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TRABECULAR BONE REMODELING AROUND IMPLANTS
by
Edward John Cheal
B.S., Michigan State University
(1980)
S.M. Mechanical Engineering,
Massachusetts Institute of Technology
(1983)
SUBMITTED TO
THE HARVARD-M.I.T. DIVISION OF HEALTH SCIENCES AND TECHNOLOGY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTORATE OF PHOLOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June, 1986
O Massachusetts Institute of Technology, 1986
Signature of Author
Certified by
___
__
Divisigf,6t Health Sciences and Technology
__
j---' '~~
r -Wirsn--- HAyes,Thesrs
uprvior
u /2vjisor
Accepted by
Professor Roger\G. Mark, ha rman
D
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Divs ion of Health Sciences and Tchnology
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TRABECULAR BONE REMODELING AROUND IMPLANTS
by
Edward John Cheal
Submitted to the Harvard-M.I.T. Division
of Health Sciences and Technology
on May 2, 1986 in partial fulfillment of the
requirements for the Degree of Doctorate of Philosophy in
Medical Engineering
Abstract
Mechanical stresses are widely assumed to influence the form and
structure of bone.
The implantation of prosthetic components into
trabecular bone are also assumed
to alter the stresses in the
surrounding trabeculae and these altered stresses are often implicated
in the pathogenesis of component loosening.
The objective of this
investigation was to examine the stress-morphology relationships for
trabecular bone around implants for which there was a controlled and
predictable alteration in the stress fields.
For this purpose two
different experimental models
were
developed using geometrically
simplified implants of various materials and surface conditions. Cobalt
chromium cylinders with a sintered-bead porous coating were implanted
unilaterally into ovine calcanei and stainless steel spheres with either
a polished surface or a sintered-bead porous coating were implanted
unilaterally into equine patellae.
The animals were maintained for
periods of 10 to 24 weeks.
Stereologic methods were then used to
quantify the morphology of the trabecular bone in the experimental
specimens and in the untreated contralateral controls.
Structural
analyses were performed using the displacement-based finite element
method to predict the stresses surrounding the implants. The finite
element models were validated
by comparing the principal stress
directions with the material orientation in the control specimens. This
assumes that the trabecular architecture was aligned with the principal
stress directions in accordance with the trajectorial theory of bone
architecture 2
A linear regression for the control equine patellae
yielded an R = 0.87.
The remodeling response was then evaluated by
comparing the stresses and trabecular alignment around the implants in
the experimental specimens. 2 A linear regression for the experimental
equine patellae yielded an R = 0.89. The two models were distinguished
by the high degree of trabecular orientation in the ovine calcanei as
opposed to the more isotropic architecture of the equine patellae. As a
consequence, the changes induced in the trabecular orientation were
greater in the equine patellae.
In general, the remodeling response
around the smooth implants was greater than that around those porous
implants which exhibited bone ingrowth.
In accordance with these
differences, the finite element models predicted greater changes in the
stresses adjacent to the smooth implants due to the nonlinear boundary
conditions. However, it did not appear that the trajectorial theory, in
-2-
its simplest form, was applicable to the remodeling induced by the
implants. In both models, the trabeculae were most often aligned with
the stress component of the greatest magnitude. However, in the equine
models, the alignment was better in regions where tension predominated.
In some regions were compression predominated, the principal material
direction was 90
to the direction of principal compression. This
suggests that cross struts may be formed to resist buckling of the
trabeculae under compression. The equine models also demonstrated that,
under certain circumstances, small changes in the stress state may
result in large changes in the principal material orientation. In
contrast, in the ovine models, the highly oriented trabeculae were more
often aligned with the direction of principal compression. The stress
changes induced by the ingrown porous implants were insufficient to
induce significant changes in the trabecular orientation. Both models
demonstrated a linear relationship between the change in bone areal
density and the change in von Mises effective stress.
Thesis committee: Wilson C. Hayes, Ph.D.
Professor of Biomechanics,
Harvard-MIT Division of Health Sciences and Technology
Augustus A. White, III, M.D., D.M.S.
Professor of Orthopaedic Surgery,
Harvard Medical School and
Harvard-MIT Division of Health Sciences and Technology
David M. Nunamaker, V.M.D.
Jacques Jenny Professor of Orthopaedic Surgery,
University of Pennsylvania,
School of Veterinary Medicine
W. Gilbert Strang, Ph.D.
Professor of Mathematics,
Massachusetts Institute of Technology
-3-
Acknowledgments
This thesis exists thanks to the efforts of my supervisor and mentor
Toby Hayes as well as the
other
has created and maintained
an
members
of my thesis committee.
exceptional
Orthopaedic Biomechanics laboratory of the
most grateful for the
opportunity
to
Toby
research environment at the
Beth
learn
Israel Hospital.
and
I am
grow under such fine
supervision.
Ernie Cravalho
deserves
much
praise
Medical Engineering and Medical Physics
me with a unique
educational
for
the
program.
opportunity
from
development of the
This program provided
which
I will reap the
benefits for the rest of my career.
Thanks also to all
of
my
colleagues
at
the O.B.L.
Brian Snyder
should probably be a co-author, considering the many ideas he shared and
the time we spent in
discussion.
Special thanks also to Tom Edwards,
Tobin Gerhart, Jason Harry, and Jeff Lotz, for their advice and support.
I have been surrounded by true professionals.
And thank-you to Anne, for giving me the strength to reach my goals,
and much more.
I
gratefully
acknowledge
the
financial
support
of
Engineering program and the National Institutes of Health.
This thesis is dedicated to my mother and my father.
-4-
the Medical
Table of Contents
Page
Abstract
2
Acknowledgments
4
Table of Contents
5
1.0 Introduction
7
1.1 Mechanical Properties of Trabecular Bone
8
1.1.1 Experimental Data
10
1.1.2 Quantitative Morphology
12
1.1.3 Microstructural Models
13
24
1.2 Adaptive Bone Remodeling
1.2.1 Wolff's Law
24
1.2.2 Implant-Induced Remodeling
31
43
1.3 Objectives
2.0 Methods
45
2.1 In Vivo Models of Implant-Induced Remodeling
45
2.1.1 Equine Patella
46
2.1.2 Ovine Calcaneus
51
2.2 Structural Analyses
56
2.2.1 Applied Loads
56
2.2.2 Material Properties
71
2.2.3 Model Development
96
114
2.2.4 Bone/Implant Interface
127
2.3 Morphologic Analyses
3.0 Results
136
136
3.1 Morphologic Analyses
-5-
3.1.1 Equine Patella
3.1.2 Ovine Calcaneus
3.2 Structural Analyses
3.2.1 Equine Patella
3.2.2 Ovine Calcaneus
3.3 Stress-Morphology Relationships
3.3.1 Model Validation
3.3.2 Relations for Equine Model
3.3.3 Relations for Ovine Model
4.0 Discussion
5.0 Conclusion
6.0 Bibliography
-6-
1.0 Introduction
It is widely assumed that mechanical stresses influence the form and
The
structure of bone.
of stress-induced bone remodeling was
concept
to adapt to the mechanical
of
More
study.
Wolff (1892).
Julius
popularized by the German anatomist
This ability
environment makes bone a fascinating subject
remodeling
the
importantly,
their
orthopaedic implants may determine
response
of
bone to
long term success or ultimate
failure.
The implantation of prosthetic
be expected to
stresses
alter
altered stresses are often
component and bone.
bone to
achieve
the
surrounding trabeculae.
These
in the pathogenesis of component
implicated
fibrous tissue interfaces between
of
One approach toward the elimination of loosening is
to design components that create
trabeculae, thereby
in
development
loosening and in the
into trabecular bone can
components
utilizing
compatible stresses in the surrounding
the
remodeling
fixation.
implant
response of trabecular bone
to
This
potential of trabecular
investigation concerns the
a controlled and predictable alteration
in the stress fields around an implant.
Two different experimental models were developed using geometrically
simplified implants of various materials
implants were surgically inserted in
patella.
Both of these
bones
trabecular bone which are ideal
internal architecture.
and surface conditions.
These
the ovine calcaneus and the equine
include large regions of highly oriented
for
study
of the normal and remodeled
The anatomy of these bones, including the tendon
-7-
and
sufficiently
are
attachments,
ligament
to
simple
allow
for
structural analysis using standard engineering techniques.
This investigation primarily
of
structural
trabecular
remodeling.
and
the
morphologic
specimens
for
the
models
bone
the development and analysis
involved
analysis
investigation
of experimental
of implant-induced
relevant literature is reviewed.
In Chapter 1 the
Special
attention was given to studies of the morphologic response of trabecular
bone to implants.
In
2
Chapter
the experimental models are described
Much effort was spent on the
and the analytical methods are developed.
property data for input to the
generation of accurate load and material
well
finite element analyses as
and manipulation techniques.
the morphology of the
specimens.
trabecular
These
bone
bone
trabecular
in the experimental and control
standard
of
statistical
the relationships between the predictions
and
models
the measured morphologic parameters.
the morphologic and structural analyses are
In Chapter 3 the results of
correlations
between
the structural predictions and
the bone architectural changes are presented.
are discussed as relevant
in the analysis of
techniques
Various
architecture.
of the finite element
methods were used to quantify
become
have
methods
the development of mesh generation
Stereologic
analysis were used to examine
presented and the
as
to
the
In Chapter 4 the results
performance of orthopaedic implants.
The results of this investigation are summarized in Chapter 5.
1.1 Mechanical Properties of Trabecular Bone
The accuracy
of
mathematical
-8-
models
of
bone-implant systems are
on
dependent
the
accuracy
of
a
function
of
various
mineral content, density,
and
contiguity ratio (a measure
and
trabeculae)
microstructural models
analyzed based on the
of
of
parameters,
physical
other
of
degree
representations.
the mechanical properties of trabecular
Numerous studies have addressed
bone as
property
material
morphological
trabecular
of
parameters such as
degree of interconnection between
the
orientation.
trabecular
results
including bone
bone
have
been
Furthermore,
formulated and
material tests.
in vitro
Previous
investigations on the mechanical properties of trabecular bone which are
of particular relevance for the present studies are reviewed.
Normal
skeletal
tissue
is
comprised
of
two
distinguished by different morphological characteristics.
of long bones are composed
over
the
metaphyses
surface of the patella, vertebrae,
trabecular bone is continuous
shell and is characterized
The
trabecular
has
bones
and the entire
Cancellous or
surface of the cortical
assumed
the
described
bone
that
compact bone.
long
inner
been
spaces.
forms
The diaphyses
a three-dimensional porous architecture.
of
calcified tissue which
bone
Compact bone also forms a
all other bones.
the
interconnecting plates and columns
It is generally
of
and
with
by
structure
of
entirely of compact or cortical bone
almost
with their characteristic Haversian Systems.
structural shell
types
as
a
lattice
of
with fluid (marrow) filled
the material properties of the
trabeculae
are
similar to those of
This was demonstrated by mechanical testing of individual
trabeculae (Townsend et al. 1975a),
but
debate continues on this issue
(Williams and Lewis 1982).
However,
it is the structural properties of
trabecular bone which
critical
for
are
-9-
the
mechanics
of normal and
resurfaced
prosthetically
joints.
and
material properties
One
of
the
the
and
simplest
structural properties
of
properties, or
function of the individual component
a
global material properties, are
structural
These
organization or architecture.
structural
bone
trabecular
determining the
parameters
critical
most
is the
bone
porosity or,
inversely, the volume density of the mineralized component.
1.1.1 Experimental Data
Numerous
studies
examined
have
of
Various measures
the
Chalmers
of
proportion
Williams and Lewis 1982).
1966;
tissue have been
mineralized
These measures
employed for correlation with the structural properties.
include volume
density
volume), apparent
volume),
and
ash
density
(ash
weight
of
related to the apparent density and
volume
tissue
(mineralized
Fortunately, the volume density
normal
per
mass
total
per
total tissue
per total tissue
tissue
volume).
trabecular bone is linearly
ash density (McElhaney et al. 1970;
Other parameters have been investigated including
Muellar et al. 1966).
the contribution of marrow and
properties, as well
tissue
(mineralized
density
of
al. 1970; Harrigan et al. 1981; Hayes
Carter and Hayes 1977; Galante et
and
properties
tests (Behrens et al. 1974;
mechanical
trabecular bone using in vitro
and Carter 1976; Weaver
structural
the
effect
of strain rate on the structural
the
viscoelastic
component
(1977)
investigated
the
as
of the structural
properties.
Carter and Hayes
trabecular bone as a function of
compressive behavior of
strain rate with and without marrow in
-
10 -
situ.
Cylindrical specimens
of
human
and bovine trabecular bone were
tested in confined compression at strain
10.0 per second.
Apparent density was measured for correlation with the
modulus and strength.
significant at
rates from 0.001 per second to
the
They found
highest
that
strain
physiologic range (Lanyon et al.
the compressive strength as a
the presence of marrow was only
which
rate,
is well
beyond the
An empirical relationship for
1975).
of the strain rate and apparent
function
density was established:
S = 68
P0.062
where S = compressive strength (MPa),
apparent density (gm/cc).
A
= strain rate (sec- 1), and p =
relationship for the compressive
similar
modulus was established:
E = 3790 0.0 6 p 3
where E = compressive modulus (MPa).
the mechanical behavior
of
bone
These relationships suggested that
is similar
to
that of fluid-filled
porous engineering materials.
Carter et al. (1979) later found that the
relationship for strength and
apparent
density
was similar in tension
and that the modulus in tension was equal to the modulus in compression.
Pugh et
al.
trabecular bone
(1973b)
over
investigated
a wide
trabecular bone was tested in
They
also
measured
the
the
frequency
a
volume
-
viscoelastic properties of
range.
viscoelastometer
fraction
11 -
and
Fresh
and defatted
from 100 to 3000 Hz.
contiguity
ratio
for
correlation with the viscoelastic properties.
tangent (the
range.
viscous
They
also
component)
found
a
the
between the contiguity ratio
was
a
unmeasurable
over the frequency
correlation
between the volume
positive
fraction and the stiffness of
this correlation
was
and
They found that the loss
specimens and a positive correlation
the
stiffness.
demonstration
of
the
They concluded that
importance
of bending
deformation with stiffening by lateral support.
1.1.2 Quantitative Morphology
The structure of
imaging
trabecular
techniques,
bone
including
are
with general texts on the
subject
been
The
the
anisotropy
and
determination
collectively
direction
of
of
scanning
electron
techniques used to quantify
referred to as stereology,
provided by Weibel and Elias (1967),
Dehoff and Rhines (1968), and Underwood (1970).
means for
studied using various
radiographs,
micrographs, and serial micrographs.
the structural parameters
has
trabecular
principle
These methods provide a
bone
density,
orientation,
and
degree of
other more
specific measures such as average trabecular width and contiguity ratio.
These data may then be used
to study
between
pathologic
trabecular
or
bone
properties are also under
to quantify normal trabecular structure and
adaptive
remodeling
structural
study
for
processes.
parameters
and
the
Correlations
mechanical
the prediction of global material
properties (Snyder 1987).
The general technique used
to
is the method of directed secants
examine trabecular bone architecture
(Saltykov 1958; Hilliard 1962).
- 12 -
This
from
method uses intersection counts
Merz
(1967)
developed
the
applied
array on plane sections.
test
to
application
trabecular
for the
bone
density and average trabecular width.
measurement of volume and surface
Whitehouse (1974)
a
the
for
technique
degree and direction of trabecular
bone
the
anisotropy.
measurement of the
Harrigan and Mann
(1984) showed that, assuming
orthotropy, trabecular bone anisotropy can
be represented by
rank
a
second
symmetric
stereology to trabecular bone include
(Merz and Schenk 1970a,
1970b;
Applications of
studies of the human iliac crests
and Sissons 1969; Whitehouse
Wakamatsu
1977), vertebrae (Whitehouse et
tensor.
al.
1971),
proximal and distal femora
(Pugh et al. 1974; Whitehouse and Dyson 1974), sterna (Whitehouse 1975),
patellae (Raux et al. 1975), canine
comparative study of a number
ribs
(Albright et al. 1978), and a
bones from man, rhesus monkey, beagle
of
and miniature pig (Beddoe 1978).
A classification of trabecular bone microstructure has been proposed
based on microscopic observations (Singh
based on
systems
of
rods,
interconnecting
plates, and interconnecting plates.
is that of open cells (rods)
The general types are
interconnecting
rods and
An alternate classification system
closed cells (plates) in an asymmetric
or
or collumnar organization (Gibson 1984;
classifications form the basis
1978).
of
Beaupre and Hayes 1985).
These
microstructural models of trabecular
bone, discussed below.
1.1.3 Microstructural Models
One approach for investigating
the
- 13 -
mechanics of trabecular bone is
These
the development and analysis of idealized microstructural models.
models
can
provide
deformation
and
insight
fracture.
into
the
mechanics
They
can
also
of
provide
trabecular bone
a
vehicle
for
investigating the relationships between trabecular bone architecture and
the global mechanical
Finally, with careful application,
properties.
the microstructural models
can
provide
microscopic stress values based on
based on an assumption of
means
calculation of the
for
the global stress values (i.e. those
behavior of an elastic material of
continuum
properties equivalent to those of trabecular bone).
Several
approaches
microstructural models.
have
been
taken
for
the
development
of
Gibson (1985), Harrigan (1985), and Townsend et
al. (1975b) used standard beam
and plate theories to develop relatively
simple models consisting of jointed trusses,
beams, and plates.
A more
complex model was developed by Klever et al. (1985) by extending a plate
and
beam
model
orientations.
with
statistical
Williams and
Lewis
distributions
based on an actual tracing from
Pugh et al. (1973a) also generated a finite
a trabecular bone specimen.
model
geometry
plate elements based on microscopic
and supporting
trabecular
the beam
(1982) used a two-dimensional finite
element analysis with the mesh geometry
element model where the
describing
bone.
developed a three-dimensional
was an idealized collection of
observations of a subchondral plate
Finally,
finite
Beaupre
element
model
and Hayes (1985)
base
on an open-
celled porous structure.
The elastic modulus of
the
trabecular
parameter for microstructural models and
- 14 -
bone tissue is an important
is under current debate.
The
most common assumption is that
to compact bone.
The only
et al. (1975a) from
the
trabecular bone material is similar
direct experimental data comes from Townsend
buckling
concluded that the modulus
studies
of
the validity of their data has been
Indirect evidence for the tissue
questioned (Williams and Lewis 1982).
modulus comes
trabecular
from
experimental
bone.
extrapolation from a
bone density.
The
data
tissue
relationship
The data
from
tissue modulus of trabecular
However, extrapolation
bone
that of compact bone.
et al. (1984)
the
macroscopic modulus of
can
between
the
be
approximated
by
modulus and trabecular
and Hayes (1977) suggest that the
is equal
data
suggests that the tissue modulus
on
modulus
Carter
the
of
They
trabecular tissue is approximately equal
However,
to that of compact bone.
individual trabeculae.
of
from
to
that of compact bone.
Williams
is roughly
a
and
Lewis (1982)
factor of 10 less than
Indirect experimental data also comes from Murray
in which
the
proximal tibia was measured.
modulus
In
of
the
cortical shell from the
the proximal metaphysis of the tibia,
as in other regions, the cortical shell is quite thin, and appears to be
simply a condensation
of
trabeculae.
indicate that the modulus of this
less than that of diaphyseal
experimental data is probably
shell
cortical
a
The
poor
data
is as
bone.
from
Murray et al.
low as a factor of ten
However, extrapolation of
predictor of the tissue modulus,
and further direct experimental data is needed.
To evaluate the applicability
the
global
stiffness
and
experimentally measured values
anisotropy.
of
strength
as
a
a proposed microstructural model,
predictions
function
are
compared
to
of apparent density and
Gibson (1985) proposed four microstructural models based on
- 15 -
four assumed classifications
of
trabecular
bone structure: asymmetric
open-celled, asymmetric close-celled, columnar open-celled, and columnar
close-celled (Figure 1.1).
She
then established the density dependence
for the global stiffness, elastic
stress, for each of
the
models
tabulated published stiffness
assumed classifications.
The
division
the
the
The
Gibson also
by the four
between open- and close-celled
modulus
predictions
1.1).
Comparison of the
and
and strength on
experimental data
Unfortunately, no statistical
straight lines corresponding to the tabulated
Also, the
simply
drawn
through
the data points on
experimental data from transverse directions
contained too much scatter
for
comparison
with the model predictions.
Nonetheless, the demonstrated correspondence
experimental data is
grouped
density of 20%.
model
functional dependencies were
log-log graphs.
data
longitudinal
showed reasonable agreement (Table
data were presented.
standard formulae.
strength
volume
functional dependencies for
between
using
and
types was assumed to be at a
volume density
buckling stress, and plastic collapse
encouraging
for
the
between the models and the
further development of such
simple microstructural models.
An alternate approach to
has been to develop a
the
structural
actual observed architecture.
element model constructed of
subchondral bone and the
the
model
was
proximal tibiae.
based
Thus,
analysis of trabecular bone mechanics
model
Pugh
plate
et
the
(1973a) analyzed a finite
representing a section of
trabecular
observations
model
al.
elements
supporting
on
which is more closely based on
was
idealized model of generic trabecular
- 16 -
from
a
bone.
The geometry of
microscopic
sections of
compromise between a highly
bone and a geometrically accurate
r
F
t
,t
F
t.
z
Figure 1.1: Microstructural models of trabecular bone. Clockwise from
top-left: asymmetric open-celled, asymmetric close-celled, columnar
close-celled, and columnar open-celled. From Gibson (1985).
- 17 -
Table 1.1
Density Dependence of Properties of Cellular Materials
(Adapted from Gibson 1985)
Property
Cubic
opened-cell
Cubic
closed-cell
Columnar
opened-cell
Columnar
closed-cell
Elastic
modulus
p
p
p
p
p
p
p
p
p
p
Elastic
collapse
Plastic
collapse
3/2
p
2
p
- 18 -
The results of the analyses suggested
model of an actual bone specimen.
the importance of bending in
a
is not surprising that
primarily by
model
bending.
purpose
was
to
and
Lewis
tracings
develop
a
of
for
applied global
strains
to
the
global
deformation
axial
correlation between area
this assumption.
constants predicted
by
model
axial deformation of the
result in a significant linear
elastic modulus thus supporting
agreement
the
For the axial
stiffness was proportional to
did
and
fraction
Reasonable
properties.
is purely
Their approach
trabecular columns.
constants assuming
This assumption is valid if the
the volume fraction of trabecular bone.
mode of global
found between the elastic
was
those measured experimentally,
and
although the models tended to overestimate the material stiffness.
is not surprising given the
assumption
above, their experimental data
A
bone was chosen to allow
elastic
material
transverse isotropy for the global
the
the global
The model was then analyzed using
establish
stiffness, they assumed that
predicting
of the observed morphology.
horizontal section through columnar trabecular
strain.
generated a two-
(1982)
a trabecular bone specimen.
method
stiffness coefficients based on analyses
for an assumption of plane
However, it
on plate elements should deform
based
Williams
dimensional model from actual
Their
bone deformation.
trabecular
of
plane strain.
This
As discussed
that the modulus of trabecular
suggests
bone tissue is much less than that of cortical bone.
Beaupre
and
Hayes
(1985)
developed
element model of an open-celled
of
trabecular
bone.
The
porous
unit
spherical void which intersects
-
19
the
-
three-dimensional
finite
foam as a microstructural model
cell
with
a
consisted
of
a body-centered
adjacent voids (Figure 1.2).
i
zY---
Figure 1.2: Open-celled porous
Beaupre and Hayes (1985).
foam
- 20 -
model
of
trabecular bone.
From
The global material properties
Note that a more general
for
this
orthotropic
geometry have cubic symmetry.
material can be generated using a
triaxial ellipsoid rather than a sphere for the void shape.
evaluated the material
the unit cube
constants
corresponding
to
evaluating the resulting forces
overestimated
the
global
by
applying defined displacements to
uniaxial
on
The authors
the
stiffness
strain
cube
and shear strain and
faces.
properties
While the model
when
compared
to
experimental data for trabecular bone (Figure
1.3), such a model can be
expected to provide
of the microscopic stress
reasonable
values in contrast to the
and beams.
rubber
predications
more highly idealized models utilizing plates
The model performed very well when applied to a natural foam
(Figure
1.4).
The
reason
trabecular bone properties may thus
for
have
of the modulus of the bone tissue.
- 21 -
the
overestimation
of
the
been due to an overestimation
COMPARISON OF FINITE ELEMENT PREDICTIONS
WITH RESULTS FROM BOVINE TRABECULAR BONE
10000 -
5000 -
4000
-
3000 -
2000 . . .-
@0.~~~~?
...
.. ~
,,.
:.;
0
·. '?
,'
1000 -
0
0t
0
.0
.
0
key:
* = bovine trabecular bone
0
0
0
0
0 = finite element results
0
i=
cortlCal bone
-
.
-
100 -
I
.
I
I
I
Density
(gm/cc)
I
.4
~II
1.0
1.0
2.0
Figure 1.3: Comparison of the predictions of the porous foam finite
element model with the results from bovine trabecular bone.
22 -
COMPARISON OF FINITE ELEMENT PREDICTIONS
WITH RESULTS FROM FOAM RUBBER
10000
5000
4000
3000
2000
1000
0.
X
key:
0
x
X
x /
11to
Ii]
100
.1
Density (gm/cc)
.4
Figure 1.4: Comparison of the predictions of
element model with the results from foam rubber.
- 23
1.0
2.0
the porous foam finite
1.2 Adaptive Bone Remodeling
Bone,
both
cortical
responds to its
and
biological
bone tissue is in
a
trabecular,
and
is a
mechanical
constant
state
living
tissue which
environment.
Furthermore,
of remodeling; under homeostatic
conditions, the resorption and formation
of
balance is maintained
components
dependent on
and
by
the
controlled
cellular
by
(1969) characterized
remodeling
internal remodeling.
Surface
resorption
or
deposition
periosteal surfaces.
osteons
are
remodeling of
replaced
many
of
physiologic
bone
remodeling
of
trabecular
as
of
calcified
Internal remodeling
or
the
lumina
bone
results
bone are in balance.
refers
to the imposed mechanical stress
particularly
intriguing.
adaptive bone remodeling
by
There
bones refers to the
on
the
endosteal
or
to a process in which
enlarged.
The
in
internal
the quantity,
From a mechanics point of
of bone can change in response
surface and internal remodeling is
are
including
Frost
either surface or
in changes
thickness, and orientation of the trabeculae.
view, the observation that the structure
long
are in turn
parameters.
being
bone
are
which
This
many
examples
of mechanically
exercise-induced hypertrophy (Jones
et al. 1977) and plate-induced osteopenia (Uhthoff and Dubuc 1971).
The
focus of this review is on the remodeling of trabecular bone, especially
in response to artificial implants.
1.2.1 Wolff's Law
It is widely accepted that
The term Wolff's Law
comes
bone
from
the
- 24 -
is a mechanically adaptive tissue.
'law of bone transformation' put
forth by Wolff (1892).
are organized along principal
trajectorial theory of
review of the
early
Thompson (1917).
adaptive bone
bone
stress
on
there
remodeling
comes
Meyer
(1867).
A
numerous demonstrations of
been
is still
and
from
architecture was provided by
bone
have
there
remodeling,
referred to as the
trajectories,
structure,
literature
While
mechanisms of bone
the first suggestion that trabeculae
However,
uncertainty over the
much
the mechanical parameters governing
the remodeling response.
Possible Mechanisms of Stress Transduction
Bone remodeling
complex set of
is a
cellular
bioelectric,
humeral factors.
The
remodeling is by what
mechanism
cellular remodeling process.
that strain, rather than
'sensed'.
been proposed, including
understanding adaptive
the imposed stress influence the
does
of
all, it is intuitively obvious
is the mechanical parameter which is
stress,
The question remains as
influences the cellular
for
question
First
is dependent on a
vascular, biomechanical, and
biochemical,
relevant
which
process
to the the mechanism by which strain
process.
remodeling
Several mechanisms have
direct vascular and diffusion
bioelectricity,
effects, and damage accumulation.
A number of
transduction of
which in turn
the
on
hypotheses
material
influence
strain
the
by
bone
remodeling
changes
activity
are based on the
in electrical potentials
of osteoblasts and osteoclasts.
Fukada and Yasuda (1957) first showed that, in vitro, dry bone acts as a
piezoelectric material.
However, it has now been well established that
- 25 -
the piezoelectric effect in vivo
and that the measured
are due to streaming
potentials in physiologically-moist cortical bone
potentials
potentials are the result
fixed charges.
of
(Gross
ion
Harrigan (1985)
are detected by the
between
junctions may
and
effectively
extracellular potentials.
cortical
bone,
the
flow
interconnections
osteocytes
and Williams 1981).
Streaming
through a porous material with
proposed that the changes in potentials
showed that electrically active
and
is insignificant due to rapid decay,
between
gap
a
While
network
this
streaming
Doty (1981)
junctions exist between osteocytes
osteoblasts.
form
bone cells.
The
for
osteocytes
detecting
with gap
changes in
is a very attractive theory for
potentials
in
trabecular
bone
are
insignificant due to the large pore size.
An alternative
effect the osteocyte
hypothesis
cell
is that
membrane
The cell membrane properties, such
material strains.
This
theory
cellular processes of osteocytes.
cyclic AMP
vitro.
in
response
to
the
material strains directly
or their extracellular components.
as
permeability, may be affected by
is attractive
considering
the long
Rodan et al. (1975) showed changes in
applied
deformations
for chondrocytes in
Justus and Luft (1970) proposed that remodeling is controlled by
changes in the solubility of hydroxyapatite.
They demonstrated that the
solubility of hydroxyapatite is a function of load and they hypothesized
that the resulting changes in calcium concentration in the extracellular
fluid influence osteoblastic and osteoclastic activity.
Another possible mechanism for the
remodeling of trabecular bone is
the continuous repair of trabecular microfractures.
- 26 -
Radin et al. (1973)
developed an experimental model for the investigation of trabecular bone
remodeling
using
intermittent
directed impulse loads roughly
equal
to
The
callus
in
present
Mechanical measurements
under
greater remodeling activity, with
the
of
majority
significant
showed
various
stages
microscopic examination of normal
concluded
that
microfractures,
absorption.
trabecular
and
The
not
Axially
animals body weight were
the
bone
viscosity,
microfractures
as
are
was
tibiae.
a
the
as
of
tibiae.
the loaded
Also, some evidence of
proximal
acts
loaded
remodeling
of
human
the
stiffening
trabecular bone following eight days of loading.
microfractures
tibiae.
specimens exhibited microfractures and
loaded
generally greater disorganization and
fracture
rabbit
one hour period once per day for
applied unilaterally via splints for a
variable duration.
of
loading
shock
on
The authors
absorber
mechanism
subsequently
found
with
for
energy
remodeled
into a
stronger and more dense tissue.
It is probable that
combination of mechanisms.
is important,
especially
While
for
present investigation will not
remodeling is controlled by a
bone
trabecular
an understanding of these mechanisms
the
treatment
directly
mechanical parameters which relate to
of
disease
states, the
address these mechanisms.
bone
The
remodeling and which may be
predicted analytically are of greater relevance.
Mechanical Parameters Governing Bone Remodeling
One approach to the role of mechanics in bone remodeling was that of
bioenergetics.
The
hypothesis
is
- 27 -
that
bone
acts
as
a
reverse
mechanochemical machine (Justus and Luft
bone drives chemical flow
and
thus
1970).
cellular excitation, which it turn
results in bone resorption or
deposition.
approach to suggest that time
is a
rather than strain, is the
Put simply, stress on
proper
Steinemann (1985) used this
natural parameter and that stress,
mechanical parameter governing bone
remodeling.
Cowin
and
remodeling.
techniques
Hegedus
Hart
to
et
(1976)
al.
predict
developed
(1984)
the
used
the
remodeling
of
application, a linear relationship
resorption as a function of
reported
good
experimental
results
results.
the
for
model
Good
optimal
appears
remodeling.
to
hold
However, the
This
dynamic
model
of
bone
model with finite element
long
assumed
bones.
In
this
for bone deposition or
strain.
Cowin et al. (1984)
predictions
agreement
was
when
compared
established
by
of
the
was
fell
They
within a
essentially an empirical approach
promise
for
theory
does
anisotropy and thus in its present
coefficient
to
proper
coefficient for each experiment.
value
relatively narrow range.
which
was
longitudinal
selection of the remodeling rate
reported that
a
the
prediction
of
long
bone
not address changes in material
development may not be applicable to
trabecular bone.
Based on a principle
(1986)
developed
a
of
structural optimization, Fyhrie and Carter
mathematical
orientation and density for a
given
formulation
general case the theory is applicable
a
state
of
for
trabecular
stress.
bone
In the most
to an orthotropic material.
The
basic premise was that trabecular bone remodels so as to form an optimal
- 28 -
structure
in
the
which
integrity
structural
structural mass is minimized.
is maximized
and the
authors assumed a quadratic form for
The
the objective function which enables the application of material failure
criteria.
energy
presentation
Included in the
an
and
density
effective
were the application of strain
The mathematics
criterion.
stress
confirmed the trajectorial theory; namely, the material axis orientation
To establish the
will correspond with the principal stress orientation.
material density, it was necessary to assume a maximum for the objective
function.
to
This is analogous
Functional relationships were developed for
energy or effective stress.
the
minimum
optimizes
on strain energy density
the formulation based
strength.
on
basis
mathematical
framework for the
framework is used
for
stress
trajectorial
the
the
the
selected
material for stiffness whereas
optimizes the material for
presentation
this
are that it provides a
theory
density-stress
of
examination
in
the
effective
of
The strengths
to
related
pointed out that the formulation based
The authors
objective function.
density
material
acceptable
a maximum value on the strain
setting
investigation
present
a
and
theoretical
relationships.
to
This
examine stress-
morphology relationships.
Investigations of Stress-Morphology Relationships
Lanyon
(1974)
provided
indirect
some
experimental
evidence
support of the trajectorial theory of trabecular bone structure.
gage rosettes were applied
The trabecular
architecture
to
in vivo
of
the
in
Strain
the surface of sheep calcanei.
sheep
calcaneus
is particularly
striking, with an orthogonal network of trabeculae which form an arch in
- 29 -
the proximal region.
The
surface lateral to
the
strain
first
gages
were
intersections
placed on the cortical
of the trabecular tracts.
Lanyon found that the principal strain directions varied throughout each
stride but were almost constant during
were little affected by
tensile
principal
the
strain
speed
with the dorsal trabeculae.
strains to
planter
internal
of
Qualitatively, the
corresponded
with
the
plantar
principal strain component corresponded
The
shear strain component increased from
regions.
architecture
provides valuable data for
locomotion.
component
trabeculae, and the compressive
the dorsal to the
the main wight bearing phase and
While
the comparison of surface
is inadvisable,
comparison
this investigation
to model predictions of internal
and surface strains.
Hayes et al. (1982)
used
two-dimensional
finite element models of
the patella to predict internal stresses due to physiologic loads.
also used stereologic methods
to
establish
the density and anisotropy
distribution for a corresponding mid-sagittal
patella.
They
section of a normal adult
A significant positive correlation was found between predicted
principal stress directions
and
measured
significant positive correlation
Mises stress and
measured
was
areal
three-dimensional finite element
significant positive
also
model,
areal
homeostasis.
Such
models
of
a
cannot
A
later studies using a
et
predicted
al.
(1984) found a
principal tensile
density but no correlation between
von Mises stress and areal density.
the examination in each case
In
Stone
between
orientations.
found between predicted von
density.
correlation
stress magnitude and measured
trabecular
These studies are both limited by
single normal patella presumably in
be
- 30 -
used
to
predict the short term
remodeling
of
response
bone
trabecular
structural
tissue mechanics will
with the same tissue architecture
specifically,
the
if
on
which
material
model
It can be
which correlate well
stresses
internal
predict
altered
very accurately represents the
which
model
an
orthopaedic implant.
mechanical environment, such as with an
assumed that a
to
subjected
the
model is based.
based
are
properties
More
on
the
stereologic measures, including modulus based on density and orthotropic
axes based on anisotropy, it would
calculated
stress
and
magnitudes
be surprising to find that the
not
correspond
orientations
with
the
measured densities and anisotropies.
1.2.2 Implant-Induced Remodeling
response
The biological
eliminate the
material
foreign
to
from
body.
the
by
formation
of
Materials used for orthopaedic
they
produce
trabecular
a
bone
dependent on the
minimal
to
an
1982).
greater
form
response,
proliferation (Cameron et
with
al.
tissue
material cannot be
sheath (Park 1979).
are biologically inert in that
The
response.
implant
conditions
Relative motion at the
tissue
body
orthopaedic
implant
thin fibrous sheath will
fibrous
implants
foreign
mechanical
For a smooth surfaced
a
the
usual response is to wall off
the
extruded or ingested by macrophages,
the material
If
attempt to
is an
materials
is at
response
least
of
partially
at the bone-implant interface.
under mechanically neutral conditions, a
adjacent
to
the
implant (Itami et al.
bone-implant interface results in a much
bone
1973).
resorption
and
fibrous
tissue
Porous-coated implants, on the
other hand, result in direct bone ingrowth if certain conditions are met
- 31 -
(Pilliar 1983).
Histologic and Morphologic Response to Orthopaedic Implants
prosthesis is the femoral
orthopaedic
The most extensively studied
component of total hip replacements.
The present femoral components are
usually constructed of a
stiff biocompatable alloy, such as
relatively
or
stainless steel, cobalt-chrome,
titanium
a
The components
alloy.
have a curved stem of variable length which is inserted into the femoral
canal.
to
Polymethylmethacrylate (PMMA) bone cement is still commonly used
secure
the
though
prosthesis,
press-fit
and
porous
ingrowth
The primary advantage of cemented
components are becoming more popular.
prostheses is the forgiving nature of the bone cement, whereas press-fit
and porous ingrowth components require greater surgical precision.
The changes in bone
following prosthetic replacement has
structure
been studied extensively,
especially
and Hierton 1982; Amstutz
et
Bocco et al. 1977;
Carlsson
Cotterill et al. 1982;
Dunn
total knee replacement
and
Gentz
1982;
1980; Charnley and Cupic 1973;
Galante
1982;
Stauffer
(Eftekhar
1983;
1971; Green et al. 1979;
et al. 1979; Reckling et al.
Pellicci
1976, 1979, 1982, 1983; Kaufer and
Tew and Waugh 1982).
Beckenbaugh and Ilstrup 1978;
1982;
al.
Harris et al. 1982; Marmor 1976;
1977; Salvati et al. 1981;
total hip replacement (Almby
for
Sutherland et al. 1982) and
Hamilton
1982;
Insall et al.
Matthews 1981; Lewallen et al. 1984;
Radiographic changes are most commonly described,
though histologic data from recovered
prostheses (Goldring et al. 1983;
Heimke et al. 1985; Linder et al. 1983; Shoji et al. 1983; Wright et al.
- 32 -
1979) or animal models (Harris et al. 1983; Hedley et al. 1982; Homsy et
al. 1972; Lanyon et al.
1981;
have
Of
been
reported.
mechanisms of implant
remodeling
response
implants is at least
Rose
et
primary
concern
loosening.
of
has
been
the pathologic
It is generally believed that the
cortical
partly
al. 1984; Spector et al. 1983)
and
trabecular
controlled
by
bone
to prosthetic
the induced stress fields.
Reduced stresses are implicated
in calcar resorption (Crowninshield et
al. 1980; Lanyon et al. 1981; Oh
and Harris 1978; Rose et al. 1984; Sih
and Matic 1981; Svensson et
1977;
al.
Tarr
et al. 1979) and elevated
stresses are implicated in distal hypertrophy (Cook et al. 1983; Rose et
al. 1984; Spector et al. 1983).
However, of critical importance to the
development of component loosening is the biologic response at the bonecement or bone-implant interface.
The
local biologic response to PMMA
bone cement may be in
due
to the release of toxic monomer
large
part
(Freeman et al. 1977; Huiskes 1980;
1979; Willert et
al.
1974)
or
Park et al. 1982; Schoenfeld et al.
the
(Andersson et al. 1972; Feith 1975;
1982) whereas the
coated
(porous
biologic
ingrowth)
exothermic polymerization process
Lindwer
response
prostheses
to
and Hooff 1975; Park et al
smooth (press-fit) or porous-
appears
to
be
more critically
controlled by the local mechanical conditions (Cameron et al. 1973; Cook
et al. 1983; Harris et al. 1983; McCarthy and Wells 1977).
damage resulting from
prosthetic
implantation
may
The vascular
also influence the
biologic response, especially in the early stages following implantation
(Gunst 1980; Huiskes 1980).
- 33 -
Cemented Prostheses
Rose et al. (1984)
presented
quantitative morphologic data from an
A cemented total hip replacement
animal model of total hip replacement.
maintained for a period of 0 to
was performed unilaterally in sheep and
12 months.
thickness, circumference, and area moment
were measured, along with the
of the
and contiguity ratio of trabecular bone
The volume fraction
cortical
The
bone.
histology
inconsistent cement
revealed
In regions of good penetration,
penetration into the trabecular bone.
the authors noted two distinct regions of trabecular bone separated by a
resorptive layer, with decreased density
the cement.
The resorptive layer
resulted in a band of bone resorption
separating the cement-bone complex
In regions of poor cement
of trabecular bone adjacent to
from
penetration,
the endosteal trabecular bone.
they noted concentric layers of
fibrous tissue adjacent to the cement and dense new bone adjacent to the
fibrous tissue.
Calcar
resorption
and
cortical thickening were also
noted consistent with clinical radiographic data.
Goldring et al.
(1983)
studied
the histological and histochemical
characteristics of fibrous tissue excised from the cement-bone interface
at the time of
surgical
biologically similar
to
revision.
They
a
lining,
synovial
adjacent to the cement layer.
capacity to produce large
which may
explain
that the membrane is
with synovial-like cells
They also found that the membrane has the
amounts
progressive
found
of prostaglandin E2 and collagenase,
bone
lysis.
They
hypothesized that
loosening of the component may increase synthetic activity which in turn
may increase bone lysis.
- 34 -
Porous-Coated Prostheses
Various materials have
been
used
to
(1970) reported
successful
bone
cylinders implanted in sheep
ingrowth as early
results
using
as
14
They
ingrowth
Hahn and Palich
porous
coated titanium
There was apparent bone
al. (1972) reported good
et
component
pyrolytic
(1969).
in
bone.
Homsy
weeks.
and
reported
Reynolds
cortical
femoral
a
polytetrafluoroethylene
canines.
and
Hirschhorn
coated
graphite
infiltration
complete
porous layer on the
a
The first report of powder metal
component surface for bone ingrowth.
methods was that of
form
with
polymer
a
porous
implanted in
by immature and mature
collagen by 3 weeks, dense mature collagen by 10 weeks, and some ingrown
mineralized bone by
15
Hedley
months.
et
al.
(1982) reported the
was given 2 weeks after surgical
presence of a fluorochrome label which
implantation in the porous surface of a femoral surface component.
Several important
parameters
have
ingrown tissue.
of
50
identified for successful
Welsh et al. (1971) found that a
bone ingrowth into a porous material.
minimum pore size
been
is necessary
microns
for mineralization of
Bobyn et al. (1980) found that the optimum pore size is
in the range of 50
to
400
microns.
They also demonstrated that bone
trabecular bone is much more rapid
ingrowth for an implant adjacent
to
than for an implant adjacent
cortical
found that the success of
to
bone
coated
fibrous tissue interface, with
Cameron et al. (1973)
ingrowth is dependent on minimal motion
at the bone-implant interface.
results for a porous
bone.
McCarthy and Wells (1977) reported poor
hip
no
prosthesis
bone
- 35 -
implanted
in canines.
A
ingrowth, was found which they
the implant-bone interface.
attributed to relative motion at
apposition
formation.
contact
between
the
They
also
for
points
coated acetabular components, good
porous
and
component
poor
of
Regions
ingrowth.
for
that,
al. (1983) found
trabecular
required
for
bone
in fibrous tissue
resulted
hypertrophy adjacent to
bone
bone-implant
discontinuous
of
cases
was
bone
apposition
noted
Harris et
contact
suggestive of stress-related remodeling.
The results of Cook et al.
that
supposition
the
the
(1983)
and Kester et al. (1984) support
critically effects the bone remodeling
prosthesis
from
transfer
stress
to bone
In the former study,
response.
porous coated hip prostheses constructed of carbon composite or titanium
Both demonstrated good fixation by 3
alloy were implanted in canines.
weeks.
The
composite
carbon
implants
trabecular bone
in
resulted
hypertrophy at the proximal stem and no calcar resorption while the more
stiff
alloy
titanium
implants
trabecular bone hypertrophy at the
resulted
in
distal
stem.
partially porous coated titanium alloy
and
adjacent
stress-related remodeling.
and
In the latter study,
hip prostheses of similar design
Trabecular bone hypertrophy in the proximal
were implanted in canines.
calcar region
resorption
calcar
the
to
also
They
porous
is suggestive of
surface
noted a fibrous tissue interface
at the smooth portion of the stem.
Several studies
have
trabecular bone adjacent to
(1982)
reported
results
noted
the
from
the
formation
porous-coated
a
porous-coated
surface replacement implanted in canines.
- 36 -
of
a
surface.
dense
layer of
Hedley et al.
cobalt-chrome femoral
They noted good bone ingrowth
complete by 11 months,
bone.
gaps
with
They also noted a
up
to
1.5 mm bridged by trabecular
bone 'plate' of dense trabecular bone adjacent
to the porous layer, with no fibrous membrane formation.
(1983)
results
reported
from
femoral
at the
bone-porous
prosthesis.
interface
and
a
porous
They noted a radiodense line
radiodense
zone
distal
to the
Histologically, they also noted a dense layer of trabecular
coating
bone around the porous
tissue ingrowth.
The
adjacent
figures
layer was separated from
the
cortical
One other interesting
that
in
some areas the dense
by very porous trabecular
bone
of load transmission for the dense
role
from this study was the presence
finding
of fibers (similar to Sharpey's
bone ingrowth and fibrous
to
indicate
bone, raising questions on the
layer.
a
with
components
polysulfone coating implanted in canines.
Spector et al.
fibers)
from areas of fibrous ingrowth
inserting into the adjacent trabecular bone.
Mixed results have been reported for human trials of porous-ingrowth
prostheses.
Engh (1983) reported a
5 year follow-up of 26 patients who
The porous coating was formed
received porous-coated Moore prostheses.
No clinical loosening was reported at
of sintered cobalt-chrome beads.
5 years.
However, one-third of the patients had radiopaque lines, which
the author interpreted
implant interface.
as
The
radiography at four years
an
indication
described
author
with
no
appearance of the radiopaque lines
experiments
of
Pilliar
trabecular bone separated
et
from
of
evidence
were
al.
the
porous
- 37 -
'healthy
bone modeling' by
of stress shielding.
The
similar to those of the animal
(1981),
tissue.
micromotion at the bone-
indicating
hypertrophied
surface by ingrown fibrous
Analytical and Experimental Models
Most analytical models of orthopaedic implants in bone have examined
the
mechanics
simplifying
interface.
of
the
whole
assumptions
bone-implant
about
the
mechanics
of
while employing
the
bone-implant
In addition, the trabecular bone is generally represented in
these models as a continuum.
bone have been developed
While microstructural models of trabecular
(see
Section
impractical for direct application
1.1.3
to
focused on the bone-implant interface
current
study,
several
above), these models are
actual bone-implant systems.
relevance to the present investigation
the
structure
Of
are finite element studies which
mechanics.
experimental
In addition, as in
studies
have
used
simple
implants, such as cylinders, to investigate the biological response as a
function of the implant material and surface characteristics.
Bone-Implant Interface Mechanics
Most analytic representations of bone-implant mechanics assume tight
bonding at
the
material
interface.
however, for smooth-surfaced
to
bone
ingrowth.
This
implants
assumption is inaccurate,
and porous-coated implants prior
Nonlinear
contact,
with
generally requires sophisticated
solution
techniques.
or
without
friction,
Several authors
have presented finite element methods
for solution of contact problems,
including the
by
detailed
presentation
Hampton and Andriacchi (1980)
presented
Bathe
and
Chaudhary (1985).
results from a two-dimensional
nonlinear contact algorithm, with Coulomb friction, applied to stainless
steel and PMMA bone cement test specimens.
- 38 -
Hipp et al. (1985) presented
two-dimensional
results from axisymmetric
bone
dental implants in compact
frictionless contact.
compressive
stresses
distribution of
for
the
They
demonstrated
are
greater
stresses
are
and
different
models
of screw threads and
cases
of rigid bonding and
the maximum tensile and
that
that
for
the
orientation
frictionless
and
contact in
comparison to rigid bonding.
parametrically investigated the influence of
Ducheyne et al. (1978)
the elastic modulus of the porous coating and the surrounding trabecular
bone
using
models.
macroscopic
two-dimensional
axisymmetric
finite
element
The geometry of the model was that of a geometrically idealized
They found that the elastic modulus of the
stem of a femoral component.
on the interface stresses.
porous coating had little effect
Increasing
adjacent
trabecular
bone, however, led to
decreased interfacial shear stresses,
suggesting
that there would be a
the elastic modulus of
the
of trabecular bone elastic modulus,
mechanical advantage for a gradient
with the maximum modulus adjacent to
to relate the stresses
predicted
the
the
by
implant.
No attempt was made
continuum elements to actual
stresses based on a trabecular bone microstructure.
Cook et al. (1982) developed
into a porous surface.
spicule with
the
The model consisted of an idealized ingrown bone
geometry
centered cubic structure
defined
by
the
porous
for
maximum free-end deflection of the
load using beam
theory.
analytical model for bone ingrowth
an
An
assuming
surface
a close-packed face(Figure
1.5).
The
spicules was solved for a transverse
'interface
element'
for
use in three-
dimensional finite element analyses was then generated using a continuum
- 39 -
2(R + B)
)d
I
2(R+B)
III
z
Figure 1.5: Model of ingrown spicule for face-centered cubic closepacked porous material. From Cook et al. (1982).
- 40 -
element which had
the
same
calculated for a specific
of
number
different degrees of ingrowth,
could
be
the
with
by
represented
The interface
spicules.
ingrown
This was
bone spicules.
modulus for the continuum element for
done by establishing an effective
ingrowth
characteristics as that
load-displacement
assumption that the degree of
the
adjusting
number
of ingrown
was then applied in three-dimensional
element
finite element models of a porous dental implant and the load-deflection
characteristics
a
combination of
formulation
were
is
to
compared
model
microstructural
an
transverse load over one
mechanics of a
relatively
metallic spheres would be
controlled problem.
the
The
microstructural model is questionable.
surface
of
the
validity
model
bone
bone
compliant
the
a macroscopic continuum
with
approach,
attractive
While
data.
experimental
of
the
assumes a uniform
However, the
spicule.
spicule surrounded by rigid
accurately described as a displacement-
more
A preferable
approach would have been to establish
the applied loads required to achieve the displaced shape which does not
violate
spheres.
the
displacement
boundary
conditions
imposed
by
the rigid
Also, no attempt was made to utilize the microstructural model
to predict microscopic stress
based
values
on
the predictions of the
macroscopic finite element models.
Idealized Models
Medige et al. (1982) presented morphologic data from an experimental
model of metastatic bone
were
implanted
quantitative
in the
stress
defects
in which
distal
metaphyses
analyses
were
large silicon rubber plugs
of
performed,
- 41 -
canines.
stress
While
levels
no
in the
surrounding trabecular and cortical
bone
low modulus rubber plugs could
be expected to transmit significant
load.
the
Radiographically,
the
which resulted in increased
in a gradual
layer
was formed around the implants,
density
adjacent to the fibrous tissue.
exhibited early microfractures
specimens
experimental
resulted
as early as 2 to 4 weeks.
Significant changes were noted
with increased trabecular bone
be elevated since the
density and cortical thickening around
Histologically, a fibrous tissue
Several of
implants
rubber
bone
increase in the trabecular
the implants.
not
should
formation and an accelerated recovery
bone
of torsional strength.
Brown et al. (1984)
an analytical and experimental model
developed
of a cylindrical metaphyseal
implant
the study of chondromalacia.
for
Their purpose was to investigate the hypothesis that chondromalacia is a
result of local elevations of the stiffness of the trabecular bone which
and
supports the subchondral plate
consisted of plastic or
The experimental model
cartilage.
metal
porous-coated
cylinders implanted in an
anterior-posterior orientation immediately beneath the subchondral plate
in the tibial plateau of
The histological evaluations indicated
sheep.
fibrous encapsulation of the porous-coated implants.
trabecular and subchondral bone
in
was greater than the corresponding
implants.
They
'corticalization' around
the
the
vicinity of the metal implants
bone
in the vicinity of the plastic
the
trabecular
implant
adjacent
describe
Two-dimensional plane strain finite element
nonlinear contact formulation for
cartilage surface.
The results
The density of the
bone
to
response
as
the fibrous layer.
models were analyzed with a
representation
of an indenter on the
demonstrated that a low modulus fibrous
- 42 -
layer around a metal implant resulted in stresses similar to those for a
Corticalization
plastic implant.
resulted in
greatly
surrounding
stresses
increased
a
in the
low modulus implant
cortical
shell.
In
summary, the formation of a fibrous tissue layer surrounding the implant
mechanically isolated the implant and resulted in elevated stress levels
trabecular
in the adjacent
bone.
bone density around the
Increased
implant further increased the stress levels.
1.3 Objectives
The objective
of
this
was
investigation
morphology relationships for trabecular
bone
to
examine the stress-
around implants for which
there was a controlled and
predictable alteration in the stress fields.
The
to
first
objective
geometrically
was
simplified
an
develop
of
implants
experimental
various
conditions implanted in the trabecular
bone
materials
model
and surface
of laboratory animals.
meet this objective two different experimental models were used.
chromium cylinders with
a
using
To
Cobalt
porous coating were implanted
sintered-bead
unilaterally into ovine calcanei and stainless steel spheres with either
a polished surface
or
a
unilaterally into equine
sintered-bead
porous
The
patellae.
coating were implanted
animals
were maintained for
periods of 10 to 24 weeks.
The second objective
specimens
experimental
specimens.
was
and
Two-dimensional
the method of
directed
to
of
perform
the
untreated
stereologic
secants
to
morphologic
analyses
measure
- 43 -
analyses of the
contralateral
control
were performed using
the trabecular bone areal
density,
the
anisotropy.
principal
trabecular
The paired data
were
orientation,
and
the
degree
of
analyzed to establish the morphology
of the trabecular bone response to the implants.
The third objective was to develop displacement-based finite element
models of the control and experimental specimens to predict the internal
stresses for correlation with the
the finite
geometry,
element
the
properties.
generation
models
applied
morphologic data.
required
loads,
establishment
and
Modeling techniques were
and
bandwidth
The development of
the
of
trabecular
the structural
bone
material
also developed for automatic mesh
reduction
as
well
as
techniques
for
representation of the cortical shell and nonlinear bone/implant contact.
The fourth objective was to validate
meet this objective the
principal
the principal
directions
material
the finite element models.
stress directions were compared with
for
the
assumed that the trabecular architecture
stress directions in
architecture.
The
accordance
influence
To
with
of
control
specimens.
This
was aligned with the principal
the
various
trajectorial theory of bone
structural
and morphologic
parameters on this alignment was examined.
The
final
objective
relationships for the
was
remodeled
The alignment of the trabeculae
based
on
accurate.
the
to
assumption
Finally, the
the trabecular bone areal
investigate
trabecular
the
bone
stress-morphology
around the implants.
and the principal stresses was examined
that
the
relationship
density
was
finite
between
element
predictions were
the effective stress and
examined to test the hypothesis
that the trabecular architecture corresponded to an optimal structure.
- 44 -
2.0 Methods
Two experimental
were
models
implant-induced remodeling of
developed
trabecular
either
implanted in the
models
were
patellae
performed
formulation and
the
of
a
trabecular
stereologic techniques.
a
or porous surface were
smooth
Structural
analyses of these
displacement-based
bone
Spherical
calcanei of sheep.
horses.
using
the investigation of
Porous coated cobalt
bone.
chromium cylinders were implanted in the
stainless steel implants with
for
morphology
finite
element
was quantified using
relations were then examined
Stress-morphology
using statistical methods.
2.1 In Vivo Models of Implant-Induced Remodeling
Two different experimental models
were
that of the spherical inclusion in
Chronologically, the first model was
the equine patella.
This
patella
primarily
is composed
model
homogeneous trabecular bone.
had
of
However,
tendon
geometry of the patellofemoral
contact
a
spherical
gave
encouraging
results,
which
geometrically simpler model of
calcaneus.
The models and
the
related
large
the
that the equine
advantage
volume
anatomy
of
relatively
of this bone, and
and ligament attachments and the
Also,
is quite complex.
load,
inclusion is particularly difficult
Despite
using finite element methods.
the
a
especially the geometry of the
the representation of
used in this investigation.
lead
these difficulties, this model
to
the
cylindrical
development
of
the
inclusion in the ovine
data are presented in chronological
order.
- 45 -
2.1.1 Equine Patella
Anatomy
The equine stifle joint corresponds to
the knee joint in the human.
The anatomy of the equine stifle
joint
(1975).
most elaborate of all articulations of
It is the
largest
and
is described in detail by Getty
the horse, consisting of patellofemoral and patellotibial articulations.
The equine patella is a
trochlea
of
the
large
femur
quadrilateral and convex.
sesamoid bone which articulates with the
(Figure
The
The patella is oriented such
2.1).
anterior
surface
is
articular surface is also quadrilateral.
that
two
quadrilateral surfaces are roughly
tibia.
The
of
aligned
the opposite corners of the
with
the
long axis of the
These corners form the base, proximally, and the apex, distally.
The articular surface has a
vertical rounded ridge which corresponds to
the groove on the trochlea of
the
femur.
The medial articular surface
is completed by the parapatellar fibrocartilage.
The patellofemoral joint is formed between the trochlea of the femur
and the articular surfaces of the patella.
of two oblique ridges
contact surface.
and
a
deep
The articular
than that of the trochlea
and
groove which provides the articular
surface
patellar ligament.
is attached
The
horse,
of the patella is much smaller
is completed medially by a supplementary
plate of fibrocartilage which curves
the trochlea and
The trochlea itself consists
to
over
the
the medial proximal ridge of
by
means
contrast
to
the human, has three
in
tibia
ligaments of insertion of the patella
- 46 -
of the medial
on the tibia (Figure 2.2).
These
of patella
Lateral head of gas
Lateral femoropat
Attachment of;
Lateral con
Lateral collatere
I of capsule
itellar ligament
ztellar ligament
ztellar ligament
Flezor digitorum
profundus
/ of tibia
cranialis
digitorum
Eztenr
di
Figure 2.1: Anatomy of the
Getty (1975).
equine
stifle
- 47 -
joint, lateral aspect.
From
EQUINE SYNDESNIOLOGY
Fer ur
la
Parapatellar
fibrocartilag
Medial ridge of trochlea
.M1edialpatellar ligament
ps femori.is
ral patellar ligament
Medial epicondyle
Medial collateral
ligament
eral collateral
ligament
.lfedial meniscus
ral meniscus
Medial condyle of tibia ral condyle of tibia
Tuberosity of tibia -
'seo,l. .pace
seous ligament
Figure 2.2: Anatomy of the
Getty (1975).
equine
stifle joint, anterior aspect.
- 48 -
From
ligaments are the lateral, middle,
and
medial patellar ligaments.
The
medial patellar ligament is the smallest
of the three and is continuous
with the the parapatellar fibrocartilage
and
the tibia on
groove.
the
medial
side
ligament extends from the
groove on the tuberosity
aligned along the line
flexion.
the
the
anterior
of
of
exerts its force to pull
of
the
patella
tibia.
action
the
of
the distal part of the
This ligament appears to be
the
patella
to
The middle patellar
quadriceps muscle group and
down the trochlear groove during
The lateral patellar ligament extends from the lateral part of
anterior
surface
of
the
tuberosity of the tibia.
patella
part
patella to the lateral
of
angle of the horse
epicondyle
allows
the
degrees of extension.
In
the upper end of
medial
the
locked in extension.
to
ridge
part
of the
be
The normal standing
held in approximately 150
trochlea, and the knee is
allows
the horse to sleep while
during flexion is that of gliding to
articular
standing position, the actual contact
a
femur.
the
of
mechanism
opposing
be
lateral
position, the fibrocartilage hooks over
Movement of the patella
groove is thought to
the
of the anterior surface of the
the
knee
this
This
different parts of the
to
A fourth ligament, the lateral femoropatellar
ligament, extends from the lateral
standing.
ends on the tuberosity of
strip
surfaces.
In the ordinary
of the patella with the trochlear
about
patella, it corresponds closely with
12-15
mm
in
width.
In the
the distal border of the articular
surface.
Implant Material and Geometry
The implants used in the
equine
- 49 -
experimental model were smooth and
The smooth spheres had a highly
porous-coated stainless steel spheres.
polished surface and an
diameter
outer
implants had a solid core
8
of
mm.
The porous coated
a final diameter of 10 mm.
The
The porous-coated implants were not
um.
purposes, it was necessary to leave
For manufacturing
perfect spheres.
10
diameter and a coating of sintered
mm
stainless steel beads of 1 mm depth for
diameter of the beads was 800
of
one area uncoated, resulting in a
plane
of 2 mm depth missing from the
sphere.
Experiment Protocol
In two
groups
three
of
spherical metallic implant
horses
each,
surgically
was
surgeon,
veterinary
Under
University of Pennsylvania.
incision was made over
the
the cortical bone in
anterior
removed
implant to the level of the
then closed routinely and
recovery stall.
anesthesia,
the insertion of the
to
boring tool was used to remove
patella.
Trabecular bone was then
by
boring
then packed over the
was
anterior cortical bone.
the
a small skin
The implant was then inserted into the
removed using a ball ended mill.
Trabecular bone
into the anterior
the New Bolton Center,
at
proximal
patella
A thin-walled
cavity.
inserted
general
middle patellar ligament.
the
smooth or porous-coated
The surgery was performed by D.
trabecular bone of the right patella.
M. Nunamaker V.M.D., a
a
allowed
animals
The incisions were
to recover in a padded
Upon arising, the animals were allowed immediate weight
bearing and maintained in a box stall
of stall rest, the animals
allowed complete freedom of
were
for one week.
turned
exercise.
- 50 -
out
to
Following one week
a 35 acre pasture and
They were observed periodically
for the next 6 months after
which
they were sacrificed.
At sacrifice,
the control and operated patellae were removed, radiographed, and frozen
for storage.
During the six month
post-operative
sequential polychrome labels
1980).
to
document
These inorganic compounds
matrix during bone formation.
label was given
using
a
were given
in
a
giving a series of labels, which each
remodeling history was recorded.
standard
injected on two sequential days,
sequence
of
the remodeling process (Rahn
are incorporated in the extracellular
By
fluoresce a different color, the
period, the animals were given
protocol
for
in which
the labels are
two sequential weeks.
blue
(calcein
blue),
Each
The labels
orange (xylenol
orange), green (calcein), yellow (oxytetracycline), and red (alizarin).
2.1.2 Ovine Calcaneus
Anatomy
The calcaneus of the
tendon of gastrocnemius
sheep
muscle
articulates
with the metatarsals.
(the
calcaneus) attaches to the
tendo
free (posterior) end of the calcaneus
(Figure
superficial digital flexor passes over a
2.3).
The
The tendon of the
bursa at the attachment of the
tendo calcaneus and continues to the digits.
The plantar ligament links
the plantar surface of the calcaneus
to the distal tarsal bones and the
proximal metatarsus.
The medial and
lateral sides of the calcaneus are
free of attachments.
The
cortical
bone
of the shaft is fairly thick.
The posterior region is composed primarily of highly oriented trabecular
- 51 -
Figure 2.3: Dissection of
lateral aspect.
the
ovine
- 52 -
calcaneus and related structures,
In the sagittal plane, the
shell.
cortical
bone and a thin external
trabeculae are arranged in two tracts which form an arch (Figure 2.4).
Implant Material and Geometry
The cylinders were composed
cylinders were
produced
by
porous-coated cobalt chromium.
of
and
Johnson
Johnson Orthopedics Division
Each cylinder had a
(Braintree, Massachusetts) for this investigation.
solid core with a 3.6
mm
the spherical
beads
was
considered
biologically
a sintered bead porous coating
final diameter of 6 mm.
with a depth of 1.2 mm for the
a
and
diameter
800
approximately
inert
The
Cobalt chromium is
um.
which
material
The diameter of
found
has
wide
application for orthopaedic implants (Park 1979).
Experiment Protocol
cylinders
The porous-coated cobalt-chromium
right calcaneus of five
Nunamaker V.M.D.,
University
of
a
Under
calcaneus was exposed using a
hole was then drilled in a
small
at
general
incision
lateral-to-medial
was
the
performed by D. M.
New
Bolton Center,
anesthesia,
the
right
and lateral approach.
A
orientation using a 6 mm
The implant was then inserted and the wound closed.
The ideal location of the
allowed
implant
normal
The animals
were
anesthesia.
The animals were kept
(three animals) or
surgery
surgeon,
veterinary
Pennsylvania.
diameter drill bit.
The
sheep.
implanted in the
were
16
weeks
(two
is shown graphically in Figure 2.5.
weight-bearing
at
recovery from
pasture for a period of 10 weeks
animals).
- 53 -
after
After this period, the
Figure 2.4: Mid-sagittal
trabecular tracts.
of
section
- 54 -
an
ovine
calcaneus
showing
Y
Figure 2.5: Schematic drawing of
implant.
an
- 55 -
ovine calcaneus with a cylindrical
control (left) and experimental (right)
animals were sacrificed and the
calcanei were removed, radiographed, and frozen for storage.
2.2 Structural Analyses
is dependent on the accuracy of
The accuracy of any numerical model
the input data.
input
Required
finite element models of the
the
for
ovine calcaneus and equine patella includes the applied loads due to the
tendons, ligaments, and joint contact
forces (equine patella only), the
material properties of the trabecular
bone
structural geometry.
The methods used to establish these input data and
the results used in the
presented
Also
are
finite
mesh
sparse symmetric matrix.
the ovine calcaneus are
and
and
generation
manipulation
techniques,
reduction of the bandwidth of a
Finally, a series of two-dimensional models of
were analyzed to develop a
These
presented.
for
technique
accurate
analyses are presented below.
element
including an original algorithm for the
simple
and cortical shell, and the
the
representation
of nonlinear
contact conditions at the bone/implant interface.
2.2.1 Applied Loads
The most significant
calcaneus is that due to the
to the
tarsals
and
articular surfaces.
in the
load
tendo calcaneus.
metatarsals
tendo calcaneus was established
finite
element
models
only
direction
from
included
- 56 -
a
attachments and
of the load due to the
dissection
the
of the ovine
This load is transferred
ligamentous
through
The location and
region
posterior
(Figure 2.3).
posterior
portion
The
of the
calcaneus.
Static
by enforcement of beam
maintained
was
equilibrium
assumptions on the distal cross-section (see Section 2.2.4).
the
Establishing
particularly difficult task.
of
to the solution
loads
applied
this
The
the
of
was
a
of this section is devoted
remainder
bone
This
problem.
patella
equine
has
a complex three-
dimensional geometry with multiple muscular and ligamentous attachments.
In
addition,
articular surface and the
certainly
contact
significant
varies
are
forces
condyles.
femoral
through
cyclically
generated
between the
While the load geometry
multiple
gait,
loading
Rather, one representative load case
configurations were not analyzed.
the actual in vivo loads cover
was selected with the understanding that
a range which includes the applied loads used here.
Load Geometry
An excised equine patella
Photographs were prepared for the six
suspended in a cubic space frame.
normal views (Figure 2.6).
the attached patellar ligaments was
with
the coordinate system of the finite
frame did not correspond exactly to
element models.
To
plots of the finite element
mesh in various orientations were generated
and overlaid on the photographs
the
space
frame
from
until
the
the mesh orientation agreed with
Angular
orientation.
coordinate systems were then
load vectors
relationship between the two systems,
a
establish
coordinate system of the space
global
The
used
rotations
the
two
for orthogonal transformation of the
space
frame
coordinate
By
trial
and
element coordinate system.
between
- 57 -
system
to the finite
error, it was found that the
41
Figure 2.6: Posterior view of an
ligaments intact.
equine patella in the space frame with
- 58 -
following Euler angle
rotations,
which
do
not conform to convention,
allowed alignment of the two systems:
***system
yz
system
= rotation about y axis to x
e = rotation about x
axis to x
= rotation about z
y
z
system
axis to x'y'z' system
The general form for an orthogonal transformation is:
x
'
(2.1)
= Ax
where:
(2.2)
X'
X =
y
(2.2)
t
all a12
A =
a13
a2 a
22 a
23
a31 a
a33J
(2.3)
32
A is the
orthogonal
transformation
direction cosines between the x
matrix
and
the
which
is composed of the
x' coordinate systems.
Then,
for the above defined Euler angles, the final transformation matrix is:
- 59 -
A =
insinsinsin-cos cosW
cosEsin*
cos sinesin-sincos'p
sin sinecos*-cos sinW
cosEcosp
cos sincosp+sin sini
coscose
-sine
sin cose
(2.4)
The Euler angles established by the superposition of the space frame and
finite element images were:
*
= 8.00; e = 9.00; and p = 11.5 °.
to
The mesh regions corresponding
ligaments and quadriceps
frame reference system.
attachments of the patellar
the
were
musculature
determined
using the space
The six normal views of the finite element mesh
of the patella in the space frame
were each overlaid on the photographs
the surface of the finite element mesh
and the regions of attachment on
were established (Figure 2.7).
A normal intact
equine
joint
stifle
was
prepared with lead shot
markers at the proximal and distal ligament attachments to establish the
three-dimensional orientation
patellar ligaments
marked,
were
patellar
including
of
markers was measured for each
the
and
degrees of flexion (Figure
2.8).
planes
The stifle joint was
with
the joint in 90
patella were measured from the two
The three-dimensional orientation of each of the patellar
ligaments was then solved using the
onto a plane.
middle, and
The marker separation distances and
angular orientations relative to the
radiographs.
All four
the proximal and distal
ligaments.
lateral
radiographed in the frontal
lateral,
the
between
distance
The
ligaments.
lateral femoropatellar ligament (see
the
medial patellar ligaments and
Figures 2.1 and 2.2).
the
of
equation for projection of a vector
For a vector v:
- 60 -
v = ai + bj + ck
where i, j, and k are unit
vectors
v
equals the magnitude of the
xy
distance between markers
on
in
the x, y, and z directions.
projection
one
(2.5)
If
of v onto the xy plane (the
radiograph)
and
e equals the angle
between the projection of v and the x axis:
2 +b2
vxy =
(2.6)
b
= tana
(2.7)
then combining Equations 2.6 and 2.7 yields:
v
xy
(2.8)
a
1 + (tan -18)
2
and b may be found from Equation 2.7.
Since data were available for two
planes one of the
could
providing a
check
vector
on
components
the
measurements.
be
solved for twice, thus
It
was
assumed
that the
quadriceps was parallel to the femur to establish the orientation of the
quadriceps musculature.
The orientations
were first established in the
space frame reference system and later transformed to the finite element
coordinate system using Equation 2.1.
- 61 -
-),
·.
·
r
..
-
·: ·.
:.
C
.·-L
·'
I
··
·
·-
·
:·..
·
·· · ·
'··..:..
Medial
Anterior
.1··
C :,'
··
Il
)
41
··1
.v
Superior
u
.
':·
· ··
·
. .,
.,'....-..
."i i..-.
i......
!:-.?...
·.
.·.
:
·:: ..... ..
. :
.::;.--..
,..
. *
.
..-
;,. .?,q
·
.
.:...
: ..
.,iI.
.
Po...s
r .....
.- ~~
..
:. -...
. . : =
:.......
~.. ... :.:..:... :.. - - . . ~..... , x,
. . .
-n.e'r.
In
f eri
...
'.',.
'
:.
~..
..
.
··..
..-.
.,,
..
Posterior
·
Inferio
.
Lateral
Figure 2.7: Attachment sites of the quadriceps musculature and the
patellar ligaments on the three-dimensional finite element model.
- 62 -
.
.
.
Figure 2.8: Anterior-posterior radiograph of an equine stifle joint with
markers at the insertions of the patellar ligaments.
- 63 -
In Vitro Load Analysis
study
An in vitro experimental
for the finite element models of
conducted to provide load data
was
equine
the
of human knee joints, co-linear
loading apparatus developed for testing
representing
loads were applied to locations
The
joints (Figure 2.9).
muscle
mass
transducer.
of
and
the
Fuji Prescale
polyethylene,
was
was
monitored
in
the
joint
uniaxial
load
space
in the
area
of
the load cells and the pressure
Calibration of
patellofemoral contact.
the
with
Sensitive Film, enclosed in sheets
Pressure
placed
the equine hip and tarsal
mechanism was clamped as a single
quadriceps
load
Making use of a
patella.
sensitive film, and data acquisition,
were accomplished as described in
The stifle
joint was loaded in 90 degrees of
Huberti and Hayes (1984).
flexion, and the quadriceps and axial loads were recorded.
Pressure films of
the
load
contact
the
femoral condyles were obtained from
typical pressure film
densities were obtained using a
acts in a direction
the
patella and the
vitro load experiments.
A
2.10.
The pressure film
photodensitometer.
The films were then
and
the
finite element mesh was overlaid
establish
the
location of the contact loads on
placed on an excised patella
the mesh (Figure 2.11).
in
in Figure
is shown
using a glass plate to
between
It
normal
assumed that the contact pressure load
was
to
This assumption is equivalent
the
to
contact surface for all locations.
assuming
friction between the contacting surfaces.
- 64 -
that
there
is no surface
Figure 2.9: Equine stifle joint in
clamp and the load cell.
the load apparatus.
- 65 -
Note the muscle
.
1
't,*
1
t; -1,
.
. I . .X4
1
i
Figure 2.10: Typical calibration film and contact pressure measurement.
66 -
1.48
] 0.85
nn
v
Figure 2.11: Patellofemoral contact
dimensional finite element model.
- 67 -
pressure
distribution
;;n
on three-
Static Equilibrium
To establish
the
final
load
applied to the finite element models
This
equations, namely the
forces
satisfying six linear independent
three
in
to be
orientations
it was necessary to satisfy static
requires
equilibrium exactly.
and
magnitudes
directions and the
orthogonal
moments about the three force axes:
(2.9)
Ax = b
where:
1
A = [F. 1
2
F
.
3
4
F.
F.
5
6
F.6
(2.10)
x = [Cil
(2.11)
b = [Fi 0
(2.12)
i =1, 2, ..., 6
For i equal to 1, 2, or 3, Fi
respectively
superscript).
for
the
is the force in the x, y, or z direction,
Similarly, for i equal
about the x, y, or z
axis,
scaler multipliers on the
vectors
load
independent
to
by
the
4, 5, or 6, F i is the moment
respectively.
independent
(indicated
load
The x vector is composed of
vectors.
composed of the forces and moments of the dependent load.
The b vector is
The system of
equations can solved by inverting the A matrix (provided the A matrix is
- 68 -
nonsingular).
The contact pressure load was made the dependent variable since only
the quadriceps and contact pressure
study.
the
For
independent
magnitudes of the
which includes the
the
defined,
load
includes
posterior
more
equations, making solution of
and
the
superior
nodes.
in six
linearly
As initially
independent
possible (a nonsingular A
equations
The calculated magnitude of the total quadriceps load provided
matrix).
a
the more anterior nodes, and one
resulted
vectors
quadriceps load were
the
and
assumed that the
variable, the quadriceps was divided
independent
into two sections, one which
it was
vectors,
loads
ligament
four
For a sixth
unknown.
load
were measured in the in vitro
load
on
check
since
solution
the
this
magnitude
was
measured
experimentally.
The solution of
a
matrix coefficients in
sixth-order
a
linear
is beyond
which
manner
system
is sensitive to the
intuition.
It was
recognized that the measurement of the vector directions for the various
thus
load components was inexact, and
static
equilibrium
directions.
A
could
computer
direction incrementally.
be
load vectors was rotated about
The load vector was then
this
a
rotated
vector in 14.4 degree increments.
resolved.
by
manipulated
program
In
the solution of the equations of
was
changing
written
procedure,
these
vector
to
change each vector
one
of the independent
perpendicular axis by an angle theta.
about
the
axis of the original load
After each rotation, Equation 2.9 was
Thus, 25 different vector orientations, forming a cone around
the original vector direction, were tried.
- 69 -
The best solution was judged
Table 2.1
Applied Loads for Equine Model
Initial
Solution
Final
Solution
LFL
569.7 N
332.0 N
10.00
LPL
152.6
302.5
10.00
MePL
596.5
455.1
9.90
MiPL
1733.1
1676.5
9.90
SQ
569.8
839.5
10.70
AQ
491.9
160.5
10.00
Component
Measured
TQ
1024.0 N
1061.8
1000.0
PFC
3115.3
3115.3
3115.3
LFL:
LPL:
MePL:
MiPL:
SQ:
AQ:
TQ:
PFC:
Lateral Femoropatellar Ligament
Lateral Patellar Ligament
Medial Patellar Ligament
Middle Patellar Ligament
Quadriceps, superior portion
Quadriceps, anterior portion
Quadriceps, total
Patellofemoral contact load
Ae: Change in angle from initial to final solution
- 70 -
to be the solution which
independent loads.
the
minimum
total load for the six
In other words, the best solution was that which, in
the least redundant
pressure load).
yielded
manner,
It was
balanced
found
that
the
the
dependent load (the contact
solution could be improved by
running the procedure several times for each of the load vectors.
The measured and calculated loads
equal
solution was
contact load was equal for
presented in Table 2.1.
to
1000
N and the patellofemoral
three solutions.
all
For
such that the total quadriceps load
this table, the results were scaled
for the final
are
The vector directions
for the initial solution were obtained from the radiographs as described
above.
The absolute angles between
are included in the table.
The
the initial and the final solutions
middle patellar ligament, which
was
angle, was essentially unchanged
by
was equal to 90 degrees for
numerical solution.
of the ligament
and
The
approximately equal to the flexion
the
solution process.
This angle
both
the in vitro experiment and the final
The total of
the independent load vectors, the sum
quadriceps
loads,
initial solution and 3766 N for
of 8.5 %.
between the quadriceps and the
angle
agreement
was
equal
to
4114 N for the
the
final solution which is a decrease
between
the measured and calculated total
quadriceps load was exceptional,
Such close agreement can not
with
be
a difference
expected
of less than 3 %.
for the unmeasured ligament
forces.
2.2.2
aterial Properties
An in
vitro
experimental
study
- 71 -
was
performed
to
determine the
material properties of
trabecular
bone
of
the
equine
patella.
Two
primary assumptions were made about trabecular bone material properties:
a) trabecular bone behaves as an
orthotropic material; and b) the local
material axis system corresponds to
anisotropy tensor
as
predicted
the
by
analysis.
Specimens were prepared
measured.
The specimens were
morphologic axis system of the
the
three-dimensional stereologic
from an equine patella their density
then tested in unconfined compression and
the data were fit to an orthotropic material model using a least squares
fit.
Specimen Preparation and Density Measurement
A normal equine patella
was
embedded
in Alumacast, a fast setting
synthetic resin, and sectioned in a manner similar to the experimental
patellae (see Section 2.3.1).
low
speed
diamond-embedded
anatomic planes.
Six cubic specimens were prepared using a
saw
The locations
bone appeared relatively
in
were
homogeneous
an
orientation
relative
to the
selected such that the trabecular
throughout
each
specimen.
Thin
sections were taken from three
orthogonal
faces and analyzed using the
automated
to
three-dimensional structural
stereologic
system
obtain
anisotropy data (see Section 2.3.1).
The specimens were sanded between
two parallel plates to obtain the final dimensions of approximately 7 mm
on each edge.
The
Vernier calipers
calculations.
and
dimensions
weighed
on
an
were accurately measured with
analytical
balance
for density
The specimens were soaked in physiologic saline overnight
prior to testing.
of testing to
specimen
The cube faces
minimize
surface
were coated with Vasoline at the time
friction
- 72 -
(see
below).
After testing
along the three
blocks at a
450
cube
axes,
the
orientation
specimens
(Figure
embedded wire saw (Figure 2.13).
were
2.12)
cut into rectangular
using a modified diamond-
The specimens were then retested along
the three rectangular block axes.
Dynamic Compression Testing
Each cube was tested in
the
wet
loop, servohydraulic materials test
state
in an Instron 1331 closed-
system.
The cubes were loaded in
unconfined compression, with the load
axial deflection monitored by
Transformer (LVDT).
directions
was
The
the
by
control.
This protocol
was
enable the measurement of
expansion
an
specimens were tested using small
in one
extensometer
of the off-axial
(Figure
2.14).
The
amplitude cyclic loading under stroke
chosen
any
by a load cell and the
Instron Linear Variable Differential
lateral
monitored
monitored
to
assure
linear behavior and to
viscoelastic stiffness component.
Prior
to testing, the specimen dimensions were measured:
W.i = specimen width along axis i
(2.13)
i = x, y, z, y', or z'
where the i subscript
axes
corresponding
corresponding to the
indicates
to
the
the
specimen
original
rectangular
following, the equations will be
cubes
blocks
axis and the measured lateral deformation
- 73 -
and
(see
presented
specimen, in which the applied compression
axes, with the unprimed
Figure
for
the
primed
2.12).
axes
In the
the first test of each
is in the direction of the x
is
in the direction of the y
z
V
z
y
x
Figure 2.12: Coordinate systems for the original cubes and the inscribed
rectangular blocks. Adapted from Snyder (1987).
- 74
j
~ ~
4i,
0
I:
0
I!
It
r
i
I
Figure 2.13: Wire saw
blocks.
apparatus
for
- 75 -
cutting the inscribed rectangular
a=.0
Figure 2.14: Trabecular bone specimen
Note the lateral extensometer.
- 76 -
prepared for compression tests.
axis.
Similar equations may be
other axes.
The applied
obtained for the measurements along the
axial compression resulted in sinusoidal axial
strain:
Da
x
exx =W sin(t) +
xx
W -
o
x
x
where
2D xa
is the
peak-to-peak
(2.14)
x
amplitude
of
the
applied
axial
deformation, Dx° is the mean
axial
of the applied deformation.
Assuming linear viscoelastic behavior, the
deformation, and
is the frequency
X
axial stress can be described as (Ferry 1970):
Da
x
xx =- W [E'sin(wt) + E"cos(wt)]+
x
where Lx° is the
mean
axial
is
used
to
conform
understanding that these are
measurement direction in
material axis.
An
a
amplitude axx,
to
the mean
to
the
general
modulus E".
descriptions
The term
with
the
described as stiffnesses since the
does
form
stress
loss
literature
better
alternate
(215)
(2.15)
load, determining two frequency dependent
functions, the storage modulus E' and
modulus
x
W
xy
is
a
not
correspond
obtained
and
by
to a principal
use of the stress
the phase angle
x between
stress and strain:
a
asin(wt +
=
=XXx
=xacos(x)s
x
+ ax
XX
X
in(wt) + a xx sin(6x
x )cos(t) +
- 77 -
Xx
(2 16)
The storage and loss moduli may then be expressed as:
E' =
a
cos(6x)
(2.17)
E" =
xxsin(6x)
(2.18)
E"
-=
E'
where
a = D a/W,
tan(x)
(2.19)
the strain amplitude.
The voltage from the LVDT, load cell, and extensometer were recorded
by computer through analog-to-digital converters.
data were simultaneously
channel
was
then
evaluated
algorithm (Cooley and
Package.
recorded
Tukey
for
using
a
1965)
one
fast
from
The three channels of
displacement
Fourier
cycle.
Each
transform
(FFT)
the IBM Scientific Subroutine
This provided the real and imaginary components of each signal
at the fundamental frequency (the
and at multiples up
to
the
frequency of the applied deformation)
data
sampling
frequency.
The purpose of
using the FFT was to effectively filter out noise which was contained in
the measured voltage and
to
calculate
the
load and the applied axial deformation.
phase angle 6.i between the
The amplitude and phase at the
fundamental frequency was calculated for each of the three signals:
A.=
Ai
2
2
+ I
R
- 78 -
(2.20)
It= tan-1 i
(2.21)
i = 1, a, or t
where A i is the amplitude, R i is the real component, Ii is the imaginary
component, and
the data channel
. is the
1
(1
for
transverse displacement).
a
load,
The
with the i subscript indicating
angle,
phase
axial
displacement, and t for
modulus,
strain ratio, and phase
for
axial
angle 6i were then calculated:
A 1 Wx B1
W B1
A1 W x
a yz a
Ex
vxy
At Wx Bt
=
conversion
repeated after repositioning
(2.23)
A txtA
W B
xx =$ 1l-
where B. are the
(2.22)
Ita
a
factors
the
for
(2.24)
each
extensometer
channel.
to
The test was
measure the lateral
deformation in the z direction, yielding vxz and a repeat measurement of
Ex and
x.
Note that the calculated modulus Ex is equal in magnitude to
the complex modulus at the frequency w:
Ex = IE'2 +
- 79 -
(2.25)
The compliances were
from
calculated
measured
the
moduli and strain
ratios:
Sxx
(2.26)
x
-v
s
xy
xy
E
was
The test protocol
behavior,
assure
the
to
assure linear material
to
and
produce
of biologic materials invariably
testing
displays nonlinear properties.
as
non-destructive,
were
Mechanical
repeatable data.
so
designed
tests
(2.27)
The
trabecular
bone specimens in this
study exhibited an initial stiffening region followed by a linear region
(Figure 2.15).
a small amplitude to avoid
never exceeded
were applied in the linear region at
The cyclic strains
2%
any nonlinearities. The maximum axial strain
assure that the testing was
was used to improve the
non-destructive.
loaded
in
of
the
yield strain) to
A preconditioning regimen
the data.
of
repeatability
was as follows: 1) the actuator
2) the specimen was
one-third
(approximately
The test protocol
was positioned to the point of contact;
compression
using a single linear ramp
function to 1.5% strain; 3) the specimen was preconditioned by applying
cyclic axial compression,
with
radians/second, such that the peak
step #2.
The
cyclic
loading
maintaining the peak load with
recorded
for
one
cycle
(two
=
ell
load
was
the
1.5
x
mm/mm
and
=
occurred at the peak load from
continued
for
two minutes while
mean level; 4) the data was
stroke
seconds)
10-3
at
a
sampling
rate
of 512
1024 data points for each channel;
points/second/channel for a total of
- 80 -
Figure 2.15: Typical load-displacement curves.
Horizontal axis: 0.05
mm/division; Vertical axis: 100 N/division or 0.005 mm/division. For
this example, E = 426 MPa and v = 0.12.
- 81 -
5) the FFT was performed and the results recorded for each channel.
extensometer was repositioned to
other lateral
direction
and
measure
the
test
the
The
transverse strain in the
protocol
repeated.
This was
repeated for each of the three original cube axes and then for the three
axes of the rectangular block.
Material Modeling
The analysis of the
material
property
assumption that trabecular bone behaves
material.
The
generalized
Hooke's
test
data was based on the
as a linear elastic orthotropic
Law
relating
stress to strain in
tensor form may be expressed as (Hearmon 1961):
ij = Sijkl kl
where i, j, k, 1 =
1, 2, or
3
(2.28)
and Sijkl are the elastic compliances
which comprise a fourth-rank tensor.
As Equation 2.28 stands there are
81 constants.
ji
However, since
ij
=
and
akl =
lk, the number of
constants are reduced to thirty-six.
Furthermore, it can be shown by a
thermodynamic argument (Hearmon 1961)
that
the
number
of
independent
anisotropic material.
constants
to
Sijkl = Sklij which reduces
21
for
the
most
general
For ease of notation, the full tensor suffixes of
the strains, stresses, and compliances can be contracted:
- 82 -
1
C11
c2
c22
c3
c33
£4
2 23
£5
2 13
£6
12
(2.29)
6..
1
a2
a2 2
aO3
a3 3
°4
a2 3
(2.30)
a1 3
a2
a61
.12.
q, r = 1, 2, or 3
Sijkl
Sqr for
q= 1,
2, or 3, r = 4, 5, or 6
(2.31)
14Sijkl
= Sqr for
Sijkl = qr for q, r = 4, 5, or 6
Note that the shear strains ci
contracted notation, if there
are
are
the engineering strains.
three
mutually orthogonal planes of
symmetry, the stress-strain relations simplify to:
- 83 -
Using the
c1
'Sll S12 S13
0
0
0
£2
S12 S22 S23
0
0
0
E3
S13 S23 S33
0
0
0
S44
0
0
E4
0
0
0
c5
0
0
0
0
S ,-
E6
0
0O
0O
0O
0
O
O
a
a2
3
a
0
DD
(2.32)
a4
a5
S66. a6
O
Note that this relationship holds only when the 1, 2, and 3 axes, of the
expanded notation, correspond to the
that the number of independent
nine.
principal material directions and
constants for an orthotropic material is
The compliance matrix in Equation
2.32 can also be expressed in
terms of the engineering constants:
1/E
1
0
0
0
1/E2 -V3 2/E3
0
0
0
1/E3
0
0
0
0
0
-v21/E2 -v31/E
-v1 2 /E1
- 13/E1 -V2 3 /E2
[S] =
3
0
0
0
0
0
0
0
0
0
0
0
1/G23
1/G31
(2.33)
0
O
0
where E. are the elastic moduli, vj..are the strain ratios, and G.. are
13
the shear moduli.
The axes
of
correspond to the
the
trabecular
material
bone
axes.
independent compliance components
specimens
Therefore,
for
necessary to transform the compliances
- 84 -
an
in
to
general
did not
solve for the nine
orthotropic material, it was
from the material axis system to
the axis system of
the
compression
tests.
The compliances compose a
fourth rank tensor, which may be transformed by the following:
S
.. =
a.ma.
a ko
a lp Smnop
S ijkl
jn
where a.. are the direction
cosines
of
the
(2.34)
(2.34)
axes for which S ijk
defined with respect to the axes
for which Sijkl are defined.
2.34 can be used
to
the
compliances
system from
material
data
provided
the
solve
for
test
assumption
that
material axes.
the
the
The
Equation
in the material axis
the
direction cosines
between the test axes and the material axes are known.
can be provided by
are
This information
three-dimensional stereologic analyses with the
structural
material
anisotropy
tests
axes
correspond
to
the
described above provide seventeen
compliances in the xyz and xy'z' axis systems, of which nine independent
compliances could be selected to solve Equation 2.34 exactly:
[a][S] 1 23 = [S]xyzy'z'
where [a] represents the
matrix
the test axes and the
material
in the material
system,
axis
composed
on
that the additional compliances
of direction cosines between
axes, [S] 12 3 represents the compliances
and
[S]xyzy'z'
compliances in the test axis systems.
be found in Hearmon (1961)
(2.35)
represents the measured
The details of the [a] matrix may
the transformation of compliances.
obtained
needed because the measurements in the
nine independent compliances.
- 85 -
in
xyz
Note
the xy'z' axis system are
axis system do not provide
The measured compliances
are inexact at
provided by the
Equation 2.35
best.
To
material
were
and
take
the
to
solved
measured structural anisotropies
advantage
tests,
expanded
experimental data, and
the
of the additional measures
[a]
and
seventeen
using
a
[S]xyzyz, matrices in
rows
least
each
squares
to include all
technique.
To
present this technique, Equation 2.35 is simplified in notation to:
Ax = b
where A represents the [a] matrix
the [S]123 matrix (9 rows by
matrix (17 rows by 1
(2.36)
(17
rows by 9 columns), x represents
1 column), and b represents the [S]xyzy'z'
column).
The
objective then is to minimize the
error:
E2 = IlAx - bl
2
(2.37)
The least squares solution is then (Strang 1976):
x = (ATA) 1ATb
where AT
(2.38)
is the transpose of A.
Results
The density, modulus, and strain
ratio
bone specimens are presented in Table 2.2.
the axis system of the
compression
data for the six trabecular
Note that these data are for
tests rather than the material axes
- 86 -
Table 2.2
Material Property Test Data
Cube Number
Property
1
2
3
4
5
6
D
[gm/cc]
0.835
0.623
1.131
0.636
0.724
0.585
Ex
[MPaJ
622.3
536.6
1495.4
187.1
936.9
209.4
E
Y
Ez
[MPa]
755.0
1026.1
2615.0
483.1
1337.9
763.1
[MPa]
939.1
890.3
3546.6
1086.2
842.8
790.8
Eyl [MPa]
420.5
1289.2
2284.1
703.6
313.7
380.2
Ez, [MPaj
368.3
191.6
1226.7
445.9
735.9
395.9
xy
0.30
0.41
0.09
0.20
0.25
0.14
vxz
0.21
0.33
0.07
0.13
0.38
0.19
vyx
0.40
0.50
0.35
0.48
0.29
0.51
vyz
0.32
0.31
0.18
0.25
0.38
0.34
v
0.50
0.34
0.24
0.26
0.33
0.41
Ivz
0.48
0.30
0.29
0.35
0.31
0.29
v
0.22
0.23
0.19
0.07
0.45
0.11
0.42
0.40
0.04
0.39
0.08
v
zx
v xzr0.38
v IX
y'x
0.14
0.31
0.26
0.16
0.03
0.17
VYr'Z
0.08
0.32
0.23
0.17
0.04
0.19
vZx
0.21
0.14
0.12
0.35
0.07
0.41
vU
0.14
0.03
0.11
0.07
0.27
0.22
D: Density
E.:
Axial stiffness
1
v..: Strain ratio
13
- 87 -
as predicted by the
stereologic
analyses.
The data demonstrate that
there is considerable variation within each specimen as well as specimen
to specimen.
Note that the modulus values represent average values from
the repeated measurements.
each test
was
repeated
transverse directions.
not change in the
Ey,
once
E
for instance, was measured twice, since
to
measure
was measured
rotated
system
direction as the factor.
ratio
in both
Figure
data
2.12).
An analysis of
for each cube with the test
In each case, at a significance level of 0.05,
the assumption that the samples
could be rejected.
strain
four times since the x axis did
(see
variance was performed on the modulus
the
Therefore,
bone of the equine patella
came
the
from populations with equal means
material behavior of the trabecular
is significantly anisotropic; it remains to
be demonstrated whether an orthotropic model is appropriate.
The measured
phase
angle
function of the direction of
2.3.
The average
phase
about 10.
for
of
the
of
the
the
loss
and
strain
(6),
as a
all
measurements
was 5.7° with a
From Equation 2.19, this indicates that, on
storage
Also, the magnitude of
of the magnitude
inclusion
of
stress
applied compression, is presented in Table
angle
standard deviation of 1.9 .
an average, the ratio
between
modulus
to the loss modulus is
the storage modulus is equal to 99.5%
complex
modulus
component
makes
(Equation 2.25).
little
Therefore
difference
for
the
compliance calculations based on linear elastic behavior (Equations 2.26
and 2.27).
A
statistical
analysis
measured phase angles
were
was
performed
anisotropic
- 88 -
to
(Table
determine
2.3).
whether the
An analysis of
Table 2.3
Phase Angle Between Stress and Strain (6)
[Degrees]
Cube Number
1
2
3
4
5
6
x
3.7
4.5
4.6
8.5
4.9
6.8
x
4.3
5.2
3.6
8.6
4.8
6.0
x
6.8
6.8
3.8
9.8
5.1
8.7
x
6.9
5.7
4.4
9.0
5.2
9.8
y
3.3
3.2
4.0
6.9
4.9
5.5
y
3.7
3.6
3.6
8.4
4.6
5.3
z
3.4
4.4
2.9
4.9
4.9
4.8
z
3.6
4.6
4.1
4.5
5.5
5.9
Y'
6.9
3.3
2.9
6.4
9.5
7.9
y,
8.1
4.0
3.0
6.8
8.3
7.6
6.6
7.1
3.8
7.5
4.6
6.9
7.8
7.0
4.0
8.5
5.7
7.9
Mean
5.4
4.9
3.7
7.5
5.7
6.9
St Dev
1.9
1.4
0.6
1.6
1.6
1.5
+
+
+
+
Direction
ANOVA
ANOVA: One way analysis of variance with the direction as the
factor. A "+" indicates that, at a significance level of 0.05,
the assumption that the samples come from populations with equal
means can be rejected. A "-" indicates that, at a significance
level of 0.05, there is insufficient evidence to reject the null
hypothesis that the samples come from populations with equal means.
- 89 -
For two of
the factor.
specimens, at a significance level of
six
the
with the measurement direction as
cube
variance was performed for each
be
rejected.
0.05, the null hypothesis could
not
those two cubes, the
between
variation
the variation within the
at a significance level of 0.05,
However, for the other four specimens,
viscoelastic behavior.
this
While
is not surprising considering the
characterization of the
the
stiffnesses,
material
in this aspect of their
anisotropy
of
the
Therefore these four specimens
rejected.
displayed a significant degree
anisotropy of
directions was no greater than
measurements for a single direction.
repeated
the null hypothesis could be
In other words, for
anisotropic viscoelastic behavior of trabecular bone is beyond the scope
of this thesis.
The three-dimensional stereology data for the six bone specimens are
summarized in Table 2.4.
normalized to
length
the
of
of
represents the axis of
similar then
the
(Equation
The
2.68).
orientation.
minimum
which
axis,
orientation,
maximum
material
this table are the axis lengths,
major
the
anisotropy ellipsoid equation
represents the axis
in
Included
is approaching
and
come
from the
major axis, d,
the minor axis, d 3,
If two axis lengths are
transverse
isotropy.
The
anisotropy ellipsoid for cube number 2, for instance, is the closest to
a prolate spheroid.
semi and minor axes are roughly equal
That is, the
and both are significantly shorter than
ellipsoid for cube number 6 is the
major
and
semi
axes
significantly shorter.
are
For a
the major axis.
The anisotropy
closest to an oblate spheroid.
roughly
equal
and
the
minor
axis
The
is
transversely isotropic material, only one
axis orientation is significant, that of the axis which differs from the
-
90 -
Table 2.4
Three-Dimensional Stereology Data
Cube Number
- -
Property
1
2
S
-
3
4
5
6
-
d1
1.00
1.00
1.00
1.00
1.00
1.00
d2
0.96
0.87
0.96
0.96
0.97
0.98
d3
0.89
0.85
0.91
0.90
0.90
0.93
alx
-0.41
-0.29
0.61
-0.33
0.46
-0.37
aly
0.17
0.62
-0.48
0.67
0.75
-0.59
alz
0.90
0.73
-0.64
0.67
-0.47
0.72
a2x
0.17
-0.52
-0.10
0.13
0.78
0.05
a2y
0.98
0.54
-0.84
0.73
-0.59
0.76
a2z
-0.11
-0.66
0.53
-0.67
-0.18
0.65
-0.90
-0.81
-0.79
-0.94
-0.41
-0.93
0.11
-0.57
-0.26
-0.14
-0.28
0.28
-0.43
0.17
-0.56
-0.33
-0.87
0.25
a3x
a3y
a3z
d : Normalized major axis length
d
2: Normalized semi axis length
Normalized minor axis length
a .: Direction cosine, ellipsoid axis i relative to global axis j
j.J
-
91 -
other two axes.
The
other
four
cubes
display
a greater degree of
anisotropy.
The above compliance and anisotropy data
orthotropic
elastic
indicated that the
determinant of
constants
system
T
(A A)
using
of
for
Equation
equations
each
of
were used to solve for the
the
2.38.
The
results
was
poorly conditioned.
six
specimens
was,
The
for all
practical purposes, equal to zero.
To investigate this
element model was
problem,
used
to
a
linear
simulate
consisted of eight 20-node
cubic
the
such
diagonals.
that
the
test
conditions.
The model
elements arranged with cubic symmetry
(four elements stacked on four elements).
oriented
elastic orthotropic finite
material
axes
The orthotropic constants
The material axis system was
were
aligned
with
the cube
were those reported by Buskirk et
al. (1981) for cortical bone.
The material tests were simulated using
applied
face
displacements
on
one
constraints on the opposite face.
symmetry for the deformation of
not aligned with the cube
of
Note
the
axes.
cube
the
that
cube
with
displacement
there were no planes of
since the material axes were
The axial stiffnesses were calculated
using the predicted stresses and the strain ratios were calculated using
the predicted nodal displacements on the lateral faces.
the cube to the
xy'z'
material axis system.
axis
results, whereas for
was
simulated
by rotation of the
The influence of friction at the faces of applied
compression was investigated
infinite friction.
system
The rotation of
using
displacement constraints simulating
Note that for the ideal case, a uniform stress field
infinite
friction,
- 92 -
the
stresses are nonuniform,
especially in the region of the applied deformation.
The predictions
of
the
element
finite
The exact solution was obtained
compliances.
comparison to the exact
using Equation 2.34 for the
were evaluated by
models
of the compliances from the
transformation
system.
material axis system to the test axis
The results of the ideal
finite element model were all within about 2 % of the exact compliances.
For the case of
infinite
friction,
the
axial compliances were within
about 5 % of the exact compliances.
These simulated test data were then introduced to Equation 2.38 with
the same result as that for the bone cube data.
fact that using nine of the
This is all despite the
compliances in the test axis systems
exact
yields the exact solution for the orthotropic constants.
It is apparent
that the two-dimensional rotation, from the xyz axis system to the xy'z'
axis system yields a poorly conditioned
amount of error,
model,
results
such
in
a
as
that
system
system of equations.
produced
equations
of
in the
further analyzed using an
additional
rotation
providing nine more rows for the A matrix.
and
orthotropic material constants
constants.
the
the y' or z' axis,
be
an
used
are
not solvable.
which
agreed
produce
- 93 -
about the y' axis, thus
of
Equation 2.38 yielded
very
well with the input
it is recommended that a three-
additional
to
The models were
The determinant of (ATA) for
solution
For future investigations,
dimensional rotation, or
which
experimental data.
a possible source of error
nonzero,
the ideal finite element
compression increases the error and is
Friction at the faces of applied
this system was
by
Any small
two-dimensional rotation about
a well-conditioned system of
simultaneous equations.
To establish material property data
models, in
light
of
the
present
relationship was established.
for input to the finite element
results,
An
a
simple density-modulus
isotropic modulus, using the average
of the measured values for the different directions, was correlated with
the measured bone density for each of the six cubes (Figure 2.16).
correlation is clearly unsatisfactory
density-modulus relationship
decided to use
this
for
Carter and Hayes (1977) or
for forming conclusions about the
trabecular
relationship,
others,
This
rather
since
bone.
than
However,
is was
the published data of
this data was drawn from the
equine patella and included multiaxial tests.
For the three-dimensional finite
experimental
assumed.
equine
patellae,
element
isotropic
models of the control and
material
properties
were
Heterogeneous properties were used based on the area fractions
measured stereologically.
densities assuming
a
The area fraction data were converted to mass
tissue
(Lindahl and Lindgren 1967).
2.16 was used
to
assign
finite element models.
density
compact
bone
of
1.8 gm/cc
The modulus-density relationship of Figure
material
The
of
properties
to
the elements of the
investigation of the relationships between
orthotropic material constants and
three-dimensional anisotropy data is
being continued by Snyder (1987).
- 94 -
Average Modulus vs. Density
rrrs
ECU
M
0
[]
d 200
U
1
U
150
5
I00
[]
M
P
50
a
2752*X - 1195
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
Density [gm/cc]
Figure 2.16: Average modulus as a function of density
trabecular bone specimens from the normal equine patella.
- 95 -
for the six
2.2.3 Model Development
Mesh Generation
The geometric data for the
the
parallel
patellae.
sections
of
finite element models were obtained from
the
embedded
ovine
calcanei
and
equine
A rough mesh was first generated from digitized outlines with
the aid of FEMGEN and other
pre-processing software.
refined using projected images of
digitizer.
the
The mesh was then
bone cross-sections on the TALOS
The two-dimensional models generally were composed of 8-node
(quadratic)
plane
stress
elements.
generally were composed of 20-node
The
three-dimensional
models
(quadratic) brick elements, but also
included an assortment of wedge elements and partially-constrained brick
elements.
Automatic Mesh Analysis
The accuracy of finite
element
poor or incorrect mesh definition.
for instance, can result in
material gap formation or
acute
angles
concentrations.
at
corner
For
a
analyses
can be greatly reduced by
Inconsistent element connectivities,
discontinuous material deformation, such as
overlap.
nodes,
large
Poor
can
element geometries, such as
result
in
three-dimensional
difficult to recognize and correct mesh
problems.
artificial
mesh,
it
stress
is
very
For this reason, two
computer programs were written, one to diagnose and one to repair threedimensional finite element connectivities.
- 96 -
The first of the computer programs tests a number of characteristics
of a finite element mesh.
Problems detected fall into two categories:
errors, such as missing nodes, and warnings, such as poor element aspect
ratios.
The
program
also
identified
element types including various
degenerated elements. The errors for which the program checked included:
1) missing nodes
or
which
nodes
for
degenerated
elements
mid-edge node with
defined inside-out,
for
nodes
different
yielding
but
have
pairs
of
a negative
following characteristics were tested
nodes at one location; 2) unused
nodes
(to
avoid
of
stress
sequence; 2) incorrect
a specific manner of
requires
maintenance
the
connectivities,
isotropy); 3) inconsistent
which share two corner
out
(ADINA
collapse of degenerated elements
collapse
are
such
of
spatial
as adjacent elements
different mid-edge nodes, or a
corner
nodes; and 4) elements
Jacobian
determinant.
The
to
produce warnings: 1) multiple
nodes;
3) poor placement of mid-edge
singularities
such
as
the
quarter-point
location):
d =
IVabi
(2.39)
Ivbc I
where A and C are corner nodes on either side of mid-edge node B, Vab is
the vector from node A to
node
B, and
produced if d was greater
than
1.2;
IVabi
> IVbcl .
A warning was
4) poor angles at mid-edge nodes,
where the optimal angle is 180 degrees:
= sin
-
Vab
.
- 97 -
180
-Vb0
(2.40)
where A and C are corner
nodes
on
warning was produced if lel was
either
side of mid-edge node B.
A
greater than 20 degrees; 5) poor angles
at corner nodes, where the optimal angle is 90 degrees:
= sin
where B and C are mid-edge
warning was produced if
-1 (Vab)X(Vac)1
Vab
vac
LVab
vac
nodes
lel
on
was
90
either
greater
(2.41)
side of corner node A.
than
A
30 degrees; and 6) poor
element aspect ratio:
IVabl
d =
(2.42)
IVcdI
where A, B, C, and D are
corner nodes and
between corner nodes and
IVcdl
nodes.
is
IVab
is the maximum distance
the minimum distance between corner
A warning was produced if d was greater than 8.
The second
of
the
computer
programs
changed
selected nodes to improve the mesh geometry.
20-node brick elements which
three-dimensional models.
eight 20-node bricks and
were
the
the coordinates of
The program was limited to
predominant element type in the
Corner
nodes
which
were shared by exactly
mid-edge
nodes
which
were shared by exactly
four 20-node bricks were automatically relocated.
positioned such that all angles at
the mid-edge nodes were relocated
approached 90 degrees and the
The corner nodes were
the corners approach 90 degrees, and
such
angle
that the adjacent corner angles
edge node approached 180 degrees.
along the edge containing the mid-
- 98 -
The corner
nodes
were
repositioned
relocation of the corner nodes
adjacent corner nodes. In
used
and mid-edge nodes.
only
contrast,
mid-edge nodes used the current
first
the
the
method for
the current locations of the
method
locations
The new location
since
for relocation of the
of the adjacent corner nodes
of a corner node was based on the
locations of the surrounding six corner
nodes, as shown in Figure 2.17,
where node T was the node to be relocated and nodes A through F were the
adjacent corner nodes.
node T was to
find
segments AC and BD.
The first
the
minimum
step to determine the new location of
length
line segment which joins line
If points G and H are defined as:
Gi = point on AC
H. = point on BD
Then the coordinates
of
points
G
and
H
can
be expressed as linear
combinations of the coordinates of A and C, and B and D, respectively:
To solve for the
unknown
Gi = Ai + l(Ci - Ai)
(2.43)
H i = Bi + m(Di - Bi)
(2.44)
constants
1 and
m,
the
vector vgh can be
expressed as:
Vgh = Hi - G i = (Bi - Ai) +
- 99 -
(Di - Bi)
m(Ai - Ci)
(2.45)
E
I
C
-B
U --
---A
F
Figure 2.17: Target corner node T and the six adjacent corner nodes.
-
100 -
For G and H to
be
the
end
points
of the minimum length line segment
joining line segments AC and BD, vgh must be normal to both
ac and vbd,
providing two equations and two unknowns:
VghvVac =0
(2.46)
Vgh'Vbd= O
(2.47)
Solution of Equations 2.46 and 2.47
yielded the coordinates of points G
and H. This process was then repeated for line segment pairs AC and EF,
and BD and EF,
to
obtain
three
along the three line segments
line
segments.
Then, the mid-points
calculated and these points were averaged
to determine the location of the target corner node.
Mid-edge nodes can be
repositioned
nodes. Shown in Figure 2.18
adjacent corner nodes (A and
is the
B),
after
target
relocation of any corner
mid-edge node (T), the two
and the eight mid-edge nodes adjacent
to the corner nodes. The first step for determining the location of node
T was to find the plane which, for all points on the plane, the distance
to node A is equal to the distance to node B:
l(x - xO) + m(y - yo) + n(z - z)
where:
=A -B
X
X
m=A -B
Y
Y
-
101 -
= 0
(2.48)
n= A
z
- B
z
x0 = (Ax + B)/2
=
YO
Z=
The next step was to take
(Az + B )/2
the cross-product between vectors from corner
node A to two adjacent mid-edge
directed towards the
(Ay + B y)/2
plane
nodes such that the resulting vector is
found
above.
This
was repeated for all
pairs, yielding four vectors:
V1 = VadXVac
(2.49)
V2 = VaeXVad
(2.50)
V3 = VafXVae
(2.51)
V4 = VacXVaf
(2.52)
The next step was to normalize and sum the vectors:
V2
V1
v5
+
Iv1I
v3
+
1V21
V4
+
Iv31
(2.53)
Iv41
The next step was to find the intersection between v5, with an origin at
point A, and the plane (Equation 2.48):
- 102 -
V5
v
(z - AZ)
(y - Ay)
(x - Ax )
5
v
5
x
(2.54)
V5
v5
5
y
z
Substitution of Equations 2.54 into Equation 2.48 yields the coordinates
of the intersection point.
This
on
node B to yield a second point
was then repeated for corner
process
the
The new location of the
plane.
of the two points on the plane, and
target node is taken as the average
thus also lies on the plane.
The methods used to determine
The establishment of the optimum
the new node locations are empirical.
mesh
geometry was not attempted since
this requires a nonlinear optimization
scheme.
to provide a method
mesh
to
improve
the
The objective here was
geometry
which, while not
producing the optimal mesh, is reliable and efficient for the generation
of a mesh
which
produces
results
acceptable
from the finite element
analyses.
These
techniques
were
applied
geometries for the finite element
mesh generation
initially
in
the
models
in
resulted
of
generation
of
the
the equine patellae.
straight
mesh
The
lines connecting the
the implant border.
The algorithm
corner nodes, with the exception
of
for mid-edge node relocation was
applied
models whereas the algorithm for
corner node relocation was selectively
applied.
Shown in Figure 2.19
application of the these
the experimental patellae.
are the mesh geometries before and after
techniques
Note that the exact implant
throughout the mesh for these
to
geometry
Most
a
was
notable
- 103 -
model of a control patella.
maintained for the models of
is
the
improvement in the
F-
i
E
r,
and the mid-edge nodes adjacent to A and B.
J
~
~~
HI
``H
G/
B,
adjacent corner nodes A and B.
node T.
T, the adjacent
Figure
Figure 2.18: Target mid-edge node
and the mid-edge nodes adjacent to A and B.
- 104 -
Y
Za x
Figure 2.19a: Sagittal section
mesh, after digitization.
of
a
- 105 -
three-dimensional finite element
Figure
2.19b: Same mesh section as in Figure 2.19a, after application ofx
Figure 2.19b: Same mesh section as in Figure 2.19a, after application of
the node relocation algorithms.
- 106 -
angles at the corner nodes
which
resulted
from relocation of the mid-
edge nodes.
Cortical Shell Representation
The cortical shell in some regions
very thin, resulting in very
element aspect
ratios
thickness of this
Very thin
would
shell,
elements
stiffness matrix.
through-thickness
poor
element aspect ratios.
have
or
can
result
The
stiffness
in
a
in
these
This
strain
These
ill-conditioning of the
corresponding
considerably
superparametric shell elements have
thin shell structures.
prohibitive number of elements.
numerical
strains.
Reducing the
either overestimating the
coefficients
are
corresponding to the other
strain energy stored
required
using
strains
of the finite element models was
larger
elements
those
results in excessive shear
components.
been
than
to the
For this reason,
developed for the analysis of
are
based on either Kirchhoff
plate theory, in which out-of-plane shear deformations are neglected, or
Mindlin plate
theory,
included.
Unfortunately,
interpolation
functions
elements
used
in
in which
these
which
these
out-of-plane
elements
differ
analyses
incompatibility, 16-node brick
nodes through the thickness
from
(Bathe
elements
of
shear
the
were
cortical
deformations are
employ
the
displacement
isoparametric
1982).
used
shell.
To
avoid
brick
this
having no mid-edge
Furthermore, the
displacements were constrained in the
direction normal to the cortical
shell surface
could
thickness.
such
This
which results when
that
no
constraint
using
strains
eliminated
elements
with
- 107 -
occur
through
the shell
the numerical ill-conditioning
poor
aspect ratios since the
stiffness terms corresponding
removed.
The
shell
to
elements
the
based
through-thickness directions are
on
through-thickness stresses are equal to
present
element
formulation
is
plate
zero.
that
overestimate the in-plane stiffnesses
The disadvantage of the
the
due
theory assume that the
elements
to
well
tend
to
the plane strain, rather
than plane stress, condition.
The amount by which the
element stiffness can
stiffness for
a
be
constrained brick elements overestimate the
calculated.
linear
isotropic
Using
rectangular
Hooke's
element
Law, the axial
in unconfined
compression is:
Ew w
yz
x -w
x
=
k
(2.55)
where E is the elastic modulus and w. are the dimensions of the element.
1
If the element is constrained
in
the
y direction, the axial stiffness
is:
Ew w
k
1
L
=
x
=
Therefore, the stiffness
in
the
increased by a factor which is a
(2.56)
(1 +
)(l - v)
direction
of
applied compression is
function of the Poisson's ratio of the
material:
k
k
x
1
(1 + v)(1 - v)
- 108 -
-
(2.57)
For example, for a Poisson
approximately 1.1.
equal
ratio
To compensate
to 0.3, the ratio above equals
this factor, the elastic modulus
for
decreased by the same factor.
of the cortical shell elements was
While
this compensation results in an underestimation of the in-plane stresses
present
in the
for these elements,
study
it was
more important to
accurately predict the trabecular bone stresses rather than the cortical
shell stresses.
For the
displacement based finite element formulation,
this requires an accurate representation of the cortical shell stiffness
rather than an accurate prediction of the cortical shell stresses.
Thus, superparametric 16-node elements, which allow strain only in the
to
element planes, were used
represent
overestimation of the in-plane
by a reduction of the
element
were used in all regions
where
thin cortical shell.
The
stiffnesses was compensated for
elastic modulus.
shell
cortical
the
These elements
shell geometry resulted in element
the
aspect ratios greater than ten-to-one.
Bandwidth Reduction
The fundamental equation for
static
finite
element analyses is of
the form:
(2.58)
Ku = r
where K is the stiffness
matrix,
unknown displacement vector.
and
in
general,
sparsely
r
is the
load vector, and u is the
The stiffness matrix is square, symmetric,
populated.
- 109 -
To
take
advantage
of these
properties, solution
algorithms
store only the matrix elements
zero element of each column.
the mean half-bandwidth
stiffness
matrix.
termed
from
the
such
as ADINA,
diagonal to the top-most non-
This is termed the half-bandwidth and thus
reflects
For
band-solvers,
the
storage
band-solvers
directly affects the solution
time
the
for
required for the entire
mean
half-bandwidth
also
the effective inversion of the
stiffness matrix.
The location of the
non-zero
elements
numbering of the finite elements.
The
column i is the difference
i
between
which is a neighbor of i (two nodes
of
K
depend solely on the
column height for a particular
and the smallest numbered node j
are neighbors if both are part of a
common element). Thus, elements should be numbered such that the average
maximum difference between neighboring nodes is minimized.
Many algorithms are available for reducing the bandwidth of a sparse
matrix.
These algorithms seek to reduce both the maximum half-bandwidth
and the
profile
algorithm
of
the
essentially
numbering each node
algorithms are
based
stiffness
passes
as
it
on
matrix.
A
inclined
plane
intersected.
The
an
is
graph
theory,
such
as
simple but effective
through
the
mesh
most commonly used
the "Cuthill-McKee"
(Cuthill and McKee, 1969) or
the "Gibbs-Poole-Stockmeyer" (Gibbs et al.
1976).
table
A
node
connection
proceeds level by level.
These
is
generated
and
the renumbering
procedures are not iterative;
the final
numbering sequence is independent of the original numbering sequence.
To further reduce the profile
of
- 110 -
the stiffness matrix an iterative
reduced the matrix profile with
algorithm was written.
This
algorithm
each iteration, rather
than
completely
mesh.
essentially
The
algorithm
renumbering the finite element
searched
for
columns
with height
greater than the original mean half-bandwidth and attempted to "swap-up"
the top-most non-zero element of that
the
total
While this concept is rather
algorithm if it were
not
number of calculations.
for
The
The change in the number
each attempted swap and only the
for
of matrix elements was calculated
swaps which reduced
column.
number
of
matrix
elements were made.
simple
this
would be a very inefficient
the
complex
logic used to minimize the
algorithm
is described in detail below in
the form of a computer program:
Two one-dimensional
1) Generate the node connection table:
arrays are used, one which contains the number of nodes to
which each node is related (the number of neighbors) and one
which contains the identities of those nodes:
nn(i) = number of neighbors, node i
nc(col i) = node neighbors, node i
2) Calculate the original
of neighbors:
n
i
j
ch(i)
pr
mh
mn
ch
pr
mh
imh
mn
imn
=
=
=
=
=
=
=
=
=
=
=
=
=
=
mean
half-bandwidth and mean number
number of columns (nodes)
column number
minimum neighbor of i
column height above diagonal, column i
matrix profile
original mean half-bandwidth
mean number of neighbors
i - j for j < i
0
otherwise;
sum of ch(i) for all i
(pr + n)/n
integer(mh)
(sum of nn(i) for all i)/n
integer(mn)
3) Find a column ii where the column height is greater than the
original mean half-bandwidth:
-
111
-
ii = n + 1;
1:
ii = ii -
1;
if ii = imh, ii = n;
if ch(ii) < imh, go to 1;
4) Set i equal to the top-most non-zero element of column ii:
i = min(nc(col ii))
5) Find a column j where j is greater than i. If column j is
related to ii (ii a neighbor of j), that column is skipped.
Start at a column which is some distance dj from column i:
j = i + dj + 1;
2: j = j - 1;
if j = i, go to 1;
if nc(col j) contains ii, go to 2;
6) Calculate the decrease
(dec):
in the
matrix profile for the swap
dec = 0;
for each k where k is a neighbor of i:
if k < i, go to next k;
if k = j, go to next k;
if k is a neighbor of j, go to next k;
set 1 = minimum neighbor of k;
if 1 < i, go to next k; (1 must = i)
set 1 = 2nd minimum neighbor of k;
if 1 > k, set 1 = k;
if 1 > j, set dec = dec + j - i
else, set dec = dec + 1 - i;
end loop;
if j is not a neighbor of i then:
set 1 = minimum neighbor of i;
if 1 > i, set dec = dec + 1 - i;
end if;
7) Calculate the increase
(inc):
in
the
matrix profile for the swap
inc = 0;
for each k where k is a neighbor of j:
if k < i, go to next k;
if k is a neighbor of i, go to next k;
set 1 = minimum neighbor of k;
if 1 < i, go to next k;
if 1 < k, set inc = inc + 1 - i
else, set inc = inc + k - i;
end loop;
- 112 -
if i is not a neighbor of j then:
find 1 where 1 is minimum neighbor of j;
if 1 > i, set inc = inc + 1 - i;
end if;
8) If the decrease in matrix profile (dec) is greater than the
increase in matrix profile (inc) then i is swapped with j, and
If not, the
the algorithm continues at step 3 (label 1).
algorithm continues for the next node j at step 5 (label 2).
Rather than updating the node connection table after each swap,
two arrays are maintained which relate the original node
numbers to the new node numbers, and vice versa.
This algorithm has
several
that the algorithm requires storage
general, this table is too
large
possible to implement the
program
of
the
to
held
a
PDP
suitable
in core memory.
In
It was
11-23 by using a virtual
While this significantly slows the
procedure, this computer was available
a
node connection table.
be
on
array for the node connection table.
limitation is that
The first limitation is
limitations.
range
procedure must be determined (dj of step
for
of
5).
use
j
at
values
no cost.
A second
for the swapping
While one choice might be
to try all columns from the right-most column (column n) to the adjacent
column (column i +
1),
the
poor
percentage of successful swaps would
The
most
efficient choice of dj was
make the algorithm quite
slow.
found to be equal to the
mean half-bandwidth when the algorithm starts,
then updated every 20 iterations to equal twice the value of the average
column separation (j - i) of the successful swaps during the previous 20
iterations.
The minimum value
of
dj
was set equal to one-quarter the
original mean half-bandwidth and the maximum
original
mean
half-bandwidth.
The
value was set equal to the
algorithm
efficiency
could
increased if the choice of j values for attempted swaps were improved.
- 113 -
be
2.2.4 Bone/Implant Interface
The mechanics of the bone/implant interface critically influence the
stresses in the adjacent trabecular
bone.
may be expected between an
and
implant
bone
porous layer for bone ingrowth
are
bone provided bone ingrowth is
complete.
following
implantation,
a
Several interface conditions
tissue.
Implants with a
tightly coupled with the trabecular
nonlinear
However, in the early stages
contact
condition
wherein this interface cannot support tensile stresses.
is
expected
Smooth implants
are never tightly coupled to the
adjacent bone since a tight mechanical
junction is not possible.
development
The
of a fibrous tissue layer
surrounding the implant is possible for both implant types which results
in
nonlinear
interface
conditions.
investigate various methods
for application in
series of
the
of
models
without a cylindrical implant,
objective
here
was
to
representing the bone/implant interface
remainder
two-dimensional
The
of
the
finite
of
the
ovine
were
analyzed
element studies.
calcaneus,
A
with and
using two different mesh
densities and a number of representations of the bone/implant interface.
Modeling Techniques
A two-dimensional mesh was
generated
normal ovine calcaneus.
The
outlines
regions were traced
and
the
finite
coordinates of the
nodal
tablet.
points
from
of
the cortical and trabecular
element
were
a sagittal section of a
mesh
established
was sketched.
The
using a digitizing
The first mesh (Model C1.1) included 57 eight-node elements and
187 nodes.
Subsequently,
a
finer
mesh
- 114 -
(Model
C1.2)
was created,
utilizing the same
elements and
71
and
tracing
more
nodes
included in both meshes
to
digitizing
methods,
2.20).
(Figure
A
but with 40 more
circular region was
for subsequent representation of the
allow
For Models C1.1 and C1.2 this area was filled with
cylindrical implant.
trabecular bone in order
represent
to
intact calcaneus prior to
the
implantation of the cylinder.
Loads were applied at the
end simulating the load applied
proximal
The direction and location of
by the Achilles tendon (tendo calcaneus).
the applied tendon load were determined from a dissected calcaneus.
Model C1.1, five nodes were loaded
loaded, all at an angle of 60
For
for Model C1.2, six nodes were
and,
degrees from the longitudinal axis of the
The circular region is described using a clock
calcaneus (Figure 2.20).
reference system, with 12-o'clock vertical as viewed in Figure 2.20.
Eighteen truss elements were attached
model.
Two truss elements were
the distal section of the
to
to each node, one parallel to
attached
the X axis and the other
parallel
to
the Y axis, and the trusses were
anchored.
trusses
were
equivalent to simple mechanical
springs.
The two
The
node
truss
rigidities
iterations, such that the
loads were distributed
tendon load was applied.
nodes
nine
in proportion
Thus
it was
were
adjusted,
through
several
remained co-linear and the shear
to
the
local
modulus when the
assumed that the bone acts as a
Bernoulli-Euler beam at the distal section.
All of the
models
two-dimensional
and elastic material properties.
The
-
115 -
assumed homogeneous, isotropic,
outer layer of elements for both
Model C.1
Model C.2
Figure 2.20: Finite element mesh
C1.2.
and
- 116 -
applied loads for models C1.1 and
models represented cortical bone
and
MPa and a Poisson's ratio of 0.3.
of the
models,
with
Poisson's ratio of
negligible.
an
0.2.
had
an elastic modulus of 15,000
Trabecular bone made up the remainder
assumed
elastic
Dynamic
modulus
effects
during
of
500
MPa and a
loading were assumed
Also, the models were assumed to be in plane stress, with a
thickness approximately equal to the diameter of the calcaneus (15 mm).
Several methods were employed
2.5 outlines the the
models.
All models subsequent to
boundary
representing the implant.
Table
significant characteristics of the two-dimensional
fine mesh corresponding to
for the
for
Model
conditions
include rigid bonding of
expected if bone ingrowth
Models C1.1 and C1.2 incorporated the
at
the
the
bone
had
bone and implant surfaces.
C1.2.
Various assumptions were made
bone-implant
to
the
occurred,
and
implant,
interface.
These
such as would be
nonlinear contact of the
Assumptions concerning implant rigidity were
also examined.
The method of representing the implant in Model C1.3 was to fill the
implant region with elements with the same elastic modulus and Poisson's
ratio as cobalt-chrome (248,000
MPa
at the bone-implant interface were
and 0.3, respectively).
The nodes
shared, thus representing bonding of
the surfaces.
For all subsequent models,
were removed.
The loads
were established from
equivalent
loads.
carried
Model
The
the
C1.2
two
elements filling the implant region
by
and
the trusses on the distal section
cases
- 117 -
these
were
trusses were replaced by
equivalent
since
static
Table 2.5
Model Characteristics for Interface Mechanics
Model
Mesh
Density
Solution
Method
Implant
Representation
C1.1
Course
Linear
Intact
C1.2
Fine
Linear
Intact
C1.3
Fine
Linear
High modulus
elements
C1.4
Fine
Linear
Interface nodes
anchored
C1.5
Fine
Linear
Truss elements
at interface
C1.6
Fine
Equilibrium
iterations
Nonlinear trusses
tangential and normal
C1.7
Fine
Equilibrium
iterations
Nonlinear trusses
45 degrees to normal
C1.8
Fine
Stiffness
reformation
Nonlinear trusses
45 degrees to normal
- 118 -
equilibrium requirements are similarly
around
the
implant
met.
anchored.
were
For Model C1.4 the nodes
bone/implant interface represents an
implant
subsequent models, two truss elements
were
equal rigidity, the results
are
the
bone.
truss
the
bone-implant
at
of infinite modulus.
For
nodes
applied to each node at the
affected
not
trusses relative to
elements
the
the
perpendicular linear truss elements of
For two
bone-implant interface.
at
Anchoring
C1.5 incorporated rigid linear
Model
contrasted with Models C1.3 and
C1.4
by the alignment of the
interface.
The
results
were
to establish that the implant was
accurately represented, especially in
of the apparent modulus of
terms
the implant.
Models C1.6 through
C1.8
Model
C1.6
had
one
orders
magnitude
of
nonlinear
truss
nonlinear
greater than in tension.
normal
and
one
linear
truss
Models C1.7 and C1.8 had two
tangential to the bone-implant interface.
mutually perpendicular
nonlinear truss elements at
The nonlinearity was such that the rigidity
the bone-implant interface.
in compression was four
incorporated
trusses
placed
45
degrees from the
surface normal (Figure 2.21).
The objective for using
the
nonlinear
the bone-implant interface soon after
the
early
stages
after
across the bone-implant
implantation of the cylinder.
implantation
there
could
Consequently, no tensile stresses
interface.
trusses was to better model
be
Once
is no
bone
119 -
ingrowth.
transmitted to the implant
ingrowth
has occurred, the
linear models should more accurately predict the stress fields.
-
In
/
?
/
/
-I-
/,I- Ii
' tI
Truss Interface Elements
i
Figure 2.21: Truss elements at the bone/implant interface.
120 -
determine the optimal method for
Several analyses were performed to
the
nonlinear
analyses.
One
This
equilibrium iterations.
method
standard
is Newton-Raphson
involves initially assuming some
method
force and displacement at each node, then redefining the displacement of
each node using the
The
modulus.
appropriate
revised utilizing the updated displacements.
nodal forces are then
This iterative process may
be expressed in equation form as:
t+AtKW. =t+At
t+t~tKu 1 = ttr
t+AtU
load
is the
superscript indicates the
iteration.
loads.
is the
f
time
nodal
(2.59)
u
(2.60)
Aui
(2.60)
nodal
force vector, K is the
displacement
the
and
The left-hand-side
fii-i
+
i-+
vector,
stiffness matrix, and u
t+At
t+Atu
1
where r is the
-
vector.
The left
right subscript indicates the
Equation 2.59 are the out-of-balance
of
This process is repeated until the difference between iterations
satisfies
a
force
prescribed
and
displacement
tolerance.
The
convergence criteria used in program ADINA are:
IIt+Atr
-
tr
t+Atfi-l1
-
tf
2
2
1
< rtol
(2.61)
tol
(2.62)
au T t+At r _ t+Atf)
Au1 T(t+Atr _ tf)
where rtol
and
etol
are
user-defined
-
tolerances.
Modified Newton-
Raphson iterations are also used wherein the stiffness matrix is updated
121
only after
convergence,
investigation
the
rather
than
at
every
iteration.
Broyden-Fletcher-Goldbarb-Shanno
equilibrium iterations was performed.
stiffness
iteration.
These methods were described
using
alternate approach was to perform
method
of
This method involves updating the
factorized
matrix
(BFGS)
For this
a
secant
approximation
at
each
in detail by Bathe (1982).
An
a single stiffness matrix reformation
with no equilibrium iterations. The original stiffness matrix, as in the
previous methods, was based on the
applied
in
a
single
undeformed geometry.
increment.
displacements, the stiffness matrix
After
was
changes
nonlinearities.
is applicable
analyses,
the
This
stiffness
matrix
regime.
BFGS
initial loading
Models
C1.6
method
and
C1.7
and
a
solving
reformed
geometry, thus accounting for
in
for
the
nodal
based on the current
elemental stiffnesses due to
when,
as
is approximately
equilibrium
single
The loads were
in the present
linear
iterations
stiffness
beyond the
were used for
reformation
with no
equilibrium iterations was used for Model C1.8.
After creating the finite element meshes, the two-dimensional models
were analyzed with the finite
Inc. Watertown, MA
02172).
The results of the ADINA computations were
post-processed using FEMVIEW
Davisville, RI
02854),
element package ADINA (ADINA Engineering,
and
(Jordan,
graphic
Apostal, Ritter Associates, Inc.
results
were
obtained for the
principle stresses.
Model Comparison
Vector plots of the
principal
stresses
- 122 -
around the circular region
were generated.
The principle
stresses
for the coarse and fine meshes
which model an intact calcaneus (Models C1.1 and C1.2, respectively) are
shown in Figure 2.22.
For Model
C1.1 the principle stresses tend to be
highest at around 2 and 5-o'clock, relative to the circle, with a stress
concentration within the circular region.
stresses remained fairly constant
For Model C1.2 the principle
the longitudinal axis, with no
along
obvious discontinuities.
When the circular region was filled with elements having the modulus
of cobalt-chrome (Model C1.3), the principle stresses around the implant
were 16%
greater in
results for Model
tension
C1.2.
and
greater in compression than the
26%
Moreover,
were highest at the implant
the compressive stress components
around
stresses were highest around 5
2
and
and 8-o'clock, while the tensile
11-o'clock (Figure 2.23).
unlike the uniform distribution encountered in Model C1.2.
largest principle stresses were compressive,
peak
tensile
stresses
were
oriented
This was
Although the
it is significant that the
normal
to
the
bone-implant
interface.
The results for the
were anchored
(Model
models
C1.4)
in
and
attached to high modulus linear
to the results
for
Model
(Model
the
the nodes around the implant
nodes
trusses
C1.3.
nonlinear trusses normal to the
to the interface
which
around
the
implant were
(Model C1.5) were very similar
Replacing
the
linear trusses with
interface and linear trusses tangential
C1.6),
the
results
yielded tensile stresses
which were about 50% greater than those for Models C1.3, 4 and 5 (Figure
2.23). Also,
the
orientation
of
the
- 123 -
compressive
and especially the
MAX - .215E-i
MIN - -. 27iE-i
. -
';i~..'
.
.
~L. Model C.1
Z
Lv
pi
kAAW
-
,A
Ac_A
296E-i
Model C.2
Figure 2.22: Principal stress vectors in the implant region, Models C1.1
and C1.2. The cross-hatches indicate compression.
- 124 -
UAV
'AMC_4
-.374E-i
Model C1.3
.
_...3..-
MAX - .394E-i
MIN - -. 403E-i
-vxi*,I .
X:X
z
k-v
X$
'`~rr·~··1A
UAY
Model Ci.6
X
ARF=-4
-. 470E-
Model C.7
Figure 2.23: Principal
C1.3, C1.6, and C1.7.
stress
vectors
- 125 -
in the
implant region, Models
tensile stresses were changed.
Results for the
normal of the
Figure
nonlinear
bone-implant
2.23.
model
with
interface
Specifically,
trusses
(Model
these
C1.7)
results
are
45
degrees to the
are also shown in
for
a
model which
performed 14 BFGS equilibrium iterations and had a relative displacement
tolerance of 0.01% and
a
relative
tolerances were less strict
Models C1.3, 4, and 5.
Figure 2.23 were also
performed (Model
about half the
obtained
of
C1.2.
different.
Also,
the
The tensile
were
of
1.0%.
If the
essentially the same as
virtually identical to those shown in
the
the
59%
stiffness
reformation method using
equilibrium
indicated
116% greater in tension and
tolerance
if 1 stiffness matrix reformation was
with
time
results for Model C1.7
Model
results
Results
C1.8),
CPU
the
force
maximum
greater
alignment
principal
of
iteration
method.
The
principle stresses that were
in compression than those for
the
stresses
principal
tended
stresses
was
to flow around the
implant directed almost tangential to the bone-implant interface.
Discussion
The results from the
mesh is insufficient.
coarser
The
non-uniform and high stresses
mesh
stresses
were
(Model C1.1) indicate that this
within the trabecular region were
encountered
on isolated nodes near
the implant region.
In contrast, Model C1.2 predicted much more uniform
stresses throughout
the
calcaneus
around the proposed implant region.
and
no
particularly high stresses
Thus the finer mesh proved superior
to the coarse mesh and, in two-dimensions, was sufficient for predicting
126 -
the stresses in the calcaneus.
All the linear
normal to the
models
implant
indicated
significant ingrowth of bone within
the
around an implant given time
for ingrowth and assuming a fibrous layer does not form.
used
to
represent
this
principle stresses and
seem
conditions,
of
displacements
Since
virtually identical.
The assumption
infinitely rigid body, and that trusses may
that the implant acts as an
be
only if there is
implant, thus these models are
the
fields
stress
tensile stresses occur
is possible
This
surface.
useful for determining the
that
Models
C1.4
Model
reasonable
C1.3,
utilized
since
the
4, and 5 were
fewer elements than
Models C1.3 and 5 it represented the most efficient, accurate method for
determining the stress fields around an implant with ingrowth.
The nonlinear models predicted markedly altered stress fields around
For
the implant.
Model
to
the
interface,
bone-implant
components normal to
this
trusses at 45 degrees to
stresses predominantly
interface.
the
the
trusses
were
normal and
principal tensile stresses were about 45
tangential to the implant, the
degrees
where
C1.6,
In
tensile
stress
contrast, Model C1.7, with
of the implant, predicted tensile
normal
tangential
indicating
to
the
implant.
Thus Model C1.7
modeling the bone-implant interface in
represents an improved method of
the early stages following implantation.
2.3 Stress-Morphology Analyses
Stereologic methods were used
to quantify the structural parameters
- 127 -
of trabecular
bone.
These
structural
density, the average trabecular width,
anisotropy.
These data were then
the finite element
models
parameters
and
include the areal
the degree and direction of
compared to the stress predictions of
using
statistical
methods
to evaluate the
stress-morphology relationships.
For practical
plane
sections
reasons
and
the
thus
stereologic
are
inherently
dimensional structural measures are
are
examined.
This
technique
measurements
were made on
two-dimensional.
Three-
possible if three orthogonal planes
was
applied
in
the
study
of
the
anisotropic material properties of trabecular bone of the horse patella.
However, the analyses of the
experimental bone sections were limited to
the planes through the implants
to
data.
were performed using an automated image
All stereologic analyses
analysis system.
the method of
The software
directed
produce
the best yield of critical
for the stereologic analyses, which uses
secants
to
predict the morphologic parameters
described below, were written by Snyder (1987).
Specimen Preparation
All
bone
sacrifice.
specimens
After
a
were
variable
dissected and the specimens
(MMA),
Alumacast
fresh
storage
were
(a synthetic
immediately
period,
the
after animal
soft tissues were
embedded in either methylmethacrylate
polymer),
polymer).
Two techniques were
analysis.
Contact radiographs of thin
series of ovine calcanei and
frozen
or
Castolite
(a styrene
used to prepare sections for stereologic
the
sections were used for the first
bone cube material property specimens
- 128 -
and
alumina
silicate
packing
was
used
for
the
remainder
of
the
specimens.
The first series of
sheep
calcanei
space by polymerized MMA which
for the preparation of very thin
allows
Sections of approximately 200 um thickness were obtained with
sections.
the diamond saw.
um
in MMA using a
This procedure results in infiltration of the marrow
standard protocol.
100
embedded
were
or
These
less
sections
on
a
were
then polished to a thickness of
metallographic
polishing
wheel.
Contact
microradiographs were prepared of the thin sections.
The second series of ovine calcanei were embedded in Castolite which
is clear and non-infiltrating.
the diamond saw.
A water
A
pick
single mid-sagittal cut was made with
was then used to clean the interstitial
spaces and 0.2 um alumina silicate
reflective filling.
This
was
surface
fluorescence microscope to
view
packed and leveled to provide a
was
the
then
photographed
through a
polychrome fluorescent labels and
the imaging microscope for stereologic analyses.
The
equine
infiltrating.
patellae
For
this
were
embedded
purpose,
in
right
Alumacast
and
made
for
patella
each
A
anatomically prior to embedding.
and
suture
is non-
left standard molds were
prepared of the femoral condyles and proximal tibia.
Alumacast was
which
the
A positive mold of
patellae
were placed
was used to mark the distal
pole which is immediately distal to the insertion of the middle patellar
ligament as a reference
the
blocks
were
for
sectioned
the
mid-sagittal plane.
sagittally
- 129 -
at
10
mm
After embedding,
intervals
using a
carborundum grinding wheel.
The
cut
surfaces
were then prepared and
imaged similar to the second series of ovine calcanei.
The cubes prepared for
material
were taken from a normal equine
sections were cut from
three
diamond saw.
lightly
100 um.
The were
property tests (see Section 2.2.1)
patella
prepared as those above.
orthogonal
faces
Thin
of each cube using the
sanded
to a thickness of approximately
Microradiographs were prepared
and imaged similar to the first
series of ovine calcanei.
Analytical Methods
The stereological techniques are
secants" (Saltykov 1958).
on
the "method of directed
In this method, a circular region of the two-
dimensional image is scanned with
lines (Figure 2.24).
based
an array of equidistant parallel test
A count is produced of the number of intersections
between bone and marrow space
The test array is then
for the particular scan line orientation.
incrementally
rotated
and the count repeated.
This procedure yields the mean path length in bone (Weibel 1967):
2AA
Lb(e) =
(2.63)
IL (O)
where AA is the bone areal density,
bone
pixels,
and
IL(O)
intersections per unit test
above.
is the
line
For random sections, a
given
test
by the ratio of bone to nonline
length),
polar
density
number of
given by the scan described
plot of Lb(O)
- 130 -
(the
yields a circle for
x
lines
test
Parallel
Figure 2.24:
architecture for the method of directed
al. (1982).
- 131 -
superimposed over trabecular
secants. Adapted from Hayes et
isotropic structures and
(Whitehouse 1974).
for partially oriented structures
ellipse
an
a
Furthermore,
three-dimensional
plot
of Lb(O)
structures and an ellipsoid for partially
yields a sphere for isotropic
oriented structures (Harrigan and Mann 1984), with the general formula:
Ax1
2
+ Bx2
+ EXX
+ Dxx
Recognizing that Equation 2.64 is the
tensor (Reismann and Pawlik 1980),
+ Fx2x3 = 1
quadratic
the
(2.64)
form of a second rank
ellipsoid may be represented in
matrix form as:
[Xl [M][X] = 1
(2.65)
where:
x1
[X] =
x2
(2.66)
x3
A DE
[M] =
Note that [M] is a symmetric
transformation law.
second
Therefore,
orientation may be found
by
D
the
B
F
(2.67)
rank tensor which obeys the tensor
degree
and directions of material
solving the eigenvalue-eigenvector problem
to yield (Snyder 1987):
- 132 -
x2
y,2
+
are
the
2= 1
-2+
2
+2d
d1
1
where di
z,2
semi-axes
(2.68)
d3
3
of
the
ellipsoid
and
x',
y',
and z'
correspond to the principal axis system.
In practice, it is not practical to scan planes at every orientation
through a structure.
forms of the equations
However, this is unnecessary if one recognizes the
describing
For a two-dimensional analysis,
Lb(G)
for two and three dimensions.
theoretically
it is only necessary to
scan at three different orientations to obtain the exact solution for an
ellipse:
Ax i
2
+ BxiY
i
+ Cyi
2
bi
(2.69)
or in matrix form:
Ax = b
(2.70)
where:
A = [xi
2
x =
XiYi
2
yi ]
(2.71)
B
(2.72)
b = [bi]
(2.73)
- 133 -
In practice, the test
count inexact.
density
is finite, making the intersection
Therefore it is more
test line density at
best fit to
line
the
greater
ellipse.
than
practical to scan at a reasonable
three
This
can
orientations and calculate a
be
done
using a least squares
technique where the error is defined as:
E2 =
b -Ax 12
(2.74)
E , with the solution being (Strang
The objective, then, is to minimize
1976):
x' = (ATA)
-1AT b
The
procedure
dimensions.
is similar
In
this
when
case,
(2.75)
solving
multiple
for
an
ellipsoid
intersection
in three
counts from three
orthogonal sections are used.
Implementation
The bone sections were analyzed
system.
The
sections
were
using an automated image processing
digitized
black and white camera mounted
on
57.2 pixels per mm.
digitizer.
For
a
the
digitized
images
to
The
A
The
by 512 pixels with 256 gray levels
typical
corresponds to about 12 pixels.
an instrumentation grade
a low magnification microscope.
analog signal was digitized into 512
by a commercial video
using
magnification used resulted in
trabecular
width of 0.2 mm, this
gray level threshold was applied to
distinguish
- 134 -
bone
and
marrow
space.
The
stereologic software performed
image
at
15°
Verification
increments
of
system
through
accuracy
180 °
has
of known
geometry
and
for
been
dimensional and three-dimensional analyses
patterns consisting of arrays of
rotation
mathematical
the
of the digitized
test
scans.
both
two-
et al. 1986).
Test
reported
(Snyder
line
for
ellipses and embedded elliptical beads
orientation
were
used
for
the verification.
Linear regression demonstrated highly significant R2 values ranging from
0.89 - 0.99 with the slopes
not
significantly different from 1 and the
intercepts not significantly different from 0.
- 135 -
3.0 Results
The
morphologic
data
from
the
experimental specimens were studied
remodeling
response
trabecular
bone
to
the
the
that the
principal
directions of
stress
material
The
implants
predictions from the finite element
were
and
equine
control
and
to characterize the trabecular bone
implants.
surrounding
of the control specimens
ovine
were
models.
then
stress
orientation.
are
using the
based on the assumption
aligned
Finally,
relations for implant-induced remodeling
evaluated
in the
The finite element models
validated
directions
changes
with the measured
the stress-morphology
were investigated based on the
assumption that the finite element model predictions were accurate.
3.1 Morphologic Analyses
The morphology of the ovine and equine specimens were analyzed using
stereologic techniques.
These
the anisotropy
for
surrounding the
ellipse
implants
control specimens.
The
and
analyses
multiple
in
present
the
provided the areal density and
regions
of
the trabecular bone
corresponding
objective
was
to
locations in the
characterize the
remodeling response especially as a function of the implant material and
surface characteristics.
3.1.1 Equine Patella
The surgical implantation
well tolerated.
The
only
of
the
the
complication
- 136 -
stainless steel spheres was
during
the
six month post-
surgical period was a
wound
infection
porous coated implant (P6).
antibiotics.
On
The
preparation
sectioning, is was noted that
in one animal which received a
animal
of
the
the
porous
was
treated with one dose of
patellae
for
embedding
coated sphere in specimen P6
was clearly loose with only fibrous tissue holding it in place.
time, there were no frank signs
sample was obtained
embedded.
for
of infection.
pathologic
All twelve patellae
in Section 2.3.1 and the
and
At this
Unfortunately, no tissue
evaluation
before the patella was
were embedded and sectioned as described
morphologic
analyses were performed using the
automated stereologic system.
Contact
radiographs
of
the
sectioned
Radiographs of a sagittal section
through
experimental patellae are shown in Figure
specimens with a smooth sphere (S1),
patellae
were
obtained.
the implant for three of the
3.1.
Included are one of the
one of the specimens with a porous
coated sphere (P7), and the specimen with the porous coated sphere which
was frankly loose (P6).
Note that increased bone densities are apparent
around each of the implants.
A fluorescent photomicrograph showing the
labeled trabecular bone adjacent to the smooth implant in specimen S2 is
shown in Figure 3.1c.
The most prominent label was the oxytetracycline,
given second to the last.
light.
There
(alizarin red).
were
also
The
This label appears yellow under fluorescent
red
bands
polychrome
corresponding
labels
vigorous remodeling response adjacent
to
confirmed
the
to
the last label
that
there was a
smooth implants and that
there was active remodeling during the late stages of the experiment.
photomicrograph showing ingrown
trabecular
spicules
surface beads in specimen P7 is shown in Figure 3.1d.
- 137 -
A
surrounded by the
It appeared that
control
Figure 3.la: Contact radiographs of a sagittal section of the
and experimental patellae with smooth surfaced implants.
- 138 -
Figure 3.lb: Contact radiographs of a sagittal section of the control
and experimental patellae with porous coated implants.
- 139 -
Figure 3.1c: Fluorescent photomicrograph of the trabecular bone adjacent
to the smooth implant in experimental patella S2.
- 140 -
Figure 3.1d: Fluorescent photomicrograph of ingrown trabecular spicules
surrounded by the sintered beads of experimental patella P7.
- 141 -
all of the
most
in the
apparent
green
The
trabecular bone.
layer
porous
spicules
ingrown
was
label
was occupied by ingrown
yellow (oxytetracycline) bands
and
(calcein)
Green
trabecular bone.
were
in the
space
void
given
and
adjacent
in the
in the
middle of the
experimental period, followed by the yellow label, indicating that there
porous layer several months after the
was active bone formation in the
surgical insertion of the implants.
The
smooth
porous
and
response,
implants.
Photographs
control
and
of
experimental
the
vicinity
of
the
which
were analyzed from the
are
shown
in Figure
patellae
on
3.2.
The
the experimental patellae using
the array of test regions shown in Figure 3.3.
around the implants.
a significant
sections
stereologic analyses were performed
This array was designed
groups at different relative locations
distinct
to provide data in two
induced
immediate
in the
especially
remodeling
spheres
coated
The "A" locations were centered a distance of 2.1
and the "B" locations were centered
mm from the bone/implant interface,
from
a distance of 6.3 mm
bone/implant interface.
the
may be examined as a function
of
their
Thus, the data
proximity to the implant.
The
analyses were also performed on the contralateral control patellae using
The
the same array.
reference
geometric
patellae was the center of
the
implant.
point for the experimental
The corresponding location on
each matched control patella was determined by superimposing the contact
radiographs.
camera and
After
video
obtaining
digitizer,
and/or inverted, as necessary,
the
the
such
images
images
that
orientation as the finite element models.
- 142 -
using
the
black and white
were
mathematically rotated
all
images were in the same
;
h·
4
r
'L-
v·
;i
e;' d!
C·
·-"
c
'j
b·
S',Cf.Ct
S1 Left
S1 Right
S2 Left
S2 Right
S3 Left
S3 Right
Figure 3.2a: Sagittal sections on which the stereologic analyses were
performed for the smooth surfaced implants.
- 143 -
P6 Left
P6 Right
P7 Left
P7 Right
^-e
I-
-··
.·, ·
"' --J h
I
LI·
,-i
3·c·:,
·
i
2CX-
rs·
P8 Right
P8 Left
Figure 3.2b: Sagittal sections on which
performed for the porous coated implants.
- 144 -
the stereologic analyses were
·
·
-
R = 2.1 mm
Y
Lx
Figure 3.3: Array of image regions
the equine patellae.
used for the stereologic analyses of
- 145 -
An example of the morphologic
a smooth surfaced
implant,
data from specimen S3, which received
is shown
in Figure
3.4.
The region of
analysis was "4A" (see Figure 3.3), and the images and results from both
the control and experimental patellae
the digitized images was 512 by
512
resolution of the displayed images
are
included.
pixels
The resolution of
with 256 grey levels.
The
in these figures was much lower and
the image was converted to black and white using a grey level threshold.
Some of the parameters which
were
measured using stereology can be
estimated by visual inspection.
The
in Figure 3.4a
homogeneous,
appears
fairly
entire area of the control patella
with
a somewhat oriented
architecture in the lower right portion and an isotropic architecture in
the
region
of
analysis.
stereologic analyses.
This
was
confirmed
by
the
In contrast, the corresponding area (Figure 3.4b)
in the experimental patella
region of remodeled
observation
is clearly
trabecular
adjacent to the implant.
bone
This
not
with
finding
homogeneous.
an
was
There is a
increased areal density
quite consistent for the
three patellae which received the smooth surfaced implants (Figure 3.2).
The extent of the
about 4 mm.
intensely
It was
this
diameter for the regions
remodeled
finding
of
region
on
which
analysis
convincing finding from this example,
was
the areas analyzed on
implants, was that
the
there
and
experimental
was
a
from the implants.
the selection of a 4.2 mm
based.
The most clear and
indeed from virtually all of
patellae which received smooth
significant
adjacent to the implants and little to
visually appeared to be
increase in bone density
no change beyond about 4 or 5 mm
This observation was confirmed by the results of the
stereologic analyses.
These data are presented below.
- 146 -
BETH ISRAEL OSPITAL-
[AREA FRACTION ANALYSIS: 1
REGION AREA = 13.94
tSTEREOLOGY ANALYSIS:]
Y
IMAGE PROCESSING LABORATORY
IMAGE TITLE: S4R4
COMMENT: Image 4
SUBREGION: 2
LENGTHSCALE: 57.18
GRAY THRESHOLD: 120
SOLID PHASE: BLK
TESTLINE LENGTH: 795.1
TESTLINE SPACING <PIXELS>: 1
AREA FRACTION = 0.443
SOLID AREA = 6.164
ELLIPSE OF MEAN INTRCPT LENGTHS:
A.*N,2+ BN4N, + CNo 2= MNINCPTLN - 2
A 6.935
B =-0. 1787
x
C = 7.6768
CORRELATION COEFF = 0.444
MAX MN INTRCPT LNTH = 0.38003
A.GLE M.AX ORIENTATION =
7 DEGS
MIN MN INIHTUP LNTH = 0.36067
PERIMETER = 52.296
PERIM/AREA = 3.7608
Figure 3.4a:
specimen S3.
Stereologic
ANGLE MIN ORIENTATION = 97 DEGS
EXTENT OF ANISOTROPY = 3.330 %
analysis
of
- 147 -
region
"4A"
from
the control
_____
___·_I
____
___·___
_
BETH ISRAEL HOSPITAL - IMAGE
PROCESSING
LABORATORY----------
[AREA FRACTION ANALYSIS:]
IMAGE TITLE: S4L4
COMMENT: Image 4
SUBREGION: 2
LENGTHSCALE: 57.18
GRAY THRESHOLD: 120
SOLID PHASE: BLK
TESTLINE LENGTH: 795.1
TESTLINE
SPACING <PIXELS>
~--"
AREA FRACTION = 0.614
REGION AREA -
SOLID AREA = 8.532
_
13.91
[STEREOLOGY ANALYSIS:]
Y
X
PERIMETER = 49.576
PERIM/AREA = 3.5652
,
-
_
_
1
__
ELLIPSE OF MEAN INTRCPT LENGTHS:
A*N42+ B*N1N 2 + CN 22= MNINCPTLN- 2
A = 3.111
B = 0.3601
C = 3.7458
CORRELATION COEFF = 0.815
MAX MN INTRCPT LNTH = 0.57135
ANGLE MAX ORIENTATION = -15 DEGS
MIN MN INTRCPT LNTH = 0.51344
ANGLE MIN ORIENTATION =
EXTENT OF ANISOTROPY =
75 DEGS
6.794 %
Figure 3.4b: Stereologic analysis of region "4A" from the experimental
specimen S3. Note implant in upper left corner.
- 148 -
there was fibrous tissue encapsulation
implant in specimen P6 in which
Again, images from both the control
3.5.
increase in bone density around the
region appeared to be limited,
from the implant.
6 mm
or
a very striking
As in the specimens with
implant.
smooth implants, the intensely remodeled
in this case, to about 5
was
There
included.
are
and experimental patellae
An example of the stereologic
3.2).
rather than bone ingrowth (Figure
analyses is shown in Figure
intense around the porous
most
was
The trabecular bone remodeling
This observation
also was supported by the results of the stereologic analyses.
The other two
porous
received
which
specimens
implants, but had
successful bone ingrowth, appeared to have undergone the least amount of
trabecular bone remodeling (Figures 3.2).
specimen P7 is shown in Figure
remodeling, as
by
determined
in this specimen, suggesting moderate
the
analyses.
stereologic
the
One other
control patella from specimen P8 was
important observation was that the
generally less dense than
There was a marginal increase in
3.6.
implant
bone density around the
Example stereologic data from
five
other
patellae.
control
This is
discussed further below.
These observations of changes in
bone density were confirmed by the
stereologic measurements of bone area
deviations of the area fractions are
for the "A" locations
based
on
implants.
the
and
observed
The means and standard
fraction.
presented
in Table 3.1.
The data
"B" locations were grouped separately
the
density
changes
in close
proximity
to the
The present objective was to examine the density changes as a
function of the distance
from
the
implants
- 149 -
and
as a function of the
BETH ISRAEL HOSPITAL - IMAGE PROCESSING LABORATORY
[AREA FRACTION ANALYSIS: I
IMAGE TITLE: P66L5
COMMENT: Image 5
SUBREGION: 2
LENGTHSCALE: 57. 18
GRAY THRESHOLD: 120
SOLID PHASE: BLK
TESTLINE LENGTH: 795.1
TEST!INE SPACING <PIXELS>:
AREA FRACTION = 0.429
REGION AREA =
SOLID AREA = 5.963
3.91
[STEREOLOGY ANALYSIS:]
Y
1
ELLIPSE OF MEAN INTRCPT LENGTHS:
A*N2+ BNIN
....
- ..
X
&..~
+ CNo2= MNINCPTLN- 2
A=
9.909
=
3.651
C = 7.8959
CORRELATION COEFF = 0.982
MAX MN INTRCPT LNTH = 0.38299
ANGLE MAX ORIENTATION = 12i DEGS
MIN MN INTRCPf LNIH = 0.30169
PERIMETER = 55.688
PERIM/AREA = 4.0048
Figure 3.5a:
Stereologic
ANGLE MIN ORIENTATION = 31 DEGS
EXTENT OF ANISOTROPY = 15.069 %
analysis
of
specimen P6.
150 -
region
"5A"
from
the control
IMAGE PROCESSING LABORATORY
[AREA FRACTION ANALYSIS:]
IMAGE TITLE: P66R5
COMMENT: Image 5
SUBREGION: 2
',EGTHSCALE:,
57 .18
GRAY THRESHOLD: 115
SOLID PHASE: BLK
TESTLINE LENGTH: 795.1
TESTLINE SPACING <PIXELS>:
AREA FRACTION = 0.738
REGION AREA = 13.91
SOLID AREA = 10.27
[STEREOLOGY AN4LYSIS: ]
Y
ELLIPSE O MEAN NTRCPT LENGTHS:
-2
AANI 2+ BNaN
+ GN0MNINCPTLN
-
1
C
A = 1.552
B =-0.3173
C = 2.0770
X
CORRELATION COEFF = 0.969
MAX MN INTRCPT LNTH = 0.81430
ANGLE MAX ORIENTATION = 16 DEGS
MIN MN INIHUHI LNH = U.bUbbU
ANGLE MIN ORIENTATION = 106 DEGS
EXTENT OF ANISOTROPY = 10.815 %
PERIMETER = 43.372
PERIM/AREA = 3.1191
Figure 3.5b: Stereologic analysis
specimen P6.
of
- 151 -
region "5A" from the experimental
_ ·
_I_
_Y______
MAGE PROCESSING LABORATORY
IAGE TITLE: P67L5
BETH ISRAEL HOSPITAL
IMMENT: Image 5
IBREGION: 2
:NGTHSCALE: 57.18
1AY THRESHOLD: 115
ILID PHASE: BLK
:STLINE LENGTH: 795.1
:STLINE SPACING <PIXELS>:
[AREA FRACTION ANALYSIS:]
REGION AREA = 13.91
[STEREOLOGY ANALYSIS:]
Y
__
___I
1
CEA FRACTION = 0.374
ILID AREA = 5.205
.LIPSE OF MEAN INTRCPT LENGTHS:
:Ni 2+ B*NiN2 + C*N22 = MNINCPTLN-2
A = 9.848
B
X
PERIMETER = 47.191
PERIM/AREA = 3.3937
Figure 3.6a:
specimen P7.
Stereologic
-2.093
C = 6.9165
RRELATION COEFF = 0.940
4X MN INTRCPT LNTH = 0.38980
IGLE MAX ORIENTATION = 72 DEGS
INMN INTRCPT NTH = 0.31337
IGLE MIN ORIENTATION = 162 DEGS
'TENT OF ANISOTROPY = 13.803 %
analysis
of
- 152 -
region
"5A"
from
the control
BETH ISRAEL HOSPITAL - IMAGE PROCESSING LABORATORY
IMAGE TITLE: P67R5
COMMENT: Image 5
SUBREGION: 2
LENGTHSCALE: 57.18
GRAY THRESHOLD: 115
SOLID PHASE: BLK
TESTLINE LENGTH: 795.1
TESTLINE SPACING <PIXELS>:
[AREA FRACTION ANALYSIS: '
REGIONI AREA = -----'13.91-"
[STEREOLOGY ANALYSIS: ]
Y
1
AREA FRACTION = 0.402
SOLID
_ . .. AREA
_ ..= _5.586
._ _
ELLIPSE OF MEAN INTRCPT LENGTHS:
A*N12+ BNiN2 + C*N22= MNINCPTLN- 2
A= 10.11
= 2.523
C= 6.1487
CORRELATION COEFF = 0.976
X
MAX MN INTRCPT LNTH = 0.41592
ANGLE MAX ORIENTATION = 106 DEGS
MIN MN INTRCPT LNTH = 0.30900
PERIMETER = 49.750
PERIM/AREA = 3.5777
ANGLE MIN ORIENTATION = 16 DEGS
EXTENT OF ANISOTROPY = 18.683 %
Figure 3.6b: Stereologic analysis
specimen P7.
of
- 153 -
region "5A" from the experimental
Table 3.1
Statistics on Bone Area Fraction Data
Equine Patella
Experimental
Control
;pec
S
Mean
Reg
SD
Mean
SD
Difference
Mean
SD
N
.
-
S1
S2
S3
P6
P7
P8
A
0.473
0.039
0.665
0.035
0.192
0.058
3
B
0.370
0.064
0.527
0.102
0.156
0.037
2
A
0.393
0.024
0.652
0.078
0.259
0.075
7
B
0.576
0.171
0.499
0.145
-0.077
0.052
7
A
0.423
0.086
0.720
0.101
0.298
0.079
7
B
0.500
0.086
0.539
0.125
0.039
0.053
5
A
0.393
0.032
0.744
0.016
0.351
0.039
3
B
0.533
0.180
0.596
0.186
0.063
0.047
4
A
0.500
0.229
0.588
0.222
0.088
0.106
6
B
0.476
0.230
0.494
0.178
0.017
0.091
5
A
0.222
0.040
0.531
0.060
0.309
0.071
5
B
0.235
0.061
0.366
0.050
0.131
0.027
4
- 154 -
different
specimens.
The
data
from
the
control
and experimental
patellae and the algebraic difference between the two for each group are
included.
All groups of data, with
specimen "S2", had, on the
the exception of the "B" regions of
average,
an
increase in the area fraction,
when comparing the experimental patella to
cases, the means of the
changes
In all
in area fraction were greater for the
confirming the above observations of
"A" regions than the "B" regions,
increased bone density in the
the control patella.
vicinity
of
the implants.
The greatest
increase in area fraction occurred in the "A" regions in specimen P6.
points was due to the criterion
The relatively small number of data
that, for the image to
homogeneous in the
be
corresponding
experimental patellae.
the trabecular bone must appear
acceptable,
locations
placement
An example of an image which
the control and
both
had the fewest number of acceptable
Specimen S1
images due to the surgical
on
of
the implant (see Figure 3.2).
was rejected due to inhomogeneity is shown
in Figure 3.7.
Statistical comparisons were
made
to
test
whether the remodeling
response was significant and to examine the possibility of grouping data
from different specimens.
fraction
data
because
These
this
was
reliable, stereologic measurement.
comparisons
the
Also,
were
simplest,
made
and
on the area
therefore
most
the area fraction is directly
related to the trabecular bone density, which is a critical parameter in
determining the mechanical properties (see Section 1.1).
To
prove
that
the
remodeling
response
- 155 -
was
significant, paired
I_
---------·-----C·------I
-- ·---.
-.__--II
-
B3ETH ISRAEL HOSPITAL - IMAGE PROCESSING LABORATORY
IMAGE TITLE: SR3
COMMENT: Image 3
_
[AREA FRACTION ANALYSIS:]
REGION AREA = 13.91
[STEREOLOGY ANALYSIS:]
Y
X
PERIMETER = 26.935
PERIM/AREA = 1.9370
Figure 3.7: Stereologic analysis
specimen S1.
SUBREGION: 2
LENGTHSCALE: 57. 18
GRAY THRESHOLD: 120
SOLID PHASE: BLK
TESTLINE LENGTH: 795.1
TESTLINE SPACING <PIXELS>
1
AREA FRACTION = 0.797
SOLID AREA = 11.09
ELLIPSE OF MEAN INTRCPT LENGTHS:
A*Ni2+ B*N
1 N 2 + CN2 2= MNINCPTLN-2
A = 0.9593
B =-0.3387
C = 0.29719
CORRELATION COEFF = 0.999
MAX MN INTRCPT LNTH = 1.9750
ANGLE MAX ORIENTATION = 76 DEGS
MIN MN INTRCPT LNTH = 0.99994
ANGLE MIN ORIENTATION = 166 DEGS
EXTENT OF ANISOTROPY = 40.640 %
of
region
- 156 -
"3A" from the experimental
-
comparisons were made with
pair
each
patellae and the area fraction from
experimental
from a location on an
the corresponding location on
of the area fraction
consisting
contralateral control patellae.
the
The
null hypothesis, that of an assumption of equal means, could be rejected
less for all of the following groups
at a significance level of 0.05 or
of paired data:
regions;
3) all
1) all
from
data
"B" regions;
S1,
2) all "A"
data from the smooth implant
4) all
S2,
and
8) all "A" regions from P6, P7, and
P7,
9) all "B" regions from P6,
6)
7) all data from the porous
S3;
implant specimens (P6, P7, and P8);
P8;
patellae;
equine
5) all "A" regions from S1, S2, and S3;
specimens (S1, S2, and S3);
all "B" regions from
the
10) all "A" regions from
and P8;
each individual specimen.
The only exceptions were the "B" regions from
specimens S3 and P7
tested
differences
in
when
the
area
bone
There were significant
individually.
fraction,
reflecting
significant
remodeling, in the immediate vicinity
of the implants for all specimens
(the "A" regions), and more distant to
the implants for four of the six
specimens (the "B" regions).
The next objective was
examine
to
from specimens which received similar
variance was performed
on
whether there was significant variation
results of these tests
were
as
A one way analysis of
implants.
change
the
validity of combining data
the
in
bone
area fraction to test
from specimen to specimen.
The
at a significance level of
follows:
0.05, 1) the null hypothesis (that of equal means) could not be rejected
for the "A" regions of specimens S1,
specimens S1, S2, and S3;
for the
"B" regions
of
2)
S2,
and S3, or the "B" regions of
the null hypothesis could not be rejected
specimens
P6,
- 157 -
P7,
and
P8;
3) there was
sufficient evidence to reject the null hypothesis for the "A" regions of
specimens P6, P7, and
P8.
latter
The
since bone ingrowth did not occur
other
two
specimens.
in specimen
Unfortuna tely,
differences between the "A" regions
two were
tested
separately
of
specimen
combining the data from
the
low
implants and also demonstrated that
also
significant
P7 and P8 when these
P6.
This conclusion was
areal densities measured for the
These
three
were
specimens
specimen
P8.
P6 but did occur in the
there
of
from
undoubtedly due to the relatively
control patella
conclusion was not surprising
tests
patellae
provided support for
which received the smooth
the morphologic data from the three
patellae which received the porous coated implants were inconsistent.
The changes
in
bone
area
fraction
for
key
for
the symbols in this figure is
Points
lying
on
area
fraction
graphically in Figure 3.9.
given in Figure 3.8.
difference between
compared to
side.
Only a
the
few
the
The
corresponding
points
lie
points displaying an increase in
difference between
the
the
on
location
below
on
density.
line
the
this
"A" locations
all
patellae
are shown
Y = X exhibited no
experimental side when
the contralateral control
line, with the majority of
This further emphasizes the
(filled
symbols)
and
the "B"
locations.
The principal material orientation,
of directed secants (see Section
as
determined using the method
2.3), was also significantly different
between the control and experimental
patellae (Figure 3.10).
Y = X, indicating no change, and Y =
X
+ 90 and Y = X - 90, indicating
the possible range of data, are included for reference.
- 158 -
The lines
While more data
KEY
S2, and S3, 'A' regions
S2, and S3, 'B' regions
'A' regions
'B' regions
Specimens P7 and P8, 'A' regions
&
A Specimens P7 and P8, 'B' regions
·
o
*
o
Specimens S1,
Specimens Si,
Specimen P6,
Specimen P6,
Figure 3.8: Key for all figures with corresponding symbols.
159 -
Change inMaterial Density
A'.
0
X.
0
/
I
1.0
A
·
0
@0
0
IA
/'7
/
A
0.6+
o
0e
CR
A
A
r-
/
/
A
0.4i
A
A A
OA
/
i
0.2-1
0.1 0.2
-.. {.. . .......... -... .- - -.
0.3
0.4
0.5
-
It - - .
......
0.6
.-.
0.7
--.[
. -. .. ..
0.8
0.9
---
H..
1.0
Area Fraction, Control
Figure 3.9: The bone area fraction in the "A" and "B" regions in the
experimental patellae as a function of the bone area fraction in the
corresponding regions in the control patellae. See Figure 3.8 for Key.
- 160 -
Change inMaterial Orientation
'7n/9uuV- ?
U
.,"
C3
0
A
A
150-
¢
@0
0
0
A
A
a
A
100-
A
00
0
0
*P
A
50
.
I
i/
0
A
A
. - --- 50
50
...
-0
100
- 50
200
150
200
Theta Amax, Control
Figure 3.10: The principal material orientation in the "A" and "B"
regions in the experimental patellae as a function of the principal
material orientation in the control patellae. See Figure 3.8 for Key.
161 -
from the "B" locations lie near
the
line
Y
= X, large changes in the
material orientation were apparent for many locations, both "A" and "B".
This further demonstrates that
significant
and
that
the
the
remodeling
measured
changes
around the implants was
in
the
trabecular bone
morphology were greater for regions adjacent to the implants.
The eccentricity of
the
definition) was different
the control patellae.
3.11.
anisotropy
when
These
comparing
data
While more data points from
X than above the line, indicating
ellipse
(see Equation 3.3 for
the experimental patellae to
are presented graphically in Figure
"A" locations lie below the line Y =
a change toward isotropy, the general
scatter of data points does not allow conclusions to be drawn.
it is possible that material
anisotropy
However,
may be sensitive to parameters
other than the implant proximity which have not yet been correlated.
- 162 -
Change in Material Eccentricity
I 1\
1.
U-
A Co
0.8-
0
CI
o
Oo
U
S
'a 1
Ob 0
O
* It I
AS
0.6;k
-&
4
0
* &
0.4- i
A
0
.
0
0
A
A
·
L,/
I
U. F
0.2
0.3
0.4
0.5
0.6
1
_1
0.7
0.8
I
I
4
0.9
Eccentricity, Control
Figure 3.11: The eccentricity of the material anisotropy ellipse in the
"A" and "B" regions in the experimental patellae as a function of the
eccentricity of the material anisotropy ellipse in the control patellae.
See Figure 3.8 for Key.
- 163 -
3.1.2 Ovine Calcaneus
implantation
The surgical
the
of
coated
cobalt chromium
There were no complications during the 10
cylinders was well tolerated.
The five experimental and five control
to 16 week post-surgical period.
calcanei were embedded
and
Contact radiographs of
mid-sagittal
sectioned
automated stereologic system.
for one of the sixteen
porous
as
sections
in Section 2.3.1.
analyzed using the
were
implant placement was unacceptable
The
so it was excluded from further
specimens
week
described
study.
The cobalt chromium cylinders resulted in significant changes in the
trabecular bone
in the
density
immediate
vicinity
of the implants.
Photographic prints from the contact radiographs for specimens C1 and C4
All four experimental specimens had extensive
are shown in Figure 3.12.
bone ingrowth into the porous
bone formation and
layer
of
within
remodeling
sintered
one
or
two
beads as well as new
millimeters of the
implants.
The trabecular bone
first analyzed to specifically
density of the trabecular bone (Figure
examine the changes in the areal
3.13).
was
morphology
A fine grid was used to provide a large number of data points at
various locations around the implants
implants.
Statistical comparisons were made between the areal densities
of the control specimens
specimens.
and at various distances from the
Paired t tests
and
the
areal
densities of the experimental
were performed to establish the significance
of the differences between the control and experimental sides.
were organized in two different formats.
The data
The first format consisted of
four data sets corresponding to four concentric annular regions with a 2
- 164 -
,
/
Figure 3.12a: Sagittal sections from
(bottom) ovine specimen C1.
the control (top) and experimental
- 165 -
Figure 3.12b: Sagittal sections from
(bottom) ovine specimen C4.
the control (top) and experimental
- 166 -
BETH SRFEL HS;P T,:'t
'
OPTHO'F-'E [ C' 81 iIECHIt
: '!
:E
L
Il'!T:'f;
l."
Titl~E
PRO:]SI
:'7r
'-.
,';'.rE
sTFr;'re:L GI :H!
F jTE ;.'.:Ti i-.'!E : O.. H C' I".t:L:.-l~
;
P
DATE
-:t? R., L.
I
!MO-,.
SlU;
I.
LH1l
[__Hi
-
11:,:".:
-rl ."0
.41I.
;-lI1
Fi0{1!' l -.I
.
.;.' FOR
FOR "32L"
I'3
1~~~~~~~~~~~~~~~~~~~~~
I
!
Figure 3.13a: Morphologic analysis
densities, control specimen C1.
- 167 -
TFi-;,HILC; LipH''
LEhIlTH
i-.C
for
the
LE
LE''EL =
= 0.0743
determination
of areal
-
!R)THtr-'E'l
BETH SF'E.!_ H':i:F'
!I1.
I IC '- ICOtECI-H:: I c .AE-"I
rATlf;:'r
I hREP t'. ', E:. : l l t.Y- .TEr l
:TEPET'L
"FEC,
DATE
t1Fi ;
Zf
J.'G
,.
,
~13"THE
C-O L: i-41FTJ~
rlONt ,'LIG
Ht:iL''-l$ -. 'LMMRY FOP "2R"
THF.:ESWHrJLD PiY LEt' EL = 7
LEN;TH SCHLE
= E0. 743
Figure 3.13b:
Morphologic analysis
densities, experimental specimen C1.
- 168 -
for
the
determination
of areal
mm radial width for each of the regions.
twelve
data
sets
corresponding
The second format consisted of
to
twelve
partitioned regions around the implant.
to test the dependence
from the implant.
of
the
equal
circumferentially
The first format was designed
density
changes on the radial distance
The second format was designed to test the dependence
of the density changes on the relative circumferential location.
The results of
Figure 3.14.
these
All
of
analyses
the
for
specimen
experimental
C1 are summarized in
specimens had a statistically
significant increase in the trabecular bone areal density within 2 mm of
the implants, but no significant changes beyond 2 mm.
circumferentially
partitioned
increase in
the
significant
difference,
trabecular
regions
bone
which
also
displayed
density.
may
be
A majority of the
Some
partially
a
significant
regions
showed no
explained
inclusion of data points which were distant from the implants.
significant decrease in bone areal
in Figure 3.14 of specimen
variation
in
the
the
The only
density occurred in the region shown
C1.
In summary, the changes in trabecular
bone areal density were limited
with some
by
to
a
density
2 mm distance from the implants,
changes
as
a
function
of the
circumferential location.
The stereologic analyses of the
as the areal density were
trabecular bone orientation as well
performed
on the experimental calcanei using
the array of test regions shown in Figure 3.15.
to provide the maximum number
adjacent to the implants.
of
The
control specimens were similarly
data
points from the trabecular bone
diameter
analyzed
- 169 -
This array was designed
of
each region was 4 mm.
with
The
care taken to properly
Distal
Distal
O
.
L_
O
0
O
a)
Q)
a)
a,
C
u)
c,(
C,
O
a.
0
0
a.
Proximal
Proximal
CIRCUMFERENTIAL
RADIAL ANALYSIS
KEY:
U
significant increase
El
Figure 3.14: Statistical analysis of
areal density, ovine specimen C1.
significant
decrease
*
ANALYSIS
implant
the changes in the trabecular bone
- 170 -
. .... .. ..
..
..
...
..
..
..
..
..
..
...
........
···
.
4
= 2.0 mm
z
,-Y
.. .. . .. . .. 1
Figure 3.15: Array of image regions used for the stereologic analyses of
the ovine calcanei.
- 171 -
orient the
test
arrays
for
paired
from
example stereologic analysis
3.16 to 3.19.
All
four
each
specimens
experimental side at this
One representative
comparisons.
specimen
had
are shown in Figures
higher areal density on the
a
in comparison to the control side.
location
It is also apparent that the density changes were primarily limited to a
distance of about 2 mm,
the radius of the analyzed regions.
was
which
This observation was confirmed by the above statistical analyses.
The
important
example
point
analyses
in Figures
the
about
morphologic
C1.
increased
The
to
3.19
changes.
demonstrate
The
an
direction of
virtually unchanged in this region
principal trabecular orientation was
for specimen
3.16
bone
density
was
reflected by an
increase in the bone area fraction but the orientation of the trabeculae
slightly removed from the
implant
remodeling in this region
was
more
In specimen C2 the
unchanged.
extensive
and resulted in a large
direction of principal material orientation.
change (54 degrees) in the
However, the significance
was
of
the
of maximum orientation in
direction
this region must be questioned due to the disorganized appearance of the
mineralized bone.
a large change in the principal
Similarly, there was
trabecular orientation for specimen C3 due to the influence of the dense
bone immediately adjacent to the
was more similar to that of specimen
in the direction of maximum
adjacent to the
trabecular orientation
slightly removed from
direction of
there
due
the
orientation
1C in that there was little change
In summary, in some regions
orientation.
implants,
to
was
little
minimal
implant.
was
This region of specimen 4C
implant.
greatly
- 172 -
remodeling
In
in the principal
change
of
the trabeculae
other regions, the principal
changed,
due
to the increased
RCTL
TCODAI
nCDTTAI
I
/L-
·_
-
IMAGE PROCESSING LABORATORY
IMAGE TITLE: 82C
COMMENT:
SUBREGION: 3
LENGTHSCALE: 34.29
GRAY THRESHOLD: 120
SOLID PHASE: BLK
TESTLINE LENGTH: 427.6
TESTLINE SPACING <PIXELS>
:]
12.47
[STEREOLOGY ANALYSIS:]
V
REGION AREA =
A
ELLIPSE OF MEAN INTRCPT LENGTHS:
A*Nj2+ B*NIN 2 + CN 22-= MNINCPTLN-2
1.900
B -4.388
I
PERIMETER == 33.659
PERIM/AREA = 2.6987
Figure 3.16a:
specimen C1.
1
AREA FRACTION = 0.51±
SOLID AREA = 6.372
A=
-
__
Stereologic
C = 4.3960
CORRELATION COEFF = 0.998
MAX MN INTRCPT LNTH = 1.2659
ANGLE MAX ORIENTATION = 30 DEGS
MIN MN INTRCPT LNTH = 0.41989
ANGLE MIN ORIENTATION = 120 DEGS
EXTENT OF ANISOTROPY = 60.260
analysis
of
173 -
region
"2"
from
the control
___ ·____·__L__ I____·___I____
_ · _·__ ___
____·___ II__ ___ __ __1_
TAL
-
______ _
____
IMAGE PROCESSING LABORATORY
IMAGE TITLE: 82E
COMMENT: 82 exp, porous
[AREA FRACTION ANALYSIS: ]
SUBREGION: 3
LENGTHSCALE: 34.29
GRAY THRESHOLD: i10
SOLID PHASE: BLK
TESTLINE LENGTH: 427.6
TESTLINE SPACING <PIXELS>:
AREA FRACTION = 0.632
REGION AREA = 12.47
SOLID AREA = 7.886
[STEREOLOGY ANALYSIS:]
ELLIPSE OF MEAN INTRCPT LENGTHS:
2 + R*NN, + C*N, 2 = MNTINCPTLN A*N1
. .. I
c
.c.
.......
...
Y
i
2
A = 0.7129
B = -1.541
X
C = 1.4548
CORRELATION COEFF = 0.997
MAX MN INTRCPT LNTH = 2.0906
ANGLE MAX ORIENTATION = 32 DEGS
MIN MN INTHCPT LNTH = 0.71816
ANGLE MIN ORIENTATION = 122 DEGS
EXTENT OF ANISOTROPY = 58.949
PERIMETER = 24.458
PERIM/AREA = 1.9610
Figure 3.16b: Stereologic analysis
specimen C1.
of
- 174 -
region "2" from the experimental
TAL - IMAGE PROCESSING LABORATORY
[AREA FRACTION ANALYSIS:]
IMAGE TITLE: 429C
COMMENT: 429 control, Co Cr
SUBREGION: 3
LENGTHSCALE: 34.29
GRAY THRESHOLD: 135
SOLID PHASE: BLK
TESTLINE LENGTH: 427.6
TESTLINE SPACING <PIXELS>:
AREA FRACTION = 0.551
REGION AREA =
SOLID AREA = 6.878
2.47
[STEREOLOGY ANALYSIS:]
Y
PERIMETER = 28.535
PERIM/AREA = 2.2879
Figure 3.17a:
specimen C2.
Stereologic
ELLIPSE OF MEAN INTRPT LENGTHS:
A*,N, 2 + R*NN + C*N,,
A = 1.544
B = -. 809
C = 2.0834
=
MINTNCPTI-N-
2
CORRELATION COEFF = 0.993
MAX MN INTRCPT LNTH = 1.0724
ANGLE MAX ORIENTATION = 37 DEGS
MIN MN INTHCPT LNTH 0.60219
ANGLE MIN ORIENTATION = 127 DEGS
EXTENT OF ANISOTROPY = 35.098
analysis
of
- 175 -
region
"2"
from
the control
-L-l-·--l------
AL - IMAGE PROCESSING LABORATORY
IMAGE TITLE: 429E
COMMENT: 429 exp, Co Cr
SUBREGION: 3
LENGTHSCALE: 34.29
GRAY THRESHOLD: 120
SOLID PHASE: BLK
TESTLINE LENGTF 427.6
TESTLINE SPACING <PIXELS>:
I__r··________
[AREA FRACTION ANALYSIS:]
-
REGION AREA = 12.47
[STEREOLOGY ANALYSIS:]
Y
x
PERIMETER = 31.437
PERIM/AREA
2.5206
__
I---W----L--·-UI
---
-----
)·I-·
---
AREA FRACTION = 0.604
SOLID AREA = 7.529
ELLIPSE OF MEAN INTREPT LENGTHS:
A*Nt2 + B*NIN2 + CN2 MNINCPTLN - 2
A= i.161
B = 0.8544
C = 2.4652
CORRELATION COEFF = 0.976
MAX MN INTRCPT LNTH = 0.98385
ANGLE MAX ORIENTATION = -17 DEGS
MIN MN INTRCPT LNTH = 0.62105
ANGLE MIN ORIENTATION = 73 DEGS
EXTENT OF ANISOTROPY = 28.440
Figure 3.17b: Stereologic analysis
specimen C2.
of
- 176 -
region "2" from the experimental
--
BETH ISRAEL HOSPITAL
-
IMAGE PROCESSING LABORATORY
IMAGE TITLE: i9iC
COMMENT: 191 control, porous
SUBREGION: 3
LENGTHSCALE: 34.29
GRAY THRESHOLD: 110
SOLID PHASE: BLK
TESTLINE LENGTH: 427.6
TESTLINE SPACING <PIXELS> : 1
amC~
LAREA FRACTION ANALYSIS: J
REGION AREA = 12.47
AREA FRACTION = 0.329
SOLID AREA = 4.098
[STEREOLOGY ANALYSIS:]
Y
ELLIPSE OF MEAN INTRCPT LENGTHS:
A*N
12+ B*NjN 2 + CN22= MNINCPTLN-2
-
A = 3.493
B = 0.7526
C = 6.2461
PERIMETER = 28.253
PERIM/AREA = 2.2653
Figure 3.18a:
specimen C3.
Stereologic
CORRELATION COEFF = 0.982
MAX MN INTRCPT LNTH = 0.53894
ANGLE MAX ORIENTATION = -8 DEGS
MIN MN INTRCPT LNTH = 0.39852
ANGLE MIN ORIENTATION = 82 DEGS
EXTENT OF ANISOTROPY = 18.966 Z
analysis
of
- 177 -
region
"2" from
the control
__
1__1
I
·
'AL - IMAGE PROCESSING LABORATORY
IMAGE TITLE: 19iE
COMMENT: 191 exp, porous
SUBREGION: 3
LENGTHSCALE: 34.29
GRAY THRESHOLD: 90
SOLID PHASE: BLK
TESTLINE LENGTH: 427.6
TESTLINE SPACING <PIXELS>
1
[AREA FRACTION ANALYSIS:]
AREA FRACTION = 0.580
_·
CII-·-)--·--·---lC--·l-----··----
REGION AREA = 12.47
SOLID AREA = 7.234
[STEREOLOGY ANALYSIS:]
Y
ELLIPSE OF MEAN INTRCPT LENGTHS:
A*Nj 2+ BNiN2 CMN22= MNINCPTLN-2
-
A = 0.9682
B = 0.7675
C = 1.5032
CORRELATION COEFF = 0.948
MAX MN INTRCPT LNTH = .141ii
ANGLE MAX ORIENTATION = -28 DEGS
MIN MN INTRCPT LNTH = 0.76618
ANGLE MIN ORIENTATION = 62 DEGS
EXTENT OF ANISOTROPY = 24.828
PERIMETER = 24.992
PERIM/AREA = 2.0038
Figure 3.18b: Stereologic analysis
specimen C3.
of
- 178 -
region "2" from the experimental
___1_____
111
_1__
__ 1___1
_______ ___ _____ _____I^_·__I_______
I_____· ___________I
_rl__l_____·_l_
I_ __
AL - IMAGE PROCESSING LABORATORY
IMAGE TITLE: 4C
COMMENT: 4 control, porous Co Cr
SUBREGION: 3
LENGTHSCALE: 34.29
GRAY THRESHOLD: 130
SOLID PHASE: BLK
TESTLINE LENGTH: 427.6
TESTLINE SPACING <PIXELS>: t
_.__._._......
.
..
lAREA FRACTION ANALYSIS: J
REGION AREA = 12.47
AREA FRACTION = 0.257
SOLID AREA = 3.210
STEREOLOGY ANALYSIS:]
ELLIPSE OF MEAN INTRCPT LENGTHS:
A*N 2 + R*NN,
Y
......
]
+ C*N. = MNTNCPTIN - 2
.......
..................
A = 2.827
B = -2.574
X
C = 3.9218
CORRELATION COEFF = 0.994
MAX MN INTRCPT LNTH = 0.7142
ANGLE MAX ORIENTATION = 33 DEGS
MIN MN INTRCPT LNTH = 0.45772
PERIMETER PERIM/AREA
Figure 3.19a:
specimen C4.
18.311
1.468i
Stereologic
ANGLE MIN ORIENTATION = 123 DEGS
EXTENT OF ANISOTROPY = 27.314
analysis
of
- 179 -
region
"2" from
the control
___
__·
____I ·_L__·___
_
___
TAL - IMAGE PROCESSING LABORATORY
IMAGE TITLE: 4E
COMMENT: 4 exp, porous Co Cr
SUBREGION: 3
LENGTHSCALE: 34.29
GRAY THRESHOLD: 120
SOLID PHASE: BLK
TESTLINE LENGTR 427.6
TESTLINE SPACING <PIXELS>: 1
[AREA FRACTION ANALYSIS:]
AREA FRACTION = 0.406
REGION AREA = 12.47
SOLID AREA
5.067
[STEREOLOGY ANALYSIS:]
ELLIPSE OF MEAN INTRCPT LENGTHS:
A*NA 2 + B*N4N
-... " ..
Y
-2
+ C*N 2=- MNINCPTLN
.........
...
A = 0.9312
B =-O.4132
C = 1.4201
X
CORRELATION COEFF = 0.959
MAX MN INTRCPT LNTH = .08ii
ANGLE MAX ORIENTATION = 20 DEGS
MIN MN INTIHCP
LNIH = 0.81766
PERIMETER = i7.173
ANGLE MIN ORIENTATION = 1iO DEGS
PERIM/AREA
EXTENT OF ANISOTROPY = 17.586
t.3769
Figure 3.19b: Stereologic analysis
specimen C4.
of
- 180 -
region "2" from the experimental
trabecular bone density immediately adjacent to the implants.
The changes in trabecular
calcanei are shown in Figure
reference.
Points above this
corresponding region of
twenty-two points lie
areal
3.20.
The
line
the
above
bone density adjacent to
the
higher areal density in the
to
the
density
areal
specimen.
in the
Twenty-one of
confirming that the trabecular
line
implants
Y = X was included for
line
control
paired
this
density for all experimental
a
had
in comparison
specimen
experimental
bone
was increased regardless of the
relative location around the implants.
The changes in the principal direction of trabecular orientation are
shown in Figure 3.21.
The lines Y
= X, indicating no change, and Y = X
+ 90 and Y = X - 90, indicating the possible range of data, are included
for reference.
Approximately
degrees of the line Y = X
in the direction
outside
of
this
adjacent to the
of
half
of
the
data
points lie within 10
indicating that there was only a small change
orientation.
range
indicating
implants
resulted
The
remainder
that
in
direction of orientation.
- 181 -
the
large
dense
of
the points lie
bone immediately
changes in the principal
Change in Material Density
1.
0
0.
0
0
0
0.
0
0
13~
0
[
0
a
O
0.
I
;
03
0.
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
Area Fraction, Control
Figure 3.20: The bone area fraction in the experimental calcanei as a
function of the bone area fraction in the corresponding regions in the
control calcanei.
- 182 -
Change inMaterial Orientation
O
O
O
D
[]
0
03
0
[o
[
-40
-20
0
20
40
60
Theta Amax, Control
Figure 3.21: The principal material orientation in the experimental
calcanei as a function of the principal material orientation in the
control calcanei.
- 183 -
3.2 Structural Analyses
The
in
stresses
control
the
predicted using finite
element
continuum models were
analyzed
The stresses in
the
and
experimental
using a displacement-based formulation.
were
characterized to provide
stresses in the experimental specimens.
reference for evaluation of the
The stresses in the trabecular
were
Two- and three-dimensional
analysis.
specimens
control
specimens
bone
surrounding the implants were also
compared between specimens to characterize the influence of the boundary
conditions at the bone/implant interface.
3.2.1 Equine Patella
The assumed boundary conditions
the bone/implant interface were
at
the experimental patellae.
based on the histology of
Nonlinear contact
was assumed for the
smooth
surfaced
implants based on the observation
of a
tissue
membrane
separating
thin
fibrous
For
trabecular bone.
two
of
implants, specimens P7 and P8,
the porous coating.
assumed for
the
For
surrounded by bone.
implant from the
patellae with porous coated
was successful bone ingrowth into
patellae, rigid mechanical coupling was
interface.
tissue
For this
three
there
these
bone-implant
displayed a thick fibrous
the
the
layer
Porous
coated specimen P6
with the specimen incompletely
specimen, it was assumed that the implant
acted as a void, with no mechanical coupling between the implant and the
adjacent trabecular bone.
The stress fields in the trabecular bone of the equine patellae were
- 184 -
altered by the presence
of
the
preliminary studies of the
stainless
cylindrical
are
conditions at
interface
bone/implant
implants.
As in the
implants in the ovine calcanei,
the stresses around the implants
the
steel
dependent on the assumed boundary
(see
Section
2.2.4).
In
general, tight mechanical bonding, as occurs with bone ingrowth, results
in stress fields which are
implant.
most
Nonlinear contact,
porous coated
implants
altered stress fields.
any implant in
which
which
prior
The
the
adjacent bone by a thick,
similar
is expected for smooth implants or
to
bony
ingrowth,
results
in greatly
stress
fields are also greatly altered for
implant
is mechanically uncoupled from the
compliant,
compliance of the tissue
to the normal condition of no
layer
not measured, it can certainly
fibrous
formed
tissue layer.
While the
in experimental specimen P6 was
be considered compliant when compared to
the trabecular bone.
The finite element
only the stresses
models
in
the
presentation and in the
relationships.
section
The
through
stereology.
The
plane
fully three-dimensional.
of
interest
However,
were considered in this
subsequent examination of the stress-morphology
plane
the
were
of
interest
implant
centers
two-dimensional
corresponded
which
stresses
were
neglecting all out-of-plane stress components.
projection of the three-dimensional
stress
analyzed
obtained
using
by simply
This corresponds to the
ellipsoid onto the plane of
interest yielding the familiar two-dimensional
a reasonable approach provided the
were
to the sagittal
stress ellipse.
This is
primary and secondary principal axes
of the ellipsoid approximately coincide with
provide a measure of the appropriateness
- 185 -
the plane of interest.
To
of the plane on which the data
were examined, the strain energy density was calculated using all of the
stress
and
components.
The average
of
ratio
energy was accounted
for
This ratio varied from model
in-plane
the
words, 80 % of the strain
other
ranging from a maximum of 0.93
model,
to a minimum of 0.62 for the
S2
for the experimental model of specimen
the in-plane stresses.
only
considering
to
only
strain energy to the
in-plane
the
In
0.80.
by
using
the locations of morphologic data for
total strain energy, evaluated at
all of the specimens, was
also
and
components
strain
control model of specimen P8.
The stresses were displayed graphically to examine the stress fields
in the control and experimental patellae (Figures 3.22 to 3.29).
the principal tensile and compressive stress
plots were used to display
components.
Each vector
Vector
direction
corresponds to the principal stress
direction and the vector length is proportional to the stress magnitude.
Contour plots were used
Cross-bars on the vectors indicate compression.
to display the
distribution
effective stress.
sagittal
section
corresponded to
system.
of
strain
density
energy
and von Mises
Two views were used for each model, the first being a
through
the
plane
the
center
was
which
of
the
This plane
implant.
analyzed
using
the stereology
The second view was an enlargment of the implant region and the
surrounding trabecular bone on the same section.
Plots from three representative model pairs are included.
pair of models was for specimen
experimental side with the
S2,
smooth
specimen P6, again including the
The first
including the control side and the
implant.
The
second pair was for
control side and the experimental side
- 186 -
with the porous coated implant
was
for
specimen
P7,
which
again
was
including
experimental side with the porous
not ingrown.
The third pair
the
side
control
and
the
coated implant with demonstrated bone
ingrowth.
In general, the stresses
each of the specimens.
on
On first examination, the stresses on the entire
sagittal section for the three
greatly.
program
stress vector for the
vector.
pointed out that an unavoidable aspect of
For
specimen
in
that
in
high
portions of
the
mesh
magnitude compressive stresses
3.22a).
corresponding vectors for the
by
was scaled to a specific
patellofemoral contact (lower right region of
Figure
solved
plot
were scaled relative to the maximum
very
vectors over the remainder of the
was partially
maximum absolute principal
the
the
vectors
S2,
occurred in the region of
the cortical shell
was
elements
length and the rest of the
(Figure 3.22) appears to vary
specimens
However, it must be
the vector plotting
control patellae were similar for
the
As
a consequence, the stress
section appear to be smaller than the
other
using
patellae.
enlarged
restricted
to
the
This technical problem
views
which include smaller
areas
of
interest.
This
eliminated from the plots high peak stresses which were not of interest.
The highest
stresses
patellae occurred in the
patellofemoral contact.
were the result of the
peak contact forces.
was probably poor.
for
both
cortical
The
high
implant
shell,
articular
control
and the experimental
especially
compressive
placement
In addition,
The
the
in the region of
stresses in specimen S2
coinciding with the plane of
the accuracy of the contact stresses
cartilage was not represented and an
- 187 -
MAX
MIN
.. ._
.
39.5
-56.
_-~
·. ·
.
_,
=
patellofemoral4 . contact . area'':+ is in
the lower
right.
:'
..
'...
.
· 1-
.
· '~r
i·
i
''X"--
~'
.
~I
p'I
control specimen
(MPa).
direction
and the S2
positive
~
·
·"
*,
.-.
The points
positivein Xtheaxis
pointsdirection.
in the distal
Y axis
anterior
The
188 -
58. 7
MAX
MIN = -45.9
. .
.-
' ¥
Il
.-
...*.',....
....
*
4
4.
·
·
'.'
I'·
control specimen P6 (MPa).-
Figure 3.22b: Principal stress
control specimen 6 (MPa),
vectors
- 189 -
on
a
agittal section of the
MAX = 36.5
MIN = -35.7
~~~~
~
C
5..
.
4.
10-1
0*
r
*
1~~~~.
*1r
*.
.
'V*.
.C
.4
*
.
By
-x
Figure 3.22c: Principal stress
control specimen P7 (MPa).
vectors
-
190 -
on
a
sagittal section of the
accurate
representation
of
nonlinear
formulation,
such
interface.
are not
surface
as
Also, in general,
reliable.
necessary to use
To
that
loads
a
generally
applied
stresses
represent
nodal
contact
are
interpolation functions (Bathe 1982).
the
a
bone/implant
adjacent to concentrated loads
surface
which
at
requires
traction
consistent
exactly it is
with the element
In the present models, consistent
loads were approximated by simply distributing each load evenly over the
surface nodes which corresponded
to
the patellofemoral contact area or
the attachment area of the ligament or muscle.
net effect but a consequence was
that
applied loads, namely the cortical
stresses in the trabecular bone,
Each load had the proper
the stresses in the areas of the
shell,
away
were not very accurate.
The
from the applied loads, were not
affected by this approximation.
High tensile stresses were seen
proximal and distal poles of all
to the action of the
anterior cortical
These shear
anterior region and at the
the specimens.
High
compressive
shell,
stresses
the
These stresses are due
quadriceps musculature, proximally, and the middle
patellar ligament, distally.
coincident tensile and
in
are
shear
principal
especially
due
stresses, indicated by high
the
stresses,
in specimen
component
of
P7
the
appear in the
(Figure 3.22c).
load which was
tangential to the surface.
The principal stresses in the
correspond to the
consistent from
implant
specimen
control
location
to
in
specimen
general flow of compression from
patellae in the region which
the experimental patellae were
(Figures
3.23).
There was a
the patellofemoral contact area to the
- 191 -
e
.·e·
:-
--
:--Ap.
'.
to
a- o-
*
k
··(I
.
.. ¥
.o,
.
MAX = 9.80
MIN = -5.20
-
...
~'..3...
:
.5e.
4
'.-.
Y
-x
Figure 3.23a: Principal stress vectors on an enlarged view of control
specimen S2 (MPa). The mesh region corresponds to the implant location
in the experimental specimen.
- 192 -
3 5.02
= -3.91
Y
'p.
k-rx
Figure 3.23b: Principal stress
specimen P6 (MPa).
vectors
- 193 -
on
an enlarged view of control
MAX = 7. 09
MIN = -4.46
,*.L----
,?.-1-
S,
.
*
XI
.I~
Ii. ,
Z
+
.. 1-4
~
-V.;~
*
-
b'~~~~~~~~~~~~~~~~~~~~~~~~~~~*
1,~~''
*
i+
xN
4.
.3~
.-'
4
.4,
.5
A.;, .
JrI
Y
,;.
Figure 3.23c: Principal stress
specimen P7 (MPa).
vectors
- 194 -
on
an enlarged view of control
anterior-proximal cortical bone.
in
the anterior region
a
highest tensile stresses were in
The
proximal-distal
stresses were less apparent for
These tensile
orientation.
S2 (Figure 3.23a) because the
specimen
implant in this specimen was deeper to the anterior surface and thus the
region shown is further removed from
tension
combination of biaxial
and
the anterior cortical shell.
This
compression results in significant
in specimen P7.
shear stresses anteriorly, seen especially
Specimen P7
had the most anterior implant placement of the three shown.
To examine the distribution of shear stress in the control patellae,
contour plots of
von
Mises
effective
Von
implant region (Figure 3.24).
stress
Mises
generated for the
were
effective stress was chosen
because this stress parameter is often applied as a failure criterion to
engineering materials.
Von
Mises
stress
the shear stress
reflects
because von Mises stress is linearly related to octahedral shear stress.
The results were comparable for the
there was a gradient
least
pronounced
placement of
the
in
from
posterior
specimen
implant
S2
in the
specimens had a von Mises stress of
region shown.
In the
three specimens shown.
most
to
anterior.
again
due
experimental
to
In general,
This gradient was
the
more posterior
patella.
All three
about 1.5 MPa in the most posterior
trabecular bone the predicted von
anterior
Mises stress was about 3.0 MPa for specimen S2, 5.8 MPa for specimen P7,
and 4.8 MPa for specimen P6.
specimen P7.
This location
The implant was placed most anteriorly in
corresponds to a point immediately anterior
to the implant in the experimental patellae.
Note that the stress
magnitudes
are
- 195 -
important in a relative sense
.·L-
*f
o.*\\
V
B
.
1. .'
.
·
.·
I
.
j
. ..
.
. i··.
.
B ..
*
7
Lu
9'.1-:--/
.
,
I.'
i.
.1
Z._
. .'I.
·o
Y
k-x
Figure 3.24a: Von Mises stress
specimen S2 (MPa).
contours
- 196 -
MAX7.61
H- 6.84
G- 6.08
F= 5.32
E= 4.56
D= 3.79
C= 3.03
B- 2.27
A- 1.51
MIN. 744
on an enlarged view of control
'C.
.. ·.'' '
..
::·- .-.
''·.
.
>
\
'..
. -'-G6
. .':
*--.A .i- ~~~~X
Figure 3.24b: Von Mises stress
specimen P6 (MPa).
-
contours
197 -
MAX6.78
H- 6. i3
5.48
F- 4.83
E 4.18
0- 3.53
*C- -~~~
2.88
B- 2.23
A- 1.58
MIN. 927
on an enlarged view of control
Z
.,
. .
I
.B>
.
.
\
.
...1
" A,,,... -,%'
...%
.
A-
t
A
MAXiO.2
H= 9. 13
G= 8.04
F= 6.94
1_1
y
i--X
to...
'.."
Figure 3.24c: Von Mises stress
specimen P7 (MPa).
:.
4
D= 4.74
C= 3.64
B= 2.54
A= 1.45
MIN.348
contours
- 198 -
on an enlarged view of control
only.
The total load
due
was set equal to 1000
N.
loads to absolute in
vivo
to
the quadriceps musculature on each model
No
attempt
was made to relate the applied
since
these data were not available.
loads
Throughout this study, the stress
stresses
were
compared
data
between
experimental patellae and
thus
were
used in a relative sense;
specimens
the
or
importance
between
control
and
of absolute magnitudes
must be minimized.
Contour plots of the strain
are shown in Figure 3.25.
The
von Mises effective stress,
density was more constant
local gradients occurred
energy
distributions were similar to those for
with
the
in the
at
density in the control patellae
exception that the strain energy
posterior region and several extreme
element
consequence of the discontinuous
boundaries.
material
This
was a direct
property distribution.
This
point is examined further in Section 3.3.2.
For the
most
part,
the
experimental patellae were
(Figures 3.26).
stresses
similar
The presence of
in
to
the
those
than one diameter from the implant.
shell of the
in the control patellae
any region of the patellae more
Stress changes in the vicinity of
the implants were apparent, especially
covering
cortical
implants could not be expected to
significantly influence the stresses in
that there was no bone
the
for
specimens
P6 and P7.
Note
the implant anteriorly in specimen P6,
seen in Figure 3.26b.
The enlarged views
significant stress
of
changes
the
implant
in the
-
region
trabecular
199 -
(Figure 3.27) revealed
bone
adjacent
to the
.
.·
.
·
·
·
.'.
.
...
~~~~~~~~~·
o
''..·
·
°''
I~~~~~~~~~
..
*.
·.
',
A.
o'
::
..
.'
-
·
-.
~~~\
·
.'
,
·
r
'..
r:~~~~~~~~~°
:::
----..
MAX.269E-1
H= .240E-i
G= .2iOE-1
F- .181iE-1
E- .151E-i
D0 .122E-1
.::::
Y
Lx
Figure 3.25a: Strain energy
control specimen S2 (MPa).
C= .924E-2
B- .630E-2
A- .335E-2
MIN. 409E-3
density
- 200 -
contours
on
an enlarged view of
j
.
.
.
.
.
*.
y
.-"
.
Figure 3.25b: Strain energy
control specimen P6 (MPa).
MAX.ii5E-i
.-.
H- . .102E-i
G- .897E-2
F- .773E-2
E- .648E-2
0- .524E-2
C= .400E-2
B- .275E-2
A- .i5iE-2
MIN.263E-3
,."
.
-X
..°I
."
density
- 201 -
contours
on
an enlarged view of
D~~~
--
*-A
.
·.
..
.·
\
-£..
-f.
.
..
\
.
8
..-.
1'.
MAX. 133E-1
H= .118E-1
G-= .103E-i
F= .886E-2
E- .739E-2
D= .593E-2
C= .446E-2
B= .300E-2
A= .153E-2
MIN .675E-4
Y
-
x
Figure 3.25c: Strain energy
control specimen P7 (MPa).
density
- 202 -
contours
on
an enlarged view of
MAX =
MIN
-·~=-·
.
*
..
,~
'.* '
..
.
- _'
.-·'.~ '.
.'qk.
-.*.
. _..
;.'
.
I
" r
'..,,
-........ 4.
vs~~~~V
S
-.
I
4
A
.
exprimna
"-'~
'","
t· ·
-
··~ ~..
.
.
a
.
· 2*
". '
'..
-.
,
""
.
~'
;
-~C:. ·
.
.
4
'..
'*"
.'
...
.'
's~
~~~~~.
.' ''
''~.
""""
·
r
-54.
-
,
l.
39.4
'
.
'tr
,.
.~
.
.\
Y
Figure 3.26a: Principal stress vectors
experimental specimen S2 (MPa).
- 203 -
on
a
sagittal section of the
r
MAXMIN
6C).3
- -4 14.3
rOXT.~~~-
C 1
Y"...
-,.
.. '
k
·
Figure 3.26b: Principal stress
experimental specimen P6 (MPa).
vectors
- 204 -
on
a
sagittal section of the
MAX = 36. 0
MIN = -36.4
1P.
.·
a
*,
jr.*. . .ip*'*~'
;*
I
-
.
.
r~
*t
*
*.
~
E2
*
4q*
4
'4r1
.u
~ ~
.1**
4
"p'
" "
..
'. '
?-.
j
4
*
*.."
'
*
,r
.I'~~~~~~~~
.*
!i
.
"1.t
~"'*
.
'
.
.
'"i.
i..
.
..~
.
Y
X
Figure 3.26c: Principal stress
experimental specimen P7 (MPa).
s
vectors
- 205 -
on
a
sagittal section of the
The stresses were altered when comparing the
implant for all specimens.
and the stresses in the
control patellae to the experimental
patellae,
experimental patellae varied greatly
between the different specimens as
bone/implant boundary conditions.
a function of the assumed
stress fields, the principal and
the
on
the influence of the implants
To examine
effective stresses in each experimental patella were contrasted to those
in the contralateral control.
The
implant in specimen S2
which
(Figure
orientation and
for
expected
the interface
were
The
The
orientation
degrees to
the
interface,
which correspond
to
both
principal tensile stresses
these
were
frictional
the
significant principal compressive
with
of
With
section, the tensile stresses at
this
on
contact.
frictionless
indicating
there
in
although the tensile stresses were
orientation was approximately tangential
result indicates that
altered
were
implant at the bone/implant interface as
associated
stresses.
surrounding the
in the other two patellae
implants,
nonlinear
exception of one nodal point
as
3.27a),
implants,
not oriented tangential to the
be
bone
trabecular
magnitude.
absolute
tended to flow around the
might
the
spherical
smooth
received
in
stresses
principal
shear
forces
were approximately 45
stresses
the
that
to
maximum shear stress
the implant surface.
stresses
between
This
at the bone/implant
the
implant
and the
adjacent trabecular bone.
The technique used to
namely three mutually
represent
perpendicular
represent friction exactly.
the nonlinear boundary conditions,
nonlinear
truss elements does not
A specific relationship between the surface
- 206 -
4AX -
4IN
~~k-;dii1-
12.0
-9.09
*°--A-o
.k.
.-%.
I·)c.
t·.j''
*
.
kL
*@s
---
-:.
A+
1'7.
At.;'
Tr.-:
.
I
-
.
Y
Figure 3.27a: Principal stress
experimental specimen S2 (MPa).
vectors
- 207 -
on
an
enlarged
view
of
MAX = 28.4
MIN = -14.8
·'
5
V
#)
.
'
.J: . T
If
d
*.
.
.
.
.
..
.
.A
.
.
4-:.O
.
.
.
'. A"...
&
&
.
A
.,,.
Ile
,
'.
.
..
.1
'
'(
¥;
2
4
.4;
Y
Z--X
Principal stress
Figure 3.27b:
experimental specimen P6 (MPa).
vectors
- 208 -
on
an
enlarged
view
of
--
Cs}
-
' -8.76
-4
:1
--'
.'
.,'
'.
.
.
.
4..4
8
IO.-
k-
a..
-
k·t.
.-i;
*56;
A4,
.
*
A
.4'
4I,
*
.
.
;
..
Y
k-x
Figure 3.27c: Principal stress
experimental specimen P7 (MPa).
vectors
209 -
on
an
enlarged
view
of
normal stress and the
was not maintained.
surface
shear
A polished
implant surface surrounded by a fibrous
tissue layer probably has a very
low coefficient of friction.
the absence of surface friction at
effect.
stress, such as Coulomb friction,
The smooth implants were
the
of the
at the bone/implant interface was
shear stresses in the trabecular bone
There were significant
control
patellae
in
implant would only have a local
entirely surrounded by bone, were not
loaded directly, and thus micromotion
unlikely.
areas
which
corresponded
stresses seen in the experimental
patellae.
had a stress concentrating effect
in
due to the presence of
However,
the
to
the shear
As expected, the implants
implant vicinity.
The error
shear stresses on the stress-morphology
surface
comparisons was probably small since the stress data were taken from the
which were analyzed with the stereology
centers of the circular regions
system.
This corresponded to a distance of 2.1 mm from the bone/implant
interface for the "A" regions and 6.3 mm from this interface for the "B"
regions (see Figure 3.3).
The contour plot of von
of shear stress at
3.28a).
the
bone/implant
The peak shear
the interface.
Mises effective stress confirms the finding
stress
However, with
occurred
the
surface
increase in von Mises
friction
stress
to
stress (Figure 3.29a).
The
of that single point, the
the implant were not excessive,
does
seen
The plot of strain energy density
at the most anterior point of
exception
von Mises stress gradients adjacent
suggesting that
interface for specimen S2 (Figure
not
account
for
the general
in the surrounding trabecular bone.
is similar
strain
energy
to the plot of von Mises
density contour plots are
included here primarily for later reference (see Section 3.3.2).
- 210 -
t>
..
.C- C-
B.
..
o,
c'I
MAX ii.
H- 9.97
G= 8.84
F- 7.70
=
Y
;
F;7
D== 5.43
C 4.30
B= 3.17
A- 2.03
MIN .898
k-x
Figure 3.28a: Von Mises stress
experimental specimen S2 (MPa).
contours
- 211 -
on
an
enlarged
view
of
.\i'*
4.
'II
1'
..
.'I
'",1,. .
Y\..
/
E"
C
,
. .
t
".,, 3.
.-1---)I
*
.
TI-
A-~-~
MAX 13. 5
H= 12.1
G= 10.8
F= 9.52
E- 8.20
D= 6.88
C= 5.57
Y
O-X
B= 4.25
A= 2.94
MIN 1. 62
Figure 3.28b: Von Mises stress
experimental specimen P6 (MPa).
contours
212 -
on
an
enlarged
view
of
,.,
..
i:4C*e/\
*-.j---<:
...-
7-
.,'
·
11
·
MAX17.0
F= 14.7
E- 12.4
D= 10.0
C= 7.69
B 5.36
A= 3.03
MIN .704
Y
kx
Figure 3.28c: Von Mises stress
experimental specimen P7 (MPa).
contours
- 213 -
on
an
enlarged
view
of
..
/
MAX. i40E-1
H- .i25E-i
G- .109E-I
F- .939E-2
E- .784E-2
Do .629E-2
C- .474E-2
B- .319E-2
A- . 64E-2
Y
kx
MIN. 907E-4
Figure 3.29a: Strain energy density
experimental specimen S2 (MPa).
- 214 -
contours
on
an enlarged view of
··
:I
.r
··
.·
''
A
Y;:
@}Z
.
e·*
L
.
..
h
A '- A,- "
-
-
.
.
*
S
.
*
.
·
·
e
..
.
-
.
.·'
*
'·
..
MAX.
.
HG'
F*
-X
ES .iO7E-i
.
..
. . .867E-2
C- .664E-2
B- .461E-2
A- .258E-2
MIN. 549E-3
. .
Figure 3.29b: Strain energy density
experimental specimen P6 (MPa).
- 215 -
IBE-1
.16BE-i
.148E-i
.i27E-i
contours
on
an enlarged view of
...
.
'.
.............
.
'·
@*
:;-
c
.''''
'.
_
;.
~
~S .AX
Ee^Jess
*
A~~v-e
·
o
.
·
'
MAX .245E-1
F-= .210E-1
E= .175E-I
Y
D= 140E-1
C .i05E-1
B= .703E-2
-x
A- .354E--2
MIN .542E-4
contours
Figure 3.29c: Strain energy density
experimental specimen P7 (MPa).
- 216 -
on
an enlarged view of
The principal stresses around the
implant
in specimen P6 were also
significantly altered when compared to the contralateral control (Figure
3.27b).
Unfortunately it is difficult to compare the vector magnitudes
to the problem with vector scaling
between Figures 3.23b and 3.27b due
mentioned above.
The peak stresses
on the section of the mesh included
in Figure 3.27b occurred on
the opposite faces of the three-dimensional
elements, parallel to those
shown,
stresses on
the
sections
either direction from the
at
a different Z coordinate.
corresponding
to
section shown
are
The
element thickness in
one
due to the flow of
high
stress around the implant region
in the medial and lateral directions.
Recall that the implant for this
specimen is represented as a void with
no mechanical coupling between the trabecular
can be seen in Figure 3.27b
bone and the implant.
It
that the principal stresses around the void
were directed tangential to the surface, as expected.
The stress magnitudes
are
plots of von Mises effective
more
easily
stress
and
compared using the contour
strain energy density for the
control (Figures 3.24b and 3.25b) and the experimental patellae (Figures
3.28b and 3.29b).
trabecular bone
There
was
surrounding
an
increase
the
implant.
bone/implant interface, indicated by the
the
plane
in the plane tangential
to
stress could
been oriented
not
vectors would be
have
apparent
which
were
in von Mises stress in the
The
shear
stress at the
von Mises stress contours, was
of
this
otherwise
not
interface.
or
The shear
principal stress
oriented tangential to the
bone/implant interface (Figure 3.27b).
The principal stresses
around
the
217 -
ingrown, porous coated implants
were generally less altered
than
those
3.27c).
seen
at
High stresses were
which were near the
coated
implants
requirements.
anterior
were
not
However,
cortical
this
by
the
(Figure 3.28c) and
energy
exception of these locations,
to the adjacent trabecular
had
due
to manufacturing
concentrating
plots
density
effect
was well
of von Mises stress
(Figure
3.29c).
With the
mechanical coupling of the implant
resulted
the
Recall that the porous
spherical
contour
rigid
bone
sharp corners of the implant
shell.
stress
localized, as demonstrated
effect, but in general,
the
perfectly
strain
of the other specimens (Figure
in some stress concentrating
least
impact
on the stresses in the
concentrating
effect
of
surrounding areas.
To compare the stress
function of the bone/implant
was calculated at a
boundary
particular
the implants, as a
conditions, the von Mises stress
location at the bone/implant interface.
It was not possible to strictly define a stress concentration factor for
these
models
patellae.
due
to
the
The stress at
interface was used
nonuniform
the
since
most
the
stress
fields
in
posterior point on the bone/implant
stress
fields
at
this location in the
control patellae were most similar from specimen to specimen.
effective stress was used, rather
usually done
in
defining
stress
complex multiaxial nature of
ratio of the von Mises
the
stress
in
Mises stress in the control patella
specimen P6, and
1.54
concentration factor for
for
a
than
Von Mises
a single stress component as is
concentration
stress
the
fields
factors,
due
to the
in the patellae.
The
experimental patella to the von
was
specimen
rigid
the control
2.04 for specimen S2, 4.84 for
P7.
spherical
- 218 -
For comparison, the stress
inclusion
in an infinite
member under uniaxial
tension,
assuming
perfect bonding, is (Peterson
1974):
2
K
For v = 0.2, Kt = 2.0.
ax
t
a
1
+
1+v
(3.1)
4-5v
For a spherical void under similar conditions,
the stress concentration factor is (Peterson 1974):
27-15v
Kt
(3.2)
14-10v
Again, for v = 0.2,
Kt
=
2.0.
Compared to these values, the stress
ratio of 4.84 for specimen P6 appears high.
This is probably due to the
fact that the bone does not surround the implant (void) in this specimen
(see Figure 3.26b).
The stress
ratios
for specimens S2 and P7 compare
well with that for a rigid spherical inclusion.
The changes in
the
examined by plotting
function
of
the
stress
the
fields
stresses
stresses
described
in
the
above may be further
experimental patellae as a
at
the
corresponding
contralateral control patellae.
The
changes
locations
in
the
in strain energy density
for all six specimens are presented in
Figure
included to indicate those data points
for which there was no change in
this quantity.
locations
of
The
the
locations
available
of
the
3.30.
The line Y = X is
data points corresponded to the
morphologic
data
from
the
stereologic
analyses. In general, the strain energy density was more greatly altered
in the regions close to the
implant (the "A" regions, shown with filled
- 219 -
Change in Strain Energy Density
0.0157
OX
I7
0.01
,f-
0.00
A.
·
j
7Iy/
X
0. 00
0.000
.
0.002
- -I
t
0.004
0.006
_ _
--
0.008
0.010
0.012
0.014
SED, Control
Figure 3.30: The strain energy density in the "A" and "B" regions in the
experimental patellae as a function of the strain energy density in the
corresponding regions in the control patellae. See Figure 3.8 for Key.
- 220 -
symbols).
Also, more
data
lie
points
above
the line than below the
line, indicating higher stresses in the experimental patellae.
The means and
standard
density are presented in
of
deviations
Table
3.2.
the predicted strain energy
higher average strain energy
A
density was predicted for all experimental patellae when compared to the
control patellae.
However, the
coefficient
mean)
the standard deviation over the
was
of variation (the ratio of
much greater for these data
This demonstrates
than for the bone area fraction data (see Table 3.1).
The above
the wide variation of stress conditions
around the implants.
contour plots of von Mises
and
strain energy density show this
changes
in bone area fraction were much
graphically.
stress
In contrast, the
more consistent, as reflected
by
the
The specimen-to-specimen variation
lower coefficients of variation.
in
strain
two specimens with ingrown porous coated
inconsistent location
proximity
of
the
of
the
implants
implants.
As
implants.
to
the
anterior
influenced the stresses surrounding the
density was also
with smooth implants and the
specimens
great, when comparing the three
energy
This was due to the
demonstrated above, the
cortical
implants.
shell greatly
Also, the number and
location of usable images varied, depending on the specimen geometry and
morphology.
Recall that
the
stress
data
presented in Table 3.2 were
taken from the locations of available morphologic data.
The changes in
Figure 3.31.
the
The angle
principal
of
stress
orientation
orientation are presented in
in this graph was that of the
principal stress which was maximum in absolute magnitude.
X + 90 and Y = X
-
90
are
The lines Y =
included to indicate the possible range of
- 221 -
Table 3.2
Statistics on Strain Energy Density
Equine Patella
[kPa]
Experimental
Control
S;pec
Reg
Mean
SD
Mean
SD
Difference
Mean
SD
-
S1
S2
S3
P6
P7
P8
A
0.736
0.074
0.960
0.720
0.224
0.657
B
1.113
0.440
1.213
0.634
0.100
0.194
A
2.231
1.147
2.889
1.312
0.658
0.997
B
3.043
2.743
3.314
2.663
0.271
0.470
A
4.544
4.729
4.546
4.498
0.002
1.034
B
2.598
3.323
2.918
3.175
0.320
0.219
A
0.607
0.165
0.788
0.467
0.181
0.302
B
0.777
0.502
0.966
0.395
0.189
0.210
A
0.967
0.757
1.636
1.595
0.668
0.936
B
1.350
2.068
2.083
3.172
0.732
1.108
A
0.640
0.656
0.943
0.554
0.303
0.168
B
0.430
0.242
0.758
0.232
0.328
0.128
- 222 -
Change in Stress Orientation
250 T
.1
200
A'P
150-l
0
100-
.
h .,
50-
r'
.IAI-SyncLj
U r
0TO
I
5
50
-1
I
100
I
150
I
-1
200
Theta Pmax, Control
Figure 3.31: The maximum principal stress direction in the "A" and "B"
regions in the experimental patellae as a function of the maximum
principal stress direction in the corresponding regions in the control
patellae. See Figure 3.8 for Key.
- 223 -
data.
For points on
the
Y
line
=
between corresponding regions in the
maximum principal stress direction
For the nine points which lie on or
control and experimental specimens.
was
near these lines,
there
principal stress.
In other
tension to compression or
a
the
these nine data points,
compression
the
in
sign of the maximum
the
the predominant stress changed from
For eight of
tension.
to
stress was compression in the
predominant
tension
control patella and
in
reversal
words,
from
there was no difference in the
X
Again, in
patella.
experimental
general, the stresses in the "A" locations were affected by the implants
more
than
the
changes
significant
This
"B" locations.
the
in
demonstrates
orientation
of
the
there were
that
stress
principal
directions due to the presence of the implants.
is presented
3.4 for definition)
the stress ellipse (see Equation
of
The change in the eccentricity
graphically
in Figure
are
apparent.
stress
was
compression,
the
dominant
negative eccentricity, for both
For the next largest
tension for both the control and
and
thus
had a
control and experimental patellae.
the
of
number
Four
For the majority of
distinct populations of data points
data points,
3.32.
data
points,
the dominant stress was
experimental patellae.
For eight data
points, as mentioned above, the dominant stress changed from compression
to tension, and for
one
generally less change
predicted
by
eccentricity of
the
the
data
in the
finite
material
Section
3.3.
the
eccentricity
element
stereology (see Figure 3.11).
is examined in
point
models,
anisotropy
reverse occurred.
of
There was
the stress ellipse, as
then
there
ellipse,
as
was
for
the
measured using
The relationship between these parameters
There
- 224 -
were
significant changes in the
Change in Stress Eccentricity
1. r
E
.
A
x
P 0.5-t
e
r
i
m
l
A
0.
e
n
t -0.
a
A
1
-1.
-1.0
-0.5
0.0
0.5
1.0
Eccentricity, Control
Figure 3.32: The eccentricity of the stress ellipse in the "A" and "B"
regions in the experimental patellae as a function of the eccentricity
of the stress ellipse in the corresponding regions in the control
patellae. See Figure 3.8 for Key.
- 225 -
stress state, as reflected
by
the
eccentricity of the stress ellipse,
when comparing the experimental specimens to the control specimens.
nine data
points,
the
tension, or vice versa,
dominant
as
stress
changed
reflected
by
3.3,
nature
from
large
For
compression to
changes in the stress
eccentricity.
As discussed in Section
the
critically influence the correlation
the measured
morphology.
magnitude of the
One
hydrostatic
stress component.
shown as a function of the
the stress state may
between the predicted stresses and
important
stress
To examine
of
parameter
component
is the relative
compared
to the shear
this parameter, the hydrostatic stress is
octahedral shear stress, for the control and
experimental patellae, in Figure 3.33.
With the exception of one data
point, the ratio of
to octahedral shear stress was
hydrostatic
always less than one in
stress
absolute
points for which this ratio
was
magnitude.
close
data points also have an eccentricity
Figure 3.32.
This
demonstrates
model predictions, there were
zero.
By definition, these
which approaches zero, as seen in
that,
areas
to
There were numerous data
according to the finite element
which approached pure shear stress
but no areas of hydrostatic compression or tension.
- 226 -
Hydrostatic Stress vs. Octahedral Shear Stress
2-r
..I J
U
,'
, I
H
S
[
-
'IC-""
0
.
[
I
0
U
U
[]
M
P
'.
o
[
aE
a
Control
Experimental
0. 64*X
.- -. 08*.X
[
-34- -
0
1
-----
2
`--i--------.------t
'--------
3
4
5
OSS (MPa)
Figure 3.33: Hydrostatic stress as a function of the octahedral shear
stress for all locations in the control and experimental specimens.
- 227 -
3.2.2 Ovine Calcaneus
The two-dimensional finite element models of the ovine calcanei were
based on
the
morphologic
data
presented
mechanical coupling was assumed between
in Section
3.1.2.
Rigid
the implants and the trabecular
bone due to the observed bone ingrowth into the porous coating on all of
the specimens.
The
established based
on
nonhomogeneous
the
isotropic material properties were
measured
areal
densities
as
described in
Section 2.2.2.
The load due to
deformations.
the
tendo
calcaneus primarily resulted in bending
The deformed mesh,
as predicted for the control specimen
C1, is shown in Figure 3.34.
Note
section,
truss
to
which
rigid
approximately co-linear.
This
elements
confirmed
sections remaining plane, at this
principal stress vectors for
that the nodal points at the distal
distal
were
that
fixed,
remained
the condition of plane
section, was maintained.
The
this
specimen
are
Most apparent were the compressive
stresses
in the anterior cortex and
the tensile
stresses
in
the
posterior
cortex
shown in Figure 3.35.
corresponding
to the
bending deformations.
The
overall
deformations
and
experimental specimens were similar to
stress
distribution
for
the
those for the control specimens.
The deformed mesh for experimental specimen
C1 is shown in Figure 3.36.
The truss elements at the distal
section were replaced by the resultant
loads calculated for the control
specimens.
The rigid body modes were
eliminated and the cobalt chromium implants were represented using truss
- 228 -
... ... . . I. . . . . . . . . . .
Z
Figure 3.34: Undeformed (solid lines) and deformed
(dotted lines) mesh
for control specimen C1.
The positive Y axis points in the distal
direction and the positive Z axis points in the
anterior direction.
- 229 -
MAX
94.4
MIN - -72.7
A
b-i
;-4
~~~~..r
.4....,.~
~~~~~~~~~"4
..
*
*.~~..4'
.4····.*
*
k~
*
"~
..
4
'.A
.
l
-0TCL
·ri·
~-
--
-
·
z
k-y
Figure 3.35: Principal stress
control specimen C1 (MPa).
vectors
- 230 -
on
a
sagittal
section of the
z
Figure 3.36: Undeformed (solid
for experimental specimen C1.
lines)
- 231 -
and deformed (dotted lines) mesh
interface
elements at the bone/implant
detail).
Section 2.2.4 for further
(see
vectors, shown in Figure 3.37, confirmed
The principal stress
that the overall mesh deformation was equivalent to that for the control
specimens
experimental
away
stresses
principal
The
specimen.
identical
essentially
were
the
from
in the
implant
to
those
in the
control specimens.
The stresses around the
in the corresponding
in the experimental specimens and
implants
regions
in
the
of the mesh.
using enlarged views of that portion
vectors for control specimen C1 are shown
general flow of
The principal stress
There was a
in Figure 3.38a.
posterior-distal
region, where the
to the anterior-proximal
occurred,
stresses
highest magnitude tensile
region.
the
from
tension
specimens were examined
control
The orthogonal compressive stresses were approximately a mirror
image of the tensile stresses with the maximum stresses in the anteriordistal region.
The stresses approached uniaxial tension adjacent to the
posterior cortex
cortex.
The
and
uniaxial
and
compressive
magnitude
around
tensile
approximately equal
in
both
stress components were either
location
indicating
that
stress
the
There
indicating significant shear stress.
same
adjacent
compression
tensile
center
the anterior
components
of
were
the implant
were no regions where the
or both compressive at the
hydrostatic
the
to
stress
component
throughout the models was small.
A small distance from the implants, the orientation of the principal
stresses in
the
experimental
corresponding region in the
specimens
was
similar
specimens.
control
232 -
to
that in the
The principal stress
MAX MIN
:: ~·a~
~=. .9.
.
93.7
-72.6
.,lpJ&'
.
~~.Cf'~ ~
lr~~~~4,
'i
4.*
4
.·
.
i·,
$
~·
Y......(...4.···.-4.-
.
4
b .
,."' t·.:
-
?:
.i.
ii
:
.
.
i
~..-~
.
^ ..
· - c..~.
.
4
- .
z
B-Y
Figure 3.37: Principal stress
experimental specimen C1 (MPa). vectors
- 233 -
on
a
sagittal
section of the
MAX t,
qlk
_
9.4
d
-10.
X
Z
k_
Figure 3.38a: Principal stress
specimen C1 (MPa).
vectors
- 234 -
on
an enlarged view of control
MAX jiI
Ha
..I
i1
,=l
'
-
8i.5
_
-CC.
f
W_onimiJi
I"__-A
Z
Figure 3.38b: Principal stress
experimental specimen C1 (MPa).
vectors
235 -
on
an
enlarged
view
of
vectors around the
implant
Figure 3.38b.
The
adjacent to the
implant,
in experimental
stresses
were
with
specimen
significantly
some
C1 are shown in
altered immediately
areas displaying higher magnitude
stresses and other areas displaying lower magnitude stresses than in the
corresponding
areas
of
the
markedly lower at around 7
the implant,
and
control
o'clock,
markedly
specimens.
higher
at
around
von
the control
specimens
experimental
the lowest stresses occurred at about
stress
in
the
2
o'clock.
This was
Mises effective stress contours for
(Figure
stresses adjacent to the implant occurred
the von Mises
stresses were
using a clock reference system for
confirmed by comparison of the
and
The
at
7
and
experimental
3.39).
The highest
about 9 and 2 o'clock and
12 o'clock.
specimen
The ratio of
to the von Mises
stress in the control specimen ranged from a high of 1.53, at 2 o'clock,
to a low of 0.45, at 7 o'clock.
The highest stress gradients occurred adjacent to the cortical shell
in
several
locations
gradients were less
in
severe
the
control
in the
higher bone densities
in
densities reduced the
material
these
specimen.
experimental
regions.
property
The corresponding
specimen
due to the
The higher trabecular bone
gradient between the cortical
shell and the adjacent trabecular bone.
The principal stress vectors in the
other three specimens are shown
vicinity of the implant for the
in Figures
3.40 to 3.42.
In general,
the orientation of the principal stresses was similar for all specimens.
The differences between the specimens
locations.
There
was
a
greater
were due to the different implant
extent
- 236 -
of
tensile
stresses in the
MAX 18.3
J- 16.6
I- 15.0
H- i3.3
G- 11.6
F- 9.97
E- 8.3i
D- 6.65
C- 4.98
B8 3.32
A- 1.66
MINO
z
k-)
Figure 3.39a: Von Mises stress
specimen C1 (MPa).
contours
237 -
on an enlarged view on control
MAX24.4
J- 22.2
I- 20.0
H- 17.8
G- 15.6
F- 13.3
E- ll.i
D- 8.89
C- 6.67
B- 4.44
z
kv
A- 2.22
MINO
Figure 3.39b: Von Mises stress
experimental specimen C1 (MPa).
contours
- 238 -
on
an
enlarged
view
on
MAX
MIN -
....
P-A
W.·3~.~:
·· · ·
~
Figure 3.40a: Principal stress
specimen C2 (MPa).
35.9
-23.0
vectors
- 239 -
on
fi
an enlarged view of control
MAX MIN -4
17.2
-23.9
-
?
Figure 3.40b: Principal stress
experimental specimen C2 (MPa).
vectors
- 240 -
on
an
enlarged
view
of
MAX MIN -
- -Z.
.-
W"... -
..0
I~~~~~~·
: "'lj
*_
I
!
/;
33.1
-18.9
.
'
.;V
-
-~~~~~~~~~~~.
.
:
i
WA;P
fav
I
Figure 3.41a: Principal stress
specimen C3 (MPa).
vectors
- 241 -
on
an enlarged view of control
.
&.
.,
I
.
MAX MIN -
...
27.7
-5.2
.;W4..
L-.
.
*
A
YY-
~~~
1A..
i;
,e---* -
*x
*
*
_i-A''
r
.
ve.
Nr
XI
LA
$
N'
I
t*-
4,
.Z
z
k-v
Figure 3.41b: Principal stress
experimental specimen C3 (MPa).
vectors
- 242 -
on
an
enlarged
view
of
...
*
II
*.
*
MAX
MIN
.
.
~W
r'''~
~~~
·
~~~~
.. s.*
w
#i?'0.
40.4
-.
5
.O..
ja
*
j
a.
31I
jl~~~~~~~~~~
....
... 0
... 0.r·.. ....
z
k_
Figure 3.42a: Principal stress
specimen C4 (MPa).
vectors
- 243 -
on
an enlarged view of control
MAX MIN .
i---*.
..
"
f,
22.9
-22.5
I
- .... .. ;
~~~~,.
.
.'
'.'I
.'
.
.
i
z
Figure 3.42b: Principal stress
experimental specimen C4 (MPa).
vectors
- 244 -
on
an
enlarged
view
of
vicinity of
the
implants
in
specimens
C3
and
C4
due
to the more
posterior and distal position of the implants.
The von Mises stress contours for specimens C2, C3, and C4 are shown
in Figures 3.43
to
3.45.
experimental specimens displayed both
All
to the control specimens, at
increased and decreased stresses, relative
various locations around the implants.
tightly coupled to
ranging from
the
trabecular
approximately
50
%
In summary, the rigid implants
resulted in stress magnitudes
bone
to
150
%
of
the
stresses in the
corresponding locations in the control specimens.
To further examine the influence of
the implants on the stresses in
the surrounding trabecular bone, the principal stress orientation in the
experimental specimens
were
plotted
as
a
function
of the principal
stress orientation in the control specimens (Figure 3.46).
orientation in this graph
maximum in
absolute
was
magnitude.
interpolated stress values
data.
that
at
The lines Y = X + 90 and
possible range of data.
the principal stress which was
The
the
data
locations
principal
implant.
in
90
were
Y
changes
Figure 3.21), the predicted changes
generally small, with most
the line Y = X. Four points lie
= X - 90 indicating that the maximum
principal stress changed from tension to
comparison to the measured
of available morphologic
orientation resulting from the
stress
orientation
and
correspond to the
on the line Y = X there was no
points lying within 5 to 10 degrees of
near the lines Y = X +
points
Y = X - 90 are included to indicate the
For data points
change in the maximum
The changes
of
The angle of
in
compression or vice versa.
In
the trabecular orientation (see
in the principal stress orientation
- 245 -
..... ..*-.... .... I..
.
*
*
. *.**.* *. ....-*
*
I
Byf
on an enlarged vie
contO~rS
Vo141ses stress
Filure 3.43na(Mfa)-
specimen
_ 246 -
o
MAX 18.6
J- 16. 9
I- 15.2
H- 13.5
6- 11.8
F- 10.2
E- 8.46
D- 6.77
C- 5.08
B- 3.38
A
1.NO
MINO
Figure 3.43b: Von Mises stress
experimental specimen C2 (MPa).
contours
- 247 -
on
an
enlarged
view
of
)·
MAX33. I
J- 30.1
I-
27.1
R-
A802
H6FEDC-
24.1
21.1
18.
15.1
12,0
9.03
A- 3.01
MINO
Figure 3.44a: Von Mises stress
specimen C3 (MPa).
contours
248 -
on an enlarged view of control
I
I
MAX27. i
J- 24.6
I-
HGFE-
22.2
19.7
17.3
14.8
12.3
D- 9.86
C- 7.39
R- A493
A- 2.46
MINO
Figure 3.44b: Von Mises stress
experimental specimen C3 (MPa).
contours
- 249 -
on
an
enlarged
view
of
··
·-
......
...........0......
.........
:
· r··~
:-~
· ·
.7/
A
'..
MAX43.2
J-
I-
H- 31.4
G- 27.5
F- 23.6
E- 19.6
D- 15.7
C- 1i.8
B- 7.85
A- 3.93
MINO
Z
k-
Figure 3.45a: Von Mises stress
specimen C4 (MPa).
39.3
35.3
contours
- 250 -
on an enlarged view of control
MAX23 .9
J- 21.7
I- 19.5
H- 17.4
6- 15,2
F- 13.0
E- 10,9
D- 8.68
C- 6.51
B- 4.34
A- 2. 17
MINO
Figure 3.45b: Von Mises stress
experimental specimen C4 (MPa).
contours
251 -
on
an
enlarged
view
of
Change inStress Orientation
6
a
[]
-60
-40
-20
0
20
40
60
Theta Pmax, Control
Figure 3.46: The maximum principal stress direction in the experimental
calcanei as a function of the maximum principal stress direction in the
corresponding regions in the control calcanei.
- 252
were small.
The von Mises stresses
specimens as
a
function
were
of
similarly plotted for the experimental
the
von
Mises
stresses
in the control
specimens (Figure 3.47).
Again, the line Y = X was included to indicate
those
which
data
points
for
experimental and control data.
below, indicating
that
the
there
More
stresses
was
no
points
were
decreased due to the presence of the implants.
- 253 -
difference
between the
lie above this line than
more
often increased than
Change in von Mises Stress
i
0O
i
0
0
l
O
O
0
O
0o
0
O
0
C3
t]
0El
5
[
6
7
8
9
10
vMS, Control (MPa)
Figure 3.47: The von Mises stress in the experimental calcanei as a
function of the von Mises stress in the corresponding regions in the
control patellae.
- 254 -
3.3 Stress-Morphology Relationships
To assess the accuracy of
the finite element model predictions, the
orientation of the principal
of the material
anisotropy
stresses
ellipses
This theory states
material axes of
specimens
the
that,
the
for
the
control specimens.
The
trajectorial theory of trabecular bone
critical assumption was that the
architecture holds for
were compared to the orientation
under
which
did not receive implants.
homeostatic
conditions, the principal
trabecular
bone
are
aligned with the principal
stress axes.
One aspect of this theory which requires detailed examination is the
dependence of the alignment
on
the
nature
stress conditions approaching uniaxial
of
the stress state.
For
tension or uniaxial compression,
it is reasonable to assume that the material will align with the maximum
principal stress, namely tension or compression, respectively.
However,
for stress conditions which approach hydrostatic tension or compression,
or pure shear, the expected
the principal stress axes
compression, a random
relationship
is less
architecture
preferred orientation.
is expected
present
bone under physiologic conditions.
For
trabeculae is expected,
with
pure
directions of principal tensile
with
no direction of
models, and is probably uncommon in
Of greater relevance is the state of
two-dimensions,
in the present models.
For hydrostatic tension or
in Section 3.2.1, a hydrostatic stress
As shown
state did not occur in the
pure shear which, in
clear.
between the material axes and
was approached for some locations
shear, an architecture with crossing
trabecular
struts
oriented
in both the
stress and principal compressive stress
- 255 -
/A,
I
I
I
A-
A-
-- i
I
Optimal structure for pure shear
has an isotropic material ellipse
Figure 3.48: The optimal structure based on a minimum weight criterion
for a stress state of pure shear (see also Hayes et al. 1982).
256 -
(Figure 3.48; see also Hayes
et
1982). However, using the present
al.
appears isotropic, and cannot be
stereologic methods, this architecture
with no direction of preferred
distinguished from a random architecture
orientation.
Furthermore, the
in a weakly oriented structure in
and compression may differ, resulting
either
the
direction
compressive stress.
of
remodeling response to tension
adaptive
tensile, stress
principal
In the case
of
will
be
less
principal
a weakly oriented structure, it is
expected that the stereologic predictions
orientation
or
accurate
for the direction of material
and
therefore
result
in poorer
correlations with the directions of principal stress.
A useful parameter for examining these issues is the eccentricity of
of the stress ellipse.
the anisotropy ellipse and the eccentricity
The
eccentricity of the anisotropy ellipse is defined as:
Id12 _ d22
EA =
where d1 is the major axis and
d2
is the minor axis.
These terms come
Equation 2.68.
For an isotropic
from the two-dimensional equivalent of
material, d 1 = d2,
perfectly
oriented
and
thus
EA
=
EA
structure,
(3.3)
dd
O.
As the material approaches a
approaches
1.
Similarly,
the
eccentricity of the stress ellipse is defined as:
IP 2
ES
p 2
=
2
1
- 257 -
if IP1I > IP21
(3.4)
EP =
2
ES
where P1 and P2 are
the
principal
one
uniaxial tension).
ES
(for
(P1
> P2).
(3.5)
An important
EA ranges from zero to one while
uniaxial
is equal
otherwise.
stresses
distinction between EA and E S is that
E S ranges from minus
2
P2
to
compression) to plus one (for
zero
for
hydrostatic tension,
hydrostatic compression, or pure shear.
After validation of the models of the control specimens, the stressmorphology relations for
implant-induced
the models of the experimental
that
the
finite
element
validation of the models of
remodeling were studied using
specimens.
predictions
the
The critical assumption was
were
control
accurate
specimens.
based
on
the
The most critical
examination of the stress-morphology relationships was then performed by
relating the changes in the
predicted
the measured morphology, as reflected
stress
fields to the changes in
by the differences between paired
locations from the control and experimental specimens.
3.3.1 Model Validation
Equine Patella
The present objective was to validate the finite element predictions
based on the assumption that
the trajectorial theory of trabecular bone
architecture holds for the control patellae.
- 258 -
The correspondence between
the principal stress
directions
and
the principal material directions
for the control patellae is shown in Figures 3.49 and 3.50.
principal stress (OpM)
is defined
maximum in absolute magnitude.
as
In
the
principal stress which was
Figure
3.49, the direction of the
principal stress component which agreed
material direction
The maximum
most closely with the principal
( p0)
is shown
as
material direction (A1).
In Figure
3.50, the direction of the maximum
principal stress
direction.
is shown
For 60
out
of
a
function
function
of
of
the principal
as
a
the principal material
71
locations, the maximum principal stress
was, in fact, the principal stress component which best aligned with the
principal material direction.
the direction of the maximum
In
other
principal
words, the difference between
stress and the direction of the
principal material orientation was less than or equal to 45 degrees.
equation form, when IPM -
A1I < 450, Op =
A linear regression was performed on
direction of
the
principal
material
independent variable since this
was
directions of principal
were
models.
stress
pM.
po as a function of
orientation
a
In
was
A1
chosen
The
as the
measured parameter, whereas the
predicted
by
the finite element
The first assumption for this regression was that the values of
the principal material orientations
were
fixed.
In other words, the
stereologic measurement of the principal material orientation was exact.
This assumption can
be
tested
through
validation
experiments of the
stereologic system, and in fact,
for
the system used, such experiments
have been carried out (Snyder et
al.
1986).
The second assumption was
that, for each principal material orientation, there was a population of
principal stress orientations
predicted
- 259 -
by
the
finite element models
Principal Stress vs. Material Orientation
Control
I
A
0@
101
54
o
A
0
A
A
A
-1.026X
____t
0
50
100
I------
150
+ 5.693
--
AT
200
Material
Figure 3.49: The principal stress direction which best aligned with the
principal material direction (Op) as a function of the principal
material direction (A1) for the control patellae. See Figure 3.8 for
Key.
- 260 -
Principal Stress vs. Material Orientation
Control
WL
a
A
A
A
A
0o
A
n
1
0
0
0
0v
·
A
0
A
0
A
0
71
-A-
-
0
50
100
150
200
Material
Figure 3.50: The maximum principal stress direction (p ) as a function
of the principal material direction (A1) for the controI patellae. See
Figure 3.8 for Key.
- 261 -
population was the same for
insufficient
be
not
assumption could
data
This
principal material orientation.
each
the
using
tested
of
number
the variance of this
Furthermore,
distribution.
normal
which had a
present
However,
points.
data
this
due
to an
an
is not
predicted principal stress orientations
unreasonable assumption for the
since there is no obvious inherent
reason for these orientations to err
in any particular direction or
any particular amount.
assumption of a
distribution
normal
range of data is limited to
GA1.
a
45
a
specifically, for
The
below.
the control patellae.
for
condition,
stress
most critical
that the trajectorial theory
was
holds
particular
=
degree deviation from the line Q
assumption underlying this regression
of trabecular bone architecture
However, the
strictly hold because the
cannot
further
is discussed
point
This
by
More
known exactly, the
principal material direction, measured
using stereology, was invariant,
and that these directions corresponded
exactly.
reasonable assumption based
Section 1.2.1).
from
This appeared to be a
literature
the
on this subject (see
This assumption is discussed further in Section 4.0.
The best fit line from the linear regression on Op0 and
in Figure 3.49.
The statistical data
indicating that the intercept of 5.69
a
p-value
of
are summarized in Table 3.3.
The
and standard deviation of 5.79 degrees,
intercept had a p-value of 0.32
slope had
A1 is shown
0.0001
degrees was not significant.
a
and
standard
The
deviation of 0.048,
indicating that the slope was not significantly different from 1.0.
The
regression had a p-value of 0.0001, standard deviation of 20.09 degrees,
and an R2 of
0.87.
In
other
principal stress direction could
%
of the variation of the
explained
by the variation of the
words,
be
- 262 -
87
Table 3.3
Linear Regression for Model Validation
Equine Patella
Slope
Value
"0O vs Al
pM vs
A1
SD
Intercept
Sig
1.026 0.048 .0001
Value
SD
Regression
Sig
SD
Sig
R2
5.693 5.790 0.330
20.09 .0001 0.870
1.051 0.080 .0001 -3.371 9.730 0.730
33.74 .0001 0.714
- 263 -
principal material direction.
While an R2 of 0.87
the standard deviation of the
is encouraging,
regression must be examined critically.
If the comparison were made on
principal
stress and the direction of the
the direction of the maximum
principal material orientation, then the maximum possible difference for
If this difference were greater than
any data pair would be 90 degrees.
parameter
90 degrees, the direction of
either
degrees thus decreasing this
difference.
closely to the
principal
deviation from
the
line
and thus the maximum possible
pair,
data
Np0
eAl'
=
Therefore, the maximum
direction.
material
possible difference for each
However, the regression was
direction which corresponded more
stress
performed using the principal
could be changed by 180
is 45
The standard
% of the data points were
68
that
deviation of 20.1 degrees indicates
degrees.
within 20.1 degrees of the best fit line, assuming a normal distribution
about this line.
data
If these
random,
were
50 % of the data points
po = A1, or 68 % within 30.6
would lie within 22.5 degrees of the line
degrees.
A stronger argument for
the
of
validity
the finite element model
predictions may be made by using the maximum principal stress directions
for the linear regression (Figure 3.50).
of 71 data
points,
the
closely with the principal
which was maximum in
regression
between
the
stress
principal
material
absolute
As mentioned above, for 60 out
direction was the principal stress
magnitude.
maximum
direction which agreed most
principal
The
results of the linear
stress
direction
principal material direction are also given in Table 3.3.
- 264 -
and the
If these data
were random, 50 % of the data
line "pM =
degrees.
A1'
The
and
68
%
standard
points would lie within 45 degrees of the
of
the
data
deviation
degrees, is much less then
61.2
of
points would lie within 61.2
the
regression,
equal to 33.7
degrees, indicating that the data were
not random.
As
mentioned
above,
the
degree
of
correspondence
between
the
principal stress and material directions may be a function of the nature
of the stress field.
To
examine
is shown graphically as
a
ellipse in Figure 3.51.
this concept, the degree of alignment
function
The
the eccentricity of the stress
Y axis represents the absolute difference
between the maximum principal
material direction (A1).
of
stress
direction (OpM) and the principal
A negative
eccentricity indicates that the
maximum principal stress was compressive whereas a positive eccentricity
indicates that the
maximum
principal
points for which the maximum
best aligned with
the
stress
principal
principal
was
tensile.
The data
stress was not the stress which
material
direction
(
•pM
# 8po) are
highlighted using filled circles.
All eleven of these data points had a
stress eccentricity of less
zero, indicating that compression was
the dominant stress.
data
points,
the
It
then
is particularly striking that, despite fewer
alignment
between
the
maximum
principal
stress
direction and the principal material
direction in regions where tension
predominated was considerably better
than
predominated.
the tensile
Furthermore, for the eleven
principal
stress
aligned
in regions where compression
data points where OpM # O,0'
more
closely
to the principal
material direction than the compressive principal stress.
that either the trabeculae
tended
to
- 265 -
align
This suggests
with the direction of the
Alignment vs. Stress Eccentricity
Control
, f%
!uu-
80- 0#0
0
*·D
T
I
fin
-
o O Theta < 45
Theta > 45
aX,
I
h
t
a
ko
2n
a
00
20V
10
-1.0
o
oo c
°
P
0--- a
ap..
a..
-CPI
p- .
0
. .
-0.5
.-
...
0.0
0.5
1.0
Stress Eccentricity
Figure 3.51: The absolute difference between the maximum principal
stress direction and the principal material direction (IPM - OAll) as
function of the eccentricity of the stress ellipse for the control
patellae.
- 266 -
principal tensile stress or,
to align at 90
degrees
to
which were very close to 90
close
the
direction of the principal compressive
The latter hypothesis was
stress.
to
-1 (indicating
that the trabeculae tended
alternatively,
supported by the several data points
degrees
that
the
and had a stress eccentricity very
stress
was
state
near uniaxial
This point is discussed further in Section 4.0.
compression).
To further examine
the
principal stresses with the
relationship
material
the alignment of the
between
the stress eccentricity is
axes,
shown as a function of the material eccentricity in Figure 3.52.
the 11 points for which
M
Again,
# Opo are highlighted using filled circles.
Three of these data points had a material eccentricity of less than 0.5,
indicating
that
anisotropic.
the
material
in
these
This point is examined
was
locations
further
not
highly
One of these 11
below.
data points had a stress eccentricity of about -0.2, indicating that the
stress state
was
demonstrates
a
approaching
poor
pure
shear.
correspondence
In
between
general,
the
measured
this graph
material
eccentricity and the predicted stress eccentricity.
The degree of alignment
also examined as a function
It was expected that the
between
the
stress
and material axes was
of the material eccentricity (Figure 3.53).
between the principal stress axes
correlation
and the principal material axes would decline as the material approaches
isotropy.
For the 11
data
points
with an angle difference of greater
than 45 degrees, there was a tendency for poorer alignment (higher angle
difference) for
lower
eccentricity of
the
material
analyzed
eccentricities.
image
- 267 -
in
Figure
For
3.54
reference, the
was
0.51.
The
Stress vs. Material Eccentricity
Control
.0T
oo
0
8 oo
0
8
0
'0"
0
0
0.5t
r
e
oD Theta < 45
0D Theta > 45
0
.0-
t
0
S
S
0
-0.5-L
0
0
Ii
*0
-. 0---.-..0.2
0.3
o
0.1.
0.4
0.5
0.5
I___
0__ O08
-efw
0.6
0.7
- --
0.8
0.9
1.0
Material
Figure 3.52: The eccentricity of the stress ellipse as a function of the
eccentricity of the material ellipse for the control patellae. The
points for which
- OA1
(IpM > 45 degrees are highlighted with filled
circles.
- 268 -
Alignment vs. Material Eccentricity
Control
nr
1UU-r'
80- t
*
0
D
0
T 60-I
h
e 40t
a
1
LVT
*
0
0
0o
i
o0
0
0
oo
0
0
00
0
o
n
v
o
oo
O
0
o0------0.3
0.2
0
o 09
0
.
0
oo
o
I
0.4
0.5
0.6
0.7
0.8
0.9
.0
Material Eccentricity
Figure 3.53: The absolute difference between the maximum principal
1) as
stress direction and the principal material direction (p
function of the eccentricity of the material anisotropy ellipse for the
control patellae.
- 269 -
IMAGE PROCESSING LABORATORY
IMAGE TITLE: S4R7
COMMENT: Image 7
SUBREGION: 2
1ENGTHSCALE: 57. 18
GRAY THRESHOLD: 130
SOLID PHASE: BLK
TESTLINE LENGTH: 795.1
TESTLINE SPACING <PIXELS>: 1
[AREA FRACTION ANALYSIS:]
REGION AREA = 13.91
AREA FRACTION = 0.384
SOLID AREA = 5.334
LSTEREOLOGY ANALYSIS:]
Y
ELLIPSE OF MEAN INTRCPT LENGTHS:
2
2
A*N
1 + B*NiN2 + CN2 = MNINCPTLN
A = 12.39
X
2
B = 2.842
C
10.469
CORRELATION COEFF = 0.979
MAX MN INTRCPT LNTH = 0.32084
ANGLE MAX ORIENTATION = 118 DEGS
PERIMETER = 56.568
PERIM/AREA = 4.0680
Figure 3.54:
specimen S3.
MIN MN INTRCPT LNTH = 0.27580
ANGLE MIN ORIENTATION = 28 DEGS
EXTENT OF ANISOTROPY = 9.598 %
"7A" from the control
Stereologic analysis of region
The eccentricity of the anisotropy ellipse equals 0.51.
- 270 -
trabecular architecture in this
oriented.
An
argument
perpendicular
could
directions
corresponds to eA1
and
example
of
be
does
not
that
there
made
trabecular
the
second
A2' as
the directions of material
orientation
was
questioned.
principal
material direction for less
to
which
However, for trabecular bone which
of
correlation between the
of
corresponds
which
isotropic architecture, the significance
be
one
this example, and therefore approached an
had a lower eccentricity than
must
were two mutually
orientation,
of
depicted by the anisotropy ellipse.
appear to be highly
It
stress
not
surprising
that the
directions and the principal
oriented
trabecular
alignment
between
bone was poor in some
instances.
Finally, the degree
of
the stress and material
axes was examined as a function of the bone area fraction (Figure 3.55).
There was a weak,
with
higher
but
area
not
tendency for better alignment
significant,
fractions.
This
is not
surprising
since,
as
demonstrated below, the control patellae displayed a tendency for higher
material eccentricities with
higher
area
fractions (see Figure 3.66).
An illustrative example from the control patella of specimen P6 is shown
in Figure 3.56.
The
dense
trabecular
bone
direction parallel to the cortical shell.
division between
the
correspondence between
trabecular
the
finite
measured morphology was very
good
more isotropic trabecular bone
lower area fractions.
As
bone
in
a
In fact, there was no clear
and
element
in
the
was highly oriented in a
the
model
cortical
The
predictions and the
such instances.
Conversely, the
control patellae tended to have
consequence,
the
data points with the
poorest alignment tended to have lower area fractions.
- 271 -
shell.
Alignment vs. Bone Area Fraction
Control
100 T
80-
T 60h
e 40-
0
t
a
0
· 00
0
0
00o
0
0
20-
0
0
0
0.i
0
-
m4--- -
0.2
0
0O
0
00
0
0o( -
o° oO
°
0 O
-
+--.-4-...:--
0.4
0.5
0
0 U W
o0O
V0
- -·
0.3
0
0.6
0.7
-
---
O.B
~........ o
0.9
1.0
Area Fraction
Figure 3.55: The absolute difference between the maximum principal
stress direction and the principal material direction (IepM - A1I) as
function of the bone area fraction for the control patellae.
- 272 -
BETH ISRAEL HOSPITAL - IMAGE PROCESSING LABORATORY
IMAGE TITLE: P66L3
COMMENT: Porous 66 Left image 3
SUBREGION: 2
'LENGTHSCALE: 57.18
GRAY THRESHOLD: 115
SOLID PHASE: BLK
TESTLINE LENGTH: 795.1
TESTLINE------SPACING
-
[AREA FRACTION ANALYSIS:J
REGION AREA = 13.91
[STEREOLOGY ANALYSIS:]
Y
X
PERIMETER = 44.782
PERIM/AREA = 3.2205
Figure 3.56:
specimen P6.
-'------
PIXELS>
I
.
.~~--'-^-~-
AREA FRACTION = 0.726
SOLID AREA = 10.09
ELLIPSE OF MEAN INTRCPT LENGTHS:
A*N12+ BN 1N 2 + CN 22= MNINCPILN- 2
A= 1.467
B = 1.787
C = 2.6966
CORRELATION COEFF = 0.961
MAX MN INTRCPT LNTH = 1.0011
ANGLE MAX ORIENTATION = -28 DEGS
MIN MN INTRCPT LNTH = 0.56498
ANGLE MIN ORIENTATION =
62 DEGS
EXTENT OF ANISOTROPY = 35. 174 %
Stereologic analysis of region "3A" from the control
The eccentricity of the anisotropy ellipse equals 0.82.
- 273
Ovine Calcaneus
The validation of the
specimens was based on
patellae (see above).
finite
element
of the control ovine
as
those for the equine
assumptions
same
the
models
The critical assumption was that the trajectorial
the control calcanei.
for
theory of trabecular architecture holds
The
correspondence between the principal stress directions and the principal
material directions for the
The principal
stress
principal material
control
component
principal stress which was maximum
was
direction (pM
Included
closely
is shown
as
For
principal material direction (A1).
stress component which
most
which
(G)
direction
best
agreed
a
function
with the
of the
out of 26 locations, the
18
in absolute magnitude ( M) was the
the principal material
aligned
with
the
best-fit
= 0)
= A1
corresponded
principal stress orientation
the
and
two
other
cloud
A
tensile principal stress components.
indicated that,
at
a
significance
argument for the model validity can
(Figures 3.12 and 3.38a).
from
a linear
The data were clearly
and
clouds,
thus the linear
to the compressive principal
of
with
The
the
corresponded to the
data
paired
level
significant difference between these parameters.
principal stress vectors
line
The cloud of data with positive angles of
regression was not justified.
stress components
for reference.
forming
groups,
separable into two
are
3.57
in Figure
regression and the line Gp
EA1
is shown in Figure 3.57.
calcanei
t test, using G0O and
of
0.05, there was no
However, the strongest
be made by visual comparison of the
trabeculae
of the control calcanei
trabeculae in the anterior portion were
- 274 -
Principal Stress vs. Material Orientation
Control
A.nn,
.LUU
!
50- t
t
0-
..
5.7'
.. "" X/
r
I
-U
L
I-C
-40I
-1.52X
/ El
,
- 5.7z I
El
_ .
i
.
_
-20
,
_ .
01
I
_
I
20
0
_
I.
40
!
60
Material
Figure 3.57: The principal stress direction which best aligned with the
principal material direction (
) as a function of the principal
material direction (A1)
for
the
cntrol
calcanei.
Al
275 -
well aligned with the
compressive
regions of the finite element
stress
models.
vectors in the corresponding
The trabeculae adjacent to the
posterior cortex were well aligned with the tensile stress vectors.
linear regression was inappropriate
there was very little
variation
because,
The
unlike the equine models,
in the directions of principal stress
which resulted in the two clouds of data seen in Figure 3.57.
The relationship between the degree
of
alignment (PM
-
A1
I)and
the eccentricity of the stress ellipse for the control calcanei is shown
in Figure 3.58.
In contrast
for which the maximum
to
the equine patellae, all of the points
principal
stress
was
not the stress which best
aligned with the principal material direction (M
stress eccentricity, indicating
that
The
better
alignment
predominated.
generally
was
All eight of the
the proximal regions where
component.
the
tension
data
points
In summary, the trabeculae
regions where
tension
was
the
state
where
pM #
where
often
compression
po came from
had a large shear
were most often aligned with the
stress
component.
dominant
aligned with the compressive stress
was the dominant stress.
in regions
stress
direction of the maximum principal
Op
) had a positive
#
stress,
However, in some
the trabeculae were
component, especially in regions of
high shear stress.
The relationship between the
eccentricity is shown in Figure
there
was
a
eccentricities.
tendency
for
degree
3.59.
worse
of
alignment and the material
Similar to the equine patellae,
alignment
with
lower
material
As expected, the alignment of the principal stress with
the trabeculae in more isotropic regions was worse than the alignment in
very highly oriented regions.
- 276 -
Alignment vs. Stress Eccentricity
Control
100r80D
t
.
0
.
T 60- th
e 40- r
t
a
20-
0
000
0
0
rN
"
0
1,
--'~~~~~~~~~~~~~~~~~~~~~
U I
-1.0
go
0
0
r
0
0.5
0.0
-0.5
0
01
1.0
Stress Eccentricity
Figure 3.58:
The
absolute
difference
between
the
maximum principal
stress direction and the principal material direction ( eM function of the
calcanei.
eccentricity
of
the
- 277 -
stress
ellipse
e I)as
a
or the control
Alignment vs. Material Eccentricity
Control
A
1U- r
0
80-
*0
D
0
T 60h
e 40t
a
0
S
o0
20-D
o
o00
0
Oo
0.65
l
F
0.70
0
0
0
0- I
0
_______
!r
.T
0.75
.
0
_0--...-____I_·_·__
1
i
0.85
0.90
0.80
l
...
......
,--. ---.
0.95
1.00
Material Eccentricity
Figure 3.59:
The
absolute
difference
between
the
maximum principal
stress direction and the principal material direction (I®pM function of the eccentricity of
control calcanei.
oAll)
as a
the material anisotropy e lipse for the
- 278 -
3.3.2 Relations for Equine Model
The present objective
predicted stresses
equine
and
patellae.
was
the
The
to
examine
measured
morphology
correspondence
directions and the
principal
patellae is shown
in Figures
3.60
principal material
direction
(CA1)
the correlation between the
between
material
and
is shown
most
material direction
Figure
3.61,
direction is shown
direction (M).
as
a
function
of
the
is important to note that for
control patellae,
the
principal
the dependent variable.
that the values of
the
The
The
principal
measure
as
Figure 3.60, the
a
function
of the
closely to the principal
the
principal material
maximum
principal stress
identical to those for the
If
the
as a function of
p.
It
stress
the
orientations were fixed.
goodness-of-fit
0.87.
In
for the control
The applied loads and the
of the experimental patellae were
patellae
assumptions
patellae are accepted, then those
chosen as the
finite element models were assumed
models
control
was
assumption for this regression was
the
of
direction
material direction was chosen as
patellae from the previous section was
finite element meshes for the
EA1
stress
first
other words, the predictions of
implant region.
In
this analysis, unlike the analysis of the
independent variable and the principal
exact.
principal stress
For 60 out of 73 locations, eo = OPM.
A linear regression was performed on
to be
the
3.61.
agreed
In
the experimental
directions for the experimental
principal stress direction which
( 0).
in
with the exception of the
of
the
analysis of the control
results
may
be extended to make the
statement that the goodness-of-fit of the predicted stress directions to
- 279 -
Principal Material vs. Stress Orientation
Exper imental
'Ann
CVU
0*
M 15
a
t
e
10
r
i
a
1 5
A
*
0o
0 0
0
~~~~
A
0
A
O
-O.815X
-50
0
50
100
150
+ 14.54
200
Stress (Dmin)
Figure 3.60: The principal stress direction which best aligned with the
principal material direction (G0) as a function of the principal
material direction (A1) for the experimental patellae. See Figure 3.8
for Key.
- 280 -
Principal Material vs. Stress Orientation
Experimental
n
d 2 f
elU
0
A
M 15
a
t
e
10
r
o
3
·0
&
o
a
i
a
A
A
I 5
0
o
*U
i.01
.
.
-50
A
50
100
150
200
250
Stress (Pmax)
Figure 3.61: The maximum principal stress direction (M) as a function
of the principal material direction (A1) for the experimental patellae.
See Figure 3.8 for Key.
- 281 -
the 'actual' stress directions for
word actual is in quotes
whereas in
considered.
this
only
single
a
the experimental patellae in
It is not
for the stress predictions for
accuracy
the control patellae was the
to
contrast
interface
representation of the bone/implant
The accuracy
conditions.
this interface were specifically examined
of the boundary conditions at
The
case was
The most critical aspect of the mechanics of
the experimental patellae.
in Section 3.1.1.
load
static
in Section 4.0.
This point is discussed further
unreasonable to expect similar
The
the in vivo stresses vary cyclically,
because
investigation
control patellae was 0.87.
the
second
was that, for each predicted
assumption
principal stress direction, there was a population of principal material
for
population was the same
assumption could
not
insufficient number of
data
orientation
orientation in any
particular
However, a
similar
was
no
differ
to
direction
be
stated
the
range
of
data
0
A1
to an
also not an
reason for the
principal stress
in
is
regards
limited
to
a normal
to a 45 degree
In summary, the objective for the
No
=
is
This
by any particular amount.
or
must
deviation from the line
the
due
data
inherent
from
caveat
distribution, in that
this
However,
there
principal material
present
the
using
points.
since
unreasonable assumption
stress orientation.
principal
each
tested
be
that the variance of this
and
distribution
orientations with a normal
control patellae was to validate the finite element predictions assuming
that the trajectorial theory
control patellae.
The
experimental patellae
accurate.
The proposed
for
critical
was
that
linear
trabecular architecture holds for the
assumption
the
finite
regression
for
the
element
was
a
predictions were
test of whether the
trajectorial theory holds for the experimental patellae.
- 282 -
analysis of the
The best fit line from the linear regression of OA1 as a function of
8pO is shown in Figure
Table 3.4.
The
3.60.
intercept
deviation of 4.29.
The
deviation of 0.034.
The
had
slope
The
statistical data are summarized in
a
had
p-value
a
regression
of
slope
different from 0,
was
the
and a standard
p-value of 0.0001 and a standard
had
a p-value of 0.0001, standard
deviation of 17.98 degrees, and an R 2 of 0.89.
that, while the
0.001
different
The statistics indicated
from
1.0
and the intercept was
for
the
experimental data was
goodness-of-fit
similar to that for the control patellae.
Similar comments may be
made
regression as those made for
To
further
examine
directions and the
the
about
the
standard deviation of the
control patellae (see Section 3.3.1).
correlation
principal
the
stress
between
the
principal material
directions, the linear regression
was performed using the maximum principal stress direction (
M) for the
independent variable rather than the direction of principal stress which
agreed most closely with
the
principal
mentioned above, for 60 out of 73
of this linear regression are
data
also
material
points, cpM = Npo.
given
deviation of the regression was 29.19
direction ( 0).
in
Table 3.4.
As
The results
The standard
which is comparable to that found
for the control patellae.
The degree of alignment
stress ellipse for the
The Y
axis
represents
as
a
function
experimental
the
of the eccentricity of the
patellae
absolute
is shown in Figure 3.62.
difference
between
the maximum
principal stress direction and the principal material direction.
of the thirteen data points
for
Eleven
which this difference was greater than
- 283 -
Table 3.4
Linear Regression for Implant-Induced Remodeling
Equine Patella
Slope
Value
SD
Intercept
Sig
Value
SD
Regression
Sig
SD
Sig
R2
A1l vs
po
0.815 0.034 .0001
14.54 4.290 .0012
17.98 .0001 0.890
Al vs
PM
0.802 0.061 .0001
13.01 7.710 0.096
29.19 .0001 0.710
- 284 -
Alignment vs. Stress Eccentricity
Experimental
I
1
An
0
D
0
T
h
e
t
a
0
.
%
0
0 0
0
0
0
0
0
0
G0
0
-1.0
-0.5
0.0
I
0.5
o
1.0
Stress Eccentricity
Figure 3.62: The absolute difference between the maximum principal
stress direction and the principal material direction (IPM - All)
1 as
function of the eccentricity of the stress ellipse for the experimental
patellae.
285 -
45 degrees had a stress eccentricity
of less than zero, indicating that
compression was the dominant stress.
For the control patellae, eleven
out
of
eleven
data
points
for
same
the
condition
had
a
stress
Also similar to the control patellae,
eccentricity of less than zero.
predominated was better than in
tension
the alignment in regions where
regions where compression predominated.
The relationship between the eccentricity
of the stress ellipse and
the eccentricity of the material anisotropy ellipse for the experimental
Six of the thirteen data points where
patellae is shown in Figure 3.63.
"PM #
po had a material
eccentricity of less than 0.5, indicating that
not highly anisotropic.
the material in these locations was
two data points for which the
with a poor alignment had
0.25).
stress eccentricity was greater than zero
low material eccentricity (less than
very
a
One of the
In general, similar to
data for the control patellae, this
the
graph demonstrates a poor correspondence between the eccentricity of the
eccentricity of the stress ellipse.
material anisotropy ellipse and the
The degree
examined as
of
a
function
Again, as for the
poorer
between
alignment
alignment
of
control
(higher
the
stress
the
material
patellae,
angle
and
material
axes was
eccentricity (Figure 3.64).
there
was a general tendency for
difference)
for
lower
material
eccentricities.
the experimental patellae as a function
The degree of alignment for
in Figure 3.65.
of the bone area fraction is shown
results for the control patellae,
there
alignment for higher area fractions.
- 286 -
In contrast to the
was
no clear trend for better
Recall
that the reason for good
Stress vs. Material Eccentricity
Experimental
1. Ir1
U- F
c0b
t
r
e
o
VO
V
O
0
0
0
0
0.5- t-
00
0o
O
0 00
0
S
00
0
0
0
o
Theta < 45
* 0 Theta > 45
0.0- t-
5
S
-0.5- i-
0
0
*
-1
n- F__
0.2
d
r_
n>
I
=
0.3
M
e ,r"-&On0-d
IN
__
0.4
l
0.5
0.6
i
0.7
0.8
__
0.9
_~~~~~~~~~~ I
1.0
Material
Figure 3.63: The eccentricity of the stress ellipse as a function of the
eccentricity of the material ellipse for the experimental patellae. The
points for which
pM - eAl
> 45 degrees are highlighted with filled
circles.
- 287 -
Alignment vs. Material Eccentricity
Experimental
100T
I
eS0
0
0
T
h
0
60
e 40-t
o
0
0
0
0
0
0
0o0
O0
3n- L
0
0%
1
0
I
i
0.2
0
0.3
.
0
a
L-
.
0
1,
0.4
0.5
Qb
0
0
80000
I
0.6
0
/
I
0
0.7
0O
00,0 -
I
0.8
0.9
--
.
1.0
Material Eccentricity
Figure 3.64: The absolute difference between the maximum principal
M - A1 I)as
stress direction and the principal material direction (
function of the eccentricity of the material anisotropy ellipse or the
experimental patellae.
- 288 -
Alignment vs. Bone Area Fraction
Experimental
4 fnn,
.UV I
D
80- t
T 60-
i
0
h
e
t
a
0
.
o o
0
0
2lo.
!0Ir
S
o
0
0
o
0o
!
t
e
o
o
o0
L° _ 1 °
%
°
o
u-I------
0.3
0.4
0
i
--
0.5
0.6
0.8
0.7
0.9
1.0
Area Fraction
Figure 3.65:
The
absolute
difference
stress direction and the principal
between
material
the
maximum principal
direction (p
- A1I)
function of the bone area fraction for the experimental patelae.
- 289 -
as
alignment in the control patellae
dense bone areas
cortical shell.
were
adjacent
In these
resulting in good
at
to
regions
correspondence
However, for the experimental
high
and
area fractions was that the
continuous
with the anterior
the trabeculae were highly oriented,
with
the finite element predictions.
patellae,
high area fractions were remodeled
many
areas
of
the data points with
adjacent to the implants.
It
can be seen in Figure 3.66 that the experimental patellae tended to have
more
data
points
eccentricities.
with
high
area
fractions
and
low
An extreme example is shown in Figure 3.67.
corresponds to image "7A" from specimen
high but virtually isotropic.
S3.
material
This image
The bone density was very
Also recall that there was a tendency for
the material eccentricity to decrease
for the "A" locations (see Figure
3.3).
not
However,
significant.
this
tendency
This is because
high in some instances.
very
dense
bone
An
pronounced
or statistically
the material eccentricity was deceptively
example
can
to
implant
adjacent
eccentricity of 0.53.
was
the
be seen in Figure 3.6b.
in specimen
P6
had
This
an
However, it is difficult to attach meaning to the
measured orientation in this
region
based
on visual inspection of the
image.
The measured area fraction
the structural properties
Section
1.2.1,
Fyhrie
of
and
state
based
critical parameter in determining
trabecular
bone.
Carter
(1986)
developed
on
and
density
structural
demonstrated that the trajectorial
optimization.
a
the
formulation for the orientation
given stress
is
of
As presented in
a
mathematical
trabecular bone for a
optimization.
The authors
theory is consistent with structural
They also pointed out that if the formulation is based on
- 290 -
Material Eccentricity vs. Area Fraction
1.07 '
f
II
13
0.8-
n
n
ra
o
M
-
0
3r
a
aci3
In
0
0
0.6-
[
U
m
.
m.
0
0.2i- 0.3
0.2
0.3
0--.4
0.4
' IL
·
Do
oControl
mExperimental
m
0
0.1
NU^
N
m
D
IO
[
i
3
KU
IL
ON [
0.4-
0.1
aw
3N
0O
o
0
I
U
DU
i
e
YJ
m
P
U a
n[
0
m
--
0.5
0.5
- 7
0.6-0..
0.6
0.7
0.9
0.8...
0.8
0.9
- -1.0
.
!.0
Area Fraction
Figure 3.66: The eccentricity of the
function of the trabecular bone areal
the experimental patellae.
- 291 -
material anisotropy ellipse as a
density for both the control and
BETH ISRAEL HOSPITAL - IMAGE PROCESSING LABORATORY
IMAGE TITLE: S4L7
COMMENT: Image 7
SUBREGION: 2
LENGTHSCALE: 57.18
GRAY THRESHOLD: 130
SOLID PHASE: BLK
TESTLINE LENGTH: 795.1
TESTLINE SPACING <PIXELS> :
[AREA FRACTION ANALYSIS:]
AREA FRACTION = 0.771
REGION AREA
13.91
SOLID AREA =
[STEREOLOGY ANALYSIS:]
Y
10.72
ELLIPSE OF MEAN INTRCPT LENGTHS:
2+ R*NANq + C*Nn 2= MNTINCPTLN -2
A*NA
.....
...
...
.c
A = 1.902
B = 0.4855E-01
C = 1.8003
CORRELATION COEFF = 0.279
MAX MN INTRCPT LNTH = 0.74643
ANGLE MAX ORIENTATION = 103 DEGS
MIN MN INTRCPT LNTH = 0.72406
ANGLE MIN ORIENTATION = 13 DEGS
EXTENT OF ANISOTROPY = 1.939
IX
PERIMETER = 45.799
PERIM/AREA = 3.2936
Figure 3.67: Stereologic analysis
specimen S3.
of
- 292 -
region "7A" from the experimental
an effective stress
the
measure
is optimized for strength,
material
whereas if the formulation
is based
optimized for stiffness.
Both
on
strain energy the material is
apparent density, for an optimum
structure,
function of an effective
measure.
between
A1 and
predict that the minimum
formulations
stress
is proportional to a power
The above linear regression
of the trajectorial theory as applied to
1 was a test
the patellae with implant-induced remodeling.
The next objective of the
the relationship between the predicted stress
present study was to test
and the bone density as reflected by the measured bone area fraction.
Two
effective
stress
investigation, von Mises
examined
stress
effective
Both measures are commonly
criteria.
were
measures
and
in the
present
strain energy density.
to engineering materials as failure
applied
One limitation of the von Mises failure criterion is that, in
three-dimensions, the failure surface is not closed.
More specifically,
this criterion allows for
infinite
strength in hydrostatic compression
and hydrostatic tension.
However,
this
the present
studies
since
data were
the
hydrostatic
Furthermore, a two-dimensional
(see Figure 3.33) and
limitation is not relevant in
a three-dimensional
limited to two-dimensions.
stress
state did not occur
hydrostatic stress state is
unlikely under physiologic conditions.
The relationship between von
fraction for both the control and
Figure 3.68.
Similarly, the
effective
Mises
stress and bone area
the experimental patellae is shown in
relationship between strain energy density
and bone area fraction is shown
in Figure
3.69.
There was roughly a
stress and bone area fraction but
linear relationship between von Mises
- 293 -
Von Mises Stress
vs. Bone Area Fraction
I
r-
oControl
mExperimental
13-
S
10-
t
r
e
· 0
s
5
_=
0-
0.1
~
0.2
I-
~ II
0.3
0.4
_,,
1
*
eOP.
i
0.5
0
.
-
0.6
E []
I _-
0.7
U
Il
0.8
I
0.9
1.0
Area Fraction
Figure 3.68: Von Mises effective stress (MPa) as a function of the
trabecular bone areal density for both the control and the experimental
patellae.
- 294 -
Strain Energy Density
vs. Bone Area Fraction
,
in-
r
U
[
10S
E
D
[
a
o
·
U
oControl
=Experimental
r-
a
_l
0$- 0 rre
_
_
0.1 0.2
a R
l
rEb,
n-P
__
I
0.3
t
-
I
l
0.4
__
I
-
;P I
--
0.5
__
I
I
El
.m
·
a
.
.
l
0.6
I-
___
i
0.7
[
N1 N
I_
_ I
l_
.
0.8
i
I-
0.9
U
-
,
1.0
Area Fraction
Figure 3.69: Strain energy density (MPa) as a function of the trabecular
bone areal density for both the control and the experimental patellae.
- 295
a linear regression would be
inappropriate
because the variance of the
data was not constant for different area fractions.
area fractions was much less
Even if a
linear
than
relationship
conclusive demonstration that
was
the
the variance at high area fractions.
appropriate,
trabecular
linear function of the measured
It should not
be
surprising
bone
will
that
not
would not be a
Recall that the isotropic
experimental patellae was based on a
bone area fraction (see Section 2.2.2).
between an effective stress measure
higher modulus
this
bone remodeling was such
that this linear relationship was maintained.
elastic modulus of the control and
The variance at low
a
and
linear relationship would result
the bone area fraction.
necessarily
have
While
higher stresses, the
circular nature of this relationship is inescapable.
The scatter of data seen in the
discontinuity of
the
elastic
strain energy density is due to the
modulus
in
the
finite element models.
Rather than a continuous distribution of material properties, there were
abrupt changes in the
elastic
modulus
from
element
to element.
The
resulting extreme local gradients in strain energy density were apparent
in
the
contour
plots
presented
in Section
3.2.1.
This
was
an
unfortunate problem inherent in the
finite element method as applied in
these studies.
be
This
problem
density, but the present models
may
solved
represented
by
increasing the mesh
a practical upper limit in
complexity for the available facilities.
The most critical test of whether the trabecular bone remodeling was
an adaptive response to the
imposed
stresses is to compare the changes
in the material anisotropy ellipse to the changes in the stress ellipse.
- 296 -
This assumes that the finite element model predictions were accurate for
both the control and the
experimental
patellae.
While an attempt was
made to validate the models of the control patellae in Section 3.3.1, no
similar validation
patellae.
patellae
It must
provided
the
was
possible
for
be
assumed
that,
accurate
models
if the
predictions,
experimental patellae also provided
then
accurate
of the experimental
models
the
of the control
models
predictions.
of
This is the
the linear regression relating
same assumption that was made for
the
A1 to
"PO
a function of the change in von
The change in bone area fraction as
Mises effective stress is shown in Figure
a linear regression is included in
regression were the same
as
the
those
3.70.
graph.
The best fit line from
The assumptions of this
for the previous linear regression.
The regression had a significance level of 0.0001 and an R 2 of 0.57.
argument can be made based on this
graph that the data from the "A" and
"B" locations come from two distinct populations.
a single data point, the data
from a linear relationship.
An
from
With the exception of
the "B" locations had less variance
This linear regression provides a critical
test of the hypothesis of Fyrie
and
Carter (1986) since these data are
an additional step removed from
the
input material property data.
stress changes were
the
result
of
properties and the changes in the
the implants.
A similar plot of
both
stress
the
the
The
changes in the material
fields due to the presence of
change in bone area fraction as a
function of the change in strain energy density is shown in Figure 3.71.
The scatter of data is inconclusive in light of the error resulting from
the discontinuity of material properties in the finite element models.
- 297 -
Change in Bone Area Fraction
vs. Change in von Mises Stress
.
U.41rF'
U
C
.
D
a
*
F
r
I
.1
i
0
0
"I,,
0
A
0
0
0
-- I
0.0-
Ara
0
A 0.2- f
r
a
.0
.
0 0
0
A
c
-- 0.108X + 0.038
-
-1
00
1
2
2
3
3
4
4
D Stress (MPa)
Figure 3.70: The difference in trabecular bone areal density between the
experimental and control patellae as a function of the difference in von
Mises stress between the experimental and control patellae. See Figure
3.8 for Key.
- 298 -
Change in Bone Area Fraction
vs. Change in Strain Energy Density
Al
^
U.4-
m
a&
0
0
0
D
* .
A 0.2-
,·
0
r
A
0 o
0
F
r
0.00
o
a
C
·
^~
-U.
11
C
L
-2
I
-1
1
0
I
-----
e
2
-------
1-
3
D SED (kPa)
Figure 3.71: The difference in trabecular bone areal density between the
experimental and control patellae as a function of the difference in
strain energy density between the experimental and control patellae.
See Figure 3.8 for Key.
299 -
was
comparison
The final
between
made
in principal
change
the
The
material orientation to the change in principal stress orientation.
orientation of the maximum principal
relationship between the change in
shown in Figure 3.72.
ApM.
As mentioned
above,
This
problem
0A1)
there
were
nine
from
data points in which the
to tension or vice
compression
by
eliminated
examining
the relationship
between the change in orientation of the principal tensile stress (p
and aA1
shown in Figure 3.73.
AA1 for reference.
As
figure includes the line A1
This
demonstrated
above,
stress directions (see Figures 3.10 and
figures that no definitive statement
about
the
correspondence
1)
=
the change in principal
than the changes in principal
greater
material directions tended to be
is
clearly two distinct populations for
changed
was
principal material axis (
the
are
There
maximum principal stress
versa.
in
and the change
stress (M)
It is clear from these
3.31).
may
be made from the present data
the
changes
between
in
trabecular
bone
orientation and the changes in principal stress orientation.
One very important aspect of the poor correspondence demonstrated in
Figure 3.73 was
that
the
anisotropy ellipse generally was
in the orientation of the
in
change
the
greater
stress
ellipse.
orientation of the material
than the corresponding change
One possible explanation is
small change in the stress state
that, under the right circumstances, a
can result in a large change in the material orientation.
selected example is shown in Figure 3.74.
the direction of principal
when the
experimental
However, by
visual
material
patella
inspection,
was
One carefully
As measured using stereology,
orientation was rotated 86 degrees
compared
the
- 300 -
to
trabecular
the control patella.
architecture
was not
Change inStress Orientation (Pmax)
vs. Change inMaterial Orientation
D
1007
0
0
I
T
h
e
t
a
50- tA
A0o
00
0a
.
S
t
r
e
A
-1004
s
-100
- ---
±
-50
..50
O
50
------------.100-
O
00
D Theta, Material
Figure 3.72: The difference in the direction of maximum principal stress
between the experimental and the control patellae as a function of the
difference in the direction of principal material orientation between
the experimental and the control patellae. See Figure 3.8 for Key.
- 301 -
Change inStress Orientation (P!)
vs. Change inMaterial Orientation
D
20T
T
h
e
t
a
10-
/
0.
A
A
-.-
A
.
O
0
A
04
0
U
A1
-
"I
I
10
-
0
S
O
r -20e
S -30+
-1:00
[
-.
-1-I --
-
-
--- .
-
50
0
-50
Theta,
Figure 3.73: The difference in
between the experimental and the
difference in the direction of
the experimental and the control
100
aterial
the direction of principal stress P1
control patellae as a function of the
principal material orientation between
patellae. See Figure 3.8 for Key.
- 302 -
BETH ISRAEL HOSPITAL - IMAGE PROCESSING LABORATORY
IMAGE TITLE: S4R4
COMMENT: Image 4
[AREA FRACTION ANALYSIS:]
SUBREGION: 3
LENGTHSCALE: 57.18
GRAY THRESHOLD: 120
SOLID PHASE: BLK
TESTLINE LENGTH: 795.1
TESTLINE SPACING <PIXELS>:
AREA FRACTION = 0.487
REGION AREA = 13.91
SOLID AREA = 6.767
[STEREOLOGY ANALYSIS:]
Y
ELLIPSE OF MEAN INTRCPT LENGTHS:
2+ B*N4No + C*No 2= MNINCPTLN-2
A*N
L
i*
C
-.-.-
X
PERIMETER - 55.881
PERIM/AREA = 4.0186
Figure 3.74a:
specimen S3.
Stereologic
1
A = 7.292
B = -3.391
C = 6.6462
CORRELATION COEFF = 0.969
MAX MN INTRCPT LNTH = 0.43672
ANGLE MAX ORIENTATION = 50 DEGS
MIN MN INTRCPT LNTH = 0.33912
ANGLE MIN ORIENTATION = 140 DEGS
EXTENT OF ANISOTROPY = 15.961 %
analysis
of
303 -
region
"4B"
from the control
___
·_____
I_
TAL - IMAGE PROCESSING LABORATORY
IMAGE TITLE: S4L4
COMMENT: Image 4
SUBREGION: 3
LENGTHSCALE: 57.18
GRAY THRESHOLD: 120
SOLID PHASE: BLK
TESTLINE LENGTH: 795.1
TESTLINE SPACING <PIXELS>:
I
[AREA FRACTION ANALYSIS: i
AREA FRACTION = 0.449
_-
REGION AREA = 43.91
SOLID AREA = 6.246
[STEREOLOGY ANALYSIS: ]
Y
ELLIPSE OF MEAN INTRCPT LENGTHS:
-2
+ - CN-2= MNINCPTLN
A*N41 2+ B*NNo
-I
C
_r
A = 7.580
A
PERIMETER = 54.109
PERIM/AREA = 3.8912
= 1.025
C
7.6414
CORRELATION COEFF = 0.891
MAX MN INTRCPT LNTH = 0.37537
ANGLE MAX ORIENTATION = -43 DEGS
MIN MN INTRCPT LNTH = 0.35084
ANGLE MIN ORIENTATION = 47 DEGS
EXTENT OF ANISOTROPY = 4.300 %
Figure 3.74b: Stereologic analysis of
specimen S3.
- 304 -
region "4B" from the experimental
tremendously different between the
appeared to be two distinct
experimental patella,
two
of oriented trabeculae.
directions
the
direction
maximum
In both cases, there
images.
In the
of orientation coincided
with the radial direction of the implant, whereas in the control patella
the
radial
direction
orientation.
The
small since
this
corresponded
predicted
was
a
changes
to
the
in
stress
"B" location.
12
X
increase
in the
magnitude
of
-3.4
experimental patella, from -3.0 to
stress vector changed by
only
component,
material
in
whereas
finite
element models
both cases, with about a
in
principal
stress
P2
in
the
The orientation of this
MPa.
rotation toward the radial
degree
stress component P1 changed
principal
It could be argued that, in the control patella,
from 0.3 to -0.5 MPa.
principal
4
The
direction of the implant.
the
a
of
at this location were
The
predicted essentially uniaxial compression
direction
secondary
direction
the
aligned
experimental
with
the
patella,
tensile
the
stress
increase
in
compression resulted in the material alignment changing to the direction
of the compressive stress component.
However, these stress changes were
relatively small, and it is questionable
as to whether any meaning can
be attached to the secondary principal stress component.
A second example of a
shown in Figure 3.75.
This
change
large
example
smooth implant in specimen S2, and
significant.
In
the
control
location was nearly isotropic,
was
in the material orientation is
from a region adjacent to the
thus the changes in stress were more
patella,
whereas
the
trabecular
bone at this
in the experimental patella the
trabeculae were clearly oriented in a direction running from the implant
to the anterior cortical shell.
The finite element models predicted an
- 305 -
___ I__·____I__
_ __._ ___ ___________
______ __ __
_ _____
_·
___I_________ _ _ ·_I____ _ ____ _ _ ___··__
__ _·_____· · l__i·__
BETH ISRAEL HOSPITAL - IMAGE PROCESSING LABORATORY
IMAGE TITLE: S2R7
COMMENT: Image 7
SUBREGION: 2
LENGTHSCALE: 57. 18
GRAY THRESHOLD: 120
SOLID PHASE: BLK
TESTLINE LENGTH: 795.1
TESTLINE SPACING <PIXELS>:
i
[AREA FRACTION ANALYSIS:
AREA FRACTION = 0.409
REGION AREA = 13.91
SOLID AREA = 5.682
[STEREOLOGY ANALYSIS: ]
Y
ELLIPSE OF MEAN INTRCPT LENGTHS:
A*N12 + B*N1N 2 + CN 22= MNINCPTLN-2
A = 6.238
B =-0.4023
X
PERIMETER = 44.860
PERIM/AREA = 3. 2261
Figure 3.75a:
specimen S2.
Stereologic
C= 6.3959
CORRELATION COEFF = 0.670
MAX MN INTRCPT LNTH = 0.40487
ANGLE MAX ORIENTATION = 34 DEGS
MIN MN INTRCPT LNTH = 0.39124
ANGLE MIN ORIENTATION = 124 DEGS
EXTENT OF ANISOTROPY = 2.179 %
analysis
of
- 306 -
region
"7A"
from the control
-C---TII----
-_-----_I^--
----- ·C·
-- U------
------
--
- - - --·---
-- I ----
·Y-·-----1-----
__-I----·.------
TAL - IMAGE PROCESSING LABORATORY
IMAGE TITLE: S2L7
COMMENT: Image 7
SUBREGION: 2
LENGTHSCALE: 57.18
GRAY THRESHOLD: 110
SOLID PHASE: BLK
TESTLINE LENGTH: 795.1
TESTLINE SPACING <PIXELS> :
1
.
[AREA FRACTION ANALYSIS:]
REGION AREA = 13.91
[STEREOLOGY ANALYSIS:]
Y
X
PERIMETER = 40.653
PERIM/AREA = 2.9235
AREA FRACTION = 0.612
SOLID AREA = 8.511
ELLIPSE OF MEAN INTRCPT LENGTHS:
A*N12+ B*NiN2 + CN 2 2= MNINCPTLN -2
A = 2.667
B = 0.8254
C =
1.9879
CORRELATION COEFF = 0.995
MAX MN INTRCPT LNTH = 0.74680
ANGLE MAX ORIENTATION = 115 DEGS
MIN MN INTRCPT LNTH = 0.59112
ANGLE MIN ORIENTATION = 25 DEGS
EXTENT OF ANISOTROPY = 14.766 %
Figure 3.75b: Stereologic analysis of
specimen S2.
- 307 -
region "7B" from the experimental
magnitude with essentially no change
approximate doubling of the stress
in orientation.
P1
=
1.6 and P2 =
MPa, and for the experimental patella, P 1 =
-1.6
0.7 MPa and P 2 =
patella, the principal tensile
control
the
For
-3.2.
were, for the control patella,
predictions
The stress
stress was well aligned with the principal material direction.
of
the
In
trabeculae.
stress
compressive
considering the isotropic nature
coincidental,
this must be considered
the
principal
the
with
aligned
well
was
patella,
experimental
the
However,
principal
material
It could be argued that the change in trabecular orientation
direction.
was an adaptive
response
the compressive stress
in
increase
the
to
component.
the remodeling induced by the implants
simplest form, was applicable to
material
orientation
encouraging results.
and
regression
linear
A
patellae.
in the equine
that the trajectorial theory, in its
appear
In summary, it did not
principal
the
eccentricity of
the
high areal density.
around
the
implants
necessarily
not
which
were
relationship was proposed between
change in von Mises stress.
hypothesis of
Fyrie
and
corresponds to an optimum
to
due
was
This
In
ellipse.
material
patellae, the alignment was
alignment
of
degree
the
the
the
contrast
change
to the control
in regions with a
presence of dense regions
highly
not
dependent on the
was
better
in areas where
Also similar to
control patellae.
tension predominated, similar to the
the control patellae,
orientation provided
better
appeared
alignment
The
stress
between the principal
oriented.
A
linear
in areal density and the
This relationship provides support for the
Carter
(1986),
structure.
- 308 -
that trabecular architecture
Finally, the change in material
appeared
alignment
alignment.
to
unrelated
Through two
the
predicted
in stress
it was suggested that,
examples,
illustrative
change
under the right conditions, small changes in the stress state can result
in large changes in the measured morphologic parameters.
3.3.3 Relations for Ovine Model
The relationships between
the
stresses and the measured
predicted
morphologic parameters were examined for the experimental calcanei.
axis
orientation of the principal material
principal stress direction which
material direction (o)
locations,
agreed
is shown
in Figure
3.76.
M).
Included
As in the validation of
the
control
linear regression between OA1 and
of principal stress orientation
was inappropriate.
o
corresponded
angles
components and the negative
A paired t test,
the
However,
data was the visually compare
(Figure 3.38b) to the
the
to the compressive stress
and
to the tensile stress
O 0, indicated that, at a
was no significant difference between
most
informative way to examine these
the predicted principal stress directions
trabeculae
anterior
®A1
The data fell
Again, the positive angles
corresponded
using
significance level of 0.05, there
these parameters.
A1 = DPo for reference.
models (see Section 3.3.1), a
into two separate groups forming two clouds.
In
For 13 out of 21
in Figure 3.76 are the best
fit line from a linear regression and the line
3.12b).
closely to the principal
pO corresponded to the principal stress which was maximum in
0p
absolute magnitude (Op=
components.
as a function of the
(eA1)
most
The
regions
in the experimental specimens (Figure
slightly
- 309 -
removed from the implants
Principal Material vs. Stress Orientation
Experimental
]°
o"
*
00
M
a
t
e
r
i
a
f 0
0o
0
0
[
I
--
-60
-40
-20
0
20
0.46X + 2.95
40
60
Stress (Dmin)
Figure 3.76: The principal stress direction which best aligned with the
principal material direction (0)
as a function of the principal
material direction (A1) for the experimental calcanei.
- 310 -
(towards the top of the figures),
similar to that in the
aligned with the
control
compressive
specimens.
stress
observation from the morphology
the dense bone
around
the
the orientation of the trabeculae was
of
components.
implants
instances this dense bone resulted
orientation
whereas
resulted
in little
trabeculae
orientation.
was
tensile stress, similar to
of
the
bone immediately adjacent to the
the
highly
organized.
The
3.1.2 demonstrated that in some
in
in
not
a
large change in the principal
other
change
instances
in
the
the
unaffected
principal
material
In summary, there was a separation of data into two groups
corresponding to the regions
to
The most important
the experimental specimens was that
example stereologic analyses in Section
material
These trabeculae were well
principal
stress
compressive
control
specimens.
implants
vectors
stress and the regions of
However, the dense
did not appear to correspond
predicted
for
the
experimental
specimens.
The relationship between the degree
of
A1
1pM
I)and
-
alignment (
the eccentricity of the stress
ellipse for the experimental calcanei is
shown in Figure 3.77.
to the control calcanei, the majority of
Similar
points where the alignment
stress
eccentricity.
patellae.
was
Again,
worse
than
45
degrees had a positive
this
was
in
control
and
experimental
contrast
For
both
the
trabeculae
were
more
often
component.
This is discussed further in Section 4.0.
The relationship between the
aligned
degree
eccentricity is shown in Figure 3.78.
- 311 -
with
of
the
to
the equine
calcanei,
compressive
the
stress
alignment and the material
Again, there was a tendency for
Alignment vs. Stress Eccentricity
Experimental
u-r
.
.
D 60T
h 40e
t
a ln%.
,L"
0
*
0
0
0
o
o0
0
0
0
0
0o
"I
$I
I
uT-
-1.0
-0.5
I
0.0
I
0.5
I
B
1.0
i.0
Stress Eccentricity
Figure 3.77: The absolute difference between the maximum principal
stress direction and the principal material direction ( pM - EAl!) as a
function of the eccentricity of the stress ellipse for the experimental
calcanei.
- 312 -
Alignment vs. Material Eccentricity
Experimental
804
S
D 60-
T
h 40e
t
0
0
0
0
a 20-i.
0
0O
0
0
0
A- LL_
V-l
0.5
0
0
-_
.__
I.--
_--_.-
0
0
-__
_ I
I.
_
-_.._.
-- -_ - _.
-
i
_
..-.
_ . _....
__
_
7
0.6
0.7
0.8
0.9
1.0
Material Eccentricity
Figure 3.78: The absolute difference between the maximum principal
stress direction and the principal material direction (IOp10 -IA1) as a
function of the eccentricity of the material anisotropy elipse for the
experimental calcanei.
- 313 -
worse alignment with lower
material
and experimental models of
the
stresses with the
principal
the
of
tended
regions
isotropic
trabeculae in more
patellae and the ovine calcanei
equine
alignment
the
demonstrated that
Both the control
eccentricities.
be
to
worse
than the
alignment in very highly oriented regions.
an effective stress measure and
To examine the relationship between
function of the
data, but
it still
applicable.
The
more
was
There
calcanei (Figure 3.79).
a
that
appeared
scatter in the experimental
single
linear relationship was
in
included
regression,
linear
was plotted as a
the control and experimental
both
for
density
areal
stress
Mises
von
the
density,
the trabecular bone
the
figure, had a
significance of 0.001, standard deviation of 1.50, and R 2 of 0.234.
value
To further examine the predictive
bone areal density was examined as
stress, the change in the trabecular
a function of the change
in
linear regression, included in the
standard deviation of 0.10, and
calcanei was equal to 0.027
the equine patellae.
Mises stress (Figure 3.80).
von
the
of the von Mises effective
The
figure, had a significance of 0.001,
R2
of
(MPa)- 1
0.18.
The slope for the ovine
in comparison to 0.108 (MPa) 1 for
However, these cannot be directly compared because
the slope is dependent on the load
magnitude.
As the load magnitude is
In summary, there was a weak linear
increased, the slope is decreased.
relationship between the change in trabecular bone areal density and the
change in von Mises effective stress.
Finally,
the
comparison
was
made
- 314 -
between
the
changes
in the
Von Mises Stress
vs. Bone Area Fraction
I
I
.
S
t
r
e
.
0
0D
.
.
.
0
O30
I
-U
I *
U
0o
s
S
a
a
·
*Experimental
.
0.2
OControl
---5.83X
0.3
0.4
0.5
0.6
0.7
+ 4.78
0.8
Area Fraction
Figure 3.79: Von Mises effective stress (MPa) as a function of the
trabecular bone areal density for both the control and the experimental
calcanei.
- 315 -
Change in Bone Area Fraction
vs. Change in von Mises Stress
A_
U .4F
o
0
0
1an
a
A 0.2r
rn
F
r 0.Ot
a
5*1:~~
oo
a
[°
U
00a
---
-2
-1
0
1
2
0.027X + 0.15
3
4
5
D Stress (MPa)
Figure 3.80: The difference in trabecular bone areal density between the
experimental and control calcanei as a function of the difference in von
Mises stress between the experimental and control calcanei.
- 316 -
orientation of the principal stresses and the changes in the orientation
of the trabeculae (Figure
reference.
As in the
these parameters.
3.81).
equine
The
models,
The changes
in
line
Y
=
X was included for
there was no correlation between
the trabecular orientation tended to
This
be greater than the changes
in the principal stress orientation.
was due to the influence of
the very dense bone immediately adjacent to
the implants as discussed above.
In summary,
implants did
not
the
response
remodeling
correspond
to
the
theory of trabecular architecture.
not
highly
organized
and
directions was unpredictable.
patellae, there was a
weak
the
influence
However,
very dense remodeled bone was
on
the
principal material
in agreement with the equine
relationship
areal density and the change in von Mises stress.
- 317 -
adjacent to the
of the trajectorial
predictions
This
linear
immediately
between the change in
Change inStress Orientation (Pi)
vs. Change inMaterial Orientation
D
jr-
13,
0
T
h 10e
t
a 5-
[]
0
L§33
0-
S
t
r
00
0
-5
m
[
e
s
s
[
0]
A i
- -Ii
, _
1
I
_
-60
-20
-40
0
20
0 Theta, Material
Figure 3.81: The difference in
between the experimental and the
difference in the direction of
the experimental and the control
the direction of principal stress P
control calcanei as a function of the
principal material orientation between
calcanei.
318 -
4.0 Discussion
The objective
of
this
investigation
was
to
examine the stress-
morphology relationships around implants of various material and surface
properties.
To
address
developed, one using
this
objective,
cylindrical
implants
other using spherical implants in equine
first objective was to
specimens based
stress
were
anisotropy.
validate
the
on
aligned
The second
the
assumption
with
two experimental models were
ovine
patellae.
calcanei and the
For each model, the
stress predictions of the control
that
the
in
the
directions
directions of principal
of
principal
material
was to examine the stress-morphology
objective
relationships in the experimental specimens based on the assumption that
the stress predictions were accurate.
One of the
equine
patellae
had
partially resolved with antibiotics.
thick fibrous tissue layer
with
no
an
This
bone
infection
which was at least
implant was surrounded by a
ingrowth.
This specimen was
included in the present study based on the assumption that the infection
was effectively isolated from the bone tissue.
It was also assumed that
the fibrous tissue effectively removed the implant mechanically and thus
was similar to a void.
It
is not known whether the infection had any
other direct effect on the remodeling
of the trabecular bone.
infections are relatively
clinical
common
important to have data on the
in
However,
orthopaedics, and it is
remodeling response in the presence of an
infection.
The assumption
that
the
trajectorial
- 319 -
theory
of
trabecular bone
While
load
parametric
would
studies
through
proper
gait may be represented by a
a
provide
complete
more
of a single loading
the cyclic loads during
selection,
single
participates in gait.
which
examination
for
representation, the justification
condition is that,
bone
any
for
true
This is particularly
challenged by
equine patella are time varying.
and
calcaneus
be
loads, and therefore the internal
considering that the applied external
stresses, on the ovine
can
specimens
control
the
for
architecture holds
load case.
"effective"
It is the
selection of this applied load which is probably the weakest link in the
analyses of the equine
loads on the equine
study
of
these
As
patellae.
patella
loads
discussed in Section 2.2.1, the
particularly
are
strengthen
might
demonstrated correspondence between
the
of
directions supports the validity
the
the
complex.
present
While further
analyses,
the
principal
material and stress
applied
loads and the finite
element predictions on the whole.
strengthened by the inclusion of
The present investigation could be
more rigorous controls.
One
appropriate control experiment would be to
drill the bone in preparation for an implant but close the wound without
inserting an implant.
a low modulus biologically inert
Alternatively,
material could be implanted.
However, an adaptive remodeling response
would be expected due to the effect of a void on the stresses.
control experiment would be to
implant
same modulus as the trabecular bone.
the
trabecular
remodeling would
A better
a porous material which has the
This would minimize the changes in
bone
stresses
due
to
the
implant.
The
be
primarily
due
to
the
biologic
effects
surgical procedure.
- 320 -
observed
of the
The ovine calcaneus proved to be
the
investigation
of
primarily due to the
a less than satisfactory model for
stress-morphology
coarse
trabecular
The selection of the regions
relationships.
This
was
architecture in the calcaneus.
for stereologic analysis required that the
trabecular bone be homogeneous and that there be a significant number of
trabeculae in each region.
The coarse architecture made this especially
difficult in the calcaneus.
Another
problem
with
the
ovine
remodeling period was too short.
to a 2
mm
distance
from
as
the
the
implants.
be
have
been
that the
In general, the trabecular
expected to correlate more closely
remodeling
experiments for the investigation of
should employ remodeling
may
The remodeling was apparently limited
architecture around implants can
with the stress state
model
periods
period
is extended.
Future
the long term response to implants
which
are
longer
than those in the
present study.
One likely
criticism
of
the
present
investigation
stress-morphology comparisons were limited
the planes which were examined
were
of the three-dimensional stresses.
was the critical plane for the
tendo calcaneus was in this
equine patellae also
were
is that the
to two dimensions.
However,
the most important planes in terms
It was clear that the sagittal plane
ovine
plane.
calcaneus since the load from the
The
approximately
maximum loads applied to the
in
the
sagittal
plane.
As
reported in Section 3.1.2, on the average, 80 % of the strain energy was
accounted for by considering only the
that by examining
only
the
most
sagittal plane.
critical
- 321 -
planes,
It can be argued
the likelihood of
valid
obtaining
correlations
stress-morphology
convincing
and
was
increased.
the
The mathematical similarities between
the
material
However, in general, this
ellipse.
between these parameters for
correspondence
A positive correlation between these parameters would
the present data.
It would also imply that the
patellae.
cortical shell of the equine
uniaxial stress would be
some locations near the anterior
for
true
was
This
of
in areas
bone
imply that the trabecular
highly oriented.
One
eccentricity of the stress ellipse to
the
the eccentricity of the anisotropy
comparison demonstrated no
comparisons,
axes, may be appropriate.
besides the orientation of the principal
such comparison is to relate
further
that
suggest
ellipse
anisotropy
Lame' stress ellipse and
trabecular bone in areas of pure shear stress would be nearly isotropic.
data in which
the
state
stress
reason that there was no
this point due to insufficient
on
inconclusive
The present models are
approached
pure
shear.
The primary
correspondence between the eccentricity of the
stress and material ellipses was that there were many instances in which
but
eccentric
the stress state was highly
the
material was not.
One
explanation is the previously discussed limitation of the examination of
a single load case.
in
The
also may significantly vary
morphologic response to such
vivo
stresses may be highly oriented but
in direction
conditions
during
the gait cycle.
The
may, very logically, result in
an isotropic architecture.
The
examination
of
important relationship.
material
The
eccentricity
of
models
- 322 -
did
demonstrate
one
the control specimens clearly
demonstrated
that
the
of alignment
degree
between
the
stress and
material directions is dependent on the degree of material eccentricity.
For trabecular bone which is nearly isotropic, the alignment tends to be
worse, since the morphologic
is subject
data
randomly
The anisotropy
oriented
and
orthogonal directions of oriented
of
trabecular
bone
is a circle both for regions with
ellipse
trabeculae
with
This
in the stereology methods applied in
also points to a weakness inherent
this study.
to greater error.
for
regions
trabeculae.
two
will
two
equal
and
Fortunately, in regions
orthogonal
trabeculae, one of these directions
with
directions
of
oriented
generally dominate.
This was
demonstrated using an example from the equine patellae in Section 3.3.2.
As the contribution from each of
and the material appears
more
the two directions becomes more equal,
isotropic,
the meaning of the predicted
direction of material orientation is diminished.
A very important finding from the
that
the
alignment
between
principal material axis
predominant stress.
was
There
models of the equine patellae was
the
maximum
better
in
were
material was aligned with the
also
regions
of
tensile stress component
patella.
One
possible
weaker in tension than
trabecular struts are formed
the
where tension was the
Stone et al. (1984) similarly
principal
a
is the
compression,
in the
explanation is that cross struts
material
three-dimensional
explanation
in
and
tensile stress component in regions where
the
for
stress
a number of instances where the
compression was the predominant stress.
found better alignment
principal
and
fact
- 323 -
formed
model
of the human
that bone tissue is
thus, to compensate, more
direction
are
direction with the
of
tension.
A second
to resist buckling of the
trabeculae which are in compression.
the importance
of
bending
and
Pugh et al. (1973a) demonstrated
suggested
effective stiffening mechanism.
This
that
cross
mechanism
struts were an
was supported by the
data points from the control patellae in which the stress state was near
uniaxial compression yet
the
The present data
direction.
both mechanisms
probably
was
material
oriented in the orthogonal
insufficient to draw a conclusion but
are
a
role
in determining the trabecular
from
the
morphology of the experimental
have
architecture.
The most striking finding
specimens were the increased trabecular
implants.
The least dramatic
areal densities adjacent to the
increase in areal density occurred around
the implants with successful porous ingrowth.
that the remodeling changes
were
at the bone/implant interface.
critically influenced by the stresses
In the equine patellae, the surface area
of the porous coated implants was
implants.
If
the
density
much
changes
response, rather than stress
adaptation,
The
a
result
of a foreign body
the changes around the porous
However, the correlation between
morphologic changes adjacent to the implants
the stress changes and the
techniques.
greater than that of the smooth
were
implants would have been much greater.
were not conclusive.
This supports the premise
This
may
stereologic
be
due
method
to limitations of the applied
requires
that
the
performed on circular regions of homogeneous architecture.
geometry results in only a
adjacent to the implants
result,
the
morphologic
essentially lost.
analyses be
The circular
small fraction of the trabeculae immediately
included
data
Furthermore,
in the
from
the
the
regions
of analysis.
bone/implant
interface
As a
is
finite element predictions at this
- 324 -
The
interface are inadequate.
assumption is no longer valid
continuum
due to the inhomogeneity of the trabecular architecture.
requires the use
of
Snyder et
of the human patella.
model
Their
the representation of pin-jointed truss
study, however, was limited
to
elements, which are clearly
inadequate
trabeculae.
these trabeculae.
to
application of structural optimization
a
One possibility is the
models.
microstructural
al. (1983) reported results from
widths of the implants
trabecular
trabecular remodeling within several
Examination of
the modeling of individual
for
More complex microstructural models of trabecular bone have
been developed (see
1.1.3),
Section
to
techniques
structural optimization
The
density and orientation.
these
but
predict
models
did not apply
changes
in trabecular
development of microstructural models of
stress adaptive trabeculae is a promising area for future research.
exhibited a region of very dense
Most of the experimental specimens
trabecular
bone
adjacent
immediately
exceptions were the two porous coated
equine patellae.
to
the
implants.
This dense region was poorly defined and discontinuous
well defined layer of
trabecular bone
compact
bone
However, there was a
around
the
smooth spheres in the
This finding was similar to the corticalization of the
around
implants
smooth
(Brown et al. 1984; Medige et al.
that there was a
only
spheres with bone ingrowth in the
around the porous cylinders in the ovine calcanei.
equine patellae.
The
plate
of
1982).
dense
described
of
the literature
Hedley et al. (1982) reported
trabecular
porous femoral surface components in canines.
important for the stabilization
in
prosthetic
bone adjacent to ingrown
This region is critically
components and thus the
mechanical parameters which govern the formation and maintenance of this
- 325 -
The present investigation did
dense layer of bone should be delineated.
not specifically address
finite element models and
was about 1
mm
region
this
the
around
thick
the
the
implants whereas the finite
smooth
mm
in each
finite
element
dimension.
models
the
circular
stereologic analyses had a diameter of
4.2
mm.
Analysis
of
dense
this
regions
for
the
As a consequence, this
However,
it is apparent from the
critical parameter controlling the morphologic
literature that the most
is the presence and magnitude of
response at the bone/implant interface
the
No
(see Section 1.2.2).
is invalid at the
requires the application of
layer
microstructural modeling techniques.
relative motion between
The continuum
portion of the images adjacent to the
dense layer occupied a very small
implants.
This dense layer
analyses.
Furthermore
microstructural
level.
limitations of both the
to
stereologic
elements were typically several
assumption implicit to
due
and
the adjacent trabecular bone
motion
should have occurred in the
implant
relative
present specimens since the implants were surrounded by trabecular bone.
of this layer around the smooth
This accounts for the minimal thickness
implants and the total lack of a compact layer around the ingrown porous
implants.
It
should
be
re-emphasized
that
the
present
investigation was
designed to study the structural aspects of the remodeling of trabecular
bone around implants.
The
objective
prove or disprove Wolff's Law.
stresses have a
critical
role
around implants, the degree to
of
this investigation was not to
the results do indicate that the
While
in determining the remodeling response
stress did not correlate with the
which
remodeling response should be interpreted
- 326 -
as either a reflection of the
model inaccuracies or an
indication
parameter.
important
that
the
exact
trabecular architecture.
do
relationship
If
the
avoid
possible
discussed
not allow for a conclusive
of
objective
the stresses and the
an investigation is to
design should be such that the
the presence of foreign materials, to
to
outside
influences
not be the only
previously
between
study Wolff's Law, then the experimental
induced remodeling is not due
may
the
Unfortunately,
limitations of the present investigation
statement as to
stress
the
realm
of
stress-adaptive
remodeling.
The results of this
should provide encouragement for
investigation
the further development of prostheses which
cement for fixation.
The morphologic response of the trabecular bone to
the implants was generally
a
positive
The only exception was due to an
For all specimens, both porous
infection in one of the equine patellae.
and smooth, there was
The
no
evidence
of
results
indicate
that
physiologic stress conditions
in terms of maintenance of
one
viable bone at the implant boundary.
implants.
do not rely on acrylic bone
net
bone resorption around the
reasonable
changes
from the
at
an
implant
Admittedly, the stress conditions
at
the bone/implant interface in the
present models were not
since
loaded.
extreme,
The experimental design
stress protection or excessive
however, provide
implants which
the
have
basis
extremes
did
for
the implants were not directly
not
stress
in stress
- 327 -
allow for the examination of
concentration.
future
expected for total joint replacements.
boundary are acceptable.
studies
This study does,
of remodeling around
conditions
such
as
may be
framework for future investigations
This investigation provides the
of stress-morphology relationships
design of the next
experiment
stress
at
conditions
trabecular bone.
One
the
and reattach the tendon to the
abductors may be
a
architecture as
well
bone/implant
interface
design
be
cylindrical
practical
as
the
The
employ implants with more severe
should
possible
insert a rectangular or
implant-induced remodeling.
in
would
and
surrounding
to detach a tendon,
implant into a metaphyseal region,
implant.
site.
normal
The greater trochanter and hip
However,
geometry
the normal trabecular
and loading conditions
should be carefully considered prior to experimentation.
- 328 -
5.0 Conclusion
The objective
of
this
investigation
morphology relationships for trabecular
there was a controlled and
conditions.
simplified
Cobalt
of
around implants for which
steel spheres with either a
various
cylinders
unilaterally
with
into
polished
coating were implanted unilaterally
then used to
quantify
experimental
specimens. and
the
performed
and surface
sintered-bead porous
ovine calcanei and stainless
surface or a sintered-bead porous
into
morphology
the
materials
a
equine patellae.
were maintained for periods of 10 to 24 weeks.
Structural analyses were
examine the stress-
experimental models were developed using
implants
chromium
coating were implanted
bone
to
predictable alteration in the stress fields.
For this purpose two different
geometrically
was
of
Stereologic methods were
the
untreated
The animals
trabecular bone in the
contralateral
controls.
using the displacement-based finite
element method to predict the stresses surrounding the implants.
There was a significant
remodeling
all of the experimental specimens.
increase in the bone areal
density
response around the implants in
The ovine calcanei had a significant
within
implants but no significant changes in
2
areal
mm of the ingrown porous
density beyond 2 mm.
The
observed remodeling in the equine patellae was a function of the implant
surface conditions.
The most
extensive remodeling occurred adjacent to
a porous coated implant which was encapsulated by a thick fibrous tissue
layer.
This implant had
a
partially
influenced the remodeling response.
around the smooth implants
was
treated infection which may have
In general, the remodeling response
greater
- 329 -
than
that around those porous
implants which
exhibited
bone
ingrowth.
differences, the finite element models
stresses adjacent to the smooth
In
accordance
with these
predicted greater changes in the
implants
due to the nonlinear boundary
conditions.
The finite element models were
stress
directions
specimens.
with
the
validated by comparing the principal
material
The critical assumption
trabecular architecture
regression between the
holds
the
the
control
specimens.
A linear
orientation and the principal
for
the ovine specimens due to a
A paired t test between the principal stress
orientation and the principal
was no significant
control
stress
material orientation was not possible
poor distribution of data.
in
was that the trajectorial theory of
for
principal
orientation
material orientation indicated that there
difference
between
these parameters.
the model predictions appeared reasonable
Furthermore,
based on visual comparison of
the principal stress vectors to the trabeculae in the control specimens.
The models of the control
0.87.
equine
In both models, the
stress component of
the
patellae
trabeculae
greatest
models, the alignment was better
were validated with an R=
were most often aligned with the
magnitude.
However, in the equine
in regions where tension predominated.
This provides support for the hypothesis that cross struts are formed to
resist buckling of the trabeculae which are under compression.
The
alignment
of
the
principal
trabecular directions in the
stresses
experimental
ovine
with
the
principal
calcanei could not be
evaluated satisfactorily.
The trabeculae
beyond 2 mm from the implants
appeared to be relatively
unaffected
the presence of the implants.
- 330 -
by
The
analyses
stereologic
to
adjacent
the
implants.
unsatisfactory
the
Overall,
model
the
for
gave unreliable
implants
bone which was formed within 2 mm
results due to the disorganized dense
of
the
ovine
calcaneus
of
investigation
proved
to
be
a
stress-morphology
relationships due to the coarse trabecular architecture.
relationships in the experimental
To evaluate the stress-morphology
patellae, the
orientation
of
the
trabeculae
were
compared with the
orientation of the principal stresses.
The critical assumption for this
comparison was that the predictions of
the finite element models of the
A linear regression between the
experimental specimens were accurate.
principal stress and material
However, there was no
principal
stress
orientation.
and
Through
orientations
correlation
the
two
between
changes
in
certain circumstances, small changes in
the
the
trabecular
principal
demonstrated a linear
relationship
between
direction
an R
= 0.89.
of
trabecular
demonstrated that, under
stress state may result in
orientation.
the
change
density and the change in von Mises effective stress.
provides support for the hypothesis
in
the changes in direction of
it was
examples,
large changes in the
resulted
Both models
in bone areal
This relationship
that the architecture of trabecular
bone corresponds to an optimal structure.
- 331 -
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