DISS. ETH Nr. 16901
New method to determine the Young’s
modulus of single trabeculae
A dissertation submitted to the
SWISS FEDERAL INSTITUTE OF TECHNOLOGY
ZÜRICH
for the degree of
DOCTOR OF SCIENCE
presented by
Silvio René Lorenzetti
Dipl. Phys. Universität Bern
Dr. phil.-nat. Universität Bern
born September 28 1974
citizen of Hallau, SH
accepted on the recommendation of
Prof. Dr. E. Stüssi, examiner
Prof. Dr. P. Niederer, co-examiner
Zürich, 2006
Abstract
Osteoporosis is one of the ten most common diseases in the world. In
Switzerland, the fracture risk of >50 year old women is 50 % and 14 % for
men respectively during lifetime. A better understanding of the stability of
the skeletal system is important.
The stability of bone depends upon material properties and geometry.
One of the major parameters of the material properties is the Young’s (elastic) modulus E. This value describes the deformation under loading.
Bone mainly consists of the two components collagen (type I) and calcium
apatite. The widely used technique to diagnose osteoporosis is dual energy
x-ray absorptiometry (DXA). With this technique the bone mineral density
(BMD) is measured. Due to the experimental technique, variations of the
calcium apatite are detectable, but changes of the collagen mesh are not
visible.
One substructure of bone is spongiosa or trabecular bone. Spongiosa is
present in the center of cubic bones like vertebra and in the ends of cylindrical
bones like femur or the ossa antebrachii. Spongiosa consists of a network of
trabeculae. A single trabecula has a diameter of 0.1 to 0.2 mm and a length
of about 1 to 2 mm. Various standard methods are known to measure the
stiffness of the entire spongiosa or bone at macroscopic scale. Furthermore
indentation tests are available to determine the elastic behavior of microscopic spongiosa samples. At macroscopic scale, the range of the Young’s
modulus given in the literature is 1 to 19 GPa. A standard test to measure the Young’s modulus of single trabeculae taking its exact geometry into
account has, to our knowledge, not been developed.
Therefore, the aim of this work is to develop a procedure to measure the
Young’s modulus of a single trabecula.
This procedure includes:
design of a testing device
sample preparation
experimental test
FE modeling and calculations
data reduction and interpretation
Based on a literature review, outlined in the first section of this work,
the following requirements were defined for the testing procedure: the sample preparation should not influence the mechanical properties, the shape
of the trabecula has to be taken into account, 3-point bending test device
with optical control of the deflection has to be built, natural network of the
spongiosa should serve as boundary condition, slow stress rate and constant
force should be applied.
In the second section of this work, the simulation of a 3-point bending
test of a single trabecula FE model is described in detail. The influence of
the shape of single trabeculae in axial direction and during a 3-point bending
test was studied. A cylinder, a double cone, a cosinusoidal and a in vitro
determined shape of a single trabecula were used as trabecular models. To
compare the models, the same length, mass and material properties were
assumed.
To simulate similar boundary conditions as in the natural network, two
blocks were attached at each side of the beam. The material of the blocks
was assumed to be similar to the material of the beam. A force of 0.5 N
was applied and the deflection, respectively the displacement was calculated.
Compared to the realistic shaped model, the mathematical models overestimated the deflection during bending test by a factor of ∼ 1.5. During
axial compression test, the displacement of the mathematical models was
underestimated by a factor of ∼ 1.55.
Based on the results of the modeling, a virtual material model taking
the shape of the trabecula into account, is proposed: a transversal isotropic
material with Eaxial = E/1.55 and Eradial = 1.5 · E. The trabecular network
can be modeled with cylindrical rods and consists of the virtual material
model.
The new method to determine the Young’s modulus of single trabeculae
is described in section three. This method takes the exact 3D geometry and
the behavior under loading into account. The preparation and the labeling
of the sample is described in detail. The best 3D information is obtained
with either in vivo labeled samples, demineralized samples or deproteinized
samples. Stacks of images were taken using a laser scanning microscope
(LSM) and a near infrared laser tunable in the range of 705 up to 980 nm.
The use of wavelengths in the near infrared allows an increased penetration
depth into the material and the laser is suitable for 2photon fluorescence
microscopy.
A special micro-positioning device was designed. This device allows sample positioning with an accuracy of ∼ 3.5 µm. A globe enables sample
rotation around the trabecular axis.
The force was applied normal to the optical axis with a thread, and the
deflection could be measured with the microscope.
Based on the image stack from microscopy, a CAD body of the trabecula
can be built using Amiraand Geomagic. FE calculations were performed
with ANSYS.
By comparing the FE modeling and the deflection of the experimental
3-point bending test, the Young’s modulus can be determined.
To obtain the accuracy of the method, two types of error calculations were
performed: First, a straight forward estimation according to Gauss which
includes the length, the force and the deflection, and second, FE calculations
with variation of the volume. The error resulting of the Gauss calculation was
dominated by the influence of the deflection. The FE calculations resulted
in a relative error of the Young’s modulus of 7.7 % assuming a relative error
of the volume of 5 %.
To validate the reconstruction of the volume, a thread with a diameter of
100 µm was imaged with the LSM. The same procedure as with the trabecula
was performed to obtain the CAD model of the thread. Compared with the
true thread volume, the reconstructed CAD body volume was 97 ± 4 %.
In summary, the expected relative error for E from the uncertainty of the
force, the length and the deflection is ∼ 5 % and from the uncertainty of the
volume is ∼ 8 %. The expected resulting error for the Young’s modulus is <
10 %.
To test the new method, one of the two main components of the bone,
the collagen was removed from a set of samples. Samples from the femur
of an adult sheep were treated with sodium hypochlorite (NaOCl) and were
studied in section four. The treatment with NaOCl is a standard method
to deproteinize bone, and therefore, to alter the Young’s modulus, without
changing the geometric properties or the inorganic part of the bone. The
analysis of a native (untreated) femural sample resulted in a Young’s modulus
of 15.0 ± 1.4 GPa. This is in the upper range of the published data in the
literature of 1 up to 19 GPa.
The treated samples showed a reduction of the Young’s modulus by a
factor of 15. Broz et al (1997) observed a decrease at macroscopic samples
by a factor of 2.7. It seems as if the Young’s modulus of our samples decreases
with the time of treatment in NaOCl.
Using DXA, no variation of the BMD would have been observed in this
type of samples. With the new method, changes of the organic part (collagen)
of the bone become visible.
It is concluded, that the new method including sample preparation, experimental 3-point bending test with optical control, determination of the
volume and FE modeling is suitable to determine the Young’s modulus of
single trabeculae.
Zusammenfassung
Osteoporose ist weltweit eine der zehn häufigsten Krankheiten. In der
Schweiz ist das Frakturrisiko für Frauen über 50 Jahren bei 50 % und bei
14 % für Männer. Deshalb ist ein vertieftes Wissen über die Stabilität des
Skeletts wichtig.
Die Stabilität von Knochen hängt von den Materialeigenschaften sowie
der Geometrie ab. Einer der wichtigsten Materialparameter ist das Elastizitätsmodul E. Dieser Wert beschreibt die Deformation unter Belastung.
Kochen besteht hauptsächlich aus den beiden Komponenten Kollagen
(Typ I) und Kalziumapatit. Um Osteoporose zu diagnostizieren wird die weit
verbreitete Röntgentechnik DXA (dual energy x-ray absorptiometry) verwendet. Mit dieser Technik wird die Mineraliendichte der Knochen gemessen.
Diese experimentelle Technik erlaubt Veränderungen vom Kalziumapatit zu
detektieren, jedoch sind Änderungen der Kollagenstruktur nicht sichtbar.
Eine Unterstruktur des Knochens ist die Spongiosa oder der trabekuläre
Knochen. Spongiosa kommt in der Mitte von kubischen Knochen wie Wirbelkörper oder an den Enden von Röhrenknochen wie Oberschenkelknochen
oder Unterarmknochen vor. Spongisa besteht aus einem Netzwerk von Trabekeln. Ein einzelnes Trabekel hat einen Durchmesser von 0.1 bis 0.2 mm und
eine Länge von 1 bis 2 mm. Verschiedene Standardmethoden sind bekannt
um die Steifigkeit von makroskopischen Spongiosa- oder Knochenproben zu
bestimmen. Zudem sind Stempeltests verfügbar um das elastische Verhalten
von mikroskopischen Proben zu bestimmen. Der in der Literatur bekannte
Bereich für das Elastizitätsmodul geht von 1 bis 19 GPa. Ein Standardverfahren für die Bestimmung des Elastizitätsmoduls einzelner Trabekel, unter
der Berücksichtigung der genauen geometrischen Eigenschaften ist unseres
Wissens noch nicht entwickelt worden.
Deshalb ist das Ziel dieser Arbeit die Entwicklung einer Messpozedur um
das Elastizitätsmodul einzelner Trabekel zu bestimmen.
Diese Messprozedur beinhaltet:
Design einer Testapparatur
Probenaufbereitung
Experimenteller Test
FE Modellierung und Kalkulationen
Auswertung und Interpretation der Daten
Basierend auf einem Literaturreview werden im ersten Kapitel dieser Arbeit die folgenden Bedingungen für die Messprozedur definiert: die Probenaufbereitung darf die mechanischen Eigenschaften der Proben nicht beeinflussen, die genaue Form der Trabekel soll einbezogen werden, Design einer
Apparatur um einen 3-Punkt Biegeversuch mit optischer Kontrolle der Auslenkung durchführen zu können, das natürliche Netzwerk der Spongiosa dient als Randbedingung, kleine Stressraten sowie eine konstante Kraft sollen
aufgebracht werden.
Im zweiten Kapitel werden Simulationen von 3-Punkt Biegeversuchen
eines einzelnen Trabekel-FE Modells im Detail beschrieben. Der Einfluss
der Form einzelner Trabekel in axialer Richtung sowie bei einem 3-Punkt
Biegeversuch wird untersucht. Als Modelle für Trabekel wird ein Zylinder,
ein Doppelkonus, ein kosinusoidales Modell wie auch eine in vitro bestimmte
Form eines einzelnen Trabekels verwendet. Um die Modelle zu vergleichen
wurden die gleiche Länge, Masse und Materialeigenschaften angenommen.
Um vergleichbare Randbedingungen wie im natürlichen Netzwerk zu simulieren, wurde auf jeder Seite des Balkens ein Block aus demselben Material
wie der Knochen angebracht. Eine Kraft von 0.5 N wirkte und die Auslenkung respektive die Verschiebung wurde berechnet. Im Vergleich zur realen
Form überschätzen die mathematischen Modelle die Auslenkung während
des Biegeversuchs um einen Faktor ∼ 1.5. Während dem axialen Kompressionsversuch, wird die Verschiebung von den mathematischen Modellen um
einen Faktor ∼ 1.55 unterschätzt.
Basierend auf den Resultaten der Modellierungen, wird ein virtuelles Materialmodell welches die Form der Trabekel miteinbezieht, vorgeschlagen: Ein
transversal isotropes Material mit Eaxial = E/1.55 und Eradial = 1.5 · E.
Das trabekuläre Netzwerk kann mit zylindrischen Balken bestehend aus dem
virtuellen Materialmodell modelliert werden.
Eine neue Methode um das Elastizitätsmodul einzelner Trabekel zu bestimmen wird in Kapitel drei beschrieben. Die Methode schliesst die exakte
3D Geometrie sowie das Verhalten unter Last ein. Die Probenaufbereitung
sowie das Labeln wird im Detail beschrieben. Die beste 3D Information
wurde entweder mit in vivo gelabelten Proben, demineralisierten oder deproteinisierten Proben erhalten. Stapel von Bildern wurden mit einem laser
scanning microscope (LSM) sowie einem infraroten Laser tunebar im Bereich von 705 bis 980 nm erhalten. Die Verwendung von Wellenlängen im
infraroten Spektrum ermöglichen eine Vergrösserung der Penetrationstiefe in
das Material. Zudem ist der Laser für 2Photonen Fluoreszenz-Mikroskopie
verwendbar.
Ein spezielles Mikropositionierungssystem wurde entwickelt. Dieses System erlaubt eine ∼ 3.5 µm genaue Positionierung der Probe. Eine Kugel
ermöglicht das Rotieren der Probe um die Achse des Trabekels.
Die Kraft wurde mit einem Faden rechtwinklig zur optischen Achse aufgebracht und die Auslenkung wurde mit dem Mikroskop gemessen.
Ein CAD Körper wurde aus einem Bilderstapel mit Hilfe von Amiraund
Geomagicgebildet. Die FE Kalkulationen wurden mit ANSYSdurchgeführt.
Das Elastizitätsmodul kann durch den Vergleich des experimentellen Biegeversuchs mit der FE Modellierung bestimmt werden.
Zwei verschiedene Fehlerrechungen wurden durchgeführt um die Genauigkeit der Methode abzuschätzen: Erstens eine analytische Fehlerrechnung
nach Gauss unter Einbeziehung der Länge, Kraft und Auslenkung und zweitens FE Kalkulationen mit Variation des Volumens. Der abgeschätzte Fehler
der Rechnung nach Gauss wurde vom Einfluss der Auslenkung dominiert.
Bei den FE Kalkulationen gibt ein relativer Fehler von ∼ 5 % des Volumens
einen relativen Fehler von ∼ 8 % für das Elastizitätsmodul.
Um die Rekonstruktion der Volumen zu validieren, wurde ein Faden mit
einem Durchmesser von 100 µm mit dem LSM aufgenommen. Dieselbe Prozedur wie bei der Rekonstruktion von Trabekeln wurde angewandt um das
CAD Modell des Fadens zu erhalten. Verglichen mit dem wahren Volumen
des Fadens war das Volumen des rekonstruierten CAD Körpers 97 ± 4 %.
Aufgrund der Unsicherheit der Kraft, der Länge und der Auslenkung
wird für E ein relativer Fehler von ∼ 5 %, und aufgrund der Unsicherheit
des Volumens ein relativer Fehler von ∼ 8 % erwartet. Der resultierende
Fehler für das Elastizitätsmodul ist < 10 %.
Um die neue Methode zu testen, wurde eine der zwei Hauptkomponenten,
die Kollagene bei einem Satz von Knochenproben entfernt. Diese Proben vom
Oberschenkel eines adulten Schafs wurden mit Natriumhypochlorit (NaOCl)
behandelt und in Kapitel vier untersucht. Die Behandlung mit NaOCl ist
eine Standardmethode um Knochen zu deproteinisieren, und somit das Elastizitätsmodul zu verkleinern ohne die geometrischen Eigenschaften zu ver-
ändern. Die Untersuchung einer unbehandelten Probe vom Femur ergab ein
Elastizitätsmodul von 15.0 ± 1.4 GPa. Das liegt im oberen Bereich der
publizierten Daten in der Literatur von 1 bis 19 GPa.
Die behandelten Proben zeigten eine Reduktion des Elastizitätsmoduls
um einen Faktor 15. Broz et al (1997) beobachtete bei makroskopischen
Proben einen Abfall um Faktor 2.7. Bei unseren Proben nimmt das Elastizitätsmodul scheinbar mit zunehmender Behandlungsdauer in NaOCl ab.
Bei der Verwendung von DXA wäre kein Unterschied in der Mineraldichte
dieser Proben zu beobachten. Mit der neuen Methode wurden jedoch die
Veränderungen des organischen Teils (Kollagene) des Knochens sichtbar.
Zusammenfassend ist es möglich mit der neuen Methode inklusive Probenaufbereitung, experimentellem 3-Punkte Biegeversuch mit optischer Kontrolle, Bestimmung des Volumens und FE Kalkulationen, das Elastizitätsmodul einzelner Trabekel zu bestimmen.
Contents
Motivation
16
1 Introduction
19
1.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2
Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2.1
Bone on nanoscopic scale . . . . . . . . . . . . . . . . . 22
1.2.2
Bone on microscopic scale . . . . . . . . . . . . . . . . 22
1.2.3
Bone on macroscopic scale . . . . . . . . . . . . . . . . 24
1.3
Conclusions for this project . . . . . . . . . . . . . . . . . . . 29
1.4
Aim of the project . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Modeling of 3-point bending test of
trabeculae
33
2.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3
2.2.1
Robustness . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.2
Models suitable for trabecular bone . . . . . . . . . . . 38
Results & discussion . . . . . . . . . . . . . . . . . . . . . . . 44
3 New method to determine E of single trabeculae
45
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.1
Preparation of the samples . . . . . . . . . . . . . . . . 45
3.2.2
Labeling using fluorochromes . . . . . . . . . . . . . . 46
3.2.3
Device for the measurements . . . . . . . . . . . . . . . 50
3.2.4
Measurement procedure . . . . . . . . . . . . . . . . . 56
3.2.5
Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.6
3-point bending in the elastic region? . . . . . . . . . . 60
3.3 Estimation of the error of the procedure . . . . . . . . . . . . 60
3.3.1
Reconstruction of the volume . . . . . . . . . . . . . . 61
3.3.2
Error calculation according to Gauss . . . . . . . . . . 62
3.3.3
Error estimation using FE calculations . . . . . . . . . 64
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 Young’s modulus of native and deproteinized samples
69
4.1 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 FE modeling of 3-point bending test . . . . . . . . . . . . . . 71
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 Conclusions & outlook
75
A Segmentation using Amira
79
B Building a CAD - Body using Geomagic
80
C Standard input file for ANSYS
82
D Samples from AO Institute Davos
84
Bibliography
85
Index
94
Index
95
List of publications 2004-2006
96
Thanks to ...
98
Curriculum Vitae
99
16
Motivation
The fracture risk of bone is influenced by several parameters. Apart from
age, gender, mobility and type of bone, the bone mineral density (BMD) and
the material properties are important factors. The mechanical properties are
influenced by the architecture, the structure and the quality of the bone
forming material.
Osteoporosis is one of the ten most common diseases in the world. According to the World Health Organization (WHO, 1993), the BMD is the
standard value to diagnose osteoporosis. BMD values which fall 2.5 standard deviations below the average for the 25 year old female are diagnosed
as ”osteoporotic”. Osteoporosis leads to a decrease in bone mass and to an
increase in the fracture risk of bone. The risk of a fracture during the life of
> 50 year old women in Switzerland is 50 % and 14 % for men respectively1 .
Considering the age of nowadays population, a better understanding of the
stability of the skeletal system is important. New methods of treatment,
studies of the development and the response of bone to interaction have to
be qualified and quantified if possible.
About 10 % of the hip fractures and > 50 % of the vertebral fractures are
thought to be atraumatic (Myers and Wilson, 1997). This epidemiology suggests that a time-dependent failure mode may be relevant to the etiology of
the atraumatic vertebral fracture (Yamamoto et al, 2006). These authors suggest that the irreversible deformities of bone may arise from time-dependent
loading.
Bone mainly consists of the two components collagen (type I) and calcium
apatite. The inorganic component calcium apatite is about 70 % of the
mass. The stability of the bone on macro- and microscopic scale depends on
1
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17
material properties (including the in- and organic constitutes) and geometry.
The same bone building materials and similar assembly of these materials
in cancellous and compact bone seem to make the mechanical parameters of
the two bone types similar at sub-millimeter scale.
The most widely used technique to measure the BMD is dual energy x-ray
absorptiometry (DXA) (Tuck and Francis, 2002). Due to this experimental
technique, variations of the calcium apatite are detectable, but changes of
the collagen mesh are not visible.
The architecture of the trabeculae in the bone spongiosa has been studied
by various experiments and model-calculations (cf. Cowin (2001)). The
spongiosa network consists of trabeculae. These bone beams have a typical
length of 1-2 mm and a diameter of ∼ 100-200 µm (see fig 1).
Figure 1: Human femur and the trajectories of the external force (Wolff,
1892).
18
The importance of trabecular architecture in maintaining the biomechanical integrity of bone tissue is well recognized. In bone architecture studies,
precise non-destructive imaging of trabecular bone, accurate characterization
of architecture, and quantification of functional measurements are paramount
to a better understanding of bone health (Borah et al, 2001).
A change in bone mass affects the architecture of the spongiosa bone.
However, it is not yet known whether a change in bone mass also affects
the mechanical properties of a single trabecula. Numerous studies have been
conducted to investigate the mechanical properties of spongiosa, but an accepted standard test procedure to measure the mechanical properties of single
trabuculae is still missing. We are interested if there is also a change in the
mechanical properties of single trabeculae due to a change in the organic part
of the bone.
Therefore, the aim of this work is to develop a new method to determine
the Young’s modulus of single trabeculae.
19
1
Introduction
The purpose of this section is first to give an overview of previous work and
second, to look at nano-, micro- and macro-scale of bone in more detail. The
focus of this work is on the microscopic scale of bone. However, part of the
conclusion of the two other scaling dimensions may affect the testing of bone
at microsopic level.
1.1
Overview
As observed by Wolff (1892), the inner architecture of bone adapts to external
influences. This can either have a positive or a negative influence on the
stability of bone.
One of the major parameters in material science is the Young’s modulus
or the elastic modulus. This parameter describes the behavior of material
under load. Standard tests to measure the elastic modulus for bone have
been developed (Reilly et al, 1974).
The mechanical and structural properties of trabecular bone samples and
the importance of trabecular architecture have been studied with µCT thoroughly by different authors (eg. Müller et al (1994), Ulrich et al (1999) and
Borah et al (2001)). Using FE analysis on a 3D open-celled model of trabecular bone, Beaupre and Hayes (1985) concluded that the material properties
of the surrounding solid phase and the stacking arrangement of the unit cells
are fundamental variables.
Differences in the mechanical properties of cortical and cancellous bone
were summarized by Guo (2001). Cancellous tissue is 20 to 30 % less stiff than
cortical bone tissue. The BMD does not explain this difference. Influence of
the structure on the mechanical properties is given by the organization and
20
orientation of lamellae and collagen material.
In cancellous bone, an osteon has a typical size of 0.2-0.3 mm. Trabeculae
have a mean diameter of 0.1 to 0.2 mm. In both types of bone, lamellae
(∼ 5 µm) are present. They run along the main axis of trabeculae, and
cylindrically around the osteocytes.
Bone is a viscoelastic material. Though, according to Linde et al (1991),
the mechanical properties hardly depend on the strain rate in the range
of 10−3 -10−2 s−1 where most experiments were performed. Hence in the
following, bone will be treated as an elastic material.
The mechanical properties of elastic material are described in linear range
by the law of Hooke:
σ =E·
whereas
σ: is the stress
E: the Young’s modulus
= ∆L/L0 , the strain, with L the length and ∆L = L2 − L1
In fact, the law of Hooke is usable for most elastic deformations as long
as the deformation is not too large, otherwise the deformation is not elastic
anymore (Landau and Lifschitz, 1991).
In general matrix form, the uni-axial stress is given by:
Tij = Cijkm · ekm
whereas
Cijkm: fourth rank tensor of the elastic coefficients
Tij : second rank cartesian stress tensor
21
ekm : second rank strain tensor
The behavior of the material under torsion (twisting due to an applied
torque) can be described with:
τ = Gγ
wheras
τ : shear stress at a point on the shaft
G: shear modulus (modulus of rigidity)
γ: angle of twist
The Poisson’s ratio ν is the ratio of transverse strain to the longitudinal
strain by uniaxial loading. This number has no dimension and is defined as:
ν=−
y
z
whereas
y,z : is the strain in y, z-direction (=∆L/L)
The three material constants E, G and ν are not independent in isotrop
materials due to the fact, that the main axis of the stress and the main axis
of the strain are the same.
E = 2G(1 + ν)
22
1.2
1.2.1
Previous work
Bone on nanoscopic scale
On a nanoscopic scale there are no differences between cancellous and compact bone. In fact, Niebur et al (2000) provide evidence that the elastic
and yield properties of trabecular tissue are similar to those of cortical bone
tissue. The small plate-shaped calcium-apatite crystals are a few tens of
nanometers (nm) long and about 2,5 nm thick. The orientation is similar to
the collagen framework.
The accumulation of in-vivo fatigue microdamage and its relation to biomechanical properties in ageing human cortical bone was studied by Zioupos
(2001). Surprisingly, the toughness properties are much more affected by
micro-crack density than either stiffness or strength.
The indentation modulus of a transversal slice from the distal end of a
humerus was found to be in the range of 2.9 to 1042 MPa (Dunham et al,
2005).
Nano-indentation tests (with depth of 1 µm) on human vertebra (Roy
et al, 1999) resulted in no significant differences in the mechanical properties
between different bone types. The types were classified according to their
porosity; 20 % as ”cortical” and 50 - 90 % as ”trabecular”.
1.2.2
Bone on microscopic scale
Keaveny et al (2001) report an anisotropy of modulus and strength of trabecular bone and suggest to perform the mechanical test in the orientation of
the bone and the sample. By analysing cylinderical bone samples of human
tibia from different locations, Goldstein et al (1983) found that trabecular
bone properties vary as much as two decades of magnitude from one location
23
to another.
In a review, Lucchinetti et al (2000) summarized the potentials and limitations of micro-mechanical testing of bone trabeculae. The range of the
Young’s modulus given in this review, is 1 GPa up to 15 GPa (see table
1). This wide range was explained to be due to difficult sample preparation,
handling and testing and furthermore due to the anisotropic and inhomogeneous bone material.
Goldstein et al (1983) showed that the modulus of the trabecular bone can
vary by 100-fold from one location to another within the same metaphysis.
This variation indicates that trabecular bone is very heterogeneous.
Three single trabeculae from dried human femural bone were loaded with
compression and tension by Bini et al (2002). The applied forces were in
the range of -0.25 up to 0.45 N. The deflection was measured using a laser
beam. The length, width and thickness of the trabeculae were estimated on
microscopic images. The irregular geometry of the trabecula was taken into
account by introducing a standard deviation of ±10 % for each size. The
calculated Young’s moduli were in the range of 1.41 to 1.89 GPa.
A tension test on single trabeculae of human vertebrae was performed
by Hernandez et al (2005). The trabeculae had a length of 3-4 mm. Small
rectangular brass holders were placed within the test claps of the substage.
The samples were glued to holders at each side. The measured typical force at
the ultimate failure point was 2-3 N. The ultimate strain was estimated based
on the force and the thickness of the trabeculae at the point of failure. It was
concluded, that the ductility of individual trabeculae varies tremendously,
can be substantial, and is weakly influenced by non-enzymatic glycation.
Litniewski (2005) proposed two methods to determine the elasticity coefficient of single trabeculae by scanning acoustic microscopy (SAM) approach.
24
The V (z) technique is the classic SAM method. The second method proposed
is using the distance between the V (z) curve first interference minimum or
maximum and the focus position. The author measured the surface wave velocity and the impedance. The pelvic bone samples were divided into three
groups: osteoporosis, osteomalacia and osteodosis with osteoporosis. The
resulting E-moduli were 26.8, 15.0 and 15.4 GPa, respectively.
The hard tissue stiffness of 28 human vertebral bodies was determined
by uni-axial compression tests (Hou et al, 1998). They found that the hard
tissue stiffness, varying from 2.7 up to 9.1 GPa, did not seem to be dependent
on age, body weight, sex or morphology. The authors concluded that the
trabecular hard tissue stiffness varies widely between individuals.
Linde and Hvid (1989) investigated the effect of different degrees of endconstraint at problematic failure of the samples at the boundary of the testing device. It was concluded that the mechanical behavior of the interface
between the trabecular bone and the surface of the testing machine is of
considerable significance for mechanical testing of trabecular bone in compression.
First micro-bending tests on single trabeculae using a 3-point bending
test, were performed by Lucchinetti (2003). The measured deflection of a
single trabecula under a load of 100 mN was about 1.1 µm. In a review,
Lucchinetti et al (2000) stated that an error of 10 % in the surface geometry,
mainly the variation of the thickness, is amplified to a 40 % error of the
elastic modulus.
1.2.3
Bone on macroscopic scale
According to Ryan and Ketcham (2005), the structure of the bone in the
femoral head of leaping primates is influenced by the locomotor forces in the
25
hip joint. Linde et al (1992) showed that the effect of specimen geometry
(length versus diameter L/D) on viscoelastic properties is less significant
than the effect on stiffness. Based on a structural mathematical model of
the viscoelastic anisotropic behaviour of trabecular bone, Kafka and Jı́rová
(1983) concluded that the viscous constituent must not be neglected but is
not supportive in the case of a static loading of 15 seconds and longer.
Cancellous bone was tested at macroscopic scale in cubes with a typical
length of five inter-trabecular spaces (Harrigan et al, 1988). This kind of
samples are highly anisotropic but can be modeled in good approximation as
a continuum. A misalignment angle of 10 degrees results in an error of 9.5
% in the Young’s modulus (Turner and Cowin, 1988).
Hobatho et al (1992) published an atlas with the mechanical properties of
the human cortical and cancellous bones based on in-vivo wave-propagation
technique. The variations of the elastic properties of osteon lamellae are
higher (40 %) at the microstructural level than those found at the macroscopic level (about 15%) for measurements performed in the same anatomical
direction.
Large scale FE model of cancellous bone were developed by various authors. Micro-computer-tomography allows to image the trabecular network
with a resolution of ∼ 30 µm. Yeh and Keaveny (1999) studied the influence
of the intra-specimen variation on the trabecular network with a 3D model including cylindrical trabeculae arranged as an irregular spaced lattice (Jensen
et al, 1990). An increase of the non-uniformity in the trabecular thickness
resulted in a decrease of the E modulus by 22 % up to 43 %. These authors
concluded that the architectural changes, which led to increasing thickness
variation, may result in more diminished mechanical properties compared
with those expected from uniform bone loss.
26
Stölken and Kinney (2003) proposed that trabecular bone is at large
scale a geometric nonlinear structure. Therefore, simulations of failure need
to include nonlinear geometric terms. A linear correlation between the bone
volume and the trabecular domain factor (TDF) was found by Tanaka et al
(2001). The TDF is a histomorphometric parameter and expresses quantitatively the structural non-uniformity by the ratio of the trabecular sizes to
the domain sites.
By comparing experimental data from uni-axial tension and compression
of human femoral neck using FE calculation, Bayraktar and Keaveny (2004)
concluded that the uniformity of apparent yield strains is primarily the result
of the highly oriented architecture that minimizes bending.
Performing an analysis of 12 rats’ femurs, Stenstrom et al (2000) found a
weak correlation of the BMD of the trabecular bone to the deflection in the
femural neck. However, a significant correlation between trabecular thickness
and femural neck bone strength was found. For cortical bone, the BMD seems
to be a good predictor of bone strength.
The development of the mechanical properties of dry bone during a week
was studied by Hengsberger et al (2001). A combination of atomic force microscopy and nano-indentation resulted in a stabilized level of the indentation
modulus after one day.
The cortical bone E modulus seems to be size-dependent; the modulus of
samples smaller than 0.5 mm was found to be significantly different compared
to samples bigger than 0.5 mm (Choi et al, 1990). A possible explanation
of this size-dependency is the presence of microstructural defects, such as
lacunae, near the surface of the specimen (Rice et al, 1988).
Trabecular network
The trabecular network was modeled as viscous with an immediate elastic
27
response by Kafka and Jı́rová (1983). In their study they used a Maxwell
body model. The trabecular bone was loaded with a step function. About
1/3 of the load is carried by the viscous matter. After about 15 s, the load
is carried by the trabecular network.
The influence of the curvature of trabeculae in a lattice on the E-modulus
was studied by Miller and Fuchs (2005). The authors compared perfectly
aligned orthogonal lattice with a random perturbation of node location lattice and a curved trabecular lattice. For these kinds of networks, even a
small curvature of the trabecular axes cause a large reduction of the effective
modulus of the trabecular network in the principal material direction.
Buckling studies of tabecular bone resulted in a variation of 24 % of the
Young’s modulus between wet and dry bone (Townsend et al, 1975).
Strain measurements of an aluminium foam using a three-dimensional
digital image correlation technique was performed by Verhulp et al (2004).
By using µCT with a resolution of 36 µm, the diameter of the specimen was
equal to 4 to 7 voxels. In this study, the aluminium trabeculae were even
thicker than real trabeculae. They suggested that the heterogenous internal
structure of the trabecular bone tissue would become visible with a higher
resolution in CT or synchrotron radiation.
The whole trabecular network can be modeled using an open-cell approach. In this type of model, each cube includes a spherical hole in the
center. Porous foam can be modeled with a similar method. The model
predict a higher stiffness of the network due to a lack of buckling of single
trabeculae (Beaupre and Hayes, 1985).
By using several phases of cortical and cancellous bone (fig. 2), Hellmich
et al (2004) suggested that hydroxyapatite, collagen, and water are tissueindependent phases, which define the elasticity of trabecular and cortical
28
bones, by their mechanical interaction.
Figure 2: Hierarchical organization of bone in terms of continuum micromechanics representation: a crystal foam (”polycrystal”) made up of hydroxyapatite (HA) crystals and non-mineral matter; b ultrastructure or solid bone
matrix made up of connected HA polycrystal matrix with cylindrical inclusions of collagen; c microstructure (cortical or trabecular bone) made up of
connected HA polycrystal matrix with cylindrical inclusions of collagen and
cylindrical micropores representing Haversian and Volkmann canals, or the
intertrabecular space (Hellmich et al, 2004).
Wet and dry bone showed similar behaviour in the response to small
amplitude mechanical excitation at frequencies from 100 to 3000 Hz (Pugh
et al, 1973a). Furthermore, the response of trabecular bone is purely elastic at
low strains and strain-rates in agreement with accepted rheological models
of cortical bone. The intertrabecular fluids and soft tissues contribute no
hydraulic strengthening or viscous component of any practical magnitude
29
for frequencies up to 3000 Hz. This results in practically negligible viscous
behavior for static and dynamic conditions.
To model the trabecular bone, a simple model with a hypothetical network of compact bone was built by Pugh et al (1973b). The authors suggested that the intertrabecular soft tissue has no influence on the mechanical
properties.
According to a literature review by Dagan et al (2004), the E-modulus of
adult human trabecular bone significantly varies by one order of magnitude.
The minimal reported value is 100 MPa and the maximal one is about 4
GPa. Even within specimen, large ranges and high standard deviations are
reported (Krischak et al, 1999). The reported E-modulus of trabecular tissue
are summarized in table 2.
1.3
Conclusions for this project
On both the macroscopic and the nanoscopic scale, standard procedures to
treat bone samples are well established. In terms of the microscopic scale of a
single trabecula, a standard method to determine the mechanical properties
is missing. Therefore, the aim of this study is to develop a method to measure
the Young’s modulus of single trabeculae. This aimed testing device should
satisfy the following requirements:
The sample preparation should not influence the mechanical properties
Exact determination of the shape of a trabecula
3-point bending test with optical control
Simultaneous measurements of the force and the deflection
The natural network as boundary condition
30
Constant value of the force
Slow stress rate
Enough time for the network to respond to loading in order to exclude
the influence of the viscosity
For the simulation of experimental bending tests, the exact determination
of the geometric parameters are essential. The influence of the force on the
deflection is low compared to the influence of the geometric properties.
1.4
Aim of the project
The aim of this project is to develop a new method to determine the Young’s
modulus taking the exact geometrical properties of single trabeculae into
account.
This includes:
1. Design a procedure to measure the Young’s modulus of a single trabecula
2. Building a measurement device
3. Definition of a standard procedure for the preparation of the samples
4. Experimental material tests
5. Analysis of the data
6. Error calculation of the procedure
7. Discussion and interpretation of the data
Reference
Runkle and Pugh (1975)
Townsend et al (1975)
Townsend et al (1975)
Ashmann and Rho (1988)
Ashmann and Rho (1988)
Rho et al (1993)
Rho et al (1993)
Mente and Lewis (1989)
Ryan and Williams (1989)
Kuhn et al (1989)
Kuhn et al (1989)
Choi et al (1990)
Protocol
Elastic Modulus
buckling
8.7 GPa (SDD3.2 GPa)
buckling
14.1 GPa
buckling
11.4 GPa
ultrasound
13.0 GPa (SDD1.47 GPa)
ultrasound
10.9 GPa (SDD1.57 GPa)
tensile test
10.4 GPa (SDD3.5 GPa)
ultrasound
14.8 GPa (SDD1.4 GPa)
cantilever bending 6.2 GPa (SDD1.8 GPa)
tensile test
0.76 GPa (SDD0.39 GPa)
4.16 GPa (SDD2.02 GPa)
3-point bending
3.03 GPa (SDD1.63 GPa)
3-point bending
4.59 GPa (SDD1.60 GPa)
dry human bone (n D9)
wet human bone (n D9)
human bone (n D53)
bovine bone (n D15)
dry human bone (n D20)
wet human bone (n D20)
human bone (n D9)
bovine bone (n D38)
human bone, old (n D29)
human bone, young (n D13)
human bone, old (n D20)
Further Information
31
Table 1: Measured values for the elastic modulus of single unmachined trabeculae and of trabecular material summarized by Lucchinetti et al (2000).
Numa
6 (6)
72 (2)
28 (28)
5 (5)
30 (1)
3 (1)
N/Ae (8)
7 (7)
12 (11)
b
Number of specimens or indentations and donors used in the study.
Module in [GPa]
c
Only range reported.
d
Specimens were dehydrated.
e
Information not available.
f
Endcaps were used to eliminate end artifacts in the apparent-level mechanical testing.
g
Endcaps were used to eliminate end artifacts in the apparent-level mechanical testing.
a
Anatomic site
Human femoral head
Human vertebra
Human vertebra
Human vertebra
Human distal femur
Method
Experiment-FEA
Nano-indentation
Experiment-FEA
Experiment-FEA
Nanoindentationd
Acoustic microscopy
Zysset et al (1999)
Human femoral neck Nano-indentation
Niebur et al (2000)
Bovine proximal tibia Experiment-FEA f
Bayraktar et al (2004) Human femoral neck Experiment-FEAg
Reference
Ulrich et al (1997)
Rho et al (1997)
Hou et al (1998)
Ladd et al (1998)
Turner et al (1999)
Etissue b
3.5–8.6 c
13.4±2.0
5.7±1.6
6.6±1.0
18.1±1.7
17.5±1.1
11.4±5.6
18.7±3.4
18.0±2.8
32
Table 2: A comparison of elastic moduli of trabecular tissue, Etissue
(mean±SD, in GPa), Bayraktar et al (2004).
33
2
Modeling of 3-point bending test of
trabeculae
2.1
Motivation
To build a powerful FE model of bone, the cortical and the spongiosa part
of the bone need to be included. For the spongiosa part, the simplest model
is a rectangular network with beams as trabeculae. In the following section
the influence of the shape of the trabecular model on a simulated bending
and compression test is outlined.
Dagan et al (2004) proposed a standard building-block model, based on
statistical data of dimensions of bovine trabeculae. They found a linear
relation between the maximal and the minimal thickness of single trabeculae,
and therefore the model is scalable with respect to thickness and length. With
this standard trabecula, the model allows to build a large-scale FE model of
the trabecular network.
The performance in 3-point bending test of standard trabeculae was compared with a cylinder, a double cone, the cosinusoidal and a measured volume
of a single trabecula.
2.2
Method
A spongiosa sample of an adult sheep vertebra was used to measure the
volume. The procedure of the sample preparation is described in the next
section. A laser scanning microscope (LSM) equipped with a 2photon laser
was used to generate an image stack of single trabeculae. The stack consists
of about 100 images and the size of the pixels is 1.11 x 1.11 µm2 . The distance
between the images is 5 µm. The following parameters were determined:
34
length L
volume V
smallest radius rmin
0.95 mm
0.0142 mm3
0.058 mm
Table 3: Parameters of the CAD volume. These values were used for all
trabecula models.
The segmentation procedure is described in detail in appendix A. The
standard procedure of the building of the CAD model can be found in appendix B.
The FE calculations were performed using ANSYS2 . As boundary condition for the bending tests, two big blocks were attached to each side of
the bone using ANSYS. The bone and the blocks were merged to a single
volume. The whole volume was assumed to be homogeneous, isotropic and
linear elastic. These assumptions are valid for small deflections. The material properties were set to common values for bone: Poisson’s ratio of 0.3 and
Young’s modulus of 18 GPa (Cowin, 2001).
For the simulation of a 3-point bending test, a force F = 0.5 N was evenly
distributed to several nodes. The deflection d is defined as the difference between the displacement of a point at the opposite side of the beam and of a
point in the center of the area beam-block (fig 3). To calculate the displacement, the average of several nodes were taken. As boundary condition, no
displacement at the outside of the blocks was assumed.
For the axial compression test (fig 8), a block with the same material
properties as the trabecula was added to the volume. A force of 0.5 N was
applied normal as a pressure on the top area of the volume. The displacement
d is defined as the difference of the displacement of nodes at the bottom of the
volume dm minus the displacement of nodes close to the boundary between
the block and the trabecula dblock (fig 3).
2
www.ansys.com
35
F
dm
dblock
Figure 3: Set-up for the simulation of the 3-point bending test. The force
is applied to several nodes. The displacement is defined as the difference
d = dm − dblock .
n
1
3
9
25
d
1.00000
0.99921
1.00000
0.99999
Table 4: Relation between the numbers of nodes to which the force was
applied and the resulting displacement. n number of nodes where the force
is applied, d normalized displacement.
36
2.2.1
Robustness
The applied force was evenly distributed to several nodes in the region next to
the center at the surface of the beam. This distribution is required because
the random selection of one single node can influence the result by 1.1 %
(Kaiser, 2006). The influence on the number of selected nodes (given in
table 4) is smaller than 0.1 %.
To estimate the robustness of the FE calculation, the following parameters
were studied:
force F
length L
radius r
boundary condition
A theoretical estimation of the displacement can be achieved using the
theorem of Castigliano (Sayir et al, 2004):
vk =
dU
dFk
(1)
where vk is the displacement at the position k, U is the total energy of
deformation of the state and Fk is the force at k. Equation (1) results in the
displacement of the 3-point bending test for a fixed beam:
L
1 Fy L3
vy
=
2
48 EπR4
(2)
Hence, the displacement varies linearly with the force, cubic with the
length L and to the power of -4 with the radius R.
37
The influence of these parameters on the deflection was studied by a simulated 3-point bending test using ANSYS. The results are summarized in
table 5. Not surprisingly, the theoretical estimation match the FE calculation.
r
L
F
normalized value
1.00
1.01
1.05
1.00
1.01
1.05
1.00
1.01
1.05
0.99
0.95
normalized deflection
1.00
0.96
0.84
1.00
1.03
1.14
1.00
1.01
1.05
0.99
0.95
Table 5: Influence of small changes (1 to 5 %) of radius r, length L and force
F on the deformation based on FE calculations by Kaiser (2006).
model
cylinder
double cone
cosinusoidal
CAD volume Fy
CAD volume Fz
d [µm]
3-point bending
8.86
9.38
8.01
5.09
5.89
axial
1.77
1.84
2.01
2.91
Table 6: Comparison of the deflection of single trabecula, in either axial
compression or 3-point bending test. d is the absolute displacement in the
direction of the force F = 0.5 N.
38
2.2.2
Models suitable for trabecular bone
Cylinder
The simplest model for trabeculae is a cylinder. The radius is given by
the length and the volume:
R=
V
Lπ
Figure 4: Result from the FE calculation of the deflection of a cylinder.
Double cone
Taking the variation of the radius into account, the double cone model
reflects the thinner part in the middle of the beam.
39
Rg is given by V , L und Rk :
Rg =
2V
− Rk2
πL
Figure 5: Double cone model.
Cosinusoidal model
A cosinusoidal model was used by Dagan et al (2004). These authors found
a linear correlation between the minimal and the maximal radius:
rmax = αrmin + β
with:
40
α = 1.3736
β = 40.9 µm
The radius r can be described as a function of the coordinate of the axis
of rotation z:
2z
arccos
r = ± cos
L
3
1
β
−
3−α−
rmin
2
rmin
2
The volume V can be calculated by straight forward disk integration.
V =
2π (2 rmid − β)2 1.375 L −
1.5 L sin ψ
ψ
+
0.125 L sin 2ψ
ψ
(1 + α)2
with
1
β(1 + α)
ψ = arccos
3−α−
,
2
2rmid − β
rmid =
rmin + rmax
2
To build the CAD model, a point cloud was generated using Matlab3 with
tave =148.5 µm. The point cloud is plotted in figure 6. The solid was then
built using the standard procedure in Geomagic.
Measured volume
Based on measured volume, a CAD model was built using Amiraand
Geomagic. The standard procedure and an estimation of the systematic
error are given in the next section.
To align the CAD model, the main axis and the center of mass were
determined and set to a new coordinate system. The irregular shape at the
edges was cut with two areas normal to the main axis. For the second time,
the main axis and the center of mass were used to set a coordinate system.
In this system, two areas were defined at the same distance from the center
3
www.mathworks.com
41
−500
0
500
100
50
0
−50
−100
−100
−50
0
50
100
Figure 6: A point cloud of the cosinusoidal model (Dagan et al, 2004).
of mass to cut the edges again. This is the reference body for the study.
The volume and the length of the CAD volume were calculated. The torsion
under load of the CAD body was calculated as αF y = 4.2and αF z = 0.8as
comparison, the torsion of the double cone model is 0.3. The definition for
small torsion γR 1 is satisfied (Landau and Lifschitz, 1991).
42
F
F
Figure 7: Deflection of the CAD model by applying the force from two different angles rotated by 90.
43
Figure 8: Deflection of the CAD model by applying the force in the axial
direction from the right. As boundary condition, the force was applied to
several nodes at the top surface, the displacement was set to x direction and
the left hand side was fixed.
44
2.3
Results & discussion
The results of the modeled 3-point bending tests are summarized in table
6. All mathematical models overestimate the deflection of the 3-point bending test by a factor of ∼ 1.5. During axial compression in contrast, the
displacement is underestimated by a factor of ∼ 1.55.
To keep a very simple model for the geometry, (eq. cylinder) the material
properties can be adjusted. Due to the axial symmetry, a virtual material
model for trabeculae can be proposed. The virtual material model can be
described as transversal isotrop material with:
Eaxial =
E
1.55
Eradial = 1.5 · E
The trabecular network can be modeled with cylindrical beams and consists of the virtual material model.
Changing the material properties may cause troubles at the nodes of the
network, but this seems to cause less trouble than adapting the radius of the
beams.
45
3
New method to determine E of single trabeculae
3.1
Introduction
The aim of the present study is to develop a method to determine the Young‘s
modulus of a single trabecula taking the exact geometry and the behavior
under load into account. The study involves the sample preparation, measurements, data reduction and the interpretation. Due to the restricted availability of human bone, trabecular bone from sheep was used. Thereby two
different types of sheep bone were used; first, in vivo labeled vertebrae provided from AO Institute (Davos, see appendix D) and second, native femur
(Tierspital Universität Zürich).
3.2
3.2.1
Method
Preparation of the samples
Initially, the frozen bones were cut into cubes (1x1x2 cm3 ) using a microbandsaw Proxxon MBS 240/E.
The cubes were then further processed using a kryostat type HM 505 N
(figure 9). The bone was removed in slices of about 5 µm. The aim was
to prepare a slice of approximately 5 mm with an intact trabecula located
on the surface. The trabecular network was cleaned from the bone marrow
with clean compressed air with 3 bar and an ultra-sound bath filled with
demineralized water. To maintain the mechanical properties of the samples,
the use of any chemicals not essential for the method as the labeling and
mechanical treatment was resigned. The chemicals used for labeling were
discussed later.
46
Figure 9: Frozen bone sample (-30) on aqueous non-permanent glue mounted
in the kryostat.
The sample holders to be used in the laser microscope are round plates
of aluminium. To minimize reflection of the light, the surface of the holder
was anodized in black color.
To glue the samples onto the sample holder, HYSOL EPOXY type 9450
was used. This glue was selected because of its stability at low temperature in
the kryostat, in the ultra wave bath and at room temperature. Furthermore,
the glue does not creep into the trabecular network.
3.2.2
Labeling using fluorochromes
Back in the 1770’s, John Hunter showed that bone is subject to both deposition and resorption using madder dye (Hall, 1992).
47
As observed by Bachmann and Ellis (1965) and Rahn (1976), bone has a
bright autofluorescence. This is a huge limitation for various applications in
bone microscopy. In terms of the present study, the light which is emitted
or even reflected by the bone is suitable for the imaging. Bone without any
labeling has a penetration depth of about 20-80 µm using a wavelength of λ
= 488 nm. This is approximately half of the diameter of a single trabecula.
The penetration depth is on the one hand depending on the intensity and
wavelength of the laser and on the other hand on the spectra of the emitted
and/or reflected light.
Figure 10: Left: Single trabecula of a native bone, length 1.5 mm, λ = 488
nm, penetration depth ∼ 70 µm. Right: the center and the far side of the
trabecula are not visible due to the limited penetration depth of the light.
The following fluorochromes are suitable for bone:
To enhance the penetration depth of bone, different fluorochromes (see table 7) which build a linkage with the bone, are suitable for labeling. Thereby,
two different labeling approaches can be used: an in-vivo labeling (see appendix D) or a labeling during the sample preparation process (table 8 and
9).
The performance of the labeling procedure can be enhanced by chemically removing either the collagen part or the calcium part of the bone.
max. λem [nm]
420– 440
610
517
624– 645
520– 560
520– 560
560
410– 550
460
550
b
1,2-bis(o-aminophenoxy)ethane-N,N,N’,N’-tetraacetic acid
4’,6-diamidino-2-phenylindol
c
2photon
d
2photon
a
fluorochrome
max. λexc [nm]
Calcein Blue
373
Xylenolorange (XO)
440/570
Calcein
494
Alizarin
530– 580
Doxycycline
390– 425
Rolitetracycline
390– 425
Tetracycline (TC)
390
BAPTAa
200– 325
700c
DAPI b
Lucifer Yellow (LY)
810 & 880d
Ref.
O’Brien et al (2002)
O’Brien et al (2002)
Suzuki and Mathews (1966)
O’Brien et al (2002)
Dhem et al (1972)
Blomquist and Hanngren (1966)
Hall (1976)
Pautke et al (2005)
Xu and Webb (1996)
Xu and Webb (1996)
48
Table 7: Possible fluorochromes for bone.
49
fluorochrome
Calcein
Alizarin
Calcein Blue
Calcein
Alizarin
Calcein
Alizarin
XO
TC
TC
Alizarin
XO
Alizarin
LY
c [mMol]
6.66
10
10
6.66
10
5
5
5
5
5
5
5
5
2.7
45
30
30
30
t
pd [µm]
Min.
30
Min.
55
Min.
35
Min.
55
2h
65
45 Min.
1.5 h
1h
60
80
1h
60
24 h
120
Table 8: Labeling of native bone. Concentration c, labeling time t in ultrasound bath and penetration depth pd in Amira; where no data is present,
the labeling was just at the surface.
The demineralization occured with ethylenediaminetetraacetic acid (EDTA)
and deproteinization with sodiumhypochlorite (NaOCl). Of course, both of
these procedures change the mechanical properties of the bone dramatically.
Therefore, the mechanical testing has to be finished before treating the samples with NaOCl or EDTA.
The performance of Lucifer Yellow (LY) on biological samples excited
with a 2photon system was studied by Selle et al (2005). These authors have
shown a detection limit of one fluorescence molecule per 0.1 µm2 on double
grating waveguide structures.
Finally, for the present study, the best results for native bone were achieved with LY, for decalcified bone with 4’,6-diamidino-2-phenylindol (DAPI)
and for deproteinized bone with tetracycline (TC) (table 10). The samples
were first treated in an ultrasound bath, and then stored in a 0.9 % saline
solution with the fluorescence added at a temperature of 5 for 2 days. Subse-
50
fluorochrome treatment
TC /Alizarin
NaOCl
TC /Alizarin
EDTA
XO / Alizarin
NaOCl
XO / Alizarin
EDTA
TC
NaOCl
TC
EDTA
Calcein
NaOCl
Calcein
EDTA
t
1 h US, 5 days
1 h US, 5 days
1/2 h US, 5 days
1/2 h US, 5 days
1/2 h US, 6 days
1/2 h US, 6 days
1/2 h US, 2 days
1/2 h US, 2 days
pd [µm]
160
70
40
50
160
70
30
20
Table 9: Labeling of bone treated with either EDTA of NaOCl. Concentration of the labeling solution was 5 mMol, labeling time t in ultrasound bath
and at a temperature of 5, penetration depth pd in Amira, where no data
is present, the labeling was just at the surface.
fluorochrome
LY
TC
DAPI
treatment
NaOCl
EDTA
c [mmol]
2.7
5
2
λexc [nm]
880
780
705
Table 10: Standard procedure reference for in vitro labeling of bone, native or
treated with either EDTA of NaOCl. λ is the 2photon-excitation wavelength
for maximal emission. The time for the labeling is 2 days.
quently, the samples were washed again in the ultrasound bath for 10 minutes
and stored afterwards in 0.9 % saline solution.
3.2.3
Device for the measurements
The device includes a laserscanning microscope, a micro-positioning system
and a force sensor.
Microscope
Due to the small sample size, a microscope is required to observe the deflection of the trabecula. The microscope used is a laserscanningmicroscope
(LSM) type Zeiss LSM 510 mounted on a confocal stand Axiovert 200 and
51
z
x
y
objectiv
optical axis
Figure 11: Definition of the coordinate system. The main axis of the bone
sample is the x-axis, the force is applied in direction of the y-axis. The z-axis
points towards the optical axis of the microscope.
52
equipped with:
He-Ne laser
Ar laser
Chameleon XR705-980 nm
The advantages of confocal laser microscopy were summarized in a review
about microdamage in bone by Lee et al (2003). Their results show:
the laser can be focused at a defined depth
thin optical section of the specimen can be taken
3D reconstruction of the section is powerful
The accuracy for the detection of micro-cracks is, according to Zioupos
(2001), approximately 10 µm.
Figure 12: Spectra of the 2photon laser of the Chameleonfamily made by
COHERENT. For higher power at wavelengths above 750 nm, the LSM was
equipped with the Chameleon XR.
53
Due to the limited penetration depth of the single photon laser system,
a 2photon laser was evaluated. Due to the longer wavelengths in the near
infrared, the penetration depth of the laser beam is increased. The advantages of 2photon lasers in optical microscopy for biological application were
summarized by Dixon (1998).
The advantages of the multi-photon microscopy compared with the classic
confocal technique are:
increased penetration depth due to longer wavelength
reduction of photobleaching
lower energy deposit because of the use of a short pulse laser
non-descanned detection (ndd) system can detect scattered fluorescence
often the samples are visible in more detail
The 2photon laser system was used for the 3D measurements of the volume. To minimize the experimental error of the measurement of the deflection, a short wavelength laser was used (λ = 488).
The ultimate limit on the spatial resolution of a microscope is set by the
light diffraction. The spatial resolution ∆x is given by (Novotny and Hecht,
2006):
∆x = 0.61
λ
na
whereas:
∆x: smallest resolvable distance
λ: wavelength of the light
na: the numerical aperture of the microscope objective according to the
Rayleigh criterion (table 11).
54
objective
magnification
Plan Neofluar
20
Epiplan Neofluar
10
Epiplan Neofluar
5
na
0.5
0.3
0.15
Table 11: Objectives used in this work. na is the numerical aperture.
Micro-positioning system
The developed micro-positioning system (MPS) device consists of a movable
xyz-table4 assembled in a globe (fig 17). The cubic bone sample is mounted
to the xyz-table (see figure 13). This set-up allows an accuracy of ∼ 3.5 µm
for the positioning of a trabecula in the focus point of the microscope. The
globe enables rotation around any angle in the focus point (fig 14); especially
around the axis of the trabecula. The MPS is also used to fix the sample
during the bending tests (see section 4).
Figure 13: xyz-table, the arrow indicates the bone sample holder. The xyztable allows movement in the range of 2x2x1 cm3 . Note the key as size
reference.
4
Note that the xyz-table is usually not aligned with the optical axis, x or the axis of
the force y.
55
Figure 14: Micro-positioning system mounted onto Axiovert 200 stand of the
Zeiss microscope. The xyz-table is positioned in the center of the globe.
Force sensor
The piezoelectric sensor was manufactured by KISTLER, type 9205 with a
sensitivity of 118 pC/N and a threshold of < 0.5 · 10–3 N. To minimize the
influence of temperature on the sensor, it was shielded with an isolating tube.
A micro-step motor with a resolution of 0.5 µm was used to apply the force
56
respectively the position of the force sensor (fig 15).
To apply the force, a nylon wire was used (fig 16). Concerning the knots,
we made good experience with Ethilon USP 5-0 and 6-0. This wire tears
apart before a simple knot is loosening. The knot was neither at the force
sensor nor close to the sample.
5
2
1
3
4
Figure 15: 1 Force sensor Kistler type 9205 equipped with a Ti hook. 2
Protection for the hook. 3 Linear stage driven by a 4 micro-step motor. 5
Thermal protection cocoon.
3.2.4
Measurement procedure
The motor and the force sensor were driven both by LabView5 . The same
set-up was used for the data acquisition. This system allows to define a target
force. In a loop, the actual force is measured and the length is controlled by
the micro-stepper. The following procedure was used to perform the 3-point
bending test including the measurement of the force and simultaneous the
triggering of the LSM:
imaging the non-deflected trabecula
input of the range of the charge amplifier, the target force and the
control parameters for the adjustment of the force
5
www.ni.com/labview
57
resetting the force sensor
activation of the micro-step motor until the target force is reached
trigger signal out for the LSM to start the acquisition of the image of
deflected trabecula
simultaneous measurement of the force
trigger signal in from the LSM after finishing the imaging
releasing the force
measuring the offset and the end of the measurement to calculate the
drift of the force sensor during the procedure
calculating the force using an average filter and the correction of the
drift
imaging trabecula to cheque that the deformation was reversible
Figure 16: Thread around a single trabecula to apply the force. The thread
has a diameter of 50 µm, λ=488 nm.
58
3.2.5
Imaging
To obtain the volume of the trabecula the following procedure was performed:
imaging a z-stack using a ∆z of ∼ 5 µm
importing the stack into Amira6
segmenting the images in Amira
building the CAD body in Geomagic7
exporting as .igs in ANSYS
The procedure for the segmentation is described in detail in appendix A.
Details for the building of the CAD model can be found in the appendix B.
The CAD model was used as body for the simulation of the experimental
bending test with ANSYS. The procedure was described in the previous
chapter 2. By iteratively adjusting the Young’s modulus within the material
model of ANSYS, the same deflection of the beam was achieved in the
simulation as in the experiment.
The images of the deflection were processed in Adobe Photoshop8 . In
the non-deflected image, the background area was selected with the magic
wand tool and deleted. The color of the image was inverted. With an opacity
of 70 %, the non-deflected processed image was copied on the deflected image.
Then the deflection was determined by counting the pixels using ImageJ9 .
6
www.amiravis.com
www.geomagic.com
8
www.adobe.com
9
http://rsb.info.nih.gov/ij/
7
59
2
1
4
3
Figure 17: Globe of the micro-positioning system. 1 aluminium globe 2 bone
sample mounted onto sample holder as part of the xyz-table 3 lens of the
LSM 4 force sensor shielded from temperature changes by foamed material.
60
force [mN]
0
745
994
1242
1492
0
deformation [µm]
0
2.7
7.2
10.8
12.6
0
Table 12: Comparison of the deformation by variant loading to check that the
3-point bending test takes place in the elastic region of the stress-strain-curve
3.2.6
3-point bending in the elastic region?
An important assumption in the FE modeling used for the proposed method
to determine E, is a linear elastic material model. Therefore, the deflection
should take place in the elastic region of the stress-strain-curve. The elastic
region can be determined by loading and releasing the beam several times.
If the samples reach the plastic region, irreversible deformation of the beam
occurs. An irreversible deformation is apparent if the unloaded positions
prior and after the test differ. The test was performed using untreated samples from femural bone of an adult sheep (fig 18). The deformation of the
beam after the testing was zero (table 12), hence, the measurements were
performed in the elastic region of the sample.
3.3
Estimation of the error of the procedure
The estimation of the error of a procedure is as important as the measurement itself, and all the relevant parameters have to be taken into account.
To qualify the reconstruction of the volume, a nylon wire was imaged and
reconstructed with the same procedure as for the bone samples. To estimate
the influence of the relevant parameters on E, two different types of error
analyses were performed. First, an error estimation according to Gauss, and
61
second FE calculation with varying input parameters where performed.
For the determination of the Young’s modulus E, the following parameters are necessary:
geometric properties of the trabecula
measured force
measured deflection
3.3.1
Reconstruction of the volume
To test the performance of the imaging system, a nylon thread was analysed.
According to the manufacturer, the diameter is 100 µm. The diameter of the
thread was determined using the LSM, λ = 488 nm, objective 10x, pixel size
0.38 x 0.38 µm2 . The measurement resulted in 96.5 ± 1.2 µm.
To measure the volume, z-stacks were taken, and the same procedure of
data reduction (with Amiraand Geomagic) as for the bone samples was
used. The reconstructed volume had a length of 400 µm. This volume was
divided along the x-axis into several pieces to be compared with the expected
volume (table 13). The reconstructed volume was used as CAD model for a
3-point bending test, and the deflection was compared with the deflection of
a cylindrical model.
In table 14, the measured volume burden is listed for two different directions, the y-axis or the z-axis. The statistic error of the deflection of the
measured volume is ∼ 8 %. This is due to the fact, that the dimensions in
the z-direction are more difficult to measure compared with the dimensions
in y, normal to the optical axis. The dimension normal to the optical axis is
measured with a good accuracy in the range of the wavelength λ. Because
62
x [µm] −x [µm]
200
200
200
0
200
100
0
200
175
175
175
25
150
50
150
75
175
10
average
% of the volume
94
103
98
89
95
99
99
99
100
97±4
Table 13: Comparison between measured volume of a thread and theoretical
volume assuming a radius of 48.25 µm.
y-axis
z-axis
relative deflection
0.964
1.088
Table 14: Relativ deflection dmeasured /dassumed between the measured thread
and the assumed cylinder model of the thread.
the deflection occurs normal to the optical axis (fig 11), the relevant dimension for the second moment of area can be determined more precise than
that one along the optical axis. Therefore, the error of the volume can be
assumed to be < 5-10 %. This is by far the biggest source of uncertainty in
the whole procedure. The influence of this uncertainty is studied in section
3.3.3. Hence, good labeling and sample preparation are the key to get precise
results.
3.3.2
Error calculation according to Gauss
Error calculation according to the Gaussian method is a powerful tool for
estimating the error propagation within an analytical function (Bevington
and Robinson, 1992).
Assuming a 3-point bending test (fig 19) with fixed areas at each side of
63
the beam, the Young’s modulus E is given by (Dubbel, 2005):
E=
kF L3
dIy
where:
k constant, depending on the boundary condition, here: 1/192
F force
L length
d deflection
Iy second moment of area
The results of a straight forward error calculation assuming a cylindrical
model and the values in table 15, according to Gauss are:
2
∂E
sF 2 = 2.75 · 1015 Pa2
∂F
2
∂E
sL2 = 2.47 · 1014 Pa2
∂L
2
∂E
sd2 = 6.86 · 1016 Pa2
∂d
where:
sF error of the force
sL error of the length
sd error of the maximal deflection
The result of the error calculation according to Gauss is E = 5.24 · 109 ±
2.67 · 108 . This is a relative error of the Young’s modulus of 5.1 %. This
64
relative error is dominated by the influence of the deflection.
k
F
L
d
Iy
adopted value
adopted error
1/192
0.5 N
1%
1 mm
1 µm
10 µm
1 µm
π(1.5 · 10−4 )4 /32
Table 15: Values used for the Gaussian error estimation.
3.3.3
Error estimation using FE calculations
If a method includes model simulation or other complicated calculations like
Fourier transformation, the variation of the input parameters can give further
information about the influence of the parameters. The variation of the
geometrical properties, especially the cross section of the sample, and their
influence on the Young’s modulus was studied using simulation of a 3-point
bending test. The set-up (fig 20) included a cylindrical beam and a block
with the same material model on each side as boundary condition. The
length was kept constant, but the radius was varied. The resulting E were
calculated (table 17). The influence on E due to area changes is linear in the
range of ± 20 %. Linear regression results in Erelative = 1.67Vrelative − 0.67.
Considering a relative error of the volume of 5 % , the relative error of the
Young’s modulus is 7.7 %.
65
rscaled /r
0.9
0.95
1
1.05
1.1
relative E
0.90
0.95
1.00
1.04
1.11
Table 16: Variation of the radius along the optical axis, and the influence on
the Young’s modulus, calculated with FE simulation.
area
0.81
0.88
1.00
1.08
1.21
relative E
0.70
0.81
1.00
1.13
1.37
Table 17: Variation of the area of the cross section, and the influence on the
Young’s modulus, calculated with FE simulation.
66
3
1
2
Figure 18: Deflection of single trabecula due to a force of 994 mN compared
with 0 N. 1 Trabecula unloaded; 2 thread lined up the direction of F ; 3
deflected trabecula with an applied force of F =994 mN in inverse colors with
a deflection d=7.2 µm. The pixel size is 1.8x1.8 µm2 , and the wavelength λ
= 780 nm.
67
F
d
L
Figure 19: Set-up for the bending test for the error calculation according to
Gauss.
F
F
r
rz
ry
Figure 20: Set-up for the bending test for the FE error calculation. Left:
Variation of the radius. Right: Variation of the excentricity of the cross
section. Bottom: Two blocks of the same material were attached to the
beam.
68
3.4
Conclusions
Due to the experimental setup, the dimension of the trabecula, in the plane
normal to the optical axis can be determined with an accuracy of ∼ λ.
The dimension in the direction of the optical axis is much more difficult to
measure. Bending occurs in the plane normal to the optical axis. This setup reduces the influence of the uncertainty of the dimension of the trabecula
on the second moment of area. To quantify the influence, the following
simulations were performed. A cylindrical beam with two blocks at each side
was testet. The length and the diameter in the plane of the bending were kept
constant whereas the radius of the cylinder in z direction was scaled. A test
showed that E is linearly proportional to radius variation (table 16). Using
a state of the art tracking algorithm, the performance of the measurement
of the deflection can be raised and the error of the deflection is below λ.
The performed measurements took place in the elastic region of the stressstrain-curve. Therefore, the assumption for the linear behavior of the Young’s
modulus is justified.
In summary, we expect from the uncertainties of the force, the length
and the deflection a relative error of ∼ 5 % and from the uncertainty of the
volume a relative error of ∼ 8 % on E. This results in a total error for the
Young’s modulus of < 10 %.
69
4
Young’s modulus of native and deproteinized
samples
In this section, the measurements of the deflection using the laserscanning
microscope (LSM), the micro-positioning system (MPS) and the force sensor are described. Samples of femural adult sheep spongiosa were step-wise
decollagenized with NaOCl to alter the elastic properties of the bone. The
results are presented at the end of this section.
4.1
Samples
The samples were taken from the collum femoris of an adult sheep. These
samples were not in vivo labeled. The sample preparation was performed
according to the standard procedure described in the previous section. To
simulate different qualities of bone, the samples were treated with sodiumhypochlorite (NaOCl, 7.5 % solution) which dissolves the organic component.
NaOCl does effectively remove collagen fibres at the 1-2 µm level of the
cortical bone structural hierarchy (Broz et al, 1997). The inorganic part of
the bone is not disturbed by this procedure (Otter et al, 1988). The Young’s
modulus of the bone changes with the time of the treatment. Therefore,
different time steps with 0, 15, 45 and 60 min of treatment with NaOCl were
performed (table 18).
First, the experimental 3-point bending test was performed (table 18).
The maximal force was adapted to the state of treatment. To prepare the
samples for the labeling, the time of treatment in NaOCl was equalized to
one hour. The labeling with TC for two days and the building of the CAD
bodies were described in the previous chapter.
70
sample
N13
N15
N17
N18
tot
F [mN]
0 min
1245
15 min
248
45 min
120
1h
147
deformation [µm]
7.0
12.7
18.7
19.1
Table 18: Results of the experimental 3-point bending test. The samples
were treated with 7 % NaOCl. tot is the time of treatment. F ist the highest
tested force. All samples were from femoral adult sheep spongiosa.
Figure 21: Mesh of the two blocks and the CAD model of trabecula N13,
based on 10000 polygons.
71
4.2
FE modeling of 3-point bending test
Based on the built CAD bodies, the 3-point bending tests were simulated with
ANSYS. As boundary condition, two blocks made of the same material were
attached on each side of the trabecula (fig 21). The force was distributed
among several nodes. By modifying the Young’s modulus, the same deflection
as in the experiment was iteratively obtained (fig 22). This procedure allows
a quantification of the Young’s modulus. The Poisson’s ratio was assumed
to be 0.3. The element type was solid Tet 10 node 187, the mesh was built
using the smart size option. A standard input file is given in appendix C.
Figure 22: Deflection in y-direction of the sample N13. The FE calculation
is typically based on 400k equations.
72
4.3
Results
The results of the 3-point bending test are listed in table 19. The Young’s
modulus for the femur of an adult sheep was found to be 15.0 ± 1.4 GPa.
This lies in the upper range of published data of 1 up to 19 GPa. With longer
treatment in NaOCl more collagen is dissolved, and the elastic properties of
the material change (fig 23). After 60 min in NaOCl, E is reduced by a factor
of 15. Thereby, the time dependence of the Young’s modulus seems to be
exponential.
18
Young's modulus [GPa]
16
14
12
10
8
6
4
2
0
lit. range
native
15 min
45 min
60 min
Figure 23: Young’s modulus E [GPa]. Comparison between the range in the
literature and the measured data. tot, the time of the treatment with NaOCl
is plotted on the x-axis.
73
sample
N13
N15
N17
N18
tot
E [GPa]
0 min 15.0 ± 1.4
15 min 6.6 ± 0.6
45 min 2.95 ± 0.27
1h
0.95 ± 0.09
Table 19: Results of the experimental 3-point bending test. The treatment
occured in 7 % NaOCl. The given error of E corresponds to the relative error
of 9.2 %. tot is the time of treatment.
tot
E [GPa]
control
1 day
3 day
7 day
21 day
12.8±1.7 10.8±1.3 10.8±0.3 6.71±0.17 4.71±1.50
Table 20: Young’s modulus [GPa] of cortical samples, 3.6*3.3*40 mm, treated
with 5.25 % NaOCl (Broz et al, 1997). tot is the time of treatment.
74
4.4
Discussion
The determined Young’s modulus of a single trabecula of femural spongiosa
of an adult sheep is 15.0 ± 1.4 GPa. This is in the upper range of the
published data of 1 up to 19 GPa.
The decrease of E in our samples treated 60 min in a 7 % NaOCl solution
is a factor of 15. Broz et al (1997) observed a decrease of the Young’s modulus
from macroscopic cortical bovine bone samples, treated for 21 days in a pH
balanced 5.25 % NaOCl solution, by a factor of 2.7 (table 20). The depletion
of E in the microscopic samples is large compared to the macrosopic samples.
This can be explained first with a different content of NaOCl in the solution
and second with a propitious ratio surface to volume to the deproteinization
of the microscopic samples.
75
5
Conclusions & outlook
The determination of the mechanical properties of bone is in the context
of osteoporosis and the aging population a major field of study. The tissue
bone, including the substructure spongiosa and corticalis is well understood
at the macroscopic scale as well as at the nanoscopic scale. However, to
our knowledge, the mechanical properties, especially the Young’s modulus,
of single trabeculae has not yet been determined accurately. Therefore, the
aim of the presented study was to develop a method to measure the Young’s
modulus.
The new method to determine E includes an experimental 3-point bending test conserving the natural boundary condition to obtain the deflection,
imaging of the labeled samples with a LSM equipped with a 2photon laser,
reconstruction of the volume of the trabecula to a CAD body, FE modeling
of the bending experiment, resulting in the Young’s modulus.
The analysis of a 3-point bending test using a FE model of a single trabecula led to the following conclusions:
1. Boundary conditions: a block of the same material as the trabecular
model to simulate the natural network gives robust solutions.
2. The force should be applied on several nodes to minimize the influence
of the randomly chosen nodes in the center of the beam.
3. Currently used mathematical models based on linear elastic material
models and cylinder, overestimate the deflection of the trabecula during
3-point bending test by a factor of ∼ 1.5 compared to our vertebra
sample from an adult sheep. For axial compression in contrast, the
displacement is underestimated by a factor of ∼ 1.55. Hence, a new
76
virtual material model is proposed. The single trabecula is modeled as
cylindrical beam with transversal isotrop material properties. Based
on the presented results, the Young’s moduli of the material can be
defined as Eaxial =
E
1.55
and Eradial = 1.5 · E.
The volume has a strong influence on the Young’s modulus and is difficult
to measure. Therefore, sample preparation including labeling, is crucial for
an accurate estimation of the trabecular volume. A nylon thread with a
diameter of ∼ 100 µm could be reconstructed with an accuracy of a few
percent. Hence, in the future, the sample preparation and measurement of
the volume are the main factors to get precise results.
In the scope of the present study, a Gaussian error calculation and a
variation of the input parameters for the simulation were performed. The
Gaussian error calculation resulted in a typical error of the force of ∼ 5 %
due to error of the deflection and less due to error of the length. Simulations
of the experimental set-up resulted in a linear relation between variation of
the cross section of the sample and E. For an anticipated error of the volume
of 5 %, the error of E becomes ∼ 8 %. Taking both error calculations into
account, the approximated relative error of the Young’s modulus is < 10 %.
During the experimental tests, only small deformations in the elastic region of the samples were observed. Therefore, Hooke’s law is valid and a
linear material model can be used for the FE calculations.
Analysis of samples of the femural spongiosa of an adult sheep resulted
in a Young’s modulus of 15.0 ± 1.4 GPa. This is in the upper range of the
published data of 1 up to 19 GPa. Hence, the new method seems to be
suitable to determine Young’s modulus.
To test the new method, one of the two main components of the bone,
the collagen, was removed from a set of samples. Samples of the femural
77
spongiosa of an adult sheep were treated with a 7 % NaOCl solution to
dissolve collagen. After one hour of treatment, the Young’s modulus was
reduced by a factor of 15. By altering the time of treatment, the influence
of NaOCl on the Young’s modulus was studied. It seems that the Young’s
modulus decreases exponentially with the time of treatment.
Using DXA, no variation of the BMD would have been observed in this
type of samples because neither the volume nor the calcium apatite is altered
during the treatment with NaOCl. With the new method, changes of the
organic part (collagen) of the bone become visible.
This sensitivity of the new method allows in future to study the influence
of collagen in the bone. Furthermore, development of the Young’s modulus of
single trabeculae during lifetime can be measured. The effect of interventions
in therapy on the material properties can be studied using animal models.
The new method including sample preparation, experimental 3-point
bending test with optical control, determination of the volume and FE calculation is suitable to determine the Young’s modulus of single trabeculae.
Outlook
Further studies can be done on the test procedure itself or on different samples. The test itself can be modified either to simplify the method for a
faster analysis or to increase the accuracy of the test. Different samples can
be analysed for a better understanding of the mechanical properties of the
bones during lifetime, and between individuals.
Some ideas are presented here:
2D analysis instead of 3D stack to speed up the procedure
motorisation of the testing device
78
including a tracking algorithm to measure the deflection
compression test for the role of calcium apatite
tensile test for the role of collagen
atlas of Eaxial and Eradial of all human bones
study of trabeculae from different orientation within one sample
methods to take samples in vivo
variant samples from different donors
79
A
Segmentation using Amira
Save data and network
Open tif images
(1) Save network as
(2) Define path
(3) Autosave for the other data
(1)
(2)
(3)
(4)
open
select all images
choose Channel 1
define size of the image and
the distance
Add 2D view
(1) OrthoSlices on tif data
Segmentation
Create point cloud
Export of the body
(1) Based on .tif data
add LabelField
(2) Magic Wand tool
select area with grayscale
for each slide
(3) Deselect what is not used
(4) close
(1) SurfaceGen on labels data
set Smoothing on
unconstrained Apply
(2) SurfaceView on labels data
(1) VRML-Export based on
surf data
(2) Define folder to save
(3) Toggle on Simple mode
(4) Export as .wrl-file
80
B
Building a CAD - Body using Geomagic
Building the CAD body
Poligon Phase
Open the wrl-file from
Amira, set number of
triangles
(1)
(2)
(3)
(4)
Select RMB Select trough
Mark the middle part
Select RMB Reverse selection
Delete
Cut the edges
Fill holes
(1) fill holes Fill All
(2) 2x Yes
(3) Ok
(1)
(2)
(3)
(4)
(5)
Relax
toggle on fix boundary
smoothness level 5 - 10
Strength 5 – 10
Ok
Smoothing the surface
(3) + (4) depending on demands
Additional corrections of the
surface
(1) Sandpaper
Strength in the range of
0.01 – 0.1
toggle on Clean
toggle on Fix Boundary
(2) LMB auf selected areas
(3) Select und delete for hard cases
Create the empty body
(1) Fix Intersection
81
Create the mesh of the body
Shape Phase
Detect Curvature
(1) toggle on Auto Estimate
(2) toggle on Symplify Contour Lines
(3) Ok
Define the mesh
Define the border
Smooth the mesh
Problem areas
areas with extreme high
density of the net
Correct the mesh (zone of
problem is the border)
(1)
(2)
(3)
(4)
Promote/Constrain
Select LMB
Deselect Ctrl + LMB
Ok
(1) Construct Patches
toggle on Optimize Vertex
Degree
toggle on Autoestimate
Ok
(2) Relax Contour Lines
(3) Relax Linear
(1) Back to Shape Phase
(2) Use Sandpaper
(3) Same procedure again (new mesh)
Repair Patches
(1) Edit Vertices
toggle on intersection paths
moves point until the red
lines disappear
(2) toggle on Display Edit Vertices
(3) Move edge points to the border
(4) Correct the angles
(5) Ok
Hard cases
other possibility to
correct the mesh
Create the CAD surface
(1) Shuffle Patches
(2) Possibility to move, delete, insert,
or merge lines
Set the nurbs
(1) Construct Grids
(2) Ok
Save the CAD body as
igs file
Fit Surface
(1) Tension set 0
82
C
Standard input file for ANSYS
Modeling a 3-point bending test including a measured CAD body.
FINISH
/PREP7
/CLEAR
A=250 B=300
Importing the CAD Body:
RECTNG,-200,200,-200,200,
/AUX15
AGEN, ,55, , , A,,, , ,1
IOPTN,IGES,NODEFEAT
RECTNG,-200,200,-200,200,
IOPTN,MERGE,YES
AGEN, ,56, , , -B,,, , ,1
IOPTN,SOLID,YES
BTOL,DEFA
IOPTN,SMALL,YES
VSBA,1,55
IOPTN,GTOLER, DEFA
VDELE, 2, , ,1
IGESIN,’N15-10000’,’igs’,’ ’
VSBA,3,56
VPLOT
VDELE, 1, , ,1
BTOL,DEFA
Defining a new coordinate system
in the center of mass:
Defining the two blocks:
LOCAL,11, 0, 439.07, 130.96,67.613,
L=A+B R=50
-2.641,15.765,2.504, 1, 1,
CYL4,0,0,5*R, , , ,L/5
DSYS,11,
VGEN, ,1, , ,A , ,, , ,1
CSYS,11,
CYL4,0,0,5*R, , , ,L/5
WPCSYS,-1,11,
VGEN, ,3, ,,-B-L/5 , ,, , ,1
WPAVE,0,0,0
VADD,1,2,3
WPSTYL, STAT
SAVE
Definition of the material:
ET,1,SOLID187
Cutting the volume at each side:
MPTEMP,,,,,,,,
83
MPTEMP,1,0
F=120000
MPDATA,EX,1,,18e3
F,8091,FY,F/5
MPDATA,PRXY,1,,0.3
F,8088,FY,F/5
MSHAPE,1,3D
F,8076,FY,F/5
MSHKEY,0
F,8090,FY,F/5
F,8096,FY,F/5
Meshing:
SMRT,10
Iteration step, variation of the
CM, Y,VOLU
Young’s modulus:
VSEL, , , , 4
/solu
CM, Y1,VOLU
MPDELE,EX,ALL
CHKMSH,’VOLU’
MPDATA,EX,1,,2.95E+3
CMSEL,S, Y
SOLVE
VMESH, Y1
FINISH
CMDELE, Y
CMDELE, Y1
Post-processing, plotting the re-
CMDELE, Y2
sults:
/POST1
Setting the boundary conditions:
PLDISP,0
/solu
/PNUM, NODE, 1
DA,18,ALL,0
PLNSOL, U,SUM, 0,1.0
DA,26,ALL,0
84
D
Samples from AO Institute Davos
Procedure of the in vivo labeling
Animal number: A12/4043
12.10.01 birth
15.6.05 63 ml CG
22.6.05 63 ml CG
29.6.05 63 ml XO
6.7.05
63 ml XO
8.7.05
obitus
Table 21: Labeling coding: CG Calcein Green, XO Xylenol Orange.
85
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Index
2photon laser, 52
in vivo labeling, 84
Amira, 58, 79
kryostat, 46
ANSYS, 34, 58, 82
labeling, 46
BMD, 16
Micro-positioning system, 54
MPS, 54
Ca-apatite, 22
multi-photon microscopy, 53
CAD model, 40
cancellous bone, 19
NaOCl, 49, 69
compact bone, 19
ndd, 53
Cosinusoidal model, 39
osteon, 20
cylinder, 38
Osteoporosis, 16
demineralization, 49
outlook, 77
deproteinization, 49
penetration depth, 53
double cone, 38
Poisson’s ratio, 21
EDTA, 49
Robustness, 36
error estimation, 60
samples, 45
fluorochrome, 46
shear modulus, 21
force sensor, 55
spongiosa, 17
fracture risk, 16
testing device, 29
Geomagic, 58, 80
Wolff, 17
Hooke, 20
Young’s modulus, 20
ImageJ, 58
Zeiss LSM 510, 50
95
List of publications
2004
Busemann H., Lorenzetti S. and Eugster O. 2004 Solar Noble Gases
in the Angrite Parent Body — Evidence from Volcanic Volatiles Trapped
in D’Orbigny Glass. Lunar and Planetary Science XXXV, Abstract #1705,
Lunar and Planetary Institute, Houston (CDROM).
Gnos E., Hofmann B. A., Al-Kathiri A., Lorenzetti S., Eugster
O., Whitehouse M. J., Villa I., Jull A.J. T., Eikenberg J., Spettelo B., Krähenbühl U., Franchi I. A. and Greenwood R. C.
(2004) Pinpointing the Source of a Lunar meteorite: Implications for the
Evolution of the Moon. Science 305, 657-659.
Christen F., Busemann H., Lorenzetti S. and Eugster O. (2004)
Mars-ejection ages of Y-000593, Y-000749, and Y-000802 (paired nakhlites)
and Y-980459 shergottite. NIPR Symp. Antarct. Meteor., XXVIII, 6-7.
2005
Lorenzetti S., Busemann H. and Eugster O. (2005) Regolith history
of lunar meteorites. Meteorit. Planet. Sci. 40, 315-327.
Eugster O. and Lorenzetti S. (2005) Cosmic-ray exposure ages of four
Acapulcoites and two differentiated Achondrites and evidence for a twolayer structure of the acapulcoite/lodranite parent asteroid. Geochim. Cosmochim. Acta, 69, 2675-2685.
Lorenzetti S. (2005) Was macht ein Physiker mit einer Flasche Falken?
Litteris et amicitiae, 106, 5.
2006
Hofmann B. A., Lorenzetti S., Eugster O., Serefiddin F., Hu D.,
Herzog G. F. and Gnos E. (2006) The Twannberg, Switzerland IIG iron:
new findings, CRE ages, and a glacial scenario. Meteorit. Planet. Sci., sup,
41, A78.
Lorenzetti S., Oberhofer K., Sprecher C. and Stüssi E. (2006)
Comparison of performances in a bending test between a real volume and
an ordinary cylindrical FE model of a singel trabecula. Abstracts 5th World
Congress of Biomechanics. 5607.
Busemann H., Lorenzetti S. and Eugster O. (2006) Noble gases in
D’Orbigny, Sahara 99555 and D’Orbigny glass – Evidence for early planetary
processing on the angrite parent body. Geochim. Cosmochim. Acta 70, 3,
5403-5425.
Eugster O., Lorenzetti S. Krähenbühl U. and Marti K. (2006) A
study of possible precompaction exposure of chondrules and a procedure for
the partitioning of the cosmogenic, radiogenic, and trapped noble gas components. Geochim. Cosmochim. Acta, submitted, .
Special Thanks to
Prof. Dr. E. Stüssi, as supervisor of this work.
Chr. Sprecher and N. Goudsouzian (AO Institute Davos) and
Prof. B. von Rechenberg (Pferdeklinik Uni Zürich) for providing the bone
samples and discussions.
Dr. H. Gerber, P. Schwilch and M. Hitz for technical support.
Dr. J. Denoth and Dr. A. Stacoff for supporting me.
C. Hauser & K. Oberhofer for assistance in the lab and with the manuscript.
P. Hess (hess innovations) for fruitful discussions.
M. Kohler (zeiss CH) for the support with the LSM.
The following students for their practical courses and bachelor works:
J. Zahnd, R. Semadeni and A. Kaiser.
The crew from the office E357.2, who made life easier.
E. Swann for stimulating discussions.
My family for the support in all kind of ways.
My girlfriend and friends for being patient with me.
Diana for the good fortune.
Curriculum Vitae
Name
Silvio Lorenzetti, Swiss citizen, Hallau (SH)
Date of birth
28. 9. 1974
Education
2004-2006
PhD thesis project at Institute for Biomechanics, ETHZ
2005
Spezialfach Fitness TUS, ETHZ
2000-2003
Dr. phil.-nat. UniBe. Dissertation: Auswurfalter und Bestrahlungsgeschichte der Meteorite von Mond, Mars und
Asteroiden anhand von Edelgasisotopenanalysen.
1995-2000
Diplom-Physiker UniBe (mathematics, astronomy)
1998
Exchange term at University of Strathclyde, Glasgow, UK
1989-1994
Kantonsschule Schaffhausen, Typus C
Experience
2005-2006
Part time teacher at Kantonsschule Im Lee Winterthur
2003
Post-doc, Weltraumforschung & Planetologie, UniBe
since 2002
Educator at Star Education, school for training and recreation
1999-2003
Assistant at Physikalisches Institut Bern
MLT OS 2/95
Awards
2006
ISB Student Dissertation Award
2002
Student Award Recipient of the Meteoritical Society
Others
Swiss Champion 2006, Winterthur Warriors, American Football
Vice Swiss Champion 2004, powerlifting up 90 kg, SDFPF
Swiss Champion 2002, powerlifting up 82.5 kg, SPC
Hunter