Cyclic Deformation and Fatigue
Behaviour in Cancellous Bone
zur Erlangung des akademischen Grades eines
DOKTORS DER INGENIEURWISSENSCHAFTEN (Dr.-Ing.)
der Fakultät für Maschinenbau
der Universität Paderborn
vorgelegte
DISSERTATION
von
M.Sc. Sebastian Dendorfer
aus Regensburg
Datum des Kolloquiums: 11.01.2008
Erster Gutachter: Prof. Hans Jürgen Maier
Zweiter Gutachter: Prof. David Tylor
Publications
S. Dendorfer, J. Hammer and H.J. Maier, Strain evolution during compressive fatigue of
bovine cancellous bone, edited by J. Hozman and P. Kneppo, IFMBE Proceedings of the
3rd European Medical & Biological Engineering Conference, EMBEC 05, November
20-25 2005, Prague, Cz, 11(1), No. 2628
S. Dendorfer, J. Hammer and H.J. Maier, Deformation behaviour of bovine cancellous
bone, Technol Health Care. 2006;14(6): 549–56
S. Dendorfer, H.J. Maier and J. Hammer, How do age and anisotropy affect the fatigue
behaviour of cancellous bone?, Studies in Health Technologies and Informatics, accepted
S. Dendorfer, H.J. Maier, D. Taylor and J. Hammer, Anisotropy of the fatigue behaviour
of cancellous bone, Journal of Biomechanics, in press
i
Contents
Contents ii
Abstract iv
Kurzfassung vi
Nomenclature viii
1 Introduction 1
1.1 Background................................. 1
1.2 Objectives.................................. 2
2 Trabecular Bone: Morphology and Mechanics 5
2.1 StructuralProperties ............................ 5
2.1.1 BoneComposition ......................... 5
2.1.2 Morphology of Cancellous Bone . . . . . . . . . . . . . . . . . . 9
2.2 Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Monotonic Loading . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 CyclicLoading........................... 17
2.3 Age-RelatedChanges............................ 21
2.4 Summary .................................. 24
ii
CONTENTS
3 Materials and Methods 25
3.1 Specimens.................................. 25
3.2 TestingConditions ............................. 28
3.3 DeformationAnalysis............................ 29
3.4 DataAnalysis................................ 29
4 Results −Mechanical Behaviour 31
4.1 Specimen Characterisation . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Monotonic Compression . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 CyclicCompression............................. 35
4.3.1 Deformation Behaviour . . . . . . . . . . . . . . . . . . . . . . 35
4.3.2 Damage Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.3 Lifetimecurves........................... 61
4.4 Age-RelatedEffects............................. 64
5 Discussion 68
5.1 ExperimentalSet-Up ............................ 68
5.2 Deformation Behaviour and Damage Mechanisms . . . . . . . . . . . . . 69
5.3 Lifetimes .................................. 82
5.4 AgeEffects ................................. 87
5.5 ClinicalRelevance ............................. 89
6 Conclusion 91
Bibliography 93
Appendix 109
iii
Abstract
The fatigue behaviour of materials is of particular interest for the failure prediction of mate-
rials and structures exposed to cyclic loading. For trabecular bone structures only a few sets
of lifetime data have been reported in the literature and structural measures are commonly
not considered. The influence of load contributions which are not aligned with the main
physiological axis remains unclear. Furthermore site and species dependent relationships
are not well described. In this study seven different groups of trabecular bone, defined in
terms of orientation, species and site were exposed to cyclic compression. In total, 108
fatigue tests were analysed with respect to lifetimes, deformation behaviour and damage
accumulation. Furthermore, damage mechanisms were derived from novel measurement
methods for the optical strain analysis on the apparent and tissue level. The lifetimes were
found to decrease drastically when off-axis loads were applied. Additionally, species and
site strongly affect fatigue lifetimes. While the characteristics of cyclic deformation were
found to be similar for all groups, large deviations were observed for the fatigue lifetimes.
Bovine specimens did reveal higher lifetimes compared to human samples and lifetimes de-
creased with increasing deviation of the specimens’ axis from the physiological bone axis.
Already small deviations cause a large reduction, whereas deviations above 45◦result in
a similar fatigue behaviour. Strains at failure were found to be dependent on specimen
orientation (with respect to the physiological bone axis). The whole cyclic deformation
process as well as damage evolution until defined failure could be shown to be a function
of normalised stress and group. The corresponding functional relationships were derived.
Damage acceleration was found to be constant for all specimens and different damage
mechanisms are acting for on-axis and off-axis groups. Likewise, load thresholds were
found, at which damage mechanisms change from low-cycle to high-cycle fatigue. Age
iv
Abstract
appeared to have a large influence on the initial modulus of the specimens. Deformation
analysis on the apparent and the trabecular level could be linked to macroscopic damage
and microdamage was found to contribute to residual strain accumulation. Concluding, the
axis of loading appears to contribute dominantly to fatigue and cyclic deformation, which
may be even more pronounced in cases of increased anisotropy (Osteoporosis). Therefore,
local morphological information has to be included in risk of fracture predictions in order
to achieve a higher reliability.
v
Kurzfassung
Das Ermüdungsverhalten von Materialien ist von grosser Bedeutung für die Beurteilung
des Versagensrisikos von Bauteilen und Strukturen die zyklischer Belastung ausgesetzt
sind. In Analogie zu technischen Bauteilen können Ermüdungsscäden auch bei Knochen
beobachtet werden. Im Gegensatz zu kortikalen Knochen, bei dem das Ermüdungsverhal-
ten weitgehend untersucht ist existieren nur sehr wenige Daten für spongiösen Knochen. In
allen verfügbaren Studien wurde die starke Anisotropie der Knochenstruktur nicht berück-
sichtigt und nur Proben entlang der physiologischen Achse verwendet. Weiter ist der Ein-
fluss von verschiedenen Spongiosa-Arten (human, bovin) sowie des Alters noch nicht gek-
lärt. In dieser Studie wurden sieben Gruppen von Spongiosaproben, insgesamt 108 Proben,
unterschiedlich bezüglich der Spezies (human, bovin), des Entnahmeortes (Wirbelkörper,
distale Femurkondyle) sowie der Abweichung von der physiologischen Hauptachse hin-
sichtlich des Verformungsverhaltens unter zyklischer Druckbelastung untersucht. Neben
der Analyse des integralen Materialverhaltens wurde ein optisches Verformungsmesssys-
tem verwendet, mit dessen Hilfe die Entwicklung von Dehnungen auf der Probenoberfläche
sowie auf Knochenbälkchenebene gemessen wurde. Proben deren Hauptachse nicht mit
der physiologischen Hauptachse übereinstimmte zeigten stark reduzierte Lebensdauern.
Ebenso haben Spezies sowie der Entnahmeort einen grossen Einfluss auf das Ermü-
dungsverhalten. Dies war auch der Fall wenn die unterschiedlichen Anfangssteifigkeiten
der Proben berücksichtigt wurden. Die Rinderproben zeigten eine höhere Ermüdungs-
festigkeit als die menschlichen Proben. Die Lebensdauern reduzierten sich mit grösseren
Abweichungen von der physiologischen Hauptachse. Bereits kleine Winkelabweichungen
resultierten in einem starken Abfall der Ermüdungsfestigkeit, bei Abweichungen grösser
45◦wurden annähernd gleiche Lebensdauern gemessen. Die Versagensdehnungen sind
vi
Kurzfassung
ebenfalls gruppenabhängig und steigen mit steigender Abweichung von der Hauptachse.
Das zyklische Verformungsverhaltens konnte ebenso wie die Entwicklung der integralen
Schädigung in Abhängigkeit der normalisierten Spannung (σ/E0) und der Gruppenzuge-
hörigkeit dargestellt werden. Die entsprechenden mathematischen Zusammenhänge wur-
den abgeleitet. Verschiedene Schädigungsmechanismen wurden für die Gruppen parallel
zur Hauptachse und die Gruppen mit Abweichungen von der Achse gefunden. Das Alter
der Spender hatte einen grossen Einfluss auf die Anfangssteifigkeit der Proben und somit
auf das zyklische Verformungsverhalten sowie die Lebensdauern. Ein weiterer Einfluss
des Alters konnte jedoch nicht gezeigt werden. Die optische Verformungsmessung konnte
erfolgreich auf die spongiösen Proben angewendet werden. Die hierbei gefunden lokalen
Dehnungskonzentrationen, sowohl auf der (integralen) Probenoberfläche als auch auf der
Oberfläche einzelner Trabekel konnten mit dem integralen Verformungsverhalten korreliert
werden. So kann gezeigt werden, dass die Entwicklung bleibender Verformungen direkt
mit der Bildung und dem Wachstum von Mikrorissen verknüpft ist. Zusammenfassend
konnte der starke Einfluss der verschiedenen Spongiosa-Arten auf die Versuchsergebnisse
gezeigt werden. Bereits geringe Abweichungen von der physiologischen Hauptachse resul-
tieren in einer starken Abnahme der Ermüdungsfestigkeit. Bedingt durch den Anstieg der
Anisotropie in älteren Knochen wird dieser Effekt noch weiter verstärkt. Deshalb ist ger-
ade bei der Abschätzung des klinischen Versagensrisikos die Berücksichtigung von Struk-
turkennwerte unabdingbar um die Genauigkeit der Methoden zu erhöhen. Mit den abgeleit-
eten Funktionen für die Entwicklung der Verformungen sowie der Schädigung und zyklis-
cher Belastung stehen wichtige Werkzeuge zur Verfügung, welche direkt in numerischen
und analytischen Studien verwendet werden können.
vii
Nomenclature
˙
ε............... Strainrate,s−1
σ. . . . . . . . . . . . . . Nominal or 1st Piola-Kirchhoff stress, MPa
σc. . . . . . . . . . . . . . Compressive strength, MPa
σf. . . . . . . . . . . . . Fatigue strength coefficient
ε. . . . . . . . . . . . . . . Nominal strain, mm/mm
εf. . . . . . . . . . . . . . Fatigue ductility coefficient
εmax . . . . . . . . . . . . Maximum strains (accumulated total strains), mm/mm
εmax f . . . . . . . . . . Maximum strains at failure (D=0.1), mm/mm
εmax t . . . . . . . . . . . Maximum strains at end of the transient, mm/mm
εres . . . . . . . . . . . . . Residual strains (accumulated residual strains), mm/mm
εres f . . . . . . . . . . . Residual strains at failure (D=0.1), mm/mm
εres t . . . . . . . . . . . . Residual strains at end of the transient, mm/mm
E0. . . . . . . . . . . . . . Initial secant modulus, MPa
N/Nf. . . . . . . . . . . Normalised cycle numbers
Nf. . . . . . . . . . . . . Number of cycles to failure (D=0.1)
a . . . . . . . . . . . . . . . Fatigue strength coefficient, Equation 4.14
viii
Nomenclature
B . . . . . . . . . . . . . . Basquin exponent
b . . . . . . . . . . . . . . . Fatigue strength exponent, Equation 4.14
BMD . . . . . . . . . . . Bone Mineral Density, g/ccm
BMU . . . . . . . . . . . Basic Multicellular Unit
C . . . . . . . . . . . . . . Fatigue ductility exponent
c, d . . . . . . . . . . . . . Regression coefficients for Equation 4.2
D . . . . . . . . . . . . . . Damage parameter: 1 −(Esec/E0)
d . . . . . . . . . . . . . . . Density, g/ccm
e, f . . . . . . . . . . . . . Regression coefficients for Equation 4.3
g, h . . . . . . . . . . . . Regression coefficients for Equation 4.4
i, j . . . . . . . . . . . . . . Regression coefficients for Equation 4.5
in vitro . . . . . . . . . Latin: in the glass
in vivo . . . . . . . . . . Latin: in the living
k, l . . . . . . . . . . . . . Regression coefficients for Equation 4.9
LM . . . . . . . . . . . . Light Microscopy
m, n . . . . . . . . . . . . Regression coefficients for Equation 4.10
MIL . . . . . . . . . . . . Mean Intercept Length
PBA . . . . . . . . . . . . Physiological bone axis
SEM . . . . . . . . . . . Scanning Electron Microscopy
ix
1 Introduction
1.1 Background
In recent years, increasing research efforts were made in the field of biomechanics and
bone mechanics. These activities are mainly driven by social and economical needs. The
economic burden (only direct costs) in Germany attributable to Osteoporosis alone was
estimated as high as euros 5.4 billion based on data from 2003 (Haeussler et al., 2007). As
demography changes with the consequence of a higher amount of elderly people as well
as the increase in the individual lifespan, life sciences will be confronted with major chal-
lenges (Borgström et al., 2007). Additionally, the level of activity of older people nowa-
days is high, which results in high requirements on implants and prostheses. The trend of
degeneration in tissue quality (in the body) with increasing age sets demand for steady im-
provements. Furthermore, the timespan in which patients are bedridden in general should
be reduced in order to improve the outcome of healing, but also to reduce health-related ab-
senteeism. Besides these rather ambitious and noble aims, from a scientist’s point of view,
the fascination of this reseach field lays in the complexity of the problems, requiring a
thorough interdisciplinary cooperation of (among others) medics, biologists and engineers.
For example, the idea of a self-maintaining (and optimising) spatial structure is especially
attractive for structural engineers and therefore one of the reasons for this work in the field
of bone mechanics.
A proper understanding of the mechanics of bones is required for multiple reasons.
The diagnosis to what extent certain bones are at risk of failing gets increasingly important
as the population of elderly people increases. In today’s clinical practise Osteoporosis and
1
Chapter 1: Introduction
therefore fracture risk is determined by measuring the Bone Mineral Density (BMD). This
integral value provides only information on the quantity of bone available, but whereas it
has been shown in numerous studies that density is related to (monotonic) strength in bone
(Gibson and Ashby, 1997), its dependency is rather weak as it neglects other components
contributing substantially to bone strength such as morphological information. Current
research tries to improve imaging techniques to allow a more precise prediction of these
parameters and therefore of bone quality (Friedman, 2006). Therefore, also knowledge
about the mechanical behaviour of bone structures with respect to bone architecture is
needed. Further, (micro-)damage is supposed to act as a stimulus for repair and adaption
processes (remodeling) in bone (Burr et al., 1985; Lee et al., 2002). Understanding how
damage in bone evolves may therefore also shed light on the underlying mechanobiology.
This knowledge is essential to investigate treatment options for bone diseases like Osteo-
porosis. Treatment options are also needed for the increasingly complicated fractures due
to declined bone quality of elderly people as well as for the so-called critical size defect,
where bone is not able to bridge the defect region. Research is conducted on artificial
bone substitutes which may be used to bridge critical defects in bones with or without the
expression of drugs. So highly porous Ti6Al4V constructs have been developed which
show apparent mechanical properties similar to cancellous bone (Li et al., 2006). Other
approaches are mainly based on hydroxylapatite and calcium, using osteoconductive and
degradable properties of the materials (Englert et al., 2007). In order to successfully apply
these materials also knowledge of the initial, unharmed bone tissue is required. Finally, of
course the further refined development of implants and prostheses takes advantage of an
improved understanding of bone mechanics.
1.2 Objectives
Bone tissue can be divided in two types, mainly differing in apparent density: high density
bone called cortical or compact bone and less dense bone showing a framework structure
named cancellous or trabecular bone. On a micro- and ultrastructural level bone exhibits
2
Chapter 1: Introduction
many of the characteristics of an engineering composite or fibre-reinforced concrete. It
is well known for engineering structures that cyclic loading may induce fatigue failure. In
fact, a vast majority of structural collapses in loaded structures are fatigue induced (Suresh,
1998). Fatigue damage is also known to cause fractures in bones (Snyder et al., 2006;
Niemeyer et al., 2006) and is associated with age-related fragility fractures (Burr et al.,
1997) as well as the loosening of implants (Bauer and Schils, 1999). Despite these clin-
ical observed fractures only few studies are available concerning the fatigue behaviour of
cancellous bone (Michel et al., 1993; Moore and Gibson, 2003b; Haddock et al., 2004;
Yamamoto et al., 2006). All of these studies were conducted using bone specimens cored
parallel to the main structural axis of the trabecular bone architecture and only few us-
ing human donor material. But loading in vivo may not always be aligned with this axis.
A change in local loading conditions, e.g. through structural implants (Fig. 1.2) or even
through habitual motion, may result in load components in off-axis directions. Further,
the influence of off-loads may be much more pronounced with increasing anisotropy of
the structure, which is one of the main characteristics of Osteoporosis. Information of
direction-dependent fatigue properties may therefore contribute essentially to our under-
standing of failure in bones and for example assist in the proper placement of implants.
Figure 1.1: Fracture and treatment of a spine injury. a) Compression fracture with torsional
components of a thoracic vertebra (T8), b) Post-operation (with kind permission
of M. Nerlich, Clinical Center of the University Regensburg), b) Stress distribu-
tion (schematically) around a pedicle screw under loading.
3
Chapter 1: Introduction
Whereas some principal mechanisms of the damage process during cyclic loading
are reported for bovine bone (Moore and Gibson, 2003a; Ganguly et al., 2004), still, a
full understanding of the failure and cyclic deformation mechanisms, especially concern-
ing human bone, is missing. Even if it has been proposed that fatigue in cancellous bone
is governed by ultrastructural mechanisms as two very different cancellous bone types
(bovine and human) did show very similar lifetime behaviour under cyclic loading con-
ditions (Haddock et al., 2004), the influence of different bone architectures on the cyclic
deformation behaviour has not been addressed yet. But especially the increasing applica-
tion of advanced numerical models (Homminga et al., 2004; Yeni et al., 2001; Hernandez
et al., 2006) as well as theoretical modelling of deformation processes in general sets a
demand for a strong experimental base of the deformation processes. Damage in cancel-
lous bone due to cyclic loading has been identified using post-testing microscopic analyses
(Moore and Gibson, 2003a). Damage evolution during loading has only been investigated
in step-wise compression experiments (Nazarian and Muller, 2004). The direct link be-
tween microscopic and macroscopic damage is still not known. Additionally, the effect of
age on the fatigue behaviour of cancellous bone remains obscure.
Therefore, the main research questions in this study were defined as:
• What is the influence of the axis of loading on the fatigue and cyclic deformation
behaviour of cancellous bone?
• How do different types of cancellous bone (species, site) and age influence the fa-
tigue behaviour?
• How do damage evolution and cyclic deformation depend on loading and orienta-
tion?
• Furthermore, methods should be established to study the underlying damage mech-
anism in order to possibly link microscopic and macroscopic damage.
4
2 Trabecular Bone: Morphology and
Mechanics
2.1 Structural Properties
2.1.1 Bone Composition
Bone tissue in mammals is present in very different configurations. The main function
of bone is to provide the mechanical stability for body and motion as well as to protect
vital parts of the body. Apart from these rather obvious facts bone is essential for mineral
homoeostasis, storing and providing minerals. At first glance, bones seem to be rather
compact and homogeneous. But on the macroscopic, microscopic and nanoscopic scale
a highly heterogeneous composite can be found. The outer layer of all bones and the
diaphysis (the mid section/shaft) of long bones are built of high density (∼1.0−1.8g/cm3)
compact (or cortical) bone. This hard shell encloses a low density (<1.0g/cm3) network
of plates and struts named trabecular (cancellous, spongy) bone (Fig. 2.1).
Marrow and cells are located in-between the trabeculae. Apart from the differ-
ing structural architecture, the tissue composition (and tissue density) is almost similar
in both, cortical and cancellous bone. It consists of a mineral (∼65 %) and an or-
ganic phase (∼35 %). The mineral component is mainly built of impure hydroxyapatite,
Ca10(PO4)6(OH)2, in form of needles, plate and rod-like crystals. Furthermore, miner-
als like carbonate, citrate, magnesium and fluoride are present, mainly for homoeostasis.
About 90 % of the organic matrix proteins are collagen type 1, organised in helix-like
fibres. Other components are various non-collagenous proteins and water. The collagen
5
Chapter 2: Trabecular Bone: Morphology and Mechanics
Figure 2.1: Bovine vertebral body sections from two different positions. A thin outer layer,
cortical bone, encloses a foam structure, cancellous bone.
fibrils possess a two-phase, viscoelastic material behaviour with a high tensile strength.
And due to its structural composition as well as its high length to diameter ratio their pri-
mary mechanical function is to withstand tension loading. The combination of the stiff
mineral and the high rupture strength of the fibres builds up an efficient composite, similar
to technical composite materials and reinforced concrete (see e.g. Cowin (2001)).
The predominant structures in cortical bone are osteons. In the middle of an osteon
lays the Haversian canal, which is surrounded by concentric layers of (collagen) lamellae.
Interconnected Osteocytes are found between the lamellae, canalaculi enable the exchange
of nutrients. Fig. 2.2 shows a section of cortical bone taken from a human femur. The
boundary between osteons and surrounding bone is referred to as the cement line. The
diameter of a osteon is about 200 µmand their longitudinal axis is mainly aligned with
the main axis of long bones. Trabeculae are built of segments of parallel lamellae (built
of fibres of collagen) preferentially aligned with the longitudinal axis of the trabeculae.
These groupings of lamellae are separated by cement lines (amorphous substance deficient
in collagen) and called trabecular packets (Fig. 2.3).
Unlike engineering material, the living body has the ability to react to altered loading
conditions and (micro-)damage, which may be induced by cyclic or isolated loadings, with
adaptation and repair processes referred to as remodeling (Frost, 2003; Lee et al., 2002)
(Fig. 2.3). A combination of bone resorbing (Osteoclasts) and bone forming (Osteoblasts)
6
Chapter 2: Trabecular Bone: Morphology and Mechanics
Figure 2.2: Microstructural composition of cortical bone. Osteons in a transverse section
through human femoral cortical bone.
cells, organised in Basic Multicellular Units (BMU), is involved in this remodeling pro-
cess. Whereas in cortical bone BMUs ‘tunnel’ through the bone with a diameter of ap-
proximately 200 µm, at a speed of 40 µm/day and resorb and build new bone in so-called
secondary osteons, in trabecular bone remodeling occurs on the trabeculae surface by re-
sorbing the old bone and filling the cavity with a new trabecular packet. Due to the large
surface to volume ratio in cancellous bone the bone turnover is much more active com-
pared to cortical bone. The turnover rate is reported as high as 26 %/year in cancellous
bone, compared to 3 %/year in cortical bone (Jess, 1990). These remodeling processes
enable bones to operate under high loading conditions, taking into account damage from
daily activities, but having the advantage of a mimimum weight design and the ability to
transform and adapt to altered loading conditions. A drastic example for a change in the
latter are space flights which are known to reduce bone mass (LeBlanc et al., 2007). But
also misuse (alcohol, smoking, imbalance in dietary intake) as well as physical inactivity in
general are known to reduce the ability to remodel (Sowers, 2000). Numerous researchers
proposed different theories on remodeling processes and models to predict these (Prender-
gast and Taylor, 1994; Fyhrie and Schaffler, 1995; Levenston and Carter, 1998; Lee et al.,
7
Chapter 2: Trabecular Bone: Morphology and Mechanics
Figure 2.3: Overview of the structural composition of trabecular bone on different scale lev-
els. a) Sketch of a vertebral body, b) low magnification microscopy of the (human
vertebral) cancellous bone structure, c) sectioned trabeculae showing the under-
lying lamenar structure, d) lamenar bundles on a fracture surface of a trabecula,
e) quantitative backscattered electron imaging of a sectioned trabecula, individ-
ual bone packets, separated by cement lines are visible, the numbers refer to the
average mineral content given in vol% within each packet (Fratzl et al. (2004),
with kind permission), f) Osteoclastic (bone resorbing) activity on the surface
of trabecula, g) basic mechanism of trabecular remodeling with bone resorbing
(Osteoclasts) and formation (Osteoblasts) cells.
8
Chapter 2: Trabecular Bone: Morphology and Mechanics
2002; Frost, 2003; Taylor et al., 2003a). But the precise mechanism by which the living
bone detects damage and in consequence adapts to loading are still unknown (Taylor et al.,
2007).
2.1.2 Morphology of Cancellous Bone
The cellular nature of cancellous bone is visible from low-magnification microscopy
(Whitehouse et al., 1971; Whitehouse and Dyson, 1974). Cancellous bone can be clas-
sified as a open (pore) polyhedral cellular solid (Gibson, 2005). Typical dimensions are:
pore sizes 0.3 - 1.5 mm, strut dimensions 0.1 - 0.3 mm. These hold regardless of the
species (Martin et al., 1998). The architecture of the trabecular network appears to be
inhomogeneously oriented. Many authors found striking evidence that bone grows in re-
sponse to loading (Wolff, 1892; Roux, 1895; Frost, 1987). That means, the architecture
of the trabeculae network follows the trajectories of the main loading direction so that
bending is minimised. The structural anisotropy may therefore depend on the ratios of
the acting principal stresses (Gibson and Ashby, 1997). Higher loading results in more
platelike trabeculae in the main loading direction with interconnecting small trabeculae
as spacers (Fig. 2.4), whereas lower loaded regions exhibit a more truss-like framework
(Fig. 2.3 b)) (Whitehouse and Dyson, 1974).
Several structural measures for the degree of anisotropy have been developed. The
most established method is the Mean Intercept Length (MIL) (Whitehouse and Dyson,
1974; Whitehouse, 1974; Harrigan and Mann, 1984). Furthermore, the Volume Orientation
(VO) (Odgaard et al., 1990), Star Volume Distribution (SVD) (Cruz-Orive et al., 1992)
and fast Fourier transformation-based measures (Brunet-Imbault et al., 2005) are, among
others, reported. Cowin defined the fabric tensor as a measure for the degree of anisotropy,
which is based on the inverse of the MIL tensor, and set up a general theory which relates
the structural anisotropy to the elasticity tensor (Cowin, 1985).
The open cell structure of trabecular bone requires a distinction between the apparent
properties, referring to the whole specimen, and the tissue properties, referring to the bone
9
Chapter 2: Trabecular Bone: Morphology and Mechanics
Figure 2.4: SEM image of a bovine vertebral cancellous bone. Plate-like structure in the main
physiological axis (vertical) direction.
tissue/trabeculae itself. The mechanical properties referred to in this study are analysed on
the apparent level. In order to fulfil the assumption of a continuum for the experimental
investigation of the apparent properties an adequate specimen size has to be chosen. The
minimum continuum length has been estimated to be about five trabecular spacings and
therefore 5 −10 mm (Fung, 1993).
2.2 Mechanical Properties
The mechanical response of cancellous bone has been characterised by numerous re-
searchers. Several approaches to model the deformation behaviour have been performed
on an analytical, numerical and experimental base. Analytical models include structural
analysis of unit cells and density-based methods assuming the cell geometry as similar at
different degrees of density (Gibson, 2005). By means of numerical methods the defor-
mation behaviour of regular or random cellular structures has been investigated (Vajjhala
et al., 2000; Schaffner et al., 2000; Makiyama et al., 2002). Recently, due to increased com-
putational power and improved imaging techniques also numerical studies on almost exact
geometries of trabecular bone samples have been performed (Yeni et al., 2001; Homminga
et al., 2004; Hernandez et al., 2006). However, most of the published work on the biome-
10
Chapter 2: Trabecular Bone: Morphology and Mechanics
chanics of trabecular bone during the last 40 years is based on experimental studies, which
do also make up the majority of the following review. Most studies discussed below deal
with cancellous bone, if not stated explicitly that other materials have been used. As mate-
rial testing on highly inhomogeneous structures like cancellous bone is quite complicated
and no real standard for the experimental boundary conditions are given in terms of spec-
imen size, rate of loading, load application and surrounding media, results from literature
are also very heterogeneous. Some of the variations in mechanical data may be ascribed to
experimental effects, introduced by ignoring the structural anisotropy, the proper boundary
conditions (e.g. end artefact errors) (Linde and Hvid, 1989; Keaveny et al., 1993, 1997)
and size effects (Taylor, 1998). But there is also a natural heterogeneity which complicates
the analysis of bone. For example, compressive stiffness was found to vary up to 100-fold
within a single proximal tibia (Goldstein et al., 1983).
2.2.1 Monotonic Loading
Monotonic trabecular bone properties have been investigated by numerous researchers
(Keaveny et al., 2001). The compressive stress-strain behaviour of cancellous bone is typ-
ical of that of a cellular solid. Following a linear regime, plastic collapse occurs with sub-
sequent failure of struts, resulting in a rather long stress plateau, and an increase of stresses
at higher strains due to densification effects (Gibson and Ashby, 1997) (see Fig. 4.3). Sev-
eral studies indicate that bending of trabeculae is the dominant deformation mode in linear
elastic deformations (van Rietbergen et al., 1995; Nazarian and Muller, 2004). Depend-
ing on the slenderness ratio trabeculae can fail either by elastic buckling or by progressive
microfracture (Townsend et al., 1975). At lower degrees of density elastic buckling domi-
nates, at higher density rates the trabeculae have lower slenderness ratios and fail by pro-
gressive microfracturing (Gibson and Ashby, 1997). Yield strength in tension was found
to be smaller than in compression (Kaplan et al., 1985). Typical values for yield strength
and modulus of cancellous bone are shown in Tab. 2.1. Strains at failure were also found
to be higher in compression than tension (Kopperdahl and Keaveny, 1998). The absolute
value of the yield strain can vary between sites (Kopperdahl and Keaveny, 1998; Morgan
11
Chapter 2: Trabecular Bone: Morphology and Mechanics
et al., 2001) whereas they seem to be almost constant within one site. Typical failure strains
(0.2% offset strain) are reported to be 1.09% for bovine tibia (Keaveny et al., 1994) and
0.79% for human vertebral cancellous bone (Haddock et al., 2004). As strains at failure
are constant, at least in bone taken from the same site, but modulus and strength may vary
by order of magnitudes, it has been suggested that the damage behaviour and therefore the
failure criterion for cancellous bone is strain-based (Moore and Gibson, 2003b). The me-
chanical performance of cancellous bone under monotonic loading seems to be rather poor,
compared to engineering material (Fig. 2.5), but in relationship to its density it reveals to
be an extraordinary composite.
The oriented architecture of the cancellous bone structure is also reflected in the
mechanical properties. The mechanical anisotropy of trabecular bone has been tested
under monotonic loading conditions and isotropy of yield strains has been suggested,
whereas modulus and strength vary with orientation (Turner, 1989; Ford and Keaveny,
1996; Nicholson et al., 1997; Bayraktar et al., 2004). Modulus as well as strength were
found to decrease in off-axis directions, depending on the structual anisotropy. Nicholson
et al. (1997) found an approximately 4-fold higher stiffness in superior-inferior direction
in human vertebral cancellous bone compared to lateral values. A recent study shows
that specimens taken from the radius with a high structural anisotropy exhibit a significant
lower value in yield strain than specimens from other sites with lower anisotropy (Mat-
suura et al., 2007). Furthermore, cancellous bone is reported to have at least orthotropic
symmetry in the elastic properties (Williams and Lewis, 1982; Turner and Cowin, 1988;
Yang et al., 1998). The multiaxial behaviour of cancellous bone has been studied and yield
criteria like the Tsai-Wu criterion have been applied to model the deformation behaviour
(Stone et al., 1983; Keaveny et al., 1999).
Due to the porous structure of cancellous bone also the influence of marrow, in terms
of pore fluid, on the mechanical behaviour may be regarded. It has been shown that this is
negligible for strain rates below 10/s (Carter and Hayes, 1977). Further, bone is known to
possess viscoelastic properties. But with a strain rate coefficient of 0.06 this dependency
is rather weak (see Equation 2.1). Under physiological loading conditions, daily activities,
12
Chapter 2: Trabecular Bone: Morphology and Mechanics
Figure 2.5: Comparison of the range of modulus and yield strength for biological and engi-
neering materials.
strain rates below 0.015 s−1are reported (Martin et al., 1998). Of course, this may not hold
for trauma situations. With increasing rate of loading yield strength as well as modulus in-
crease, whereas strain at failure decreases. Therefore, bone tissue can withstand in trauma
situation higher loadings, but does also fail in very high speed impacts in a very brittle
manner and complex fracture patterns are observed in these cases. The transition from
physiological frequencies (0.5-2 Hz) to high frequencies (30 - 125 Hz) were observed to
affect the fatigue strength in experimental situations by a factor of 1.33 (Taylor, 1998).
Moisture also influences the mechanical behaviour of bone and is an essential boundary
condition in the experimental setup. Wet bone exhibits an increased ductility whereas dry
bone has a higher modulus and yield strength.
The determination of density - mechanical properties correlations is the scope of
numerous research activites as bone density measurements can be applied in clinical praxis.
As structural cell modelling suggests both elastic modulus and yield strength depend on
13
Chapter 2: Trabecular Bone: Morphology and Mechanics
Table 2.1: Compressive modulus and compressive strength of cancellous bone
Anatomic site Modulus, MPa Strength, MPa
Human vertebra 70 to 90 1.37 to 1.86 (Yamada, 1970)
Human proximal tibia 489 ±331 2.22 ±1.42 (Rohl et al., 1991)
Human proximal tibia 445 ±257 5.33 ±2.93 (Lotz and Hayes, 1990)
Human vertebra 67 ±44 2.45 ±1.52 (Mosekilde and Mosekilde, 1986)
Human vertebra 291 ±113 2.23 ±0.95 (Kopperdahl and Keaveny, 1998)
Bovine tibia 2690 ±900 (Bowman et al., 1998)
apparent density, even if the precise relationship remains controversial. The measurement
of Bone Mineral Density (BMD) is a non-invasive standard procedure in clinics. Different
methods have been developed in order to measure the density of minerals in bones (e.g.
X-ray, computed tomography or ultrasound). Correlation between elastic modulus and
density were found to be linear (loaded in physiological axis, from one site) (Kopperdahl
and Keaveny, 1998), quadratic (Rice et al., 1987) or cubic (Carter and Hayes, 1977). The
latter authors’ relationships are frequently used in the following forms:
σc=68 ×˙
ε0.06 ×d2(2.1)
E=3790 ×˙
ε0.06 ×d3(2.2)
˙
ε: strain rate, s−1
d: density, g/ccm
σc: compressive strength, MPa
E: modulus, MPa
Recently, there seems to be evidence that no universal modulus - density relation-
ship for on-axis loading is present (Morgan et al., 2003). Correlations between mechanical
properties and BMD were found to be highest if loads were applied aligned with the main
physiological axis (Augat et al., 1998). Alternative formulations for the prediction of the
mechanical properties which include microstructural information are also under investi-
gation (Rajapakse et al., 2004). Despite the fact that most of these relationships show
14
Chapter 2: Trabecular Bone: Morphology and Mechanics
good correlations (from a statistical point of view), the quality of the data seems to be
unacceptable for precise fracture prediction. Therefore, attempts are made to increase the
predictability power with morphological information. Several authors used successfully
different combinations of volume fraction and measures for structural anisotropy in order
to increase the predictability strength of the elastic properties (Hodgskinson and Currey,
1990; Zysset and Curnier, 1996; Yang et al., 1998; Zysset, 2003). Furthermore, ultrasonic
techniques were applied to measure the elastic constants in cancellous bone (Ashman and
Rho, 1988), having the advantage of excluding many experimental artefacts but lacking the
opportunity to predict the mechanical behaviour precisely in the in vivo situation. Taken to-
gether, approaches exist to determine the mechanical behaviour non-invasively but today’s
knowledge leaves plenty of space for improvements.
Finite element models developed from micro computer tomography have been ap-
plied to detect failure mechanism and local strain fields (Homminga et al., 2004; Hernan-
dez et al., 2006). Microdamage initiation was found to occur prior to apparent yield using
image-based finite element models with local principal strains of 0.46 −0.63% in com-
pression (Nagaraja et al., 2005). Further numerical studies demonstrated also that damage
is already visible below the apparent compressive yield strain. The authors suggest that
local tissue yielding can in fact initiate at very low apparent strains and that the appar-
ent mechanical properties are degenerated through these localised effects (Morgan et al.,
2005).
The behaviour of microcracks in bone is a highly active research field (Hazenberg
et al., 2007). Most of the research has been addressed to cortical bone. It has been shown
that fracture toughness increases with crack length (Vashishth, 2004). Multiple mecha-
nisms were suggested to contribute to this effect. Microstructural barriers were found to
arrest crack growth. Cracks below a critical length are supposed to be stopped, whereas
above a certain threshold crack energy is large enough for further propagation and initi-
ation of macroscopic damage. Microcracks of less then 100 µmwere found to arrest at
cement lines, which surround the osteons, whereas medium microcracks of 100 −300 µm
were deflected at the cement lines’ boundary. Longer microcracks (greater than 400 µm)
15
Chapter 2: Trabecular Bone: Morphology and Mechanics
did cross the cement lines (Mohsin et al., 2006). Furthermore, bridging of microcracks
(through collagen fibres) (Yeni and Fyhrie, 2003; Nalla et al., 2003) as well as diffuse
damage in front of larger microcracks (Vashishth, 2004) are supposed to raise the fracture
toughness with increasing crack length. The exact connection between microstructural
composition of trabecular bone and its mechanical behaviour is still under discussion. A
mechanical role of the cement lines with respect to microfractures in trabecular tissue has
been suggested (Choi and Goldstein, 1992), as well as the importance of the non-uniform
mineral distribution in trabeculae on the apparent mechanical properties was emphasised
(van der Linden et al., 2001). The microstructural properties itself, lamellar-level elastic
modulus, hardness and extracellular matrix have been suggested not to depend on age, gen-
der or increased bone mass maintenance (known to occur in heavier individuals) (Hoffler
et al., 2000). Likewise, studies on crystal length and crystal thickness in the mineral phase
of the bone tissue did not reveal significant differences between normal and osteoporotic
trabecular bone (Rubin et al., 2003). But nevertheless, more data is needed to reveal the
exact relationship between the micro- and ultrastructural composition and the mechanical
properties.
It has been suggested that damage in human cancellous bone is at least of orthotropic
nature. Transverse damage components for on-axis (0◦) specimens were found to be on
average 1.9% of the damage components aligned with the stress axis, whereas transver-
sal oriented specimens (90◦) did reveal transverse components of 6.5% (Bredbenner and
Davy, 2006). In contrast, other attempts to characterise the damage effects in bovine tib-
ial trabecular bone used high-resolution finite element models resulted in much higher
transverse damage values for the off-axis specimens and much higher magnitudes of the
damage parameters (up to 36% of the loading axis components) (Bredbenner (2003) as
cited in Bredbenner and Davy (2006)). Although damage was found to be anisotropic in
cancellous bone, the material coordinate system did not lose alignment with the structural
orientation (fabric tensor) (Liu et al., 2003).
16
Chapter 2: Trabecular Bone: Morphology and Mechanics
2.2.2 Cyclic Loading
The majority of failure cases in mechanical engineering resulting from mechanical load-
ing, are fatigue-induced (Suresh, 1998). In contrast to failure under monotonic loading (in
ductile materials), where failure is associated with macroscopic plastic deformation un-
der cyclic load exposure, fractures appear almost sudden and unexpected without visible
distinctive deformation. Additionally, this may happen at stress levels much below the
monotonic yield strength. Therefore, severe accidents may result, which did cause serious
damage in the past. The main stages of damage evolution during fatigue following the
description of Mugrabi (1985) are shown in Fig. 2.6.
Figure 2.6: Main stages of fatigue damage evolution (Mugrabi, 1985).
The cyclic deformation process is started with a transient, characterised by cyclic
softening/hardening during the first few load cycles followed by cyclic saturation. In this
region the structural response on the loading is stabilised. As strains localise cracks are ini-
tiated which do subsequently propagate and finally cause macroscopic failure. Lifetimes
are frequently analysed using the concept of S-N (Woehler) curves. Basquin (1910) es-
tablished a relationship between the stress amplitude ∆σ/2 and the number of cycles to
failure Nf:
∆σ
2=σf(2Nf)B(2.3)
σf: fatigue strength coefficient
B: Basquin exponent
17
Chapter 2: Trabecular Bone: Morphology and Mechanics
Some decades later, Manson (1954) and Coffin (1954) found that cyclic damage in
metallic materials is caused by plastic strains:
∆ε
2=εf(Nf)C(2.4)
εf: Fatigue ductility coefficient
C: Fatigue ductility exponent
Therefore, Equation 2.4 is valid in the low-cycle fatigue regime, as high stress is
necessary in order to cause plastic deformation whereas Basquin’s relationship is governed
by elastic deformations and is used in high-cycle, low stress fatigue. Details are not further
discussed here as most of the published work in this field refers to metallic materials and is
therefore not directly applicable to bone tissue. Comprehensive information on cyclic de-
formation and fatigue can be found in standard textbooks (Hertzberg, 1995; Suresh, 1998).
Cyclic failures of bones due to the accumulation of fatigue damage are clinically
known as overuse injuries or stress fractures which are observed in athletes (Snyder et al.,
2006; Niemeyer et al., 2006) and military recruits (Sormaala et al., 2006) and are also
associated with age-related fragility fractures (Burr et al., 1997). Cyclic loading-induced
damage accumulation is also reported to weaken vertebrae (Burr et al., 1997) and is often
associated with loosening of implants (Bauer and Schils, 1999). It has been suggested
that microdamage induced due to fatigue loading does initiate remodeling processes (Burr
et al., 1985; Lee et al., 2002).
The investigation of fatigue failure has a rather long history in traditional material
sciences, starting in the late 18th century (Schuetz, 1996). King and Evans (1967) were
the first to measure lifetime (S-N) curves for (cortical) bone in the sixties. Subsequently,
different authors analysed mechanical fatigue concerning compact and trabecular bone.
Whereas the fatigue strength of cortical bone is extensively reported in the literature (Carter
et al., 1981; Zioupos and Casinos, 1998; Yeni and Fyhrie, 2002; O’Brien et al., 2003; Taylor
18
Chapter 2: Trabecular Bone: Morphology and Mechanics
et al., 2003b; George and Vashishth, 2006) only few data is available for trabecular bone
(Michel et al., 1993; Moore and Gibson, 2003b; Haddock et al., 2004; Yamamoto et al.,
2006). Most studies concentrate on bovine and human trabecular bone specimens cored
in the main (on-axis) direction and analysed in cyclic compression (with varying mean
stress). The studies showed a considerable scatter for the same bone type between different
research groups which may also result from differing experimental boundary conditions in
terms of specimen size, surrounding media and embedding.
In order to reduce the large scatter of the results applied stresses are usually nor-
malised by the initial modulus of each specimen, which is also a standard procedure in
the testing of cellular solids. Analysing this normalised stress (σ/E0) as a function of the
cycles to failure results in Basquin type relationships (Equation 2.3). A more recent ap-
proach showed that a normalisation of the applied stress by (bone) volume fraction and
the axial fabric eigenvalue (Cowin, 1985) can improve the estimation of fatigue life for
cancellous bone in axial cyclic compression (Rapillard et al., 2006). Maximum strains at
failure as well as accumulated residual strains (Fig. 4.4) were found to depend linearly on
the applied normalised stress (Moore and Gibson, 2003b). A fatigue endurance limit at
a normalised stress level of σ/E0=0.0035 has been suggested based on microdamage
evaluation in bovine trabecular bone (Moore and Gibson, 2003b), whereas modeling of the
fatigue behaviour hints at a lower endurance limit as low as σ/E0=0.0024 (Ganguly
et al., 2004).
The characteristics for fatigue in cancellous bone were found to be comparable to the
principles derived for metals. Beside the earlier mentioned three stages of deformations,
stress strain hysteresis were found to move along the strain axis during cyclic compression.
This observed cyclic creep effect is also associated with a decrease of the secant modulus
and a broadening of the hysteresis loops (Haddock et al., 2004; Moore and Gibson, 2003b).
The physical effect of the translation has been associated with damage in form of micro-
cracks, which do not fully close during unloading in bovine trabecular bone (Moore and
Gibson, 2003a). The exact influence of creep remains somehow controversial (Bowman
et al., 1998; Moore et al., 2004), but recently there seems to be some evidence that creep
19
Chapter 2: Trabecular Bone: Morphology and Mechanics
effects are negligible in low-cycle fatigue tests (Moore et al., 2004). Nevertheless it was
suggested that creep effects may be dominant on physiological load levels and be therefore
related to non-traumatic vertebral fractures (Yamamoto et al., 2006) whereas, higher load-
ings may result in residual strains due to microcracking. The observed decrease in secant
modulus of equine and canine cortical bone exposed to cyclic loading was also shown to
be mainly due to the formation and growth of microcracks (Burr et al., 1998; Fleck and Ei-
fler, 2003). The same mechanism was observed in cancellous bone, the reduction in secant
modulus was also found to be related to increasing microcracking and fractured trabecu-
lae (Moore and Gibson, 2003a). Increasing strains during fatigue testing were observed
to be associated with a reduction in secant modulus for both cortical and trabecular bone
respectively (Moore and Gibson, 2003b; Ganguly et al., 2004; Cotton et al., 2005).
Cotton et al. (2005) found that damage rate (the rate of change in secantmodulus
per load cycle) is a good predictor of fatigue life for human cortical bone exposed to pure
tensile fatigue. The authors took the (constant) damage rate from the rather linear damage
behaviour between 10 % and 90 % of the specimen’s lifetime and showed that correlations
between this damage rate and cycles to failure were better than for normalised stress vs. cy-
cles to failure. The damage rate did also appear to be a function of normalised stress σ/E0.
Fatigue damage in cancellous bone has been quantified with numerical and experimental
means. 2-D and 3-D cellular solids were modeled and analysed with finite element meth-
ods (Guo et al., 1994; Schaffner et al., 2000; Makiyama et al., 2002). The models failed
to predict the real response of cancellous bone under cyclic loading but showed principle
mechanisms, which may also hold for the biological material. So the failure of only few
trabeculae resulted in a largely reduced modulus and macroscopic damage was initiated
by the subsequent failure of only few struts. Furthermore, crack growth and propagation
was observed to be the primary failure mechanism for low-stress, high-cycle fatigue of tra-
becular bone, while the primary failure mechanism for high-stress, low-cycle fatigue was
suggested to be creep deformation and fracture (Guo et al., 1994).
Ganguly et al. (2004) did model the reduction in secant modulus during compression
fatigue loading of bovine cancellous bone as a function of the maximum strain. They ap-
20
Chapter 2: Trabecular Bone: Morphology and Mechanics
proximated upper and lower bounds for this reduction within normalised stress ranges. The
authors suggest that damage is governed by maximum strain, and determined the relation-
ships between strain and reduction in secant modulus with monotonic compression tests,
whereas the secondary residual strain accumulation rate was governed by the normalised
stress and was taken from cyclic compression experiments. Their model is successful in
predicting the experimental data for a given stress range, but these ranges were rather
coarsly chosen. Additionally to the exact applied stress level also the initial/transient phase
of the cyclic deformation model has been neglected.
A study on the comparability of the fatigue behaviour of bovine and human can-
cellous bone suggests that fatigue damage occurs at a ultrastructural level as these very
different bone types exhibit a rather similar deformation behaviour if the differences in
monotonic yield strain are included in the analysis (Haddock et al., 2004).
Of course, all mechanical data on the fatigue behaviour of bone which is established
on an experimental basis do not take into account any biological reactions, such as remod-
eling/healing. So it remains unclear how this repair process influences the overall outcome
as there are two contrary ideas. Firstly, these healing processes may prolong fatigue life as
microdamage is repaired, but on the other hand, as osteoclastic resorption is preliminary
to repair, resorption lacunae may cause local stress concentrations, which can also cause
damage acceleration (Martin et al., 1997; Hernandez et al., 2006).
2.3 Age-Related Changes
Age affects the biomechanical and morphological properties of bone resulting in an in-
creased risk of bone fractures with advanced age (Kanis et al., 2004). With increasing
age remodeling activity loses its equilibrium with damage and bone quality decreases. A
mismatch between bone resorption and formation results in diminishing bone density. The
reason for this mismatch may be both mechanically- and biologically-induced. For the di-
agnosis of severe bone mass reduction (Osteoporosis) density measurements (DXA-based)
are used and classifications based on density data from young females is applied to deter-
21
Chapter 2: Trabecular Bone: Morphology and Mechanics
mine the severity of the disease (the World Health Organisation (WHO) defined a higher
reduction in bone density than 2.5 standard deviations below bone peak mass as Osteo-
porosis (W.H.O., 1994)). But not only the amount of bone available (density) is important,
moreover, bone architecture and bone quality have a major impact on the risk of fracture.
The determination of this quality is in the focus of many current research activities (Comp-
ston, 2006; Nagaraja et al., 2007). Peak bone mass and strength were found to be highest
at an age of 20 to 30 years and bone mass and strength was found on average to be higher
in men than women (Mosekilde, 2000). Density decreases with age (Fig. 2.7, 2.8).
Mosekilde (1989) reported an approx. 50 % reduction in vertebral cancellous bone
density between 20 and 80 years. The compressive strength of human femoral cancellous
bone revealed a decrease by 8.5 % each decade (McCalden et al., 1997). For both sexes an
extreme decline in vertebral bone strength with ageing could be observed. Whereas men
show a certain compensatory increase in bone size with age, no cross-sectional adaptation
could be found for women (Mosekilde, 2000). The architecture of cancellous bone struc-
tures does also change with age, e.g. in aging tibial bone trabeculae were found to align
more strongly with the primary direction-parallel to the longitudinal loading axis (Ding
et al., 2002). In general, in cancellous bone structures trabeculae are removed, starting
with struts transversally oriented to the physiological load axis and therefore increasing the
degree of anisotropy. Women show, especially at ages older than 50 years (menopause), a
Figure 2.7: Human vertebral can-
cellous bone, male 46
years, main axis in
horizontal direction
(SEM).
Figure 2.8: Human vertebral can-
cellous bone, male 80
years, main axis in
horizontal direction
(SEM).
22
Chapter 2: Trabecular Bone: Morphology and Mechanics
higher tendency for the perforation of these horizontal (normal to the main physiological
axis) struts of the trabeculae network (Mosekilde, 1989, 2000). Trabecular number was
found to be reduced in females, whereas in males, strut thickness was reduced (Mullender
et al., 2005). Despite these structural changes overall vertebral cancellous bone density
was found to be equal for both sexes and compressive stress appeared to be independent
of gender (Ebbesen et al., 1999). The microcrack density was observed to increase ex-
ponentially with age in human femoral cortical bone (Schaffler et al., 1995). Despite this
increase, longer fatigue life in cortical bone could be associated with higher initial crack
density when modulus variability was taken into account (Sobelman et al., 2004). Fatigue
life in cortical bone decreased exponentially with age, and old bone exhibited a different
development of damage in terms of modulus degradation than younger bone (Diab et al.,
2005). It has been shown that microdamage may be higher in cancellous rather than cor-
tical bone. Wenzel et al. (1996) found a microcrack density of almost 5 microcracks/mm2
in human vertebral bone, but in contrast to the findings on cortical bone (Schaffler et al.,
1995) they did not find a significant increase with age. Furthermore, fracture toughness
of bone was found to decrease with age whereas its microhardness increased. It has been
suggested that changes in bone fracture toughness may not be necessarily reflected in its
mineral density, porosity, elastic modulus, yield strength and ultimate strength (Wang et al.,
1998). The same researchers did not find any significant changes in BMD, elastic modulus
and yield strength as a function of time. By contrast (Zioupos and Currey, 1998) found
a steady and significant decrease with age for all these mechanical measures. For exam-
ple, elastic modulus fell by 2.3 % per decade of later life from its value of 15.2 GPa at 35
years of age. Similar declines were also observed for yield strength and fracture toughness.
The found deterioration in the elastic properties of the material reduced the critical stress
intensity level required to initiate a macrocrack which was preceded by less damage and
after initiation needed less energy to propagate (Nalla et al., 2004; Cotton et al., 2005). An
increased anisotropy of bone resulting from age effects does also increase the anisotropy
of the compressive yield strength. It is unclear how this influences clinically fracture eti-
ology (Keaveny et al., 2001). In summary, the overall effect of age on bone quality (as a
23
Chapter 2: Trabecular Bone: Morphology and Mechanics
predictor of fracture risk) is still controversial.
2.4 Summary
Despite the availability of the described data some issues remain unsolved. Whereas the
fatigue behaviour of cancellous bone has been described for bovine and human trabecular
bone for uniaxial cyclic compression in the main physiological axis, the relationship be-
tween off-axis loading and lifetime has not been addressed so far. The influence of species
and therefore very different architectures has only been partly described. Likewise, the
way deformation and damage evolve during cyclic compression with respect to differing
load axes and species remains unsolved. Furthermore, the influence of age on the fatigue
behaviour of cancellous bone is unclear. Especially the mechanical consequences of an
increased anisotropy remain unsolved. The fact that many osteoporotic fractures occur
during a fall emphasises the need for a proper understanding of the effect of off-axis load-
ings on the mechanical properties of cancellous bone. Further, the relationships between
localised damage and macroscopic structural integrity are still unknown.
24
3 Materials and Methods
3.1 Specimens
In order to characterise the behaviour of trabecular bone under monotonic and cyclic load-
ing specimens were prepared using only fresh frozen bones. The use of human donor
material was subject to an ethics proposal, which was approved by the ZKS (Zentrum für
Klinische Studien) of the Clinical Centre of the University Regensburg. Bovine material
was supplied by a butcher. Human cadaveric material was obtained from the Institute for
Pathology, Clinical Centre Augsburg.
Bovine bone specimens were used for preliminary studies and for two series of fa-
tigue tests. As bovine bone is frequently used for the mechanical characterisation of tra-
becular structures (Michel et al., 1993; Moore and Gibson, 2003b) this data may be used in
order to facilitate comparison with literature. Furthermore, bovine trabecular bone exhibits
a much stronger structure compared to human bone and therefore represents an upper limit
of mechanical performance. In total, 27 bovine lumbar spine segments (each L1-L5) were
prepared. The animals were 18 to 24 months old. Human bone specimens were dissected
from lumbar spine segments (L1-L5) and the distal femoral condyle from 9 donors with an
age from 46 to 80 years and a mean of 62.2±10.7 years (8 male, 1 female). No addi-
tional information about the pathological history was available. Vertebrae with deformed
endplates were excluded from the study. Specimens used for the fatigue series were cored
in different orientations with respect to the physiological whole bone axis. In order to de-
fine a local coordinate system for each bone the endplates were chosen as reference for
the vertebral body and the distal condyle for the femurs (see Fig. 3.1). The specified angle
25
Chapter 3: Materials and Methods
refers to the misalignment with the physiological bone axis (PBA) (0◦= coincident with
PBA). Bovine vertebral samples were drilled in 0◦and 90◦. Human vertebral specimens
were dissected in 0◦, 22◦, 45◦, 90◦, and femoral specimens also in 0◦(Fig 3.3).
Vertebrae were separated and cores were drilled in the frozen state and then cut to
length (Discotom-2, Struers). The bovine specimens were ground at low speed (grit size:
1200) to a rectangular cross section in order to facilitate a rigorous measurement of the two
dimensional surface strains, without distortion of the out of plane deformations. For the
same reason, marrow was retained for the bovine specimens. Human specimens were kept
cylindrical and marrow was fully removed with compressed air and water as in these series
deformations at the trabeculae level were of major interest. The resulting cross sections
were: Cylindrical: Diameter 11.2 mm ×length 15 mm (human) and cubic: 7.75 mm ×
7.75 mm ×15 mm (bovine). Specimen dimensions are likely to be large enough to assume
continuum behaviour (Harrigan et al., 1988). By microscopic analysis no damage caused
by the preparation process was identified. The specimens were again frozen at −20◦C.
Prior to testing, each specimen has been fully thawed for five hours in 0.9 % NaCl solu-
tion. For testing, specimens were embedded in specimen holders with methylmethacrylate
(KEM 15, ATM, stiffness: 5000 MPa) in order to guarantee a homogeneous load appli-
cation and avoid end trabeculae breaking, which results in a toe region in the stress strain
curve and therefore causes uncertainties in the strain data. These effects are named end
artefact errors (Keaveny et al., 1997). End artefacts refer to damage occurring in the speci-
men’s ends due to inhomogeneous boundary conditions. All mechanical experiments were
accomplished with a servo-hydraulic testing system (MTS 810 System, Teststar II, 5 kN
load cell). In order to supply almost physiological conditions experiments were performed
in tempered (37◦C) 0.9% NaCl solution. All bovine specimens and partly the human spec-
imens were scanned for Bone Mineral Density (BMD). BMD was measured along the load
axis in 2 mm slices (Stratec XCT-900).
26
Chapter 3: Materials and Methods
Figure 3.1: Specimen orientations, schematically. The coring angles refer to the physiological
coordinate system of each whole bone. The endplates of the vertebrae and the
proximal condyle of the femur were chosen as references, the 0 degree direction
is in alignment with the physiological axis.
Figure 3.2: Specimen embedding rig.
Figure 3.3: Typical samples from different groups of cancellous bone. a) human femoral 0◦,
b) human vertebral 45◦, c) human vertebral 90◦(all SEM).
27
Chapter 3: Materials and Methods
3.2 Testing Conditions
Monotonic compression tests were performed on bovine specimens using a constant (phys-
iological) strain rate ˙
εof 0.015 s−1. All fatigue tests were performed under load control
with the same strain rate ˙
εof 0.015 s−1(0.9−3 Hz) expect for the bovine tests where
the frequency of loading was fixed for reasons of optical deformation measurement at 1
Hz. The specimens were preloaded to a small load level (approx. 1 −2% of the static
failure load, −2 to −50 N) to ensure full contact (Bowman et al., 1998; Moore and Gibson,
2003b). Prior to fatigue testing, ten cycles from −50 N to −300 N for the bovine bone
and from (approximately) −1 N to −5 N for the human bone were applied to determine
the initial modulus of the specimen. A triangular wave form from the preload level to a
prescribed load value was applied until catastrophic failure occurred. Therefore, a varying
mean stress was operating in the different specimens. In order to reduce the scatter of the
results the load level was determined by relating the applied stress to the initial modulus of
each specimen, which is a common practice in the mechanical testing of spongeous struc-
tures (Gibson and Ashby, 1997). Detailed information of the load level range is given in
Table 4.1. The initial modulus for the post-analysis was corrected subsequent to the fatigue
test because the low load levels of the preconditioning cycles did not always allow for an
absolutely exact determination of E0. Therefore, the secant modulus of the first cycle was
determined. Some specimens reveal a slight increase in stiffness in the first few cycles. In
these specimens, which were mainly low load −high cycle specimens, secant modulus of
the fifth cycle was taken for the analysis. All specimens were stopped as an integral strain
of −1.33% was reached. At this value stiffness is already reduced to a large degree but
structural integrity is mostly still given, sufficient for microscopic postprocessing. Failure
was defined as a ten percent reduction in secant modulus in order to facilitate comparison
with literature data (Bowman et al., 1998). Maximum and minimum force and displace-
ment values were measured continuously, full hysteresis loops were recorded with more
than 400 data values in at least logarithmic manner. Post-testing microscopic analysis was
conducted by means of Scanning Electron Microscopy (SEM) and Stereomicroscopy.
28
Chapter 3: Materials and Methods
3.3 Deformation Analysis
The integral deformation was measured using the deformation sensor of the test machine,
measuring the grip to grip motion. Extensive preliminary studies assured the accuracy of
the test setup. Surface strains were measured on two sides of the specimens with the same
optical deformation measurement system (Aramis, GOM mbH) for the bovine specimens.
This setup allowed for a continuous two-dimensional analysis of surface deformations.
Concerning human specimens, mostly single trabeculae were analysed. The optical defor-
mation measurement system is based on the correlation of a pattern of grey values. There-
fore, prior to testing a stochastic colour pattern has to be applied. In this work, mostly a
pattern of black and white speckles was applied using pressurised dispensers, also other
methods were investigated (e.g. black light-(UV-A-)based marker paint). The accuracy
of the system depends on the resolution of the CCD-Cameras (1600 ×1200 pixel), the
measurement volume and the applied pattern and is specified to be in the range of 0.01%
strain. Especially the pattern is a critical parameter: it has to be applied in a very accurate
manner. In the framework of this study different approaches were investigated for the setup
of the optical deformation measurement. Because of the highly porous stucture of cancel-
lous bone 3D-methods were found to be rather unstable. Most measurements were done
with 2D-methods, neglecting deformations in the depth direction. Therefore, for continu-
ous surface strain analysis specimens with a rectangular cross section (bovine 0◦and 90◦)
were used, whereas microdamage has been detected on single trabeculae (human groups)
which were aligned with the analysis plane.
3.4 Data Analysis
The force displacement data was normalised using nominal (1st Piola-Kirchhoff) stress and
nominal strain. In order to take into account the stiffening effect of the embedding material,
initial height of the specimens was reduced by one half of the layer (1 mm on each side) for
strain calculation, therefore resulting in an initial gauge length of 14 mm. Secant modulus
29
Chapter 3: Materials and Methods
was determined for all hysteresis loops by:
Esec =σmax −σmin
εmax −εmin
(3.1)
The cycle to failure was defined as the cycle where secant modulus has decreased by 10%
in relation to the initial modulus. Applied maximum compression stress was normalised
with the inital modulus for each specimen (σ/E0). Maximum strains were defined as the
total strain (elastic and residual) components, (accumulated) residual strains as the inelastic
deformation (see Fig. 4.4). Various strain and deformational components were differenti-
ated with respect to the number of cycles, the details are shown in the results section.
Regression analyses were performed using least square methods. In some cases (which
are indicated in the results) a robust algorithm was chosen which takes data outliers into
account in order to facilitate reasonable fits. The significance level for statistical tests was
chosen to be p<0.05. All analyses were performed using Matlab (R14, The Mathworks).
Surface strains were analysed using different methods, which are stated in detail in the
corresponding results section.
30
4 Results −Mechanical Behaviour
4.1 Specimen Characterisation
Bone Mineral Density (BMD) was measured for most of the specimens in order to find rel-
evant correlations of the mechanical data with this clinically important measure. Especially
for the bovine specimens also the homogeneity along the specimens’ axis was a major con-
cern as the cancellous bone structure shows a relatively large architectural deviation from
the physiological bone axis in the middle of the vertebrae (Fig. 2.1). Thus, specifically the
two aspects were: a) the homogeneity of the specimens and b) density correlations with
mechanical data. The homogeneity of the bovine specimens was verified with 2 mm sliced
QCT scans and quantitative stereology. Both measures were found to be reasonably con-
stant for a specimen length of 15 mm. This was valid from both ends (superior and inferior
directed) of the bovine vertebra (Fig. 4.1). So only this section of the cores was taken for
further analyses.
Bone Mineral Density measurements on 75 specimens from all groups (bovine and
human) lead to a correlation of
E0=0.0072 ×BMD2.09,R2=0.68.(4.1)
E0: Initial secant modulus, MPa
BMD: Bone Mineral Density, g/ccm
By taking more detailed analyses correlations for the single groups showed the fol-
lowing strengths: bovine 0◦:R2= 0.69; bovine 90◦:R2= 0.30; human 45◦:R2= 0.33;
31
Chapter 4: Results −Mechanical Behaviour
human 90◦:R2= 0.44. These values were determined with linear functions, as they showed
the best fits for these tests in contrast to the combined fit for all specimens (Fig. 4.1).
4.2 Monotonic Compression
Typical for a cellular solid (Gibson and Ashby, 1997), the monotonically loaded cancellous
bone specimens’ behaviour was found to be linear (in stress-strain space) at the beginning
of the compression (Figure 4.3). The linear regime is followed by a decreasing slope, in-
dicating failure and plastic collapse, and an increase (densification) at higher strain values.
While the magnitude of the maximum compressive stress varies up to tenfold between the
groups, the characteristics of the deformation behaviour are similar. Even if the initial de-
formation behaviour appears to be rather linear, non-linearities can be found by analysing
the deformations with a higher local resolution. The surface deformations of the specimen
reveal these inhomogeneous components. Localised strain concentrations can be found al-
ready at a small percentage of the maximum stress. These strain concentrations do further
localise and increase in value until failure occurs in the same region. Macroscopic damage
appears in the form of deformation line across the whole specimen cross section (Fig. 4.3).
Due to the limited number of available samples no statistical analyses could be applied to
the monotonic compression tests. Nevertheless, in the preliminary studies strains at failure
were found to be almost constant for bovine samples.
33
Chapter 4: Results −Mechanical Behaviour
Figure 4.3: Monotonic compression of a human 0◦, a human 90◦and a bovine vertebral 0◦
bone specimen. The surface von Mises strains at two different stages were anal-
ysed at the bovine sample. b) 0.44 MPa, c) 26 MPa, d) after failure; (The human
specimens were compressed to fixed strain values for further investigations, thus
only the linear regime and the maximum stress peak is visible. The indicated (sur-
face) strain values are not valid on the tissue level).
34
Chapter 4: Results −Mechanical Behaviour
4.3 Cyclic Compression
Altogether, 108 specimens were tested in cyclic compression. A summary of the results of
the cyclic compression experiments of all specimen groups is given in Table 4.1. Cycles to
failure for all specimens in all groups ranged from 1 −1,000,000. Seven specimens did
not fail and were stopped at a minimum of 240,000 cycles.
Table 4.1: Summary of groups.
Group N E0[MPa] Load range σ/E0, %
Bovine 0◦31 2709 ±548 0.39 −1.47
Bovine 90◦12 1227 ±725 0.33 −1.06
Human vertebral 0◦14 447 ±117 0.22 −1.00
Human vertebral 22◦8 159 ±11 0.23 −0.68
Human vertebral 45◦13 111 ±68 0.17 −1.25
Human vertebral 90◦15 98 ±78 0.20 −1.25
Human femoral 0◦14 1031 ±461 0.23 −0.82
N: Number of specimens
4.3.1 Deformation Behaviour
The (integral) cyclic deformation behaviour of trabecular bone follows the principle mech-
anisms of fatigue, which were derived for metallic materials as well as bone tissue (Michel
et al., 1993; Moore and Gibson, 2003b; Ohrndorf et al., 2006) both intra- and intergroup-
wise. It can be characterised by increasing residual strains (cyclic creep), which causes a
shift of the hysteresis loops along the strain axis and a broadening and increasing nonlin-
earity of the hysteresis. Furthermore, the secant modulus, defined as the slope between the
minimum and maximum stress level, decreases with increasing cycle numbers (Fig. 4.4,
Equation 3.1).
The cyclic deformation process reveals the three classical stages of fatigue: a tran-
sient behaviour characterised by rapid strain increase within the first load cycles, a sat-
uration of strains and a fast increase of deformation (softening) near catastrophic failure
(Fig. 4.5). This holds qualitatively for all groups.
35
Chapter 4: Results −Mechanical Behaviour
Figure 4.4: Stress strain hysteresis loops (human femoral 0◦specimen, σ/E0=0.0060,
Nf=300. Residual strains are defined as the accumulated plastic deformations
(shift along strain axis), maximum strains are defined as total strains (residual
strains superimposed with the elastic part of the strains).
Figure 4.5: Maximum and residual strains for a bovine vertebral 0◦specimen.
36
Chapter 4: Results −Mechanical Behaviour
Analysis of the deformation behaviour
The integral (along the load axis) cyclic deformation process can be separated in several
phases: deformations in the initial transient region (εt), evolution of the deformations
during the pseudo saturation regime (εs), deformation at the defined failure criterion
(εf) and increasing deformations prior to macroscopic failure (Fig. 4.6). The first two
stages were analysed with respect to groupwise differences in the deformation behaviour
and related to the load level σ/E0in order to provide information on the evolution of
deformations until failure and the accumulated deformations at a point prior to failure.
Therefore, deformations in the inital region are linearized and the deformation rates in the
saturation regime are determined.
Figure 4.6: Cyclic deformation of a bovine 0◦specimen. Shown is the evolution of the max-
imum strains and the differentiated maximum strains with respect to the number
of cycles as well as the characteristic points at which the deformational behaviour
was analysed.
37
Chapter 4: Results −Mechanical Behaviour
Initial transient region
Maximum strains as well as accumulated residual strains were analysed at the end of the
transient behaviour. Higher normalised loads σ/E0resulted in higher deformations at the
end of Stage I. While maximum strains showed a linear dependency on the applied maxi-
mum load (σ/E0), residual strains exhibited only very weak trends. The regression analysis
results of the maximum strain vs. load were further assessed by an analysis of covariance
test in order to find out statistically relevant differences between the specimen groups. Sta-
tistically significant differences were observed for the bovine groups and human 90◦. The
test between human 0◦and human 90◦only just failed statistical significance with a p-value
of 0.06. Despite the failed statistical significance between all groups results are shown in
a groupwise manner. Maximum strains at the end of the transient went up with increasing
misalignment of the samples’ orientation with respect to the main structural axis of the
bone intergroupwise and with increasing load magnitude intragroupwise, respectively.
εmax t =c×(σ/E0)+d(4.2)
εmax t: accumulated maximum strain at the end of Stage I
c, d: regression coefficients (see Table 4.2)
Table 4.2: Regression coefficients for Equation 4.2
Group c d R2
Human vertebral 0◦0.89 0.003 0.86
Human vertebral 22◦0.89 0.001 0.89
Human vertebral 45◦1.06 0.001 0.78
Human vertebral 90◦1.67 0 0.89
Human femoral 0◦0.97 0 0.92
Bovine vertebral 0◦0.82 0.002 0.62
Bovine vertebral 90◦0.85 0.001 0.84
All analyses were performed with the robust option.
38
Chapter 4: Results −Mechanical Behaviour
Figure 4.7: Maximum strain at the end of the initial transient as a function of load σ/E0for
the human vertebral groups.
The functional relationship between the number of cycles in the initial transient phase
and the load level σ/E0was best represented by exponential approaches (Fig. 4.8).
Nt=e×exp−f×(σ/E0)(4.3)
Nt: Number of cycles in Stage I
e, f: Regression coefficients (see Table 4.3)
No statistically significant difference was found among the human vertebral groups
(even if a trend for differences exists), whereas the bovine groups behaved markably dif-
ferent. At similar load levels, bovine specimens exhibited higher absolute cycle numbers
for the transient. Therefore the human vertebral groups were taken as one common group
as well as the bovine vertebral groups.
The relation of the number of cycles in the transient and the number of cycles to failure
can be seen in Table 4.4. The fraction of lifetime within the transient is going up with
increasing deviation from the physiological axis. At the same site, statistical analyses
39
Chapter 4: Results −Mechanical Behaviour
Table 4.3: Regression coefficients for Equation 4.3.
Group e f R2
Human vertebral 7.7×105−1864 0.91
Bovine vertebral 9.1×107−2074 0.94
Both analyses were performed with the robust option.
Figure 4.8: Number of cycles at the end of the initial transient as a function of load σ/E0.
Regression curves are shown for the combined human vertebral and combined
bovine vertebral group.
revealed significant differences between the human vertebral 0◦and the human vertebral
45◦and 90◦group. Irrespective of the species and site, 0◦groups were found to have the
smallest relative number of cycles within Stage I.
Saturation Region
The evolution of maximum and accumulated residual strains within the saturation region
(Stage II) was analysed with respect to the load level σ/E0. For this reason, the cyclic
deformation curves for the two strain values of each specimen were differentiated with re-
spect to the number of cycles (comp. Fig. 4.6). The mean strain evolution rate was used
for the regression analysis. Bovine groups showed once again a remarkably different be-
40
Chapter 4: Results −Mechanical Behaviour
Table 4.4: Fraction of number of cycles in Stage I (Nt) to number of cycles to failure (Nf).
Group Nt/Nf
Human vertebral 0◦0.16 ±0.07
Human vertebral 22◦0.27 ±0.18
Human vertebral 45◦0.42 ±0.23
Human vertebral 90◦0.47 ±0.26
Human femoral 0◦0.17 ±0.13
Bovine vertebral 0◦0.13 ±0.05
Bovine vertebral 90◦0.18 ±0.15
haviour. Groupwise differences were also obtained for the human specimens (Fig. 4.9). At
similar load magnitudes, bovine cancellous bone exhibited smaller strain evolution rates,
which increased in the off-axis group, nevertheless being smaller than the rates in all hu-
man groups. Among the human groups, smallest rates were observed in femoral cancellous
bone, whereas in the vertebral groups strain evolution rates increased with increasing spec-
imens’ orientation angle. Power-law relationships were established for the groups.
dεmax
dN =g×(σ/E0)h(4.4)
g, h: Regression coefficients (see Table 4.5)
Table 4.5: Regression coefficients for Equation 4.4.
Group g h R2
Human vertebral 0◦2×1027 14.07 0.96
Human vertebral 22◦2×1018 10.2 0.96
Human vertebral 45◦9×1011 7.5 0.87
Human vertebral 90◦8×1016 9.53 0.89
Human femoral 0◦2×1014 8.8 0.88
Bovine vertebral 0◦2×1021 12.45 0.91
All analyses were performed with the robust option.
41
Chapter 4: Results −Mechanical Behaviour
Figure 4.9: Rate of residual strain evolution as a function of the load level σ/E0. Regression
curves are shown for the human vertebral 0◦, human vertebral 90◦as well as the
bovine 0◦group. The dotted lines refer to the supposed behaviour beyond the data
range.
Strains at failure
Failure was defined as a ten percent reduction in secant modulus. The accumulated residual
strains as well as the maximum strains were analysed at this point. Both appear to be
linearly dependent on the applied load σ/E0.
εf=i×σ/E0+j(4.5)
i, j: Correlation coefficients (see Table 4.6)
A linear relationship between maximum strain at failure and the applied load pro-
vided an accuracy in the order of R2= 0.80 when all data, irrespective of the specific group
was analysed. Considering the groups separated, the two groups human vertebral 0◦and
human vertebral 90◦represent the upper and lower limits of all specimens (Fig. 4.10), the
correlations increased to R2= 0.93 and R2= 0.85 respectively. Off-axis specimens tend
to have larger strain values at failure. The analysis of covariance (ANOCOVA) between
42
Chapter 4: Results −Mechanical Behaviour
Table 4.6: Regression coefficients for Equation 4.5.
Group i j R2
Human vertebral 0◦−1.07 −0.001 0.93
Human vertebral 22◦−1.13 −0.001 0.92
Human vertebral 45◦−1.37 −0.001 0.84
Human vertebral 90◦−1.52 −0.001 0.85
Human femoral 0◦−1.22 −0.001 0.85
Bovine vertebral 0◦−1.23 −0.001 0.84
Bovine vertebral 90◦−1.01 −0.002 0.86
Figure 4.10: Maximum strains at failure for the two bounding groups human 0◦and hu-
man 90◦.
43
Chapter 4: Results −Mechanical Behaviour
these two groups showed a significance level of p=0.02 and was therefore statistically
significant. The other groups failed statistical significance. Detailed information about the
relationships is given in Tab. 4.6. Similar relationships were found for the accumulated
residual strains with a lower correlational power. As described in Equation 4.14 a power-
law relationship between cycles to failure (Nf) and the normalised load ( σ/E0) can be
observed. Therefore, also a power-law can be observed by correlating the strains at failure
(εmax f ,εres f ) with Nf.
4.3.2 Damage Mechanisms
In this chapter, quantitative and qualitative analyses of damage and its evolution during
cyclic loading are described in order to reveal the principle mechanisms of cyclic failure
in cancellous bone structures. Quantitative analyses are shown for the integral damage
evolution and certain aspects of two-dimensional damage. The aim of two dimensional
(surface) analyses of the damage process was to define the methodology which different
scale levels of damage can be detected with and to identify the nature of damage rather
than a strict statistical analysis, which is beyond the scope of this work. The main question
concerning quantitative damage analyses was if a single variable (load σ/E0) is capable of
predicting damage evolution.
Integral damage
An integral damage parameter can be defined by the decrease of the secant modulus of the
hysteresis loops.
D=1−Esec/E0(4.6)
Different levels of structural degeneration can therefore be associated with the alteration
of this parameter. While a value of 0 is associated with the undamaged initial state, at
D=1 the structure has lost all load bearing capability. Fig. 4.11 shows the onset and
the evolution of 1 −D(which is generally acknowledged in the literature) under cyclic
loading for bovine vertebral specimens cored in two different directions (group 1 cored
44
Chapter 4: Results −Mechanical Behaviour
along the main physiological axis; group 2 cored in the medial-lateral direction of the
vertebral body). For all load levels (σ/E0) the physiologically loaded specimens (group 1)
clearly show retarded onset of damage.
Reduction of the normalised secant modulus (and therefore an increase of D) with
respect to percentage lifetime appeared to be different among the groups. The bovine ver-
tebral 0◦and 90◦samples hardly reached normalised lifetime (N/Nf) values greater than
1.2 (as failure is defined at D=0.1 also normalised lifetimes > 1 are possible). Likewise,
specimens from human groups aligned with the physiological axis (human vertebral 0◦and
femoral 0◦) failed shortly after N/Nf=1.0. In contrast, off-axis groups revealed a much
higher resistance. With increasing misalignment up to 8-fold normalised lifetimes were
reached in some cases (Fig. 4.12).
The overall damage evolution for a single specimen could be approximated with a
quadratic function for all groups. This holds at least in the range D<0.2. While the qual-
itative course of damage evolution was similar for the bovine groups and human vertebral
0◦, 22◦as well as human femoral 0◦with an positive quadratic behaviour, some specimens
from the human vertebral 45◦group and all specimens in the human vertebral 90◦group
revealed a negative quadratic functional relationship. Fig. 4.13 shows the different damage
behaviour.
As D appeared to be quadratic dependent on the number of cycles, damage rate
(∆D
∆N) changes in a linear manner and damage acceleration is found to be constant for each
specimen:
∆2D
∆N2=const.(f or one specimen)(4.7)
If the damage acceleration and the initial damage rate are known, damage evolution
can be determined theoretically.
D=∆D
∆Ninitial
×N+1
2×
∆2D
∆N2×N2(4.8)
Damage acceleration was found to be function of the normalised load σ/E0. The
45
Chapter 4: Results −Mechanical Behaviour
Figure 4.11: Evolution of damage (1−D) in two different groups of bovine trabecular bone
where specimens of each group were loaded with equivalent σ/E0. Group 1:
cored along physiological axis, Group 2 cored perpendicular (medial-lateral) to
the physiological axis.
Figure 4.12: Reduction in normalised secant modulus (1−D) as a function of percentage
lifetime for the groups human vertebral 0◦and human vertebral 90◦.
46
Chapter 4: Results −Mechanical Behaviour
Figure 4.13: Qualitative course of damage in the different groups. a) all groups except human
vertebral 90◦and parts of human vertebral 45◦, their course is shown in b).
relationship could be best approximated with a power law function:
∆2D
∆N2=k×(σ/E0)l(4.9)
k, l: Regression coefficients (see Table 4.7)
While damage acceleration was positive for most of the specimens, also a few were
observed to have negative damage acceleration values. This effect could be increasingly
observed with larger deviation from the physiological axis. In detail, human vertebral 22◦
Table 4.7: Regression coefficents for Equation 4.9.
Group k l R2
Human vertebral 0◦7×1040 20.36 0.90
Human vertebral 45◦♦3×1014 7.5 0.94
Human vertebral 90◦2×1032 16.5 0.90
Human femoral 0◦4×1029 15.55 0.93
Bovine vertebral 0◦8×1033 18.84 0.89
♦correlations only for positive damage acceleration values
correlations for −σ/E0
47
Chapter 4: Results −Mechanical Behaviour
contained 2 specimens with a negative trend out of 8 analysed, human vertebral 45◦4 out
of 11, human vertebral 90◦10 out of 12 with the remaining two having values close to zero.
Therefore, damage slows down in these specified samples with increasing cycle numbers
whereas damage goes up more rapidly in the others. The initial damage rate also revealed
groupwise power-law dependencies on normalised loads. Analysing the damage rates at
the defined failure D=0.1 results in another powerlaw dependency (Table 4.8).
∆D
∆Ninitial =m×(σ/E0)n(4.10)
m, n: Regression coefficients (see Table 4.8)
Table 4.8: Regression coefficents for Equation 4.10.
Group m n R2
Human vertebral 0◦5×1020 10.54 0.81
Human vertebral 45◦9×1011 6.63 0.85
Human vertebral 90◦2×1016 8.39 0.88
Human femoral 0◦3×1013 7.41 0.88
Bovine vertebral 0◦3×1016 9.6 0.77
Modeling the course of damage as a function of the normalised load levels can be
done through substituting Equation 4.10 and 4.9 in Equation 4.8:
D=m×(σ/E0)n×N+k×(σ/E0)l×N2(4.11)
Fig. 4.15 shows the experimental data for two human vertebral 90◦specimens and the
computed data for equivalent load levels. Even if the determined relationships are strong,
an immense scatter compared to the specimen-specific data was observed. While the rela-
tionships work well for the on-axis groups and the human vertebral 90◦group, correlations
with experimental data from the human vertebral 45◦group were poor as specimens failed
either with positive or negative damage acceleration.
48
Chapter 4: Results −Mechanical Behaviour
Figure 4.14: Damage acceleration as a function of load for the human vertebral 0◦, human
vertebral 45◦and bovine vertebral 0◦groups.
Figure 4.15: D as a function of number of cycles for human vertebral 90◦computed with
Equation 4.11 and experimental data for two load levels σ/E0.
49
Chapter 4: Results −Mechanical Behaviour
Damage acceleration was also assessed for its dependency on cycles to failure. Merg-
ing the three 0◦groups in one common regression analysis with regard to cycles to failure
leads to the strongly correlated powerlaw (R2=0.98):
Nf=0.13 ×∆2D
∆N2−0.55
.(4.12)
The other groups could not be included in this analysis as they also contained speci-
mens with negative damage acceleration.
Figure 4.16: Behaviour of cycles to failure as a function of damage acceleration (∆2D
∆N2). One
common regression curve was established for the three groups aligned with the
main physiological bone axis.
Poisson’s ratio
Surface strains were measured on two sides of the specimens for the bovine specimens.
This setup allowed for a continuous two-dimensional analysis of the surface deformations.
The surface strain components in longitudinal and transverse direction were integrated over
the whole surface in order to observe Poisson’s ratio defined as
50
Chapter 4: Results −Mechanical Behaviour
ν=−RR(εtransverse)dxdy
RR(εlongitudinal)dxdy.(4.13)
The optical surface strain analysis revealed two distinctive deformation modes for
the bovine vertebral 0◦and 90◦groups characterised by a variation in Poisson’s ratio (es-
tablished using Equation 4.13) during each test. For the group 0◦specimens Poisson’s
ratio went up with increasing cycle number, indicating a relative increase in the amount of
transverse strain. Exactly the opposite behaviour was observed for group 90◦specimens:
cell deformation increased more rapidly in the longitudinal (along the load axis) direction
and thus Poisson’s ratio continuously decreased with cycle numbers. This characteristic
behaviour was found for almost all specimens, with varying absolute values of the rate of
change (Fig. 4.17 shows two typical examples). The large scatter between the specimens
did not allow for a quantitative evaluation of this effect. (A second method to quantify
Poisson’s ratio was also established: binarisation techniques were applied to the images of
the optical deformation measurement system. In a self-written routine average extension
in the middle section of the specimen was computed and coupled with the longitudinal
reduction in the specimens’ height. Therefore, the more traditional Poisson’s ratio rela-
tionship could be established. Likewise, the large inter-specimen scatter did not allow for
a quantitative evaluation as transverse deformation appeared rather inhomogeneously.)
Strain localisation
As shown for the monotonic compression behaviour of cancellous bone (Fig. 4.3) also
strain localisations were found in specimens exposed to cyclic compression. Optical de-
formation analyses were utilised to reveal these effects.
Two examples of the evolution of strain concentrations observed during various
stages of cyclic compression are given in Fig. 4.18 and Fig. 4.19. In both cases, strains
are rather stochastically distributed at the beginning of the saturation region (Fig. 4.18b).
As accumulated residual and maximum strains evolve, strain concentrations do further lo-
calise (Fig. 4.18c−d). This is followed by the formation of a slip line through the specimen
51
Chapter 4: Results −Mechanical Behaviour
Figure 4.17: Evolution of Poisson’s ratio for two specimens with almost equivalent lifetimes
(bovine 0◦, bovine 90◦).
(Fig. 4.18e−f). The strains appear to be highly non-uniformly distributed. The initiation
of catastrophic failure with respect to local strains undergoes the following stages: strain
localisations right from the beginning; further strain ‘hot spots’ formations all-over the
specimen as integral strains increase; localisation of strains at few regions with initiation of
a macroscopic crack through the spongy structure. In the bovine 0◦specimen of Fig. 4.20
an example for the connection between strain concentrations and structural failure is given.
Fig. 4.20a) shows the distribution of transverse strain components at approximately 18 %
of the overall lifetime. Each pair of positive and negative transversal strain components
represents the deformation of a cell and its trabeculae, respectively. Many highly strained
cells are visible distributed all-over the specimen’s surface. In Fig. 4.20c) the von-Mises
strains close to the cycle of macroscopic failure are shown. One distinctive region of strain
localisation (indicated by the circle) from the earlier detected pattern plays a major role in
the initiation of the fracture line whereas regions with initially higher deformations do not
contribute to catastrophic failure. The structural element where failure initiates has been
located as a trabecula which was orientated in alignment with the specimens’ stress axis
(Fig. 4.20b).
52
Chapter 4: Results −Mechanical Behaviour
Figure 4.18: Evolution of surface von-Mises strains on a bovine 0◦specimen, σ/E0= 0.0060,
Nf= 1550. (Remark: The absolute strain values are not valid due to distortions
in the pattern which are caused by marrow and structural cell collapse).
Figure 4.19: Evolution of surface von-Mises strains on a bovine 0◦specimen, σ/E0= 0.0077,
Nf= 240. (Remark: The absolute strain values are not valid due to distortions in
the pattern which are caused by marrow and structural cell collapse).
53
Chapter 4: Results −Mechanical Behaviour
Figure 4.20: Strain distribution on a bovine vertebral specimen 0◦(Load σ/E0= 0.00414),
a) transverse strains at cycle 10,000, b) SEM image of initial failure region after
macroscopic failure, c) von-Mises strains at cycle 185,000 (Dendorfer et al.,
2005).
Quantitative Assessment of Surface Strains
Surface strains at the bovine specimens were processed in order to analyse the stochastic
strain concentrations in a quantitative manner. On-axis, longitudinal and perpendicular,
transverse strains were analysed and the total surface area was classified in steps of 1 per-
cent strain bounds from −10 to 10 percent at various stages of each experiment (to study
the influence of these rather arbitrary bounds, an analysis of various bounds has been un-
dertaken, and the strain bounds shown below were found to be the most suitable). The
percentage of strained area within a certain bound was plotted versus the number of cy-
cles for each specimen (Fig. 4.21). The major strain class is from 0 to -1% (compressive).
Its magnitude decreases in the course of the experiment and subsequently more areas are
strained with higher strain values. All specimens were merged in a further step by analysing
the strained areas at failure (10% reduction in secant modulus) and correlating these values
with the corresponding normalised load ratios σ/E0. Weak linear correlations between
the strain classes −1 % to −2 % and −2 % to −3 % and the applied load were observed.
The tensile components were found to have the tendency to decrease with increasing nor-
54
Chapter 4: Results −Mechanical Behaviour
malised load. In the bovine 0◦group tensile components do rapidly decrease at a load level
of 0.0075 (Fig. 4.22). The specimens transversally loaded to the physiological direction
(bovine 90◦group) did not exhibit this boundary (Fig. 4.23).
Figure 4.21: Accumulated strained areas as percentage from the total area within different
strain classes as a function of the number of cycles of a bovine 0◦specimen,
σ/E0= 0.0075, Nf= 1500.
Trabeculae Level
Strains at the trabeculae level reveal residual components during cyclic loading (Fig-
ure 4.24b-e). These are already visible after few load cycles. Increasing the applied loading
from almost zero (preload level) (Figure 4.24c) to maximum load in the first cycle causes
strain concentrations in the middle of the transverse-oriented trabeculae (Figure 4.24e).
Strains between preload level (Figure 4.24b) and peak load (Figure 4.24d) in the 100th cy-
cle (about 15% of the overall lifetime of the specimen) did not differ significantly. There-
fore, the load bearing capacity of the trabecula has been reduced by damage accumulation.
Damage occurs in the middle section in form of (micro-)cracking (Figure 4.25). The (de-
structive) magnitude of the strains lies in the range of 4 −5%. The analysis of trabeculae
55
Chapter 4: Results −Mechanical Behaviour
Figure 4.22: Percentage of accumulated strained area from total area within different strain
classes at the failure for the bovine 0◦group.
Figure 4.23: Percentage of accumulated strained area from total area within different strain
classes at the failure for the bovine 90◦group.
56
Chapter 4: Results −Mechanical Behaviour
orientated in 45◦to the load axis revealed bending of the struts and strain concentrations
at the joints. Examples for macroscopic damage of cyclic loaded specimens are given
in Fig. 4.25, 4.26. In the plate-like structure of the bovine vertebral bone, damage can
be found in the transverse struts as well as in the plates. The struts fail mainly due to
bending and cracking occurs along the laminary structural compostion of the trabeculae.
Furthermore, packages likely from former remodeling activity were found to burst from
the trabecula (Fig. 4.25). Failure in the structural elements aligned with the stress axis fail
mostly due to compression fractures, perpendicular to the laminar compostion (Fig. 4.26).
Also fractures of whole trabeculae (Fig. 4.27, Fig. 4.28) were found.
57
Chapter 4: Results −Mechanical Behaviour
Figure 4.24: a) Stress strain hystereses (human femoral 0◦specimen, σ/E0=0.0060, b) −e)
von Mises Strain distribution on a single trabecula during different stages of the
experiment.
58
Chapter 4: Results −Mechanical Behaviour
Figure 4.25: (Micro-)Cracking of transversely loaded trabeculae (SEM).
Figure 4.26: Damage in bovine vertebral 0◦cancellous bone, σ/E0=0.0041,Nf=185,000,
integral strain at examination εmax =1.33 %, the stress axis lies in the vertical
direction (SEM) (Werz, 2005).
59
Chapter 4: Results −Mechanical Behaviour
Figure 4.27: Fracture of a trabecula, bovine vertebral specimen, load axis is in vertical direc-
tion, specimen has been embedded and sectioned (LM).
Figure 4.28: Rupture of a trabecula joint of a bovine vertebral specimen, load axis is in verti-
cal direction, specimen has been embedded and sectioned (LM).
60
Chapter 4: Results −Mechanical Behaviour
4.3.3 Lifetime curves
Analysing the relationship between the applied load and the cycles until the (defined) fail-
ure results in lifetime curves. For the runout specimens, the cycle number at abort time
of the experiment was taken for the analysis. The lifetime curves can be expressed ac-
cording to a power-law and can therefore be shown in log-log plots as linear functions.
Lifetime curves were established for all groups separately. Stress versus cycles to fail-
ure regression analyses resulted in very weak correlations. While the groups aligned with
the physiological axis showed reasonable trends, off-axis groups behaved rather arbitrarily
(Fig. 4.29). Using the normalised stress levels σ/E0(as load value) results in strongly cor-
related power-law relationships for all groups. (Remark: σ/E0is used to establish lifetime
data, σrefers to the applied maximum compressive stress rather than the stress amplitude.)
σ/E0=a×Nb
f(4.14)
Nf= Cycles to failure
a, b: Regression coefficients (see Table 4.9)
The intersection point at the σ/E0axis as well as the gradients of the curves ap-
pear to be different between the groups (Fig. 4.30). Lifetimes were highest for the bovine
vertebral specimens loaded in physiological direction and lowest for the human vertebral
bone loaded at 90◦off the physiological axis. A reduction in lifetime was observed for
Table 4.9: Regression coefficients for Equation 4.14.
Group i j R2
Human vertebral 0◦0.0098 −0.109 0.94
Human vertebral 22◦0.0091 −0.111 0.97
Human vertebral 45◦0.0096 −0.121 0.89
Human vertebral 90◦0.0087 −0.119 0.88
Human femoral 0◦0.0103 −0.108 0.97
Bovine vertebral 0◦0.0133 −0.094 0.91
Bovine vertebral 90◦0.0105 −0.105 0.95
61
Chapter 4: Results −Mechanical Behaviour
the off-axis groups, both in human and bovine. In the human vertebral groups a larger
deviation from the main structural axis of the cancellous bone results also in lower life-
times. But despite the observed differences, a certain loss of information is associated
with the normalisation process (σ/E0) as it blurs the deviations in the fatigue behaviour
between the groups. In order to clearly point out the consequences of non-physiological
loading on fatigue lifetime the data was normalised further by the mean modulus of each
group (Fig. 4.31). Relating this data specifically to the specimen orientation angle (with
respect to the physiological bone axis) results in Fig. 4.32. This modified form of a S-N
diagram with the introduction of an anisotropy-based pseudo stress reveals the magnitude
of the load bearing capability with respect to the anisotropy. While lifetimes are decreas-
ing rather rapidly within a narrow band of the physiological bone axis, they remain almost
constant at higher (deviation) angles.
Figure 4.29: Cycles to failure as a function of peak stress σfor human vertebral 0◦,45◦and
90◦groups.
62
Chapter 4: Results −Mechanical Behaviour
Figure 4.30: Cycles to failure as a function of σ/E0for bovine (BV: bovine vertebral, the
number refers to the angle between the specimen axis and the physiological bone
axis) and human (HV: human vertebral, HF: human femoral) groups. In order
to enhance the clarity of the representation only the point data of the bounding
groups was included. For comparison with the literature an additional curve
taken from Haddock et al. (2004) is plotted in the graph.
Figure 4.31: Lifetime curves for human vertebral 0◦,45◦and 90◦groups. Load values were
modified by the mean initial modulus of each specimen group.
63
Chapter 4: Results −Mechanical Behaviour
Figure 4.32: Stress values for the corresponding cycle numbers (Fig. 4.30) were modified by
the mean initial modulus of each specimen group and plotted as a function of the
angle between physiological and specimen axis. (The dashed lines are not based
on regression analysis.)
4.4 Age-Related Effects
The influence of age on the mechanical behaviour of cancellous bone is addressed in this
chapter. For this reason, human vertebral cancellous bone data was analysed with respect
to donor age. No sex-specific differences could be analysed due to vast majority of male
donors.
The initial secant modulus of the specimens was found to be highly dependent on
donor age (Figure 4.33). For all specimen groups the modulus decreased with increasing
age. This holds with the exception of the specimens from the 46 year old male donor of the
0◦group, which showed a slightly weaker mechanical performance than the overall trend,
as well as one specimen from a 72 year old male donor whose value was exceptionally high.
In both cases, the bone samples did not show any (macroscopic) conspicuous features.
The relationships between the decline of the initial secant modulus and the donor’s
age are represented most suitably as quadratic and cubic polynomials (thus exponential
64
Chapter 4: Results −Mechanical Behaviour
relationships provide good correlations). While the principle change appears to be similar
concerning the different groups, the relative decrease of the modulus appears to be more
pronounced in the 45◦and 90◦groups compared to the 0◦group. In these off-axis groups
mean modulus drops within 20 years (60 −80 years) by more than 70 percent from (mean)
approx. 180 MPa and 160 MPa to 50 MPa and 20 MPa, respectively, whereas the group
aligned with the physiological axis revealed a decrease of 25 percent from approx. 440
MPa to 330 MPa.
The characteristic cyclic deformation behaviour showed no remarkable change with
increasing age. All specimens followed the stages of deformation described earlier in fa-
tigue of cancellous bone with an increasing residual deformation, decreasing secant modu-
lus and increasing hysteresis loops during cyclic deformation. Damage evolution, defined
as a decrease in secant modulus, did also not appear to be different with increasing donor
age. Figure 4.34 shows the evolution of this scalar damage variable as a function of the
percentage of total fatigue lifetime for four 45◦specimens from two 61 year old and two
80 year old specimens. Damage increases rapidly within a small percentage of lifetime,
this transient is followed by a lowered, constant damage rate which increases at approx. 75
to 85 % of the total lifetime indicating macroscopic failure. Similar results can be obtained
for the other specimens/groups.
Neither maximum strains at failure nor accumulated residual strains revealed a sig-
nificant influence of donor age on the results. Specimens with different donor ages showed
no tendencies in higher of lower (integral) deformations at failure. Likewise, no influ-
ence of donor age on the tendency of earlier or later failure could be observed once initial
stiffness was taken into account. Age showed no influence on the probability distribution
along the lifetime curves (Figure 4.35). The other groups behaved similar to the 45◦group
already shown.
65
Chapter 4: Results −Mechanical Behaviour
Figure 4.33: Initial (undamaged) secant modulus as a function of donor age for different spec-
imen groups. The groups differed in their coring direction with respect to the
physiological axis.
Figure 4.34: Evolution of damage in terms of a decreasing secant modulus as a function of
the percentage of fatigue life. The numbers refer to the age of the donors. The
lifetimes of the plotted specimens (45 degree group) ranged between 1,600 and
11,000 cycles.
66
Chapter 4: Results −Mechanical Behaviour
Figure 4.35: Lifetime (S-N) curve for the 45◦group. The loads are normalised by the mean
initial modulus. The numbers refer to the age of the donor.
67
5 Discussion
This study focuses on the investigation of the mechanical behaviour of cancellous bone
under cyclic compression loading with specific respect to the trabecular bone structure, i.e.
orientation of loading axis, species and site. Especially the influence of load components,
which are not aligned with the main physiological bone axis has not been addressed yet
and is an important issue in understanding bone failure following unphysiological load-
ing (implants, misuse). Besides architecturally dependent lifetime correlations also the
deformation and damage behaviour of different cancellous bone structures have been in-
vestigated and the influence of load magnitude and direction are shown. The data has been
analysed with respect to donor age in order to provide information about the influence of
age on lifetime and deformation behaviour of cancellous bone.
5.1 Experimental Set-Up
In this study, no quantitative measure for architectural features was used. Much effort was
put into ensuring the proper alignment with respect to the physiological coordinate system.
An analysis of the main axes of a group of ten specimens by means of µCT, analysing the
fabric tensor, as well as stereological analyses of microscope images revealed a rather good
agreement of the core direction with the main axis of the cancellous structure. As the data
was only analysed for few specimens, results were not included in this work. Any inaccu-
racies concerning the resolution of deformations of the samples under cyclic loading due
to the experimental setup were carefully analysed and minimised in preliminary studies.
The contribution of the methylmethacrylate layer to the overall deformation of the sample
68
Chapter 5: Discussion
in terms of a superimposed strain can be regarded as negligible. A certain stiffening effect
in the bone boundary layers has been taken into account in the strain calculations. In con-
trast to most studies on cancellous bone in the literature, which use external extensometers
for deformation measurement, the strain data of this study was registered by the internal
displacement sensor, i.e. by measuring the grip to grip motion. Nevertheless, the precision
of the strain data is assured, for the fixations are clamped in a force-fit manner to the test
rig, where slipping is absolutely prevented for the load range of this study. Additionally,
individual deformations of all fixing parts aligned within the ‘gauge section’ of the internal
displacement sensor can be neglected due to their high mechanical stiffness. Furthermore,
the occurrence of any deformation of these fixing parts were finally analysed and verified
by means of external extensometers and by optical deformation measurements. For ex-
ample, line deformations on the specimens’ surface were analysed and the deformational
results were extrapolated to the whole specimens’ length (as a small percentage was not
visible for the optical system). Thus, it is assured that the resolution of strains by the
internal deformation measurement system is absolutely reasonable in this study. Secant
modulus was chosen instead of other methods (Moore and Gibson, 2001) as its determina-
tion appears to have a much higher accuracy (Cotton et al., 2005). Like in all experimental
fatigue studies on bone up to now no biological effects in terms of remodeling processes
could be included in this study and their influence remains unclear.
5.2 Deformation Behaviour and Damage Mechanisms
Characteristics of Deformation
The principle characteristics of the cyclic deformation behaviour for all groups were found
to correspond with the data given in the literature. In accordance with the relations found
for bovine tibial (Bowman et al., 1998; Moore et al., 2004) bone and human vertebral
bone (Haddock et al., 2004; Rapillard et al., 2006) loaded in the physiological axis of
the trabecular architecture stress −strain hysteresis was found to become increasingly
nonlinear exhibiting an increased area and a shift along the strain axis. The shift of the
69
Chapter 5: Discussion
hysteresisis referred to as cyclic creep effect in traditional material sciences (e.g. Schijve
(2001)) (but of course may be more related to the phenomena observed in concrete (e.g.
Jirasek and Bazant (2002)) rather than metallic materials). It has been discussed in the
literature whether or not creep does contribute to fatigue in cancellous bone (Bowman
et al., 1998; Moore et al., 2004; Yamamoto et al., 2006). It has been suggested that creep
may play a role in habitual loadings, in experimental data, which uses higher loadings,
microcracking is dominant (Yamamoto et al., 2006). In the range of this work residual
strains were found to be linked to microdamage initiation and growth (Fig. 4.24). Rather
large, localised strains were found, even at low load magnitudes (Fig. 4.3), which may
also hint at the formation of microcracks. Additionally, quantitative surface strain analysis
showed, even at low load levels, relatively large areas with higher strains, so that - at least
in the range of this work - microcrack initiation and growth may be the major cause for
the evolution of residual strains during cyclic loading, even if the observed mechanisms
are not validated with statistical methods. Furthermore, the definition of an elastic regime
for cancellous bone may be partly rejected. The terms ‘Young’s’ or ‘Elastic modulus’
appear to be imprecise as strain localisations in trabecular bone, exposed to monotonic
compression, can be found already at very early stages of the experiment, although the
stress −strain curve indicates strictly linear elastic behaviour (Fig. 4.3). Deformations
in trabecular bone may therefore, even at low magnitude, localise and in consequence be
coupled with damage. The secant modulus was found to increase slightly for a part of
the specimens during the first few load cycles. In these cases, the secant modulus of the
fifth cycle was chosen for the normalisation of the load (σ/E0). This increase may be
induced by stress redistributions in the volume directly beneath the surface to compensate
the loss of adjacent trabeculae due to the specimen preparation. Another point may be
early inelastic deformation caused by bending and the formation of certain plastic hinges
of unfavourably arranged trabeculae. The superposition of both effects causes the observed
almost instantaneous random pattern of locally enhanced strain areas, which directly result
in an integral loss of the specimen height. Extensive experimental effort has verified that no
embedding or other experimental inaccuracies did cause this effect. The definition of the
70
Chapter 5: Discussion
damage criterion as a ten percent reduction in secant modulus was chosen in accordance
with literature (Bowman et al., 1998; Haddock et al., 2004) and did also perform very well
in the preliminary studies on bovine vertebral cancellous bone. Therefore it was chosen for
the context of this work.
Detailed Cyclic Deformation Behaviour
The cyclic deformation behaviour of cancellous bone can be described, in analogy with the
literature on the fatigue behaviour of fibrous composite laminates, as a tripartite process.
Starting with stage I, where deformations increase rapidly and damage is initiated, the de-
formation process stabilises through stage II and in stage III deformations increase leading
to catastrophic failure. The definition of damage as a ten percent reduction in secant mod-
ulus shifts failure within stage II for all groups. Therefore, stage III is only partly analysed
and discussed in the section concerning damage.
Stage I: Initiation
In this initial phase of fatigue behaviour, deformations and damage are initialised and pro-
ceeding stabilisation of the deformation behaviour can be observed. The stabilisation of
the deformational process is prolonged in the off-axis direction groups. Both strain com-
ponents, maximum and residual, respectively, at the end of the transient exhibited a linear
dependency on the applied load (σ/E0). The off-axis groups showed increasing strain val-
ues with growing deviation from the physiological axis. Even if no (statistically relevant)
groupwise differences were observed for the number of cycles within Stage I as a func-
tion of the applied load, trends for a varying behaviour can be supposed from Fig. 4.8.
Despite this rather weak trend, the percentage of the number of cycles with respect to the
cycles to failure is increasing in the off-axis groups. Thus, with rising deviation from the
main physiological axis strain deformations as well as the percentage of lifetime during
Stage I is increasing. This may be the result of a growing lack of the load-bearing opti-
mum, which the structure has been adapted to during remodeling processes. Therefore, the
71
Chapter 5: Discussion
highly adapted structure of the bovine 0◦group possesses the shortest (relative) initiation
duration. As Stage I is also associated with the initiation of microcracks, the prolonged
stabilisation process also contributes to this. In the load adapted structures, aligned with
the 0◦direction, the following mechanism may be present: Microdamage is initiated and
microcracks stop relatively rapidly on ultrastructural, collagen fibrils or microstructural,
cement lines, boundaries, whereas microcracking in specimens loaded in non-optimised
directions may proceed longer until these barriers are reached, depending on the orienta-
tion and location (see Trabecular Composition, Fig. 2.3). Furthermore, the trabeculae are
increasingly loaded in bending, which results in stress concentration at the trabecular joints
and may also produce a higher amount of microdamage.
Stage II: Constant Deformation Rate
The initial transient region is followed by a region with a rather constant deformation rate.
This deformation rate was assessed for group differences and analysed as a function of
loading. Not all specimens of each group could be used as very high loaded specimens
did not reveal a region with constant deformation. The rate of integral strain evolution
depends on both, species and direction. In the bovine 0◦the lowest deformation rates (with
respect to normalised load) were observed, whereas rates increase at bovine 90◦. Similar
behaviour with an offset on the load axis was found for the human groups. The off-axis
groups revealed higher deformation rates. Interestingly, a species-dependent intersection
point of the deformation rate regression curves seems to exist. The corresponding load
values (σ/E0) are approximately 0.007 −0.008 for bovine and 0.0045 −0.005 for human
vertebral bone, respectively. The human 45◦group exhibits some deviation above the
threshold, in the bovine 90◦group the bearable maximum loads are reduced compared
to the bovine 0◦group. Therefore, deformation rates increase rapidly beyond a certain
load level. Fig. 5.1 shows the deformation rate curves for these two groups of bovine
and human bone. Data taken from the literature (Ganguly et al., 2004) on bovine tibial
bone shows a similar behaviour, lying between the bovine vertebral and human goups in
this study. The behaviour beyond the data range has been assumed. These results point
72
Chapter 5: Discussion
to a change in the deformation mechanism at the specified load level for each species.
Below this load threshold, architectural features accelerate or decelerate the deformation
rate which may correspond to increasing microcrack density and length, but most of them
being stopped at the materials’ internal boundaries. Above the specified load level micro-
and ultrastructural barriers may be skipped and therefore their orientation with respect to
the stress axis becomes less important. The rates for the maximum strains showed to be
very similar, with almost identical regression coefficients. Even if the data is afflicted by
remarkable scatter the derived trends are rather clear.
Strains at Failure
Strains at failure were found to be dependent on the specimen group. Even if statistical
significance could only be established for the differences between the human vertebral 0◦
and human vertebral 90◦groups, the results indicate the influence of structural orienta-
tion as well as species on the deformations at failure. Therefore, an isotropic strain-based
failure criterion may not be able to predict fatigue failure. One explanation for this differ-
ence can be found in structural cell deformation models. The evolution of Poisson’s ratio
(Fig. 4.17) revealed predominating transversal deformations in the bovine 0◦group, which
correspond to bending and/or buckling of the longitudinally oriented trabeculae. There-
fore, the main deformational components are oriented in the transversal direction and the
longitudinal components are less activated. The cell deformation in transverse loaded spec-
imens (90◦) is also dominated by bending of the long struts and thus, large deformations in
the longitudinal direction of the specimen accumulate. The observed differences in strains
at failure may therefore be caused by geometrical effects as only uniaxial deformational
components are used. The analysis of Poisson’s ratio was done by integrating the (surface-
)strains across the surface for both, transversal and longitudinal components. Even with
this advanced method only trends could be observed. Alternatively, the optical data was
used to calculate Poisson’s ratio in a more traditional sense with extension in the middle
of the specimen. This analysis did not result in valuable findings, the damage mechanism
in this highly non-uniform structure can can only hardly be described with integrally mea-
73
Chapter 5: Discussion
Figure 5.1: Stage II deformation rates (residual strains) as a function of the applied load
σ/E0.
sured deformations as for example cells may collapse either in one or the other direction.
In analogy to these findings other researchers reported similar results using extensome-
ter measurements for the orthogonal deformation components of trabecular bone, which
were also found to be very difficult and were only partly successful (Bredbenner and Davy,
2006).
Average Cyclic Deformation
By combining the derived relationships for stage I and stage II, cyclic deformation curves
can be analysed and established for a set of “average specimens” as the equations are based
on regression analyses. All relationships only depend on the applied normalised stress and
group. Thus, with the combination of Equation 4.14 (rearranged with respect to cycles
to failure), Equation 4.2, Equation 4.3 and Equation 4.4 (with g, h for maximum strains),
the deformation behaviour can be modeled for experiments stopped in stage II. Fig. 5.2
shows an example for the computed strains of a human 0◦specimen and the comparison to
two experimental data sets for specimens which were loaded with equivalent magnitudes.
Due to the linear connections between the computed points the smooth appearance of the
experimental curves is missing. Unsurprisingly, the computed data represents a mean curve
74
Chapter 5: Discussion
between the experimental data. The derived equations may be used in order to calibrate
and create continuum-based numerical models dealing with cyclic deformation behaviour
of cancellous bone. Strain values at the end of stage II, the beginning of the transient to
macroscopic failure, were also analysed, but a relatively large scatter in the results was
obtained. Therefore, this data was not used for further analyses.
Damage Evolution
The results indicate that damage evolution is dependent on species and structural orienta-
tion (with respect to the integral stress axis). The integral loss of structural integrity can be
described by a scalar damage parameter D, relating the actual secant modulus to the initial,
unharmed secant modulus of the specimen. While it has been shown for cortical (Cotton
et al., 2005) and cancellous bone (Moore and Gibson, 2003b; Ganguly et al., 2004) that in-
creasing strains during fatigue testing were associated with a reduction in secant modulus,
the effect of the axis of loading has not been addressed so far in the literature. Likewise,
direct comparison of the fatigue behaviour between human and bovine cancellous bone has
only been discussed sparsely (Haddock et al., 2004).
Specimens loaded in the off-axis direction reached lower overall cycle numbers at
comparable load levels for both human and bovine samples (Fig. 4.30, 4.11). In order to
assess the qualitatively observed differences in loss of stiffness as a function of number
of cycles for the various groups (Fig. 4.11) in a more quantitative manner, D vs.Nfdata
was normalised with respect to percentage lifetime. And the average loss of stiffness was
analysed at various percentage values of the lifetime for the specimens (Fig. 4.12). Sam-
ples with only few cycles were removed from the analysis as serious damage was already
induced during the first load cycle. As Nfis associated with the defined failure criterion of
D=0.1 rather than catastrophic failure also N/Nfvalues beyond 1.0 are possible. While
only minor differences in Dwere observed for normalised cycle numbers N/Nfbelow
1.0, at higher normalised cycle numbers a deviation between the groups becomes visible.
Bovine samples hardly reached higher N/Nfnumbers as a stiffness loss of 10 percent was
almost suddenly followed by structural failure of the samples. This does not hold for the
75
Chapter 5: Discussion
Figure 5.2: The cyclic deformation of a human vertebral 0◦specimen is computed with an
assumed load of σ/E0= 0.0033 using the equations derived in Section 4. Two
sets of experimental data loaded with equivalent magnitude are added.
human groups. Here higher N/Nfvalues were reached with magnitudes up to 5- to 10-
fold for human vertebral 90◦samples. The relative lifetimes reached rose with increasing
misalignment of the structural axis. Likewise, remarkable differences in the way damage
evolved was observed between the groups. For all specimens, a quadratic function did
approximate the course of damage very well, at least for the region below twofold nor-
malised lifetimes. Therefore, damage rate is linear and damage acceleration constant for
cancellous bone (within the framework of this study). The two parameters have been de-
termined by derivation of Dwith respect to the number of cycles. Damage evolution was
found to be quadratic positive for the bulk of the specimens with exception of the human
90◦and a part of the human 45◦specimens. Therefore, damage decelerates with increas-
ing cycle numbers for the latter group of specimens, whereas damage rate increases for
the others. This does also imply that damage increases more rapidly in the initial phase
in the 45◦/90◦specimens. In their unoptimised structure (with respect to the stress axis)
weak structural elements may fail suddenly and microcracks may grow relatively long until
they are stopped. Once the structure has found some kind of equilibrium, the damage rate
76
Chapter 5: Discussion
decreases. The elements/struts transversally oriented to the physiological axis may be the
main cause for this effect as it is mainly observed in the human vertebral 90◦group. Like-
wise, stress concentrations in the trabeculae joints are supposed to be more critical in the
45◦group, but this is not reflected in the data, which hints at the above mentioned findings.
Specimens in the 45◦group did fail either with positive or negative damage acceleration
manner. This may depend on the real orientation of the trabecular architecture as in this
study structural orientation is defined with respect to the physiological bone axis. There-
fore, the structure may be not loaded exactly in 45◦and either transversal or longitudinal
trabeculae are mainly loaded and the corresponding damage acceleration characteristics
may be activated. Furthermore, local bone structure in general is very inhomogeneous.
Despite the negative damage acceleration in the 45◦/90◦specimens this does not lead to
the conclusion that these structures exhibit a higher resistance against damage. Obviously,
the bearable stress levels are much lower (Fig. 4.31) and therefore their absolute damage
resistance is heavily reduced. (Critical) damage in samples, which are loaded along their
optimised structural axes appear to take much longer to be initiated (Fig. 4.11), but once
started, damage evolves rapidly to catastrophic failure. The critical level of structural de-
generation and loss of stiffness is much lower compared to the 45◦/90◦samples. In the
extremal case of the highly organised and optimised structures of the bovine 0◦group al-
ready a damage level of D=0.1 is critical.
Cotton et al. (2005) found that damage rate is a good predictor of fatigue life for
human cortical bone exposed to pure tensile fatigue. Damage rate has been defined in their
work as the linear fit to the damage curves within 10 % and 90 % of the specimens’ life-
time. Therefore, damage rate has been determined as constant and early damage in the
inital transient has been neglected. Ganguly et al. (2004) modelled the reduction in secant
modulus during compression fatigue loading of bovine cancellous bone as a function of
the maximum strain. They approximated upper and lower bounds for this reduction within
normalised stress ranges. It was also suggested that damage is governed by maximum
strain and the relationships can be determined with monotonic compression tests, whereas
the secondary residual strain accumulation rate (stage II) is governed by the normalised
77
Chapter 5: Discussion
stress. In this work, the evolution of the residual (and maximum) strains during stage II
and damage propagation were approximated as functions of normalised stress. Damage
curves were approximated with quadratic fits for the whole course of each specimens’ life-
time. Hence, also the initial damage processes were included in the analysis. The main
advantage of this approach is that only one parameter (σ/E0) is needed for the analysis of
the deformation and damage process. One weakness of this method is obviously the mag-
nification of any inconsistency of the variables through the integration process. Therefore,
well-defined structural responses, as bovine 0◦, human 0◦and 90◦specimens supplied,
result in reasonable damage curves, whereas data from the 45◦specimens, which exhibit
a rather random behaviour with respect to damage acceleration, as discussed above, are
only restrictedly suitable for damage predictions. In analogy to (Cotton et al., 2005) one
variable for the evolution of damage - in this study acceleration - was found to strongly
correlate to the number of cycles to failure. This holds irrespectively of site and species
if the samples were loaded along the main physiological axis. Therefore, damage accel-
eration may be a very good predictor for lifetime, even if there will be hardly a chance
to determine this factor in clinical praxis. This relationship may be more valuable for the
(numerical) modeling of the fatigue behaviour of cancellous bone and theoretical analyses
of the damage process.
As the critical level of stiffness loss was found to be different between the groups also
the definition of the damage criterion with a 10 percent reduction in stiffness is somehow
arbitrary. Nevertheless, a common measure for failure was applied in this study in order
to enable a direct comparison with literature data. Specimens were loaded up to a fixed
integral strain level to make post analysis of the structures’ architecture possible. That’s
why the catastrophic failure cycle was not available. Furthermore, sensitivity analyses
were conducted with different failure criteria: D=0.25 (25 % reduction in Esec), D=0.50
and end of stage II. Results were similar to the obtained findings with D=0.10, but with
weaker correlations concerning each of the relationships.
78
Chapter 5: Discussion
Damage Mechanisms
The appearance of (surface) damage in cancellous bone was analysed in this study at dif-
ferent scale levels. The main goal was to identify methods for an analysis and to show the
underlying damage mechanisms.This study does not claim a rigorous statistically verified
analysis of these principles. This lies beyond the scope of this survey and is a field of
further research.
The optical deformation measurement system used in this study was found to be
capable to analyse strains on the specimen as well as the trabecula level. Even if some
researchers applied optical deformation measurements on biological tissues - for example
pattern recognition algorithms to track the motion of a random speckle pattern (Sanghavi
et al., 2004) or optical strain measurement method (microdisplacements) (Kim et al., 2005)
- the application to trabecular bone as well as the analysis of strains at various scales, from
specimen to trabeculae level, is novel and shows the potential of these measurements. E.g.
through a rigorous statistical analysis of trabeculae strain evolution essential information
for precise models of damage in cancellous bone may be provides, which can be used for
the increasingly used micro-CT-based finite element computations and may also contribute
largely to crack growth models necessary for both, remodeling and damage, respectively.
Due to the highly inhomogeneous (apparent) surface of the specimens analysis was in most
cases restricted to two dimensions. The influence of the third deformational component
was minimised by aligning the object with the analysis plane. Therefore, its contribu-
tion may be negligible. The character of the optical system restricts the measurement of
strains to the surface of the specimen. Therefore, some deformational components may
be included in the analysis of whole surface strains, which originate from deformations at
sectioned trabeculae (through the preparation process), not representing the precise struc-
tural response. Strain magnitudes at the apparent level are therefore always afflicted with
some degree of uncertainty. This does not hold for the analyses of trabeculae deformations.
In these cases, trabeculae in the volume (second or third row) of the specimen were mon-
itored. Absolute strain values at the trabecula level are therefore much more accurately
79
Chapter 5: Discussion
comparable to the apparent specimen level. Nevertheless, surface strain data is found to
be very useful in order to explain the evolution of overall damage in the specimen and, as
damage is supposed to initiate on the surface of the structure, to detect regions of damage
initiation and propagation.
Besides the experimental investigation of the feasibility of these kinds of measure-
ments also principle damage mechanisms could be obtained. Damage was found to ap-
pear very localised within cancellous bone specimens under monotonic and cyclic loading.
Thus, strain localisations in trabecular bone, exposed to monotonic compression, were
found at very early stages of the experiment, although the stress −strain curve indicates
strictly linear elastic behavior. Comparable results were obtained with numerical stud-
ies, which demonstrated that damage is already visible below apparent compressive yield
strain and local tissue yielding initiates at low apparent stress levels (Nagaraja et al., 2005;
Morgan et al., 2005). Taken together, the results do reject the assumption of a linear elas-
tic regime in cancellous bone which may result from pure bending deformation modes of
the structural elements. Deformations in trabecular bone may therefore, even at low mag-
nitude, localise and in consequence be coupled with damage. Interestingly, local strain
concentrations could be detected in the region of later macroscopic damage initiation at
very low apparent stress levels (Fig. 4.3). Therefore, flaws (which may also be unfavorable
structural elements) are supposed to be pre-existing and if so may also be determined as a
measure for bone quality in further studies.
Cyclic loading of trabecular bone results in highly non-uniform deformations.
Analysing the integral material behaviour, damage appears in the form of increasing resid-
ual strains, often referred to as cyclic creep, an increasing hysteresis area and a decrease in
secant modulus. The quantitative measures for these values depend highly on the trabecu-
lar architecture. These effects are linked to several (local) damage mechanisms. Singular
cell deformations, resulting in bending of trabeculae, are observed within the first few cy-
cles. In combination with the non-uniform loading situation at the single trabecula level
(Fig. 5.3) struts fail even at low fractions of the overall lifetime (Fig. 4.24). Therefore,
trabeculae are subsequently removed from the load-bearing part of the structure. This is
80
Chapter 5: Discussion
visible from the appearance of strain ‘hot spots’ across the specimens’ surface. This sup-
ports the results from Keaveny et al. (1994) that damaged trabecular bone may be stress
protected by redistributing stress to undamaged regions. A critical density of local damage
may cause a large increase in crack growth rate, failure of whole trabeculae (initiation of
the fracture line) and successive failure of the whole structure in a relatively narrow failure
line across the specimen (Fig. 4.18, 4.19). Damage of single trabeculae appears in form of
very localised strains, which are linked to microcrack initiation and propagation. As micro-
cracks progress and trabeculae fail, respectively, residual strains accumulate and the secant
modulus decreases. Therefore, the observed cyclic creep effect is a result of (microcrack)
damage. Friction along the cracks’ surfaces may appear in the broadened hysteresis loops.
The highly reduced lifetimes of non-physiological loaded specimens are supposed to be a
result of architecturally induced stress concentrations, which result in an acceleration of
the above mentioned damage mechanism.
Usually studies on trabecular bone exhibit a large scatter in results, for which the
heterogeneity of local deformations across the specimen might be an explanation. As this
inhomogeneity is (without further refined analyses of the structure) a stochastic value for
the bone sample, an adequate specimen size has to be chosen in order to average this effect.
But as size is limited (donor bone size, homogeneity of local structure) a decrease in scatter
can only be gained with more detailed information.
As deformations and damage appear very localised in trabecular bone structures, in-
+ε
F1
F2
Figure 5.3: Schematic structural cell deformation
81
Chapter 5: Discussion
tegral approaches may lead to a high amount of uncertainties in deformation and fracture
predictability. This may be even more pronounced as structural inhomogeneity increases
as it appears in osteoporotic structures. The accumulation of residual strains may be the
result of microcracking of individual trabeculae. In order to give more accurate quantita-
tive measures specimen size, tissue quality and, as damage appears local, local structural
information should be included in the standard procedures.
5.3 Lifetimes
A traditional approach to lifetime analysis is to establish S-N or lifetime curves. Relating
the maximum stress magnitude to the number of cycles to failure for the groups in this work
did not result in useful correlations (Fig. 4.29). In accordance to what has been shown in the
literature for bovine tibial (Bowman et al., 1998; Moore et al., 2004) and human vertebral
bone (Haddock et al., 2004; Rapillard et al., 2006), single power-law relationships could be
established for each group by applying a normalisation of the load with the initial modulus
of each specimen.
Lifetimes with respect to a normalised load σ/E0varied to a large degree between
the groups. The group with the highest initial stiffness (bovine 0◦) revealed the best perfor-
mance, whereas in comparison human bone specimens displayed reduced initial stiffness
and reduced lifetimes. Non-physiological (off-axis) load application was correlated with a
decrease in fatigue strength for both bovine and human bone, respectively. Lifetimes de-
creased with decreasing mean initial stiffness of the groups. Therefore, bovine specimens
performed superior to human specimens. Among human samples femoral bone showed
the highest fatigue strength. It should be noted that almost no differences in fatigue life-
time were observed between the human 45◦and 90◦group. In both cases, the trabecular
structure is not optimised for the applied loading. This can result either in shear stress con-
centration at the strut joints or in failure of the weak transversely oriented trabeculae, also
the initial stiffness of both groups is comparable. Similar behaviour has been reported on
anisotropic fibre reinforced composites (Hertzberg, 1995), where strength reduced dras-
82
Chapter 5: Discussion
tically for changes in orientation close to 0◦but was similar at 45◦and 90◦. Therefore,
in analogy to engineering methods in composite materials, detailed information about the
local bone quality and structure is essential for an improved risk of failure estimation of
cancellous bone.
The remarkably higher lifetimes observed for the bovine vertebral 0◦groups may be
explained by the different architectural structure of this high dense bone type. The trabec-
ular structure is organised in a plate-like manner, aligned with the main loading (0◦) di-
rection, whereas in the directions normal to the load axis strut-like elements can be found
(Fig. 2.4). In the bovine vertebral 90◦group the latter elements are aligned with the in-
tegral stress axis. Therefore, lifetimes decrease, but at lower levels the decrease is less
pronounced. One explanation may be the combination of the short buckling length of the
struts and the rather stiff “floor” plates. While at high load levels struts fail in compression
and initiate macroscopic failure, lower stress levels result in a more elastic type of deforma-
tion, mainly causing fatigue in the plates. Furthermore, the species-dependent differences
may not only be a result of differing architectural structures, but also of the differing donor
age. All bovine bones were obtained from young cows (the age of the cows matched the
age of a 20-year-old man/woman) with an appropriate bone quality, whereas human donors
were middle- to old-aged where quality may be decreased.
The lifetime curves (σ/E0vs. Nf) do not exhibit a common endurance limit. In the
literature, an endurance limit of approx. σ/E0= 0.0035 has been suggested for cancellous
bone, which has been derived from finite element studies (Guo et al., 1994) and low cycle
fatigue tests (Moore and Gibson, 2003b). Ganguly et al. (2004) determined a theoretical
endurance limit for bovine bone between σ/E0=0.0024 −0.0032. While this limit may
exist for the bovine data, human specimens failed at much lower load values (e.g. 0.0017
for a human 45◦specimen), so the endurance limit is supposed to be smaller for human
bone. The in-vivo apparent strain of human vertebral trabecular bone during moderate ac-
tivity has been estimated to be about 2300 µm(Kopperdahl and Keaveny, 1998). Therefore,
the fatigue data of this study does also, at least partly (as the literature deals with on-axis
strains), map physiological load magnitudes. Of course, no biological reaction in terms of
83
Chapter 5: Discussion
remodeling processes is included in the in-vitro experiments. However, the influence of
these repair/optimisation processes on the fatigue behaviour remains still controversial in
the literature. Beside the well-known repair function the influence of resorption cavities
preceding repair may contribute in a negative manner to the structural integrity (Hernan-
dez et al., 2006). So it remains unclear if the experimental fatigue data matches a ‘safe
engineering design’ or not.
As the initial modulus plays a major role in establishing lifetime curves for spon-
geous structures, the question arises if this structural parameter is sensitive enough to rep-
resent the initial mechanical potential of the structure with regard to cyclic loading. The
lifetime curves exhibited large differences between the groups, even if the initial modulus
was included. Likewise, two groups with similar initial modulus (bovine vertebral 90◦and
human femoral 0◦) behaved differently. Therefore, initial modulus seems to be a good es-
timator for the fatigue properties of cancellous bone, as correlations for the lifetime curves
were found to be very strong, but does fail to fully describe the mechanics of the structure.
Haddock et al. (2004) compared data from bovine tibial cancellous and human vertebral
cancellous bone. They suggest that one single lifetime curve works well for the combined
set of data if the monotonic yield strain is taken into account, which is slightly higher for
bovine cancellous bone. The applied load is therefore normalised to a percentage applied
strain with respect to yield strain. Fig. 5.4 shows the merged specimen data for five groups
of cancellous bone. For the static yield strain the yield strains at the 0.2% offset (ε0.2)
from few monotonic compression tests were taken. The overall fit reveals a rather strong
correlation of R2=0.90, but of course in this approach the information of the lifetime
coefficient (b) is missing which decreases the accuracy at higher lifetimes/lower applied
loads. Thus, this approach does not seem usefully contribute to the discussion. Further
attempts to improve the inter- and intra- groupwise scatter with density (BMD) values did
not succeed.
The lifetime data has been investigated in order to reveal differences in the low-cycle
(LCF) and high-cycle regime (HCF), respectively, in order to obtain a combined Basquin-
Coffin-Manson relationship. Fig. 5.5 shows best fit results on the data from three groups.
84
Chapter 5: Discussion
Figure 5.4: Merging specimens from different groups (bovine vertebral 0◦,90◦and human
vertebral 0◦,90◦and human femoral 0◦together by normalisation with monotonic
yield strain.
This means, the boundary between LCF and HCF has been shifted until the specimens of
one group were divided in two subgroups with best linear fits in the log-log plot of the
lifetime diagram. The relationships are rather poor due to the relatively low amount of
specimens within each subgroup analysed. Nevertheless, the transition between the two
deformation modes is rather well-defined. For the human vertebral groups a normalised
stress level of approximately 0.0045 seems to cause a change in mechanism, whereas in
the bovine groups a value of approximately 0.007 −0.0075 was obtained. Surface strain
analysis on the bovine specimens revealed a normalised stress threshold of the same mag-
nitude, where areas exhibiting certain strain values did disappear (Fig. 4.22). Furthermore,
in the analysis of the deformation rates during stage II (saturation regime) corresponding
normalised stress values were found for human and bovine specimens, where deformation
rates were independent of groups. Taking these results together, a change in the damage
mechanism can be obtained at approximately σ/E0=0.0075 for bovine vertebral bone
and σ/E0=0.004 −0.0045 for human vertebral bone.
The real effect of the axis of loading with respect to the main physiological axis on
85
Chapter 5: Discussion
Figure 5.5: Separating the groups bovine vertebral 0◦(BV0), bovine vertebral 90◦(BV90),
human vertebral 0◦(HV0) and human vertebral 0◦(HV90) with respect to low
cycle and high cycle fatigue regions.
the fatigue behaviour is blured in the normalised (σ/E0) lifetime data. In order to clearly
point out the consequences of non-physiological loading on fatigue lifetime the data is
normalised by the mean modulus of each group (Fig. 4.31). The so spread lifetime curves
are further processed to get an engineering “design diagram”. Load values are taken at
certain cycle numbers and plotted as a function of the specimen orientation angle (with
respect to the physiological bone axis). This modified form of a S-N diagram with the
introduction of an anisotropy-based pseudo stress clearly reveals the magnitude of the load-
bearing capability with respect to anisotropy. Even small deviations from the physiological
axis result in highly reduced bearable stress. Therefore, the oriented, optimised cancellous
bone structure is highly sensitive to changes in the load directions. This effect may have
direct clinical implications because structural implants in most cases alter local loading
conditions in terms of induced stresses acting in non-physiological load axes.
Furthermore, it should be mentioned that the relationships between the applied load
and cycles to failure showed comparably good correlations in the off-axis groups which
indicates that damage occurs in a reproducible manner. This confirms the assumption that
86
Chapter 5: Discussion
a damage mechanism related to microstructural properties in terms of microcrack initiation
and propagation is operative. The architectural differences cause local stresses which may
shorten and thus accelerate the initiation and propagation phases.
Comparing our data for the same bone type (human vertebrae 0◦) with the literature
(Haddock et al., 2004) showed a good agreement (Fig. 4.30). Another study on human
vertebral cancellous bone, cored along the main physiological axis (Rapillard et al., 2006)
observed similar trends, but slightly different values (see Table 5.1). Earlier studies did
not include embedding of the end surfaces and reported a much higher intercept on the
load axis (Michel et al., 1993). Even if the studies shown in Table 5.1 used some simi-
lar experimental techniques, direct comparison is limited due to variations concerning the
experimental boundary conditions, i.e. surrounding media, temperature, specimen fixing,
strain rate etc. In both cases, specimens were smaller than those used in this study (diam-
eter 8.3 mm and 8.0 mm compared to 11.2 mm), also boundary conditions (embedding)
were applied in a different manner. Additionally, Rapillard and co-workers used a tem-
pered fluid bath, whereas Haddock et al. tested at room temperature and samples were kept
wet with physiological solution. The concluding evaluation, where the differences origi-
nate, remains unclear, but may be a combination of experimental boundary conditions and
of course donor-specific differences.
5.4 Age Effects
The data of the cyclic deformation experiments was analysed with respect to donor age
in order to reveal the influence of age on the fatigue properties of cancellous bone. The
Table 5.1: Coefficients for lifetime curves Equation 4.14.
a b
This work 0.0098 -0.109
Haddock et al. (2004) 0.0093 −0.1179
Rapillard et al. (2006) 0.0121 −0.0808
87
Chapter 5: Discussion
initial secant modulus of the specimens was found to be highly dependent on donor age.
The modulus decrease was much more pronounced in the off-axis groups, this increasing
mechanical anisotropy indicates also an increased degree of structural anisotropy with age,
findings that correspond with data from Nicholson et al. (1997), who showed that there
is relative conservation of stiffness in the axial direction compared with the transverse
direction.
Initial stiffness is used as a normalisation factor in fatigue analysis of cancellous
bone (Bowman et al., 1998; Moore and Gibson, 2003b) and therefore the corresponding
relationships are directly related to its magnitude. Consequently, fatigue lifetime is also
highly dependent on age and decreases more pronouncedly in the off-axis orientations.
This implies that old bone is much more sensitive to (cyclic) failure loads in general but
particularly to loads which are not coincident with the physiological main axis. There-
fore, for instance the right implant anchorage (which may result in unphysiological local
stress) is much more important in older bone. Also changes in the physiological load flow
through a change in habitual tasks increase the risk of fatigue fractures for elderly people.
Furthermore, these findings suggest that the integrally measured value for bone strength in
current routine clinical practice, BMD (Friedman, 2006), may be of decreased predective
value concerning older bone as the structural anisotropy becomes increasingly important.
While for human cortical bone modulus degradation profiles were found to be dif-
ferent regarding younger and older bones (Diab et al., 2005), the data in this study did
not reveal an effect of age on damage evolution. Additionally, age did not show any fur-
ther influence on lifetime curves and deformations at failure once initial stiffness has been
included. As old bone is known to have reduced quality e.g. in terms of increased mi-
crodamage (Schaffler et al., 1995), one could expect older bones to reveal a tendency for
reduced lifetimes. At least in the range of the data presented there is no evidence for this,
once initial stiffness is included. There may be different explanations for these findings.
Firstly, initial stiffness is a highly sensitive parameter of tissue quality, which has also ex-
planatory power for cyclic loadings; secondly, initial (micro-)damage may not be critical
for fatigue failure; and thirdly, the range of donor age is too limited to reveal significant
88
Chapter 5: Discussion
results. O’Brien et al. (2003) found for bovine cortical bone that bone fails by extensive
propagation of a few cracks, not the proliferation and coalescence of microcracks, and
therefore initial microdamage is not a significant factor in fatigue failure, which may also
be true for cancellous bone. Additional microdamage quantification and analysis is needed
to confirm this assumption. It has been suggested that damage rates increase with age for
cortical bone (Cotton et al., 2005). Whether this is also true for cancellous bone could not
be proved with this study due to the limited amount of data.
The deformation mechanism for static failure is reported to depend on the cancel-
lous bone density and therefore age. At lower density values elastic buckling dominates,
whereas at higher density levels a lowered slenderness ratio of the trabeculae causes failure
by progressive microfracturing (Townsend et al., 1975; Gibson and Ashby, 1997). While
these findings hold for static loadings, cyclic loadings much below the static failure values
did not result in these critical deformations and may therefore not so strongly depend on
the slenderness ratio, which could explain the similarity of the deformation behaviour.
Even if the age range of the donors is limited in this study, it represents the age-
group, where bone quality becomes an important issue. The limited number of data points
does not allow for an exact determination of the relationships between initial modulus and
age, but nevertheless the trends can be shown and are valid.
Concluding, initial modulus and therefore lifetimes were found to be highly depen-
dent on age. The decrease in both with increasing age was much more pronounced in
specimens which were not aligned with the main physiological axis.
5.5 Clinical Relevance
The high dependency of fatigue lifetime on the axis of loading may have direct clinical
consequences. For instance, fracture treatment through implants should preserve the phys-
iological load axis. In the simple example of a pedicle screw placement in a vertebral body
(Fig. 1.2) screw insertion in strictly vertical direction may lead to highest stability in the
89
Chapter 5: Discussion
long run. In general, implant anchorage should lead to mostly on-axis loading and regions
of low structural anisotropy should be preferred. But the results do also indicate that above
a certain deviation from the main axis (Fig. 4.32) fatigue strength is almost equal and there-
fore, if alignment with the physiological direction can not be achieved, other factors may
be more important for the choice of implant anchorage. The need for a inclusion of struc-
tural information in the determination of the mechanical performance of cancellous bone
has been stressed by recent studies (Ulrich et al., 1999; Stauber et al., 2006; Matsuura et al.,
2007). This study does support this concept with mechanical data showing the large influ-
ence of the axis of loading on the fatigue behaviour of cancellous bone. So, especially for
the case of osteoporosis, which incorporates an increased structural anisotropy, it appears
to be essential to determine morphological parameters which enhance the predictability
of clinical diagnosis. It could be shown in this work that BMD could predict the initial
stiffness at least to some degree, when specimens are loaded along their structural main
axis, but fails to assess the behaviour of non-axis loaded specimens. Bone Mineral Den-
sity alone is therefore not suitable to predict bone strength in trauma treatment situations
and provides only weak predictability for the fracture risk. The development of techniques
to improve the non-invasive estimation of density and structure of trabecular bone, e.g.
quantitative magnetic resonance imaging- (MRI-)based (Lammentausta et al., 2006), are
essential.
90
6 Conclusion
This study shows for the first time the relationship between the cyclic deformation and
fatigue behaviour and axis of loading (anisotropy), site, species and age of cancellous
bone. Furthermore, damage mechanisms were derived from novel measurement methods
for strain analysis on the apparent and tissue level. For this reason, cyclic compression tests
were performed on different groups of cancellous bone varying in specimen orientation
angle with respect to the physiological bone axis, site and species. Surface strains were
analysed by means of optical deformation analysis.
While the characteristics of cyclic deformation were found to be similar for all
groups, large deviations were observed for the fatigue lifetimes. Bovine specimens did
reveal higher lifetimes compared to human samples and lifetimes decreased with increas-
ing deviation of the specimens’ axis from the physiological bone axis. Already small
deviations cause a large reduction, whereas deviations above 45◦result in a similar fatigue
behaviour. Strains at failure were found to be dependent on specimen orientation (with
respect to the physiological bone axis). The whole cyclic deformation process as well as
damage evolution until defined failure could be shown to be a function of normalised stress
and group. The corresponding functional relationships were derived. Damage acceleration
was found to be constant for all specimens and different damage mechanisms are acting
for on-axis and off-axis groups. Likewise, load thresholds were found, at which damage
mechanisms change from low-cycle to high-cycle fatigue. Age appeared to have a large in-
fluence on the initial modulus of the specimens. Deformation analysis on the apparent and
the trabecular level could be linked to macroscopic damage and microdamage was found
to contribute to residual strain accumulation. Concluding, the axis of loading appears to
91
Chapter 6: Conclusion
contribute dominantly to fatigue and cyclic deformation, which may be even more pro-
nounced in cases of increased anisotropy (Osteoporosis). Therefore, local morphological
information has to be included in risk of fracture predictions in order to achieve a higher
reliability.
92
Bibliography
Ashman, R. B., Rho, J. Y., 1988. Elastic modulus of trabecular bone material. J Biomech
21 (3), 177–181.
Augat, P., Link, T., Lang, T. F., Lin, J. C., Majumdar, S., Genant, H. K., 1998. Anisotropy
of the elastic modulus of trabecular bone specimens from different anatomical locations.
Med Eng Phys 20 (2), 124–131.
Basquin, O., 1910. The exponential law of endurance tests. Proceedings of the American
Society for Testing and Materials 10, 625–30.
Bauer, T. W., Schils, J., 1999. The pathology of total joint arthroplasty.ii. mechanisms of
implant failure. Skeletal Radiol 28 (9), 483–497.
Bayraktar, H. H., Gupta, A., Kwon, R. Y., Papadopoulos, P., Keaveny, T. M., 2004.
The modified super-ellipsoid yield criterion for human trabecular bone. J Biomech Eng
126 (6), 677–684.
Borgström, F., Sobocki, P., Ström, O., Jönsson, B., 2007. The societal burden of osteoporo-
sis in sweden. Bone 40 (6), 1602–1609.
Bowman, S. M., Guo, X. E., Cheng, D. W., Keaveny, T. M., Gibson, L. J., Hayes, W. C.,
McMahon, T. A., 1998. Creep contributes to the fatigue behavior of bovine trabecular
bone. J Biomech Eng 120 (5), 647–654.
Bredbenner, T., 2003. Damage modelling of vertebral trabecular bone. Phd thesis, Mechan-
ical & Aerospace Engineering , Case Western Research University, Cleveland, OH.
93
BIBLIOGRAPHY
Bredbenner, T. L., Davy, D. T., 2006. The effect of damage on the viscoelastic behavior of
human vertebral trabecular bone. J Biomech Eng 128 (4), 473–480.
Brunet-Imbault, B., Lemineur, G., Chappard, C., Harba, R., Benhamou, C. L., 2005. A new
anisotropy index on trabecular bone radiographic images using the fast fourier transform.
BMC Med Imaging 5 (1), 4.
Burr, D. B., Forwood, M. R., Fyhrie, D. P., Martin, R. B., Schaffler, M. B., Turner, C. H.,
1997. Bone microdamage and skeletal fragility in osteoporotic and stress fractures. J
Bone Miner Res 12 (1), 6–15.
Burr, D. B., Martin, R. B., Schaffler, M. B., Radin, E. L., 1985. Bone remodeling in re-
sponse to in vivo fatigue microdamage. J Biomech 18 (3), 189–200.
Burr, D. B., Turner, C. H., Naick, P., Forwood, M. R., Ambrosius, W., Hasan, M. S.,
Pidaparti, R., 1998. Does microdamage accumulation affect the mechanical properties
of bone? J Biomech 31 (4), 337–345.
Carter, D. R., Caler, W. E., Spengler, D. M., Frankel, V. H., 1981. Uniaxial fatigue of
human cortical bone. the influence of tissue physical characteristics. J Biomech 14 (7),
461–470.
Carter, D. R., Hayes, W. C., 1977. The compressive behavior of bone as a two-phase porous
structure. J Bone Joint Surg Am 59 (7), 954–962.
Choi, K., Goldstein, S. A., 1992. A comparison of the fatigue behavior of human trabecular
and cortical bone tissue. J Biomech 25 (12), 1371–1381.
Coffin, L., 1954. A study of the effects of cyclic thermal stresses on a ductile metal. Trans-
actions of the American Society of Mechanical Engineers 76, 931–940.
Compston, J., 2006. Bone quality: what is it and how is it measured? Arq Bras Endocrinol
Metabol 50 (4), 579–585.
94
BIBLIOGRAPHY
Cotton, J. R., Winwood, K., Zioupos, P., Taylor, M., 2005. Damage rate is a predictor of
fatigue life and creep strain rate in tensile fatigue of human cortical bone samples. J
Biomech Eng 127 (2), 213–219.
Cowin, S., 2001. Bone mechanics handbook. CRC Press Inc., Boca Raton.
Cowin, S. C., 1985. The relationship between the elasticity tensor and the fabric tensor.
Mechanics of Materials 4 (2), 137–147.
Cruz-Orive, L., Karlsson, L., Larsen, S., Wainschtein, F., 1992. Characterizing anisotropy:
a new concept. Micron and Microscopica Acta 23, 75–76.
Dendorfer, S., Maier, H. J., Hammer, J., 2005. Strain evolution during compressive fa-
tigue of bovine cancellous bone. IFMBE Proceedings of the 3rd European Medical &
Biological Engineering Conference, Prague, Czech Vol. 11(1).
Diab, T., Sit, S., Kim, D., Rho, J., Vashishth, D., 2005. Age-dependent fatigue behaviour
of human cortical bone. Eur J Morphol 42 (1-2), 53–59.
Ding, M., Odgaard, A., Linde, F., Hvid, I., 2002. Age-related variations in the microstruc-
ture of human tibial cancellous bone. J Orthop Res 20 (3), 615–621.
Ebbesen, E. N., Thomsen, J. S., Beck-Nielsen, H., Nepper-Rasmussen, H. J., Mosek-
ilde, L., 1999. Age- and gender-related differences in vertebral bone mass, density, and
strength. J Bone Miner Res 14 (8), 1394–1403.
Englert, C., Angele, P., Fierlbeck, J., Dendorfer, S., Schubert, T., Müller, R., Lienhard, S.,
Zellner, J., Nerlich, M., Neumann, C., 2007. [conductive bone substitute material with
variable antibiotic delivery]. Unfallchirurg 110 (5), 408–413.
Fleck, C., Eifler, D., 2003. Deformation behaviour and damage accumulation of cortical
bone specimens from the equine tibia under cyclic loading. J Biomech 36 (2), 179–189.
Ford, C. M., Keaveny, T. M., 1996. The dependence of shear failure properties of trabecular
bone on apparent density and trabecular orientation. J Biomech 29 (10), 1309–1317.
95
BIBLIOGRAPHY
Fratzl, P., Gupta, H. S., Paschalis, E. P., Roschger, P., 2004. Structure and mechanical
quality of the collagen-mineral nano-composite in bone. J. Mater. Chem. 14, 2115–2123.
Friedman, A. W., 2006. Important determinants of bone strength: beyond bone mineral
density. J Clin Rheumatol 12 (2), 70–77.
Frost, H. M., 1987. Bone "mass" and the "mechanostat": a proposal. Anat Rec 219 (1),
1–9.
Frost, H. M., 2003. Bone’s mechanostat: a 2003 update. Anat Rec 275A (2), 1081–1101.
Fung, Y., 1993. Biomechanics: mechanical properties of living tissues. Springer Verlag,
New-York.
Fyhrie, D. P., Schaffler, M. B., 1995. The adaptation of bone apparent density to applied
load. J Biomech 28 (2), 135–146.
Ganguly, P., Moore, T. L., Gibson, L. J., 2004. A phenomenological model for predicting
fatigue life in bovine trabecular bone. J Biomech Eng 126 (3), 330–339.
George, W. T., Vashishth, D., 2006. Susceptibility of aging human bone to mixed-mode
fracture increases bone fragility. Bone 38 (1), 105–111.
Gibson, L. J., 2005. Biomechanics of cellular solids. J Biomech 38 (3), 377–399.
Gibson, L. J., Ashby, M. F., 1997. Cellular solids: structure & properties. Cambridge Uni-
versity Press, Cambridge 2nd.
Goldstein, S. A., Wilson, D. L., Sonstegard, D. A., Matthews, L. S., 1983. The mechan-
ical properties of human tibial trabecular bone as a function of metaphyseal location. J
Biomech 16 (12), 965–969.
Guo, X. E., McMahon, T. A., Keaveny, T. M., Hayes, W. C., Gibson, L. J., 1994. Finite
element modeling of damage accumulation in trabecular bone under cyclic loading. J
Biomech 27 (2), 145–155.
96
BIBLIOGRAPHY
Haddock, S. M., Yeh, O. C., Mummaneni, P. V., Rosenberg, W. S., Keaveny, T. M., 2004.
Similarity in the fatigue behavior of trabecular bone across site and species. J Biomech
37 (2), 181–187.
Haeussler, B., Gothe, H., Goel, D., Glaeske, G., Pientka, L., Felsenberg, D., 2007. Epi-
demiology, treatment and costs of osteoporosis in germany–the bone eva study. Osteo-
poros Int 18 (1), 77–84.
Harrigan, T. P., Jasty, M., Mann, R. W., Harris, W. H., 1988. Limitations of the continuum
assumption in cancellous bone. J Biomech 21 (4), 269–275.
Harrigan, T. P., Mann, R. W., 1984. Characterization of microstructural anisotropy in or-
thotropic materials using a second order rank tensor. Journal of Materials Science 19,
761–767.
Hazenberg, J., Taylor, D., Lee, T., 2007. The role of osteocytes and bone microstructure in
preventing osteoporotic fractures. Osteoporos Int 18 (1), 1–8.
Hernandez, C. J., Gupta, A., Keaveny, T. M., 2006. A biomechanical analysis of the effects
of resorption cavities on cancellous bone strength. J Bone Miner Res 21 (8), 1248–1255.
Hertzberg, R., 1995. Deformation and fracture mechanics of engineering materials. John
Wiley and Sons, Inc.
Hodgskinson, R., Currey, J. D., 1990. The effect of variation in structure on the young’s
modulus of cancellous bone: a comparison of human and non-human material. Proc Inst
Mech Eng 204 (2), 115–121.
Hoffler, C. E., Moore, K. E., Kozloff, K., Zysset, P. K., Goldstein, S. A., 2000. Age, gender,
and bone lamellae elastic moduli. J Orthop Res 18 (3), 432–437.
Homminga, J., Van-Rietbergen, B., Lochmueller, E.-M., Weinans, H., Eckstein, F.,
Huiskes, R., 2004. The osteoporotic vertebral structure is well adapted to the loads of
daily life, but not to infrequent "error" loads. Bone 34 (3), 510–516.
97
BIBLIOGRAPHY
Jess, W., 1990. The skeletal tissue. in book: Cell and tissue biology: A textbook of histol-
ogy, edited by: Weiss, l. Urban & Schwarzenberg, Baltimore.
Jirasek, M., Bazant, Z. P., 2002. Inelastic analysis of structures. John Wiley and Sons, Ltd,
Chinchester, England.
Kanis, J. A., Johnell, O., Oden, A., Borgstroem, F., Zethraeus, N., Laet, C. D., Jonsson,
B., 2004. The risk and burden of vertebral fractures in sweden. Osteoporos Int 15 (1),
20–26.
Kaplan, S. J., Hayes, W. C., Stone, J. L., Beaupre, G. S., 1985. Tensile strength of bovine
trabecular bone. J Biomech 18 (9), 723–727.
Keaveny, T. M., Borchers, R. E., Gibson, L. J., Hayes, W. C., 1993. Theoretical analysis of
the experimental artifact in trabecular bone compressive modulus. J Biomech 26 (4-5),
599–607.
Keaveny, T. M., Morgan, E. F., Niebur, G. L., Yeh, O. C., 2001. Biomechanics of trabecular
bone. Annu Rev Biomed Eng 3, 307–333.
Keaveny, T. M., Pinilla, T. P., Crawford, R. P., Kopperdahl, D. L., Lou, A., 1997. System-
atic and random errors in compression testing of trabecular bone. J Orthop Res 15 (1),
101–110.
Keaveny, T. M., Wachtel, E. F., Guo, X. E., Hayes, W. C., 1994. Mechanical behavior of
damaged trabecular bone. J Biomech 27 (11), 1309–1318.
Keaveny, T. M., Wachtel, E. F., Zadesky, S. P., Arramon, Y. P., 1999. Application of the
tsai-wu quadratic multiaxial failure criterion to bovine trabecular bone. J Biomech Eng
121 (1), 99–107.
Kim, D. G., Brunski, I. B., Nicolella, D. P., 2005. Microstrain fields for cortical bone in
uniaxial tension: optical analysis method. Proc Inst Mech Eng 219 (2), 119–128.
98
BIBLIOGRAPHY
King, A. I., Evans, F. G., 1967. Analysis of fatigue strength of human compact bone by the
weibull method.
Kopperdahl, D. L., Keaveny, T. M., 1998. Yield strain behavior of trabecular bone. J
Biomech 31 (7), 601–608.
Lammentausta, E., Hakulinen, M. A., Jurvelin, J. S., Nieminen, M. T., 2006. Prediction of
mechanical properties of trabecular bone using quantitative mri. Phys Med Biol 51 (23),
6187–6198.
LeBlanc, A. D., Spector, E. R., Evans, H. J., Sibonga, J. D., 2007. Skeletal responses to
space flight and the bed rest analog: a review. J Musculoskelet Neuronal Interact 7 (1),
33–47.
Lee, T. C., Staines, A., Taylor, D., 2002. Bone adaptation to load: microdamage as a
stimulus for bone remodelling. J Anat 201 (6), 437–446.
Levenston, M. E., Carter, D. R., 1998. An energy dissipation-based model for damage
stimulated bone adaptation. J Biomech 31 (7), 579–586.
Li, J. P., Li, S. H., Van Blitterswijk, C. A., de Groot, K., 2006. Cancellous bone from
porous ti6al4v by multiple coating technique. Journal of Materials Science: Materials in
Medicine 17 (2), 179–185.
Linde, F., Hvid, I., 1989. The effect of constraint on the mechanical behaviour of trabecular
bone specimens. J Biomech 22 (5), 485–490.
Liu, X., Wang, X., Niebur, G. L., 2003. Effects of damage on the orthotropic material
symmetry of bovine tibial trabecular bone. J Biomech 36 (12), 1753–1759.
Lotz, J. C., Hayes, W. C., 1990. The use of quantitative computed tomography to estimate
risk of fracture of the hip from falls. J Bone Joint Surg Am 72 (5), 689–700.
Makiyama, A. M., Vajjhala, S., Gibson, L. J., 2002. Analysis of crack growth in a 3d
voronoi structure: a model for fatigue in low density trabecular bone. J Biomech Eng
124 (5), 512–520.
99
BIBLIOGRAPHY
Manson, S. S., 1954. Behaviour of materials under conditions of thermal stress. NACA TN
2933.
Martin, R. B., Burr, D. B., Sharkey, N. A., 1998. Skeletal tissue mechanics. Springer Ver-
lag, New York.
Martin, R. B., Gibson, V. A., Stover, S. M., Gibeling, J. C., Griffin, L. V., 1997. Residual
strength of equine bone is not reduced by intense fatigue loading: implications for stress
fracture. J Biomech 30 (2), 109–14.
Matsuura, M., Eckstein, F., Lochmueller, E.-M., Zysset, P., 2007. The role of fabric in the
quasi-static compressive mechanical properties of human trabecular bone from various
anatomical locations. Biomech Model Mechanobiol Jan, in print.
McCalden, R. W., McGeough, J. A., Court-Brown, C. M., 1997. Age-related changes in the
compressive strength of cancellous bone. the relative importance of changes in density
and trabecular architecture. J Bone Joint Surg Am 79 (3), 421–427.
Michel, M. C., Guo, X. D., Gibson, L. J., McMahon, T. A., Hayes, W. C., 1993. Compres-
sive fatigue behavior of bovine trabecular bone. J Biomech 26 (4-5), 453–463.
Mohsin, S., O’Brien, F. J., Lee, T. C., 2006. Osteonal crack barriers in ovine compact bone.
J Anat 208 (1), 81–89.
Moore, T. L., Gibson, L. J., 2001. Modeling modulus reduction in bovine trabecular bone
damaged in compression. J Biomech Eng 123 (6), 613–622.
Moore, T. L., Gibson, L. J., 2003a. Fatigue microdamage in bovine trabecular bone. J
Biomech Eng 125 (6), 769–776.
Moore, T. L., Gibson, L. J., 2003b. Fatigue of bovine trabecular bone. J Biomech Eng
125 (6), 761–768.
Moore, T. L., O’Brien, F. J., Gibson, L. J., 2004. Creep does not contribute to fatigue in
bovine trabecular bone. J Biomech Eng 126 (3), 321–329.
100
BIBLIOGRAPHY
Morgan, E. F., Bayraktar, H. H., Keaveny, T. M., 2003. Trabecular bone modulus-density
relationships depend on anatomic site. J Biomech 36 (7), 897–904.
Morgan, E. F., Yeh, O. C., Chang, W. C., Keaveny, T. M., 2001. Nonlinear behavior of
trabecular bone at small strains. J Biomech Eng 123 (1), 1–9.
Morgan, E. F., Yeh, O. C., Keaveny, T. M., 2005. Damage in trabecular bone at small
strains. Eur J Morphol 42 (1-2), 13–21.
Mosekilde, L., 1989. Sex differences in age-related loss of vertebral trabecular bone mass
and structure–biomechanical consequences. Bone 10 (6), 425–432.
Mosekilde, L., 2000. Age-related changes in bone mass, structure, and strength–effects of
loading. Z Rheumatol 59 Suppl 1, 1–9.
Mosekilde, L., Mosekilde, L., 1986. Normal vertebral body size and compressive strength:
relations to age and to vertebral and iliac trabecular bone compressive strength. Bone
7 (3), 207–212.
Mugrabi, H., 1985. Disclocations in fatigue. Dislocations and properties of real materials,
Book No. 323, The Institute of Metals, London, 244–261.
Mullender, M. G., Tan, S. D., Vico, L., Alexandre, C., Klein-Nulend, J., 2005. Differences
in osteocyte density and bone histomorphometry between men and women and between
healthy and osteoporotic subjects. Calcif Tissue Int 77 (5), 291–296.
Nagaraja, S., Couse, T. L., Guldberg, R. E., 2005. Trabecular bone microdamage and mi-
crostructural stresses under uniaxial compression. J Biomech 38 (4), 707–716.
Nagaraja, S., Lin, A. S. P., Guldberg, R. E., 2007. Age-related changes in trabecular bone
microdamage initiation. Bone 40 (4), 973–980.
Nalla, R. K., Kinney, J. H., Ritchie, R. O., 2003. Mechanistic fracture criteria for the failure
of human cortical bone. Nat Mater 2 (3), 164–168.
101
BIBLIOGRAPHY
Nalla, R. K., Kruzic, J. J., Kinney, J. H., Ritchie, R. O., 2004. Effect of aging on the
toughness of human cortical bone: evaluation by r-curves. Bone 35 (6), 1240–1246.
Nazarian, A., Muller, R., 2004. Time-lapsed microstructural imaging of bone failure be-
havior. J Biomech 37 (1), 55–65.
Nicholson, P. H., Cheng, X. G., Lowet, G., Boonen, S., Davie, M. W., Dequeker, J., der
Perre, G. V., 1997. Structural and material mechanical properties of human vertebral
cancellous bone. Med Eng Phys 19 (8), 729–737.
Niemeyer, P., Weinberg, A., Schmitt, H., Kreuz, P. C., Ewerbeck, V., Kasten, P., 2006.
Stress fractures in the juvenile skeletal system. Int J Sports Med 27 (3), 242–249.
O’Brien, F. J., Taylor, D., Lee, T. C., 2003. Microcrack accumulation at different intervals
during fatigue testing of compact bone. J Biomech 36 (7), 973–980.
Odgaard, A., Jensen, E. B., Gundersen, H. J., 1990. Estimation of structural anisotropy
based on volume orientation. a new concept. J Microsc 157 (Pt 2), 149–162.
Ohrndorf, A., Krupp, U., Christ, H.-J., 2006. Metallic open-cell foams–a promising ap-
proach to fabricating good medical implants. Technol Health Care 14 (4-5), 201–208.
Penzkofer, R., 2005. Quantitative charakterisierung struktureller und experimenteller ein-
flussgroessen auf das wechselverformungsverhalten von spongiosa. Diplomathesis, Uni-
versity of Applied Sciences Regensburg.
Prendergast, P. J., Taylor, D., 1994. Prediction of bone adaptation using damage accumu-
lation. J Biomech 27 (8), 1067–1076.
Rajapakse, C. S., Thomsen, J. S., Espinoza Ortiz, J. S., Wimalawansa, S. J., Ebbesen,
E. N., Mosekilde, L., Gunaratne, G. H., 2004. An expression relating breaking stress
and density of trabecular bone. J Biomech 37 (8), 1241–1249.
Rapillard, L., Charlebois, M., Zysset, P. K., 2006. Compressive fatigue behavior of human
vertebral trabecular bone. J Biomech 39 (11), 2133–2139.
102
BIBLIOGRAPHY
Rice, J. C., Cowin, S. C., Bowman, J. A., 1987. On the dependence of the elasticity and
strength of cancellous bone on apparent density. Journal of biomechanics 21 (2), 155–
168.
Rohl, L., Larsen, E., Linde, F., Odgaard, A., Jorgensen, J., 1991. Tensile and compressive
properties of cancellous bone. J Biomech 24 (12), 1143–1149.
Roux, W., 1895. Gesammelte abhandlungen ueber die entwicklungsmechanik der organis-
men. W. Engelmann, Leipzig.
Rubin, M. A., Jasiuk, I., Taylor, J., Rubin, J., Ganey, T., Apkarian, R. P., 2003. Tem analysis
of the nanostructure of normal and osteoporotic human trabecular bone. Bone 33 (3),
270–282.
Sanghavi, P., Bose, D., Kerrigan, J., Madeley, N. J., Crandall, J., 2004. Non-contact strain
measurement of biological tissue. Biomed Sci Instrum 40, 51–56.
Schaffler, M. B., Choi, K., Milgrom, C., 1995. Aging and matrix microdamage accumula-
tion in human compact bone. Bone 17 (6), 521–525.
Schaffner, G., Guo, X.-D. E., Silva, M. J., J., G. L., 2000. Modelling fatigue damage accu-
mulation in two-dimensional voronoi honeycombs. International Journal of Mechanical
Sciences 42, 645–656.
Schijve, J., 2001. Fatigue of structures and materials. Kluwer Academic Publishers, Dor-
drecht, Nl.
Schuetz, W., 1996. A history of fatigue. Eng. Fracture Mechanics 54, 263–300.
Snyder, R. A., Koester, M. C., Dunn, W. R., 2006. Epidemiology of stress fractures. Clin
Sports Med 25 (1), 37–52, viii.
Sobelman, O. S., Gibeling, J. C., Stover, S. M., Hazelwood, S. J., Yeh, O. C., Shelton,
D. R., Martin, R. B., 2004. Do microcracks decrease or increase fatigue resistance in
cortical bone? J Biomech 37 (9), 1295–1303.
103
BIBLIOGRAPHY
Sormaala, M. J., Niva, M. H., Kiuru, M. J., Mattila, V. M., Pihlajamaeki, H. K., 2006. Bone
stress injuries of the talus in military recruits. Bone 39 (1), 199–204.
Sowers, M. F., 2000. Lower peak bone mass and its decline. Baillieres Best Pract Res Clin
Endocrinol Metab 14 (2), 317–329.
Stauber, M., Rapillard, L., van Lenthe, G. H., Zysset, P., Mueller, R., 2006. Importance
of individual rods and plates in the assessment of bone quality and their contribution to
bone stiffness. J Bone Miner Res 21 (4), 586–595.
Stone, J. L., Beaupre, G. S., Hayes, W. C., 1983. Multiaxial strength characteristics of
trabecular bone. Journal of biomechanics 16 (9), 743–747.
Suresh, S., 1998. Fatigue of materials. University Press Cambridge 2nd.
Taylor, D., 1998. Fatigue of bone and bones: an analysis based on stressed volume. J
Orthop Res 16 (2), 163–169.
Taylor, D., Hazenberg, J. G., Lee, T. C., 2003a. The cellular transducer in damage-
stimulated bone remodelling: a theoretical investigation using fracture mechanics. J
Theor Biol 225 (1), 65–75.
Taylor, D., Hazenberg, J. G., Lee, T. C., 2007. Living with cracks: damage and repair in
human bone. Nat Mater 6 (4), 263–268.
Taylor, D., O’Reilly, P., Vallet, L., Lee, T. C., 2003b. The fatigue strength of compact bone
in torsion. J Biomech 36 (8), 1103–1109.
Townsend, P. R., Rose, R. M., Radin, E. L., 1975. Buckling studies of single human tra-
beculae. J Biomech 8 (3-4), 199–201.
Turner, C. H., 1989. Yield behavior of bovine cancellous bone. J Biomech Eng 111 (3),
256–260.
Turner, C. H., Cowin, S. C., 1988. Errors induced by off-axis measurement of the elastic
properties of bone. J Biomech Eng 110 (3), 213–215.
104
BIBLIOGRAPHY
Ulrich, D., van Rietbergen, B., Laib, A., Ruegsegger, P., 1999. The ability of three-
dimensional structural indices to reflect mechanical aspects of trabecular bone. Bone
25 (1), 55–60.
Vajjhala, S., Kraynik, A. M., Gibson, L. J., 2000. A cellular solid model for modulus
reduction due to resorption of trabeculae in bone. J Biomech Eng 122 (5), 511–515.
van der Linden, J. C., Birkenhager-Frenkel, D. H., Verhaar, J. A., Weinans, H., 2001.
Trabecular bone’s mechanical properties are affected by its non-uniform mineral distri-
bution. J Biomech 34 (12), 1573–1580.
van Rietbergen, B., Weinans, H., Huiskes, R., Odgaard, A., 1995. A new method to de-
termine trabecular bone elastic properties and loading using micromechanical finite-
element models. J Biomech 28 (1), 69–81.
Vashishth, D., 2004. Rising crack-growth-resistance behavior in cortical bone: implica-
tions for toughness measurements. J Biomech 37 (6), 943–946.
Wang, X., Masilamani, N., Mabrey, J., Alder, M., Agrawal, C., 1998. Changes in the
fracture toughness of bone may not be reflected in its mineral density, porosity, and
tensile properties. Bone 23, 67–72.
Wenzel, T. E., Schaffler, M. B., Fyhrie, D. P., 1996. In vivo trabecular microcracks in
human vertebral bone. Bone 19 (2), 89–95.
Werz, T., 2005. Rasterelektronenmikroskopische bruchcharakterisierung biologischer ma-
terialien nach zyklischer wechselbelastung. Diplomthesis, University of Applied Sci-
ences Regensburg.
Whitehouse, W. J., 1974. The quantitative morphology of anisotropic trabecular bone. Jour-
nal of Microscopy 101 (2), 153–168.
Whitehouse, W. J., Dyson, E. D., 1974. Scanning electron microscope studies of trabecular
bone in the proximal end of the human femur. J Anat 118 (3), 417–444.
105
BIBLIOGRAPHY
Whitehouse, W. J., Dyson, E. D., Jackson, C. K., 1971. The scanning electron microscope
in studies of trabecular bone from a human vertebral body. J Anat 108 (3), 481–496.
W.H.O., 1994. World health organisation: Assessment of fracture risk and its application
to screeening for postmenopausal osteoporosis. Report of a W.H.O. Study Group, World
Health Organisation, Geneva, Switzerland.
Williams, J. L., Lewis, J. L., 1982. Properties and an anisotropic model of cancellous bone
from the proximal tibial epiphysis. J Biomech Eng 104 (1), 50–56.
Wolff, J., 1892. Das gesetz der transformation der knochen. Hirschwald, Berlin.
Yamada, H., 1970. Strength of biological material. Williams & Wilkins, Baltimore.
Yamamoto, E., Crawford, R. P., Chan, D. D., Keaveny, T. M., 2006. Development of resid-
ual strains in human vertebral trabecular bone after prolonged static and cyclic loading
at low load levels. J Biomech 39 (10), 1812–1818.
Yang, G., Kabel, J., van Rietbergen, B., Odgaard, A., Huiskes, R., Cowin, S. C., 1998. The
anisotropic hooke’s law for cancellous bone and wood. J Elast 53 (2), 125–146.
Yeni, Y. N., Fyhrie, D. P., 2002. Fatigue damage-fracture mechanics interaction in cortical
bone. Bone 30 (3), 509–514.
Yeni, Y. N., Fyhrie, D. P., 2003. A rate-dependent microcrack-bridging model that can
explain the strain rate dependency of cortical bone apparent yield strength. J Biomech
36 (9), 1343–1353.
Yeni, Y. N., Hou, F. J., Vashishth, D., Fyhrie, D. P., 2001. Trabecular shear stress in human
vertebral cancellous bone: intra- and inter-individual variations. J Biomech 34 (10),
1341–1346.
Zioupos, P., Casinos, A., 1998. Cumulative damage and the response of human bone in
two-step loading fatigue. J Biomech 31 (9), 825–833.
106
BIBLIOGRAPHY
Zioupos, P., Currey, J. D., 1998. Changes in the stiffness, strength, and toughness of human
cortical bone with age. Bone 22 (1), 57–66.
Zysset, P. K., 2003. A review of morphology-elasticity relationships in human trabecular
bone: theories and experiments. J Biomech 36 (10), 1469–1485.
Zysset, P. K., Curnier, A., 1996. A 3d damage model for trabecular bone based on fabric
tensors. J Biomech 29 (12), 1549–1558.
107
108
Appendix
Appendix
Group Load (σ/E0) Cycles to failure E0
Human vertebral 0◦0,0091 1 574
0,0100 1 713
0,0086 5 469
0,0045 2600 533
0,0089 3 483
0,0044 100 420
0,0035 25000 215
0,0046 120 389
0,0022 711000 400
0,0057 25 372
0,0033 22000 403
0,0029 450000 508
0,0032 17000 430
0,0027 150000 350
Human vertebral 22◦0,0068 2 170
0,0036 5600 165
0,0042 1300 135
0,0026 32500 167
0,0035 9800 150
0,0063 35 162
0,0048 145 160
0,0023 180000 165
Human vertebral 45◦0,0058 15 98
0,0017 450000 170
0,0075 12 111
0,0109 1 233
0,0036 2400 140
0,0046 1600 230
0,0031 40000 68
0,0070 25 27
0,0025 3400 120
0,0027 7200 93
0,0078 1 30
0,0036 11000 48
0,0037 1700 73
109
Appendix
Group Load (σ/E0) Cycles to failure E0
Human vertebral 90◦0,0063 5 20
0,0037 300 70
0,0021 95000 130
0,0125 1 44
0,0049 27 58
0,0050 100 70
0,0098 1 178
0,0040 900 151
0,0045 950 88
0,0020 100000 206
0,0031 18000 82
0,0021 550000 288
0,0036 16000 51
0,0059 30 22
0,0042 350 15,6
Human femoral 0◦0,0031 12000 845
0,0029 47000 600
0,0082 1 1118
0,0045 1950 1720
0,0078 7 1600
0,0071 1 1395
0,0026 210000 1440
0,0030 27500 1008
0,0052 525 1262
0,0050 540 1280
0,0029 250000 230
0,0029 75000 308
0,0023 1010000 601
0,0060 300 890
110
Appendix
Group Load (σ/E0) Cycles to failure E0
Bovine vertebral 0◦0,0100 220 2746
0,0125 31 1880
0,0075 180 2170
0,0068 14000 2330
0,0080 125 2600
0,0075 1500 2500
0,0086 210 3300
0,0125 9 1917
0,0039 570000 2700
0,0063 500 2340
0,0060 1550 2250
0,0070 425 1580
0,0070 2000 2700
0,0077 240 2470
0,0057 7600 2145
0,0056 9200 2604
0,0055 8100 2050
0,0049 42000 3250
0,0147 3 1636
0,0102 5 1957
0,0093 42 2573
0,0061 1000 1900
0,0044 132000 3082
0,0051 83000 83000
0,0130 6 2004
0,0084 100 2500
0,0065 180 2559
0,0063 3900 3562
0,0056 17000 3205
0,0103 6 2370
0,0108 6 2715
Bovine vertebral 90◦0,0100 1 550
0,0100 1 850
0,0062 220 860
0,0043 5900 1120
0,0068 500 503
0,0106 3 1520
0,0058 1400 3080
0,0050 4600 1700
0,0082 5 470
0,0074 50 1550
0,0078 35 1038
0,0037 60000 1480
111
