm). The
narrow bracket has twice the frictional force because the normal force
is two times that of the wide bracket. Therefore, the wide bracket has
less friction during phases II and III of space closure.
Narrow brackets may show faster tooth movement
initially; therefore, it may be assumed that they have less
friction, but this concept is wrong. The tooth movement
in this case is not directly related to the friction. The
reason narrow brackets seem to show initial faster tooth
movement during sliding mechanics is due to the play
between the bracket slot and the wire in phase I of sliding mechanics (Fig 16-22). With the same amount of
play (clearance) between the bracket and the wire, the
narrow bracket can tip (rotate) more during phase I of
space closure. In this phase, the friction comes only from
the normal force of the ligature mechanism. To find the
frictional force, we must use a moment (couple) that
produces vertical forces. The formula is
Smaller cross-section wires may have more clearance
between the wire and the bracket and therefore may have
an extended phase I (no friction). Also, these wires have
lower wire stiffness and associated lower normal forces
during other phases of canine retraction. But remember
that the lower friction found in small round wires is not
caused by the smaller contact area.
FF = µ × N = µ × 2M
W
where F F is frictional force, N is normal force, M is
moment at the bracket, and W is bracket width.
Figure 16-23 compares two brackets: a narrow 2-mm
bracket and a wide 4-mm bracket. Let us suppose both
teeth need a counterclockwise moment of 1,000 gmm
for translation. The narrow bracket requires equal and
opposite 500-g forces (500 g × 2 mm = 1,000 gmm), and
the wide bracket needs 250-g forces (250 g × 4 mm =
1,000 gmm). The narrow bracket has twice the frictional
force because the normal force is two times that of the
wide bracket. Therefore, the wide bracket has less friction
during phases II and III of space closure.
Is Friction Always Bad?
Orthodontists may commonly think of frictional forces
as bad. In reality, however, they are not always bad. Let
us use canine retraction as an example. In Fig 16-24, a
200-g distal force is applied along with a counterclockwise moment of 1,000 gmm. If a 5:1 moment-to-force
(M/F) ratio is delivered to the bracket of the canine as
the applied force, it would be expected that the canine
would tip back with a center of rotation approaching
the apex (Fig 16-24a). Let us now calculate the frictional
forces (Fig 16-24b). The effective force is reduced to 94
g. Is this good or bad? The plan was controlled tipping
with an M/F ratio of 5; however, translation occurred.
Not only has the force magnitude changed, but so has
the M/F ratio. The new ratio of 10.6 could translate the
canine, since the force has been reduced. So the effective
force system might be better if less tipping is desired.
The negative aspect of friction is that it makes our appliances less predictable. There can be a large difference
between the applied force system and the effective force
393
16
The Role of Friction in Orthodontic Appliances
FIG 16-24 (a) A 5:1 M/F ratio is delivered
to the bracket of the canine, and canine tipback with a center of rotation approaching
the apex is expected. (b) The calculated effective force is reduced to 94 g. The effective
force system might be better if less tipping
is desired. The negative aspect of friction is
that it makes our appliances less predictable.
a
b
a
b
FIG 16-25 An effective force of 200 g is needed for canine retraction. Because the sum of all frictional forces is 300 g (a), a total applied
force of 500 g must be used in order to override the 300 g of frictional force (b).
system. Perhaps in some situations, friction is so great
that there is no effective force at all. Sometimes teeth do
not respond because of tooth-bone ankylosis; sometimes,
it could be the appliance that is ankylosed.
500 g. Let us assume a coefficient of friction (µ) of 0.2
and a 4-mm bracket width.
Overriding Friction
FF (occlusal view) = 200 g × 4 mm × 2 × 0.2 = 80 g
4 mm
If the clinician knows all of the frictional forces, force
can be added during canine retraction to compensate
for the frictional forces. This is called a friction override.
An example is shown in Fig 16-25a. An effective force of
200 g is needed for canine retraction. An M/F ratio of 6
is estimated for the tipping phase around the center of
rotation at the apex in the facial view, and an M/F ratio
of 4 is estimated to prevent rotation of the canine in the
occlusal view. The ligature tie has a normal force of
394
FF (facial view) = 200 g × 6 mm × 2 × 0.2 = 120 g
4 mm
FF (ligature tie) = 500 g × 0.2 = 100 g
∑FF = 120 g + 80 g + 100 g = 300 g
Because the sum of the frictional forces is 300 g, 500 g
must be applied in order to produce an effective force of
200 g (Fig 16-25b). This override is only for the tipping
at the apex, which is phase II of canine retraction. More
Friction and Anatomical Variation
FIG 16-26 Vibration in the mouth could relieve some frictional forces.
Liew et al2 demonstrated a 60% to 85% reduction of frictional force
using O-rings and round wire.
(an additional 80 g) is needed for phase III translation,
as a rough estimate.
Unfortunately, clinically it is not always accurate or
practical to calculate the frictional forces to estimate
the override needed. The frictional force is continually
changing during the different phases of retraction. It is
difficult to measure ligation force, and it can change.
Anatomically, teeth vary in morphology and support.
The coefficient of friction is difficult to determine, and
other factors can be present. However, the principle of
the override can be a useful clinical concept.
Thorstenson and Kusy1 showed that a conventional
twin bracket with a metal ligature tie with a 200-g normal
load produces about 30 g more frictional force during
retraction compared with a self-ligating bracket. If this
is known, an override could be easy and practical. A
30-g overload is added to the applied load when canine
retraction is initiated (phase I). The major problem that
confronts the clinician is that most of the time a thorough understanding of frictional forces is not possible.
Still, some average friction data could be helpful. More
helpful would be more predictably designed orthodontic
appliances. They do not have to be frictionless; known
friction is acceptable with an override.
O-rings and round wire (Fig 16-26). O’Reilly et al3 also
demonstrated a 19% to 85% friction reduction in both
rectangular and round wires.
Different phenomena may operate to reduce the magnitude of friction. The horizontal component of occlusal
forces can produce lateral tooth displacement that can
loosen the ligature tie or O-ring. Thus, vibration or tooth
displacement could be an important factor in eliminating
the frictional force from the ligation mechanism. The
frictional forces produced in response to tipping during
sliding of a tooth along an archwire are an entirely different phenomenon, because it is the elastically bent wire
that produces the normal forces, not the force from ligation. Occlusal forces may not relieve the friction unless
the chewing force is placed in a direction to temporarily
reduce the normal force between the wire and the bracket.
This suggests once again that friction from the ligation
mechanism may not be as important as friction from
tooth-moving forces—the forces from the elastically bent
wires. One of the main advantages of a self-ligating
bracket is that the ligation mechanism produces less
normal force in the passive state of the wire. This advantage may be minimized because vibratory forces seem to
be successful in reducing friction from conventional
ligature ties or O-rings.
Occlusal Forces, Vibration,
and Friction
Friction and Anatomical Variation
It could be theorized that vibration in the mouth could
relieve some frictional forces. This certainly is a commonly
observed phenomenon in laboratory friction. Liew et al2
showed a 60% to 85% reduction of frictional force using
Patients could have identical brackets, malocclusions,
and wires and still not have the same frictional forces
based on anatomical variation in root length and alveolar and periodontal support. Let us only consider the
395
16
The Role of Friction in Orthodontic Appliances
FIG 16-27 Patients could have identical brackets and wires but
not have the same frictional forces based on anatomical variation in
root length and alveolar and periodontal support. (a) A typical tooth
with average periodontal support as a reference. (b and c) Teeth with
shorter roots, with their CRs closer to the bracket. Here, the M/F ratios
are low with subsequent low frictional force. (d) A tooth from an adult
showing alveolar bone loss has the largest distance to the CR and
would have the greatest friction during translation.
a
b
c
d
translation phase during canine retraction for the four
teeth in Fig 16-27. To translate the teeth, a force must
be placed through the CR (yellow arrows). That force is
usually replaced at the bracket level with a force and a
couple (red arrows). The magnitude of this couple is the
force times the distance from the bracket to the CR. Thus,
the greater the M/F ratio, the higher the vertical normal
forces that create the frictional force.
The tooth in Fig 16-27a is a typical tooth with average
periodontal support as a reference. The CR is away from
the bracket; therefore, a high M/F ratio at the bracket
is required. This moment produces much friction, as
discussed in this chapter. The teeth in Figs 16-27b and
16-27c have shorter roots, with their CRs closer to the
bracket. Here, the M/F ratios are low with subsequent
low frictional force. Root resorption (see Fig 16-27c)
is certainly unwanted, but it does have the advantage
of minimizing the friction produced at the level of the
bracket.
The tooth from an adult showing alveolar bone loss
(Fig 16-27d) has the largest distance to the CR and would
have the greatest friction during translation. Clinically,
the tooth might not move so rapidly by translation, and
we would be disappointed in the response. We might
blame the poor response on the age of the patient and
biologic factors, but perhaps the greater frictional force
is the real culprit.
Anchorage and Friction
The claim is sometimes made that larger friction at the
molar can lead to prevention of anchorage loss during
space closure in an extraction case. Let us consider for
simplicity just two teeth—a canine and a first molar (Fig
16-28a)—in which a chain elastic is used. All of the forces
are acting on the same line of action. The applied force is
slowly increased from 0 g (red arrows). The forces are equal
396
and opposite on the molar and canine (Newton’s First
Law). Also, the frictional force (purple arrows) increases
with equal and opposite force (Newton’s Third Law).
Once the applied force overcomes one of the maximum
static friction forces at one of the two brackets, sliding of
a tooth along the archwire can occur. There are two possible interfaces, either at the molar or the canine bracket;
sliding will only occur at the interface that overcomes the
lower maximum static friction force. Let us assume for
now that the lower maximum static friction force is at
the canine bracket (Fmax at the molar > Fmax at the canine).
The canine slides because the applied force is greater than
the maximum static friction force at the canine bracket
(Fig 16-28b). On the molar, however, the applied force is
less than the maximum static friction force, and therefore
sliding does not occur at that interface. The sliding occurs
only at the interface with the smaller frictional force,
where the applied force is greater than the frictional force.
Even if the sliding occurs at only one interface (ie, at the
canine in this case), both teeth can still move; the canine
can move distally, or the molar can move mesially, with
sliding occurring at the canine bracket interface.
This is easily seen with a simple experiment. Place
a ruler or any rigid rod on your fingers, as depicted in
Fig 16-29a. Let us assume that the ruler is a wire and
the fingers are the brackets. The coefficient of friction
between the ruler and the fingers is the same; therefore,
the frictional force will be proportional to the normal
force. Now move the fingers closer together very slowly,
still keeping static equilibrium. Let us assume that the
left finger has less maximum static friction force to begin
with. Only the left finger starts to slide toward the center
of the ruler. As the left finger moves to the center, the
normal force will be increased (Fig 16-29b). Therefore,
sliding is stopped and the right finger starts to slide.
The sliding occurs alternately between the left and right
fingers, and the two fingers will finally meet at the center.
The normal force changes as the ruler is placed off-center,
Anchorage and Friction
a
b
FIG 16-28 The applied force is slowly increased from 0 g (red arrows). (a) The forces are equal and opposite on the molar and canine (Newton’s
First Law). Also, the frictional force (purple arrows) increases with equal and opposite force (Newton’s Third Law). (b) Once the applied force
overcomes one of the maximum static friction forces at one of the two brackets, sliding of a tooth along the archwire can occur.
a
b
FIG 16-29 (a) Place a ruler or any rigid rod on your fingers. Slowly move the fingers closer together, still keeping static equilibrium. Only one finger will start to slide toward the center of the ruler. As the finger moves to the center, the normal force will
be increased. Therefore, sliding is stopped and the other finger will start to slide. (b) The sliding occurs alternately between the
left and right fingers, and the two fingers will finally meet at the center.
a
b
FIG 16-30 An omega stop was placed immediately anterior to the molar tube for visualization of infinite friction. (a) Sliding will not occur at
the molar tube. (b) However, sliding does occur at the canine bracket, and the applied force at the molar will be reduced by the frictional force
at the canine. The differential friction at the two interfaces never produces differential space closure.
and only the finger at the contact with less friction slides.
Note that both fingers never slide at the same time. Therefore, as long as the friction is higher at the molar either by
higher normal force from a tight ligature or by a higher
coefficient of friction, it never slides. But just because it
is not sliding does not mean that it is not moving. In the
experiment above, the sliding occurs alternately between
the left and right fingers, but the movement of the fingers
is continuous. Suppose the left finger is glued to the ruler
and the sliding will occur on the right finger only; both
fingers still move, and the glued left finger does not feel
higher resistance to moving due to higher friction. The
same is true with a wire. The higher-friction side does
not feel higher resistance to moving.
In Fig 16-30a, an omega stop was placed immediately
anterior to the molar tube for visualization of infinite
friction. Sliding does not occur at the molar tube, but
sliding does occur at the canine bracket, and the applied
force at the molar will be reduced by the frictional force
at the canine (Fig 16-30b).
397
16
The Role of Friction in Orthodontic Appliances
prevents the maxillary incisors from retracting, and the
maxillary molar is now free to slide anteriorly. A differential coefficient of friction is possible, but differential
frictional force is not possible. Even with differential
coefficients of friction, differential space closure will not
occur.
In some clinical applications, archwires are required
to slide through many brackets. Under these conditions,
friction is very complicated, and hence, predictive modeling is difficult.
FIG 16-31 In the special situation where deep bite is present, anchorage loss may occur where higher maximum friction is at the
canine. This anchorage loss is not due to higher friction at the canine,
however; it is rather a result of the deep bite preventing the maxillary
incisors from retracting, meaning the maxillary molar is free to slide
anteriorly.
It is not too surprising that friction does not usually
influence anchorage loss. Note the clinical situation in Fig
16-28. The key to understanding is to remember that the
archwire is in equilibrium. Without friction, the applied
forces from the chain elastic sum to zero force on the wire.
There are two possible frictional forces on the wire: the
molar pushing the wire forward and the canine pushing
the wire backward. They must sum to zero for equilibrium, and hence, both must be equal and opposite in
magnitude. This is independent of the interface (gate),
where sliding can occur, as explained above. In short,
Newton’s First Law of equilibrium does not allow the
possibility of differential frictional forces. Differential
frictional forces acting from the molar and the canine
would accelerate both wire and patient into outer space.
The maximum static friction force may be different
for the canine and the molar; however, the magnitudes
of frictional forces are always equal and opposite. Under
these conditions, the friction does not influence the
anchorage. In Fig 16-29, the ruler is different than an
orthodontic wire, where moments can be present at each
end along with the vertical force; however, the same principle is true.
Now let us assume that the sliding interface is at the
molar, so friction is very high at the canine. With these
boundary conditions, en masse retraction of the incisors
and canine can still occur. The forces on the molar and
anterior segment will still be equal and opposite; therefore, less friction at the molar tube never produces more
anchorage loss. In the special situation where deep bite
is present, anchorage loss may occur (Fig 16-31), but not
because of higher friction at the canine. The deep bite
398
Reducing Friction During Space
Closure
Space can be closed using sliding mechanics even if there
are frictional forces. The problem with friction is that
it makes the force system more unpredictable. There
are a number of approaches that can be employed to
reduce frictional forces and make the force system more
predictable.
We have already discussed bracket design and the use
of wider brackets and lower ligation forces. Some cases do
not require translation, and then tipping can be allowed.
Tipping and suitable rotation such as distal-in canine
rotation can require less friction, because less moment
means less frictional force. If the force is placed closer to
the CR, it is not necessary for the archwire to produce
the anti-tip and antirotation moments, and subsequent
friction will be eliminated. The applied force can be placed
more apically by an extension arm or by an equivalent
force system at the bracket from an additional wire or
spring. Apical levers and lingual placement of the force
can readily be utilized. The spring to store and release
energy can be part of the canine retraction spring and its
apical extension (Fig 16-32). To eliminate or minimize the
friction from canine retraction, rotational forces from a
chain elastic or a coil spring can be attached on the lingual
surface of the canine (Fig 16-33).
If an auxiliary retraction spring or loop is used, activations can be placed to minimize tipping and rotation
during canine retraction three-dimensionally so that the
sliding archwire can deliver a smaller frictional force. An
archwire is still present to give positive control with minimal friction (Fig 16-34).
En masse space closure requires sliding of the archwire
at the posterior brackets. Because the mesial force is
buccal to the CR of the posterior teeth, molars tend to
rotate mesial in (Fig 16-35). The use of a buccal archwire
can barely prevent this side effect, and friction will be
produced. Lingual or transpalatal arches can preserve
Reducing Friction During Space Closure
FIG 16-32 Apical levers and lingual placement
of the force can readily be utilized for reduction
of friction.
FIG 16-33 A chain elastic or a coil spring can
be attached on the lingual surface of the canine
to reduce the friction in the occlusal view.
FIG 16-34 If an auxiliary retraction spring or loop
is used for canine retraction, three-dimensional
activations can be placed to minimize tipping and
rotation. The archwire delivers additional vertical
and lateral normal forces for control.
FIG 16-35 En masse space closure requires
sliding of the archwire at the posterior brackets. Because the mesial force is buccal to the
CR of the posterior teeth, molars tend to rotate
distal out.
FIG 16-36 Lingual or transpalatal arches can
preserve arch form without producing friction
from a wire observed in the occlusal view.
FIG 16-37 Space closure can be accomplished
without sliding or friction mechanics by a frictionless spring. All needed anti-tip and antirotation moments are bent and twisted into the
springs.
arch form without producing friction from a wire
observed in the occlusal view (Fig 16-36).
Finally, space closure can be accomplished without
sliding or friction mechanics by a so-called frictionless
spring. In Fig 16-37, canine retraction springs were used.
All needed anti-tip and antirotation moments are bent
and twisted into the springs. No sliding on an archwire is
required. With sliding mechanics, the required moments
are obtained by perpendicular normal forces from the
archwire, inevitably producing friction. With frictionless
springs, the same forces and moments may be required
and are present, but because no sliding occurs, there is
no friction (see also Fig 13-13).
399
16
The Role of Friction in Orthodontic Appliances
b
a
FIG 16-38 (a and b) Frictional forces produce a component
of force that is parallel to the archwire. (c) As the wire deactivates, the brackets are leveled. While the wire deactivates,
not only vertical force but also significant horizontal force is
produced by the frictional force (purple arrows).
c
a
b
Friction During Initial Alignment
and Finishing
Frictional forces can be present and influence results at
all stages of treatment from leveling to finishing. Two
effects that occur with lighter alignment arches merit
mention. Frictional forces produce a component of force
that is parallel to the archwire (Fig 16-38). Sometimes this
is good, and other times it is bad. The positive effect of
mesiodistal forces due to friction is the opening of space for
tooth alignment. Many patients have moderate crowding,
and an increase of arch length is desirable. If the wire
does not have any friction, there would be no horizontal
forces; with moderate friction, the wire will open space by
400
FIG 16-39 The positive effect of frictional
forces. (a) There would be no mesiodistal
forces to open the space for alignment without friction. (b) Adequate frictional forces
provide mesiodistal forces to adjacent teeth
for alignment.
pushing teeth laterally, causing an increase in arch length
(Fig 16-39). It is a well-known principle that teeth cannot
be aligned or rotated unless there is enough space for
them. Because there are limitations in the ability of a main
archwire to sufficiently increase arch length, auxiliary or
secondary wires such as coil springs, intrusion arches, and
bypass arches can be used to increase arch length. If there
is adequate space, low friction in an archwire is desirable.
The negative effect of friction during leveling is that the
wire may not be free to slide mesially or distally through
the brackets; therefore, the desired buccal forces are not
free to express themselves. Horizontal frictional forces
prevent deactivation of the wire (Fig 16-40). Large deflections of the wire that cannot be recovered to the original
shape because of friction prevent the slide of the wire in
Friction During Initial Alignment and Finishing
FIG 16-40 The negative effect of frictional forces. With infinite friction, the wire may not be free to slide mesially or distally through the
brackets; therefore, the wire cannot deactivate unless the interbracket
distance increases. The desired buccal forces are not free to express
themselves.
a
b
c
FIG 16-41 (a to c) A reverse articulation of a maxillary lateral incisor is treated by a Ni-Ti overlay wire. It is important to allow sliding at the tie.
Note that the overlaid Ni-Ti wire has a hook on each side (circles) and that elastics are activated with light force in the direction of the axis of
the wire. The mesiodistal forces at the wire will thereby unlock the friction, allowing full labial force expression to the lateral incisor.
the bracket slot. If full deactivation does not occur and
the wire does not slide spontaneously, it can be removed
and retied. Leaving a wire in place to deactivate it can
open space and relieve the offending friction; however,
these mesiodistal forces may not be efficient or wanted.
A reverse articulation of the maxillary lateral incisor
is treated by a nickel-titanium (Ni-Ti) overlay wire (Fig
16-41). If the ligature is too tight, the Ni-Ti wire cannot
fully deactivate. It is important to allow sliding at the tie.
Note that the overlaid Ni-Ti wire has a hook on each side
(circles in Figs 16-41a and 16-41b) and that the elastics
are activated with light force in the direction of the axis
of the wire to overcome the frictional force from the
ligature. The mesiodistal forces to the wire will thereby
unlock the friction and will allow full labial force expression to the lateral incisor (Fig 16-41c). Another approach
is to remove the Ni-Ti overlay and retie the ligature to
eliminate the unwanted longitudinal forces.
Tying of archwires into irregular teeth can either
increase the arch length or reduce the arch length, even
when identical forces are applied, because of friction.
To simplify this explanation, let us consider a cantilever force system with a single force delivered at the free
end. Figure 16-42 shows an intrusion arch with a V-bend
placed anterior to the molar tube. Its configuration after
initial intrusion will also produce flaring of the incisors;
however, let us not consider this effect. We could assume
that the intrusion force is acting at the CR of the anterior
teeth. Because it is a cantilever, the location of the V-bend
is not very important. It can be placed at many locations
further anteriorly along the intrusion arch to produce
identical intrusive forces; yet different configurations
would produce varying amounts of horizontal force from
the described friction effect during deactivation.
In Fig 16-42a, it is assumed that the wire and the bracket
do not have any friction. An occlusal activation force
brings the intrusion arch to the level of the incisor brackets; the wire is allowed to freely slide through the molar
tube so that the wire just touches the labial of the incisor
brackets, exerting no horizontal force. After being tied to
the incisors with a ligature, the wire will initially produce
only an intrusive force; no labial or lingual (horizontal)
forces are possible.
Next, infinite friction on the molar tube is assumed; the
wire is not free to slide through the tube after tying of the
incisors. Where does the friction come from? The large
moment and force (red) acting at the molar tube generate
high frictional forces. In Fig 16-42b, the same deactivated
401
16
The Role of Friction in Orthodontic Appliances
a
b
FIG 16-42 (a) An intrusion arch with a V-bend placed anterior to the
molar tube. It is assumed that the wire and the bracket do not have
any friction. After the wire is tied to the incisors with a ligature, the
wire will initially produce only an intrusive force. No labial or lingual
(horizontal) forces are possible. (b) Infinite friction on the molar tube
is assumed. The same deactivated shape is placed with the above
method. The incisors will gradually feel a labial component of force
as the intrusion proceeds. (c) When a gradual curvature rather than a
sharp V-bend is formed into the intrusion arch, a lingual component
of force will be produced.
c
a
b
shape is placed with the above method. Initially the force
at the incisors will feel only intrusion because no horizontal force is present, but as the archwire deactivates,
it becomes straighter and longer; hence, the incisors will
gradually feel a labial force as the intrusion proceeds. The
shape at full deactivation is a straight line. Note that it
is longer horizontally than the curved activated shape.
Figure 16-42c has a gradual curvature rather than
a sharp V-bend formed into the intrusion arch. With
this shape during deactivation, the intrusion arch will
get shorter, and a lingual force will be produced. Different configurations initially could deliver the same force
system, but if friction is present, over time the force system
can change significantly when new horizontal forces are
produced. Of course in reality, the frictional force is finite;
still, the same phenomenon will be observed. The example given was for a cantilever, but this effect can operate
402
FIG 16-43 (a) The canine requires distal
movement and rotation into an extraction
site. A very simple single distal force at the
bracket produces the desired force system.
(b) Tying the wire into the bracket would
potentially make the situation worse due to
the friction. The rotation is in the desirable
direction, but the CR may move to the mesial
direction.
between other bracket geometries involving forces and
moments at each bracket.
Brackets and archwires are needed for threedimensional control where both forces and couples are
required. In some situations, only a single force is needed,
and an archwire placed into a bracket can complicate the
mechanics not only by added moments but also by added
unavoidable frictional forces. The canine in Fig 16-43
requires distal movement and rotation into an extraction
site. A very simple single distal force at the bracket without an archwire produces the force system that we want
(Fig 16-43a). From the occlusal view, wire engagement
is not needed for the rotational moment. Tying the wire
into the bracket would potentially make the situation
worse (Fig 16-43b). The frictional mesial force added
to the clockwise (or mesial-out) moment could move
the canine CR to the mesial, which is not indicated. The
Conclusion
FIG 16-44 Intrabracket forces occurring between the wings of a
bracket can be a significant source of friction even in the passive
wire. One example is wedging, where a ligature tie is too tight or a
wire cross section is too large.
a
b
c
FIG 16-45 (a) Wedging occurs even with undersized wire. A curvature such as a curve of Spee can fill up the bracket slot. (b) A permanent
deformation of the wire between two brackets can be caused by heavy mastication. (c) An unnoticed nick by a pliers must be considered as
another potential source of significant friction.
additional moment can also restrict effective distalization
of the canine, slowing down retraction. The clinical case
in Fig 16-11 showed the undesirable effect of moving the
CR to the wrong position because of friction.
Conclusion
This chapter has discussed the role of classic friction
in understanding the biomechanics of an orthodontic
appliance. Basic formulations have been presented to
give clinicians a rational basis for how frictional forces
operate so they can more efficiently use any appliance.
These simple formulas must not be assumed to fully
give all the restraining forces to sliding. It is far more
complicated. Tribologists, specialized engineers who
study friction in depth, still debate its effects, principles,
and mechanisms. There are also many research papers
regarding friction in the orthodontic field. It is interesting to find that some results are contradictory even
though the research methods are similar. This is because
there are too many variables to control in the research of
static and kinetic friction. Saliva, for example, acts as a
lubricant or adhesive depending on the materials used.
Even at low forces where little permanent deformation
or wear occurs, classic theory may be too limited. The
measurement of coefficients of friction is difficult and
may not be reproducible, hence giving potentially inconsistent values in the literature. The force system in vivo is
continually changing over time as teeth displace. Force
decay is inherent in our wires and appliances. Actual
loading conditions may be different and many times more
complicated than envisioned. We have discussed here
only four stages of canine retraction. If the arch is not
fully leveled before retraction, the force system will be
different than described in this chapter.
Resistance to sliding can involve more than classic
engineering formulas if heavier loads are present in the
mouth. Wires or even brackets can be permanently
deformed, wear, and undergo abrasion. Here prediction
or calculation of sliding resistance becomes very difficult.
Our discussion of friction has mainly described interbracket distance activity that generates forces between
brackets. Intrabracket forces occurring between the
wings of a bracket can also be significant sources of
friction even in the passive wire. Wedging was briefly
mentioned in Fig 16-20c; wedging (Fig 16-44) occurs
when a ligature tie is too tight or a wire cross section is
slightly larger than the slot size. Wedging is common
even when undersized wires are used. Examples are given
in Fig 16-45. Archwires are sometimes placed with a
curvature such as a curve or reverse curve of Spee,
which can fill up the bracket slot (Fig 16-45a). A permanent deformation of the wire between two brackets can
result from heavy mastication (Fig 16-45b). Any small
bends or torque inside the bracket, possibly as the result
of a nick by a pliers, must be considered as another source
of significant friction (Fig 16-45c).
403
16
The Role of Friction in Orthodontic Appliances
Other forces from the cheeks, lips, and tongue may
influence the cyclic unlocking of friction. Briefly we
have discussed the importance of cyclic and occlusal
forces in frictional force reduction. Unfortunately, our
understanding makes for poor prediction because of the
complexity involved. Nevertheless, an understanding of
classic friction and the formulas that underlie it can go a
long way to explain much that is seen clinically and help
clinicians in the selection and design of the individualized
orthodontic appliances for their patients.
References
1. Thorstenson GA, Kusy RP. Resistance to sliding of self-ligating
brackets versus conventional stainless steel twin brackets with
second-order angulation in the dry and wet (saliva) states. Am J
Orthod Dentofacial Orthop 2001;120:361–370.
2. Liew CF, Brockhurst P, Freer TJ. Frictional resistance to sliding
archwires with repeated displacement. Aust Orthod J 2002;18:
71–75.
3. O’Reilly D, Dowling P, Langerstrom L, Swartz ML. An ex-vivo
investigation into the effect of bracket displacement on the resistance to sliding. Br J Orthod 1999;26:219–227.
Recommended Reading
Burstone CJ. Biomechanical rationale of orthodontic therapy. In:
Melsen B (ed). Current Controversies in Orthodontics. Chicago: Quintessence, 1991;131–146.
Burstone CJ. Precision lingual arches: Active applications. J Clin Orthod
1989;23:101–109.
Burstone CJ. Self-ligation and friction: Fact and fantasy. Presented at
the 37th Moyers Symposium on Effective and Efficient Orthodontic
Tooth Movement, Ann Arbor, MI, 26 Jan 2011.
Burstone CJ. The segmented arch approach to space closure. Am J
Orthod 1982;82:361–378.
Burstone CJ, Hanley KJ. Modern Edgewise Mechanics Segmented
Arch Technique. Glendora, CA: Ormco, 1986.
Burstone CJ, Koenig HA. Creative wire bending—The force system from
step and V bends. Am J Orthod Dentofacial Orthop 1988;93:59–67.
404
Burstone CJ, Koenig HA. Force systems from an ideal arch. Am J
Orthod 1974;65:270–289.
Burstone CJ, Koenig HA. Optimizing anterior and canine retraction.
Am J Orthod 1976;70:1–19.
Choy K, Pae EK, Kim KH, Park YC, Burstone CJ. Controlled space
closure with a statically determinate retraction system. Angle Orthod
2002;72:191–198.
Gottlieb EL, Burstone CJ. JCO interviews Dr. Charles J. Burstone on
orthodontic force control. J Clin Orthod 1981;15:266–268.
Iwasaki LR, Beatty MW, Randall CJ, Nickel JC. Clinical ligation forces
and intraoral friction during sliding on a stainless steel archwire. Am
J Orthod Dentofacial Orthop 2003;123:408–415.
Kusy RP, Whitley JQ. Coefficients of friction for arch wires in stainless
steel and polycrystalline alumina bracket slots. Am J Orthod Dentofacial Orthop 1990;98:300–312.
Nägerl H, Burstone CJ, Becker B, Kubein-Messenburg D. Centers of
rotation with transverse forces: An experimental study. Am J Orthod
1991;99:337–345.
Park JB, Yoo JA, Mo SS, et al. Effect of friction from differing vertical
bracket placement on the force and moment of NiTi wires. Korean J
Orthod 2011;41:337–345.
Ronay F, Kleinert MW, Melsen B, Burstone CJ. Force system developed
by V bends in an elastic orthodontic wire. Am J Orthod Dentofacial
Orthop 1989;96:295–301.
Smith RJ, Burstone CJ. Mechanics of tooth movement. Am J Orthod
1984;85:294–307.
Tanne K, Koenig HA, Burstone CJ. Moment to force ratios and the
center of rotation. Am J Orthod Dentofacial Orthop 1988;94:426–431.
Tanne K, Nagataki T, Inoue Y, Sakuda M, Burstone CJ. Patterns of
initial tooth displacements associated with various root lengths and
alveolar bone heights. Am J Orthod Dentofacial Orthop 1991;100:66–71.
Tanne K, Sakuda M, Burstone CJ. Three-dimensional finite element
analysis for stress in the periodontal tissue by orthodontic forces. Am
J Orthod Dentofacial Orthop 1987;92:499–505.
Thorstenson GA, Kusy RP. Comparison of resistance to sliding between
different self-ligating brackets with second-order angulation in the dry
and saliva states. Am J Orthod Dentofacial Orthop 2002;121:472–482.
Timoshenko S, Goodier JN. Theory of Elasticity, ed 2. New York:
McGraw-Hill, 1951.
Problems
Disregard all forces out of the plane of the diagrams. For problems 1 through 4, 600 g of normal force is acting on the
canine and the first molar. The coefficient of friction is 0.2.
1. What is the maximum static friction force?
2. 100 g is applied at the canine by a coil spring from
canine to molar. How much force is applied at the
molar? Calculate the magnitude of the frictional forces
at each tooth.
3. Calculate the magnitude of force to override the friction.
4. How much force does each tooth feel in phase I of
space closure when 300 g of force is applied at the
canine?
405
Problems
For problems 5 through 7, 600 g of normal force is acting on the canine.
5. 1,200 g of ligature force is applied at the molar. How
much force would the canine and molar feel in phase
I of space closure?
6. Assume that the normal force of the molar tube is
completely removed. How much force would the canine
and molar feel in phase I of space closure?
7. Assume that the canine bracket from problem 6 is
replaced with a ceramic bracket and the coefficient of
friction between the wire and bracket is 0.5 at the
canine. 300 g is applied at the canine. How much force
would the canine and molar feel?
8. The canine and the first molar are to be translated.
Calculate the maximum static friction force on the
canine and molar. How much applied force is needed
to translate the teeth? Where would sliding occur?
9. A 10:1 M/F ratio at the bracket will translate the molar
and canine. A 100-g force is placed at the extension
hooks (lever arms). What is the effective force on the
molar and canine?
406
17
Properties and
Structures of
Orthodontic Wire
Materials
A. Jon Goldberg / Charles J. Burstone
“Men are only so good as their technical developments allow them to be.”
— George Orwell
The design of an orthodontic appliance includes consideration of shape and the amount
of wire, but this chapter discusses an additional important design factor: the material
used in the appliance. Clinically, every appliance could be described as a spring or a
series of springs. Springs have three characteristics: stiffness, maximum force, and
range. On a stress-strain level, linear materials have three properties that relate directly
to these three clinical phenomena: modulus of elasticity (E), yield strength (YS), and the
E/YS ratio. Stainless steel and β-titanium have stress-strain curves in the elastic range
that are linear. Most nickel-titanium (Ni-Ti) wires have a more complicated relationship
between their clinical properties and their underlying stress-strain curves. β-titanium
wires are only about 0.42 the stiffness of stainless steel. Along with low-stiffness
Ni-Ti, variable-modulus orthodontics is possible where the material is varied rather
than the cross section. Superelastic Ni-Ti wires can provide the advantage of unique
properties, including large elastic deflections, relatively constant forces, and thermal
or shape-memory effects. Esthetic wires are typically coated metal wires. However, a
newer generation of esthetic wires is being developed that are clear. Materials used
are fiber-reinforced composites and self-reinforced polymers.
407
17
Properties and Structures of Orthodontic Wire Materials
T
he design of orthodontic appliances continues to
grow in sophistication to achieve more effective and
predictable control of force systems. Force systems
result from a combination of the wire and bracket design
and the wire’s material properties. The properties of a
material are determined by its composition, its atomic
structure, and the microstructural mechanisms responsible for deformation. This chapter describes the properties
of the metal alloys widely used in orthodontics as well as
fiber composites and the future potential for orthodontic
wires based on polymers.
The three primary mechanical characteristics of an
orthodontic wire are range, stiffness, and maximum force
(or moment). Range is the maximum distance over which
the wire can be deflected and recover to apply force.
This characteristic is most important in the early stages
of treatment, where larger range allows engagement of
teeth with greater malalignment. The forces generated by
a wire are proportional to its stiffness and the amount of
deflection. For the early to middle stages of treatment,
low stiffness is desirable because it imparts biologically
favorable low and continuous forces. Low stiffness also
allows use of wire cross sections that can fill and fully
engage the bracket, allowing for early three-dimensional
control.
During initial deflection, the following linear relationship exists among the primary wire properties. While
not exact for all clinical situations, this relationship is
important and useful.
Range
Maximum force (or moment)
=
(maximum deflection)
Stiffness
In addition to range, stiffness, and maximum force,
there are other important features of a wire. Formability
is important not only for the clinician but also for the
manufacturer to form arch shapes and to customize appliances. Of course, the wires must be biocompatible and
stable in the oral environment. Finally, there is continued
interest in orthodontic wires that are esthetic. This last
feature is the primary motivation for the introduction of
the composite and polymer-based archwires described
at the end of the chapter.
408
Mechanical Behavior and
Relationships
Force-deflection curves
The mechanical characteristics of an orthodontic wire
are most conveniently represented with one of two related
graphical forms: the force-deflection (F/∆) curve or the
stress-strain curve. A force-deflection curve is useful clinically because it illustrates the force generated with
deflection of a given wire. F/∆ curves are not meant to
be clinical simulations, but the graph is applicable to
various clinical loading conditions. The curves are typically measured in a laboratory with simplified conditions,
such as loading in the center of a wire supported at its
ends (three-point bending) or loading one end of a wire
that is gripped at the opposite end (free-end cantilever).
To facilitate comparisons between laboratories, standard
methods for testing orthodontic wires have been developed, such as American National Standards Institute/
American Dental Association (ANSI/ADA) Specification
No. 32 and International Standards Organization (ISO)
Specification No. 15841. In addition to the loading conditions, an F/∆ curve is dependent on the span length and
cross-sectional dimensions of the wire test sample. For
the same wire geometry, the curves are very helpful in
comparing different orthodontic materials.
A typical F/∆ curve for stainless steel, cobaltchromium (Co-Cr; Elgiloy), or β-titanium (β-Ti) is
shown in Fig 17-1. For all materials, the initial force
imparted by a wire is proportional to the initial deflection, meaning the beginning of the graph is linear, as
indicated by segment O-YP. Materials or appliances that
exhibit force proportional to deflection are said to follow
Hooke’s law. The slope of the linear region (F/∆) is the
measure of a wire’s stiffness. The steeper the slope, the
greater the stiffness. For clinical situations where high
stiffness is desirable, such as when stabilizing the teeth
into a segment or during the later stages of treatment, a
steeper slope is preferred. For initial alignment and leveling stages of treatment, flatter slopes that produce lower,
more continuous forces for each unit of deflection are
desirable. Any deflections in the linear region are elastic.
Accordingly, during unloading due to tooth movement,
the wire will follow the solid YP-O segment back to the
origin. For stainless steel, Co-Cr, and β-Ti, this is the
working region of the wire. The area under the loading
curve is the mechanical energy applied to the wire, and
the area under the unloading curve is the energy released
Mechanical Behavior and Relationships
Elastic
deformation
Plastic
deformation
Elastic
deformation
Plastic
deformation
Maximum
Force, F (g)
x
a
O
Ultimate strength
YP
YP
Stress, σ
(MPa)
Offset yield
strength
Elastic limit
Proportional limit
x
E
0.2%
F/∆
Stiffness
b
d
O
b
d
Deflection, ∆ (mm)
Strain, ε (mm/mm)
FIG 17-1 A typical F/∆ curve for stainless steel, Co-Cr, or β-Ti wire.
The stiffness of a wire is indicated by the slope of the linear region, F/∆.
The maximum force in the elastic region is represented by point YP.
Unloading in this region follows the solid blue line back to the origin,
O, with no permanent deformation. Wires loaded into the region of
plastic deformation will follow the dashed blue line during unloading
and sustain permanent deformation. Failure occurs at point x. The
distance from O to b represents the range or maximum elastic deflection of a wire. The distance from b to d is a measure of the plastic
deformation that the wire can sustain, or its ductility or formability.
FIG 17-2 A typical stress-strain curve for stainless steel, Co-Cr, or
β-Ti wire. These curves represent the inherent material properties of
an alloy. The slope of the linear region is a measure of the modulus of
elasticity (E), the material’s stiffness. Point YP indicates the initiation
of yielding, or the region where plastic deformation begins to occur. It
may be measured by the proportional limit, elastic limit, or offset yield
strength. Failure occurs at point x. The distance from b to d is a measure of the maximum plastic deformation, or ductility or formability.
by the activated wire. In the elastic range, stored energy
is released without any loss.
As a wire is deflected further, it eventually reaches
the end of the linear region (point YP in Fig 17-1). For
stainless steel, Co-Cr, and β-Ti, this is associated with
initiation of the movement of microstructural features
called dislocations (see section later in the chapter titled
“Dislocation-dependent alloys”). The movement of dislocations is not reversible, so any deflection due to this
mechanism is not recoverable; this is a region of plastic
deformation as indicated in Fig 17-1. Unloading in the
region of plastic deformation follows a line parallel to
O-YP but intersects at a distance from the origin, which
quantifies the extent of permanent deformation (dashed
blue line in Fig 17-1).
The point YP is important because it corresponds to the
amount of force—point a in Fig 17-1—that the wire can
sustain before any permanent deformation occurs. Additionally, the corresponding point on the deflection axis
(point b) is a measure of the amount of elastic deflection
the wire can sustain, or its range, the distance from O to b.
Referring again to Fig 17-1, as deflection increases
beyond point YP, the force continues to increase to the
maximum that the wire can support. This point is often
labeled as the maximum or ultimate force (or moment).
Continual deflection beyond the ultimate force will
eventually result in fracture at point x. The amount of
plastic deflection that a wire can sustain (segment b-d) is a
measure of the wire’s formability. Stainless steel, β-Ti, and
Co-Cr (in the appropriate condition) have large regions
of plastic deformation in their F/∆ curves and therefore
are very formable clinically and can be easily shaped into
various configurations.
Stress-strain curves
While F/∆ curves are clinically useful for comparing
wires, the graphs are dependent on wire cross section
and length. Normalizing the values facilitates analysis
of the inherent material properties, which is useful for
both clinical and engineering evaluations. Each specific
appliance configuration will have a unique F/∆ ratio even
if the wire alloy in different configurations is the same;
normalizing gives universal wire material properties independent of design. Dividing force by the cross section of
the wire and dividing deflection or deformation by the
original length results in a stress-strain curve. A typical stress-strain curve is shown in Fig 17-2; its shape is
similar to the F/∆ curve. Stress-strain curves are usually
measured with tensile loading. Stress, often represented
by o, is a measure of force/area and can be expressed in
MPa (megapascals), psi (pounds/square inch), or other
409
17
Properties and Structures of Orthodontic Wire Materials
comparable units. Strain (ε) is a measure of the deformation per original size of the sample and is expressed
as mm/mm, inch/inch, or dimensionless units. Because
of its importance, the slope of the initial linear region of
a stress-strain curve has a specific name: the modulus
of elasticity or Young's modulus, which is abbreviated as
E. In a stress-strain curve, the point at which the wire
changes from elastic to plastic deformation is called the
yield point or yield stress and is designated YP or YS,
respectively. There are different definitions of where on
the curve the transition from linear to nonlinear behavior
occurs. The point at which the line changes from linear
to nonlinear is the proportional limit. The precise point
at which the material transitions from elastic to plastic behavior is the elastic limit. Because it is difficult to
experimentally detect these values precisely, a common
method is to construct a line parallel to the linear region
but offset by a predetermined amount of typically 0.2%.
The intersection of this line with the stress-strain curve is
the offset yield strength. In the stress-strain curve shown
in Fig 17-2, the distance from O to b is equal to the ratio
of YS/E. This is a useful relationship because it explains
not only why a lower modulus imparts lower forces for
a given deflection but also why for a given YS, a lower E
increases working range (O-b).
The area under the linear region of the graph, or the
area enclosed by points O, YP, and b, is a measure of the
resilience or elastic energy stored in a wire when deflected
to the YP. It is sometimes used to indicate the elasticity
of a wire, although the ratio YS/E is a more clinically
relevant measure of the working range. The total area
under the stress-strain curve is a measure of a material’s
toughness. Extremely high-strength wires will have a
high YP but limited plastic deformation (segment b-d),
resulting in a lower toughness, which is the quantitative
measure of brittleness.
The F/∆ and stress-strain curves, and all of the derived
properties, are based on test methods where the sample
wire is loaded continuously until failure. However, in
clinical use, appliances are exposed to multiple cycles
of force or stress below the YP. This loading condition
can result in a cumulative effect that causes failure, even
if the maximum load is never exceeded. This is known
as fatigue.
The modulus of elasticity is an inherent physical
property of the material that is not changed by physical
stimulus such as bending, torsion, or even heat treatment.
Annealing with high temperature will reduce the YP but
not the modulus.
410
Relationships between material
properties and flexure behavior
The similarity between stress-strain curves and F/∆
curves can be quantified with very important and useful
relationships between the engineering material properties
and the clinical flexure behavior of orthodontic wires.
While the following formulas are strictly correct only with
small deflections and within the initial linear regions of
the curves, they can estimate beyond these conditions and
illustrate the dependence of orthodontic force systems on
material properties and wire geometry. The stiffness of a
wire in flexure, or its F/∆ rate in an F/∆ curve, is related
to the modulus of elasticity (E) of the material and the
wire geometry by the following relationship:
F/∆ = EI 3
KL
where I is the moment of inertia and is related to the
cross-sectional size, shape, and direction of bending;
K is a geometric factor related to the configuration and
loading conditions; and L is the wire length.
The maximum bending moment (Mmax ) that a wire can
generate is calculated by the following:
Mmax =
YS
C/I
where YS is the yield strength, C is the cross-section
radius, and I is the moment of inertia. Because I varies
exponentially with cross-sectional dimensions and therefore has a significant effect on appliance stiffness, it is
convenient to define a cross-section stiffness number.
These values provide a convenient comparison of wires
or appliances of different wire diameters, relative to a
clinically useful baseline. Cross-section stiffness numbers
and the related material stiffness numbers are described
in chapter 18, where useful tables of these values are also
provided.
Crystal Structure and Phase
Transitions
All metal alloys used in orthodontics are crystalline, consisting of very specific arrangements of atoms. An example of
one atomic pattern that is possible in Ni-Ti alloys is shown
in Fig 17-3a. This pattern is referred to as body-centered
Composition and Properties of Orthodontic Alloys
FIG 17-3 Examples of two atomic lattice arrangements that can be
adopted by metals: (a) body-centered cubic (BCC) arrangement; (b)
end-centered monoclinic arrangement.
a
cubic (BCC). It is a square arrangement with an atom in
each corner, forming a cubic shape with an atom in the
center of the cube. Another arrangement known as endcentered monoclinic is shown in Fig 17-3b. Schematic
diagrams showing the patterns of crystals are called unit
cells. Other atomic arrangements are possible and involve
changing the lengths of or relative angles among the axes
of the unit cell or different positioning of interior atoms,
such as on the base or the faces of the unit cell. In nature,
there are only 14 possible arrangements, referred to as
the Bravais lattices. All crystalline materials adopt one
of these specific lattices, but many materials, such as
memory alloys, can transition between different lattices
with temperature, processing conditions in the manufacture of the wire, or stress during clinical application.
The different lattice arrangements of an alloy can also be
referred to as phases. The intrinsic mechanical properties
of an orthodontic wire are dependent on both its composition and the phases that are present. Some alloy phases
are particularly important and are assigned names. For
example, the phase that is stable at higher temperatures
is sometimes referred to as austenite or the austenitic
phase. Most phase changes require diffusion of the atoms
over distances of many unit cells. However, there are
phase transitions that result from stress or temperature
changes that cause shifts of only one atomic distance to
adjacent positions in the lattice. These are often referred
to as martensite or the martensitic phase. Importantly for
orthodontics, because the atoms in the martensite phase
are so close to their position in the previous phase, the
atoms can shift back and thereby appear to remember
their original position. Contemporary orthodontic alloys
make use of different types of phase transitions.
b
Composition and Properties of
Orthodontic Alloys
The four broad categories of metal alloys used in orthodontics include stainless steel, Co-Cr, β-Ti, and Ni-Ti.
In addition to composition and phase or lattice-pattern
arrangements, it is useful to discuss the alloys according
to the atomic mechanisms responsible for their unique
mechanical characteristics. The properties of the first
three categories of alloys are dependent on microstructural features called dislocations. Most of the Ni-Ti alloys
derive their mechanical characteristics from phase transitions. The compositions and mechanical characteristics
of the various alloys are described below, followed by
an explanation of the dislocation-dependent and phase
transition–dependent microstructural mechanisms.
Dislocation-dependent alloys
Stainless steel
In the mid-1900s, stainless steel replaced gold-based
alloys as the most widely used orthodontic wire alloy.
Today, it remains one of the standard alloys for clinical
use because of its overall mechanical properties, low
coefficient of friction, and low cost.
Stainless steel is iron alloyed with chromium, nickel,
and less than 1% carbon. As with all orthodontic alloys,
the properties are dependent on both composition and
atomic structure, the latter being determined by the
processing to form the wire. Orthodontic stainless steels
are most commonly AISI 304 series alloys containing 18%
411
17
Properties and Structures of Orthodontic Wire Materials
Table 17-1 Mechanical properties of orthodontic wire materials
Engineering term
Modulus, E (GPa)
Yield strength, YS (MPa)
YS/E (×10–3)
Clinical term
Stiffness
Strength
Range
Formability
Stainless steel
159–200
1,200–1,930*
8.69
Low–high*
Co-Cr
150–211
1,400
7.78
High†
β-Ti
Superelastic
Ni-Ti
Martensitic Ni-Ti
68–72
960–1,170
15.4
High
See F/∆ curve‡
450–600
Very high§
Not formable
33||
1,655
50.2
Not formable
*Strength and formability depend on the amount of cold working during wire drawing. More cold working increases strength but decreases formability.
†
Formability of Co-Cr alloys is high in the softened, non–heat-treated condition.
‡
Modulus varies with activation (see Fig 17-6).
§
Range is very high and varies with activation.
||
Modulus of the initial linear region of the stress-strain curve.
chromium and 8% nickel and are sometimes referred to
as 18-8 stainless. The chromium imparts corrosion resistance and, along with the nickel, stabilizes the austenitic
atomic lattice structure. The high mechanical properties
of stainless steel are primarily due to the cold working
that occurs during drawing of the wire to clinically relevant cross-sectional dimensions. High amounts of cold
working produce wires with very high strength and
high elastic springback, but formability may be limited.
Therefore, most commercial wires have a history of only
moderate work hardening to minimize brittleness. Figure
17-2 is representative of the shape of the stress-strain
curve for stainless steel. The stiffness or modulus (E) of
stainless steel is approximately 180 GPa, the highest of all
the orthodontic alloys and comparable to that of Co-Cr.
Depending on the amount of cold reduction during wire
drawing, the YS can vary from 1,200 to 1,930 MPa, with
a corresponding decrease in formability. The mechanical
property values are summarized in Table 17-1.
Cobalt-chromium
Co-Cr orthodontic alloys have a long history of use in
orthodontic therapy. Their unique feature is the ability to
be easily formed into a desired shape while in a softened
condition, then strengthened with a short, in-office heat
treatment to develop properties more effective for force
application. The alloy, originally developed for watch
springs, was introduced as Elgiloy, but other brands are
now available, such as Colboloy (G&H Orthodontics).
The wires are available in several starting, softened
conditions that can be readily formed. After forming to
the desired shape, the wires are strengthened with heat
treatments lasting typically 5 to 10 minutes at 480oC
(896oF). The heat treatments increase YS and springback
412
but decrease formability, with the extent of the effects
dependent on the starting condition of the wire. Heat
treating does not alter the modulus of elasticity, so the
forces for any given activation are not changed. Representative values of the mechanical properties are shown in
Table 17-1. After heat treatment, the flexure properties of
Co-Cr wires are comparable to those of stainless steel. The
ability to have both good formability and high strength is
useful, but Co-Cr wires are less popular now because after
heat treatment their properties are comparable to those
of stainless steel, preformed arches are readily available,
and heat treatment involves an added step.
-titanium
Titanium has been an important structural metal in various industries for over 50 years because of its high ratio
of strength to weight, corrosion resistance, and biocompatibility. The primary industrial titanium is alloyed with
6% aluminum and 4% vanadium. In dentistry, the success
of implants is due to the ability of bone to grow against
and maintain intimate contact with commercially pure
(99%) titanium. The structural and implant applications
use titanium that is partially or completely in its hexagonal close-packed or alpha lattice arrangement. However,
as an orthodontic archwire, the YS and elastic range of
alpha titanium provides little benefit over stainless steel.
However, with the addition of molybdenum, titanium
acquires the BCC lattice or β-Ti structure. The modulus of
β-Ti is about 35% less than the alpha form of titanium and
about 42% of that of stainless steel and Co-Cr orthodontic
alloys. Accordingly, for the same deflection, the forces
delivered by a β-Ti wire are 42% of those of stainless steel
or Co-Cr wires of the same dimension. With proper cold
working to increase strength, the elastic range of β-Ti
Composition and Properties of Orthodontic Alloys
FIG 17-4 Typical shape of the F/∆ curves for martensitic and austenitic Ni-T