Diss. ETH No. 16223
Volumetric spatial decomposition
of porous microstructures
A framework for element based
analysis of trabecular bone
A dissertation submitted to the
Swiss Federal Institute of Technology Z¨ urich
for the degree of
Doctor of Science
Dipl. Phys. ETH
born 9thOctober, 1974
citizen of Zetzwil AG
accepted on the recommendation of
Prof. Dr. Ralph M¨ uller, examiner
Prof. Dr. Philippe Zysset, co-examiner
Table of Contents
2Volumetric spatial decomposition17
2.1 Volumetric spatial decomposition of trabecular bone . . . . . . . . . . 19
2.2 A sensitivity analysis of the volumetric spatial decomposition . . . . . 45
3Local morphometry 63
3.1Age-related changes in trabecular bone microstructures . . . . . . . . 65
3.2 Contribution of rods and plates to bone quality and competence . . . 93
3.3Limitations of morphometry . . . . . . . . . . . . . . . . . . . . . . . 113
4Idealized computational models129
4.1A finite element beam-model for efficient simulation . . . . . . . . . . 131
4.2Specimen-specific beam models . . . . . . . . . . . . . . . . . . . . . 149
A Mathematical background175
B Thinning algorithms 179
C Point-classification algorithm183
iiTable of Contents
The present work would not have been possible without the help of several persons
to whom I am thankfully indebted.
First of all, I would like to thank Prof. Dr. Ralph M¨ uller who initiated this project
and gave me the possibility to carry out the present work. I always appreciated his
conceptual advice, constructive criticism, and his very detailed and helpful reviews.
The countless stimulating and very interesting discussions shaped my thoughts and
founded the basis of the present work. He always gave me the freedom to choose
my way and supported me generously in all my decisions. I am very glad to have
had such generous doctoral advisor.
I would also like to thank my co-referee Prof. Dr. Philippe Zysset who helped me
to improve the thesis both, linguistically and scientifically through his careful and
helpful review. Furthermore, I would like to thank him for providing the mechanical
data of the human trabecular bone samples in our common project.
Moreover, I would like to thank Dr. G. Harry van Lenthe for his great work in
building the beam finite element models from the spatially decomposed bone
structures and for solving all the finite element models.
I owe special thanks to Martin Huber for his diploma work, in which he explored
the feasibility of beam FE models.
My thanks also go to Ara Nazarian who introduced me to the secrets of IGFA
during my enjoyable visit in Boston. I also enjoyed the brain heating statistical
discussions for our common project.
I am very grateful to Laurent Rapillard for the assessment of the mechanical data
of the human trabecular bone samples.
I would also like to thank Prof. Dr. Steven K. Boyd for his collaboration on our
common project and for the nice ski-touring days we could share in the Swiss Alps.
Furthermore, I would like to express my gratitude to Dr. Philipp J. Thurner for
the fruitful discussions and fun we had during the first part of my thesis where we
shared the office.
Thomas Kohler deserves special thank for providing many useful tools on OpenVMS
and for supporting the whole group in computer questions. I am very glad to have
had such a patient and supportive friend to share the office with during the second
part of my thesis.
I would also like to thank the Scanco Medical team, especially Dr. Bruno
Koller, Dr. Andres Laib, and Dr. Stefan H¨ ammerle for their support on using the
µCT systems and for providing the opportunity to incorporate my software into IPL.
Moreover, I would like to thank Antonio Bongulielmi for his support on OpenVMS
questions and for constructive tips and tricks on how to use the OpenVMS Debugger.
Particular thanks go to the members of the Institute for Biomedical Engineering
at the Moussonstrasse who accompanied me along my way and who provided a
relaxing working atmosphere, which turned this place into more than just a working
place. I enjoyed the many interesting and funny talks, and of course the “Gipfeli”
and cakes during the coffee breaks.
Furthermore, I would like to express my gratitude to all my friends who took me
out for mountaineering, ski-touring, climbing, sailing, dancing, having a beer,...
and gave me a good time beside from work.
I am very grateful to my parents, who always supported and believed in me during
all these years, and who made my studies possible.
Lastly, I am very much indebted to Esther Th¨ urig for her support in statistical
questions, her assistance in using R, and for her love and encouraging words over
all these years.
Finally, the financial support of the Swiss National Science Foundation (SNF) and
the Swiss Federal Institute of Technology (ETHZ) is gratefully acknowledged.
Bone mineral density measures are currently the standard for fracture risk assess-
ment in osteoporotic patients. Nevertheless, since it was recognized that bone den-
sity leaves a rather large variation in bone strength unexplained, many attempts have
been made to also include trabecular structure in the bone strength prediction. Con-
sequently, many morphometric indices were devised to characterize trabecular bone
architecture. Where most of these methods analyzed bone samples as a whole, in
this thesis a new framework was devised and implemented to analyze bone samples
on an elemental level. With this method, for the first time, trabecular bone struc-
tures could be spatially decomposed into their volumetric rod and plate elements.
Based on this volumetric spatial decomposition two applications were presented.
First, the extracted elements were analyzed using conventional morphometric mea-
sures, a method that was referred to as local morphometry. Second, the extracted
elements were converted to idealized computational models for a fast and accurate
prediction of bone mechanical properties.
Local morphometry revealed large differences in the rod/plate composition of
the structures from different anatomical sites. It was suggested that the strength
of dense plate-like structures was determined by a few “major elements” spanning
through the whole structure, whereas the strength in loose rod-like structures was
determined by the relative arrangement, quality and shape of a whole set of ele-
ments. These site differences were also reflected in different age-related remodelling
mechanisms. Where in lumbar spine bone loss was mainly expressed as a loss of rods,
in femoral head bone loss was expressed by perforation of large plates followed by a
transformation of plates to rods and by loss of interconnecting stabilizing trabecular
elements. It was also suggested that these interconnecting elements played a key role
in fracture initiation in inhomogeneous bone samples. Furthermore, local morphom-
etry was able to accurately predict bone mechanical properties. A multiple linear
regression model combining mean trabecular spacing (Tb.Sp), mean slenderness of
the rods (?Ro.Sl?), and the relative rod volume fraction (Ro.BV/BV) accounted for
90% of the variance in Young’s modulus as computed by conventional finite element
method. From these results it was concluded that local morphometry was a helpful
tool to improve the understanding of bone quality and the relative importance of
local structural changes with age and in the determination of bone strength.
Idealized computational models could be created directly from the volumetric
spatially decomposed images by converting the trabecular elements to beams in
a corresponding beam finite element model. Although theses models were highly
idealized, the apparent elastic properties of a set of human trabecular bone samples
were equally well predicted by these models as compared to conventional voxel FE
models (R2= 0.97). The big advantage of beam-FE models over conventional
voxel based FE models is the tremendous reduction in number of elements which
goes along with a tremendous reduction in computation time (up to a factor of
10’000). The strong reduction in CPU time opens up ways for research that was
not possible before, such as the routine assessment of mechanical properties of large
bone specimens or even whole bones.
In conclusion, a new framework for element based analysis of trabecular bone
structures was introduced. Local morphometry and idealized computational models
may become important tools to explain age- and disease-related changes in bone
quality and the competence of bone. With upcoming in vivo high-resolution imaging
systems idealized computational model have the potential to become a standard in
routine bone failure prediction.
Die Knochendichtemessung ist zurzeit das Standardverfahren, um das Fraktur-
risiko in osteoporotischen Patienten abzusch¨ atzen. Da aber erkannt wurde, dass
die Knochendichte einen relativ grossen Teil der Varianz der Knochenst¨ arke nicht
erkl¨ aren kann, wurden viele Versuche unternommen, auch die trabekul¨ are Struktur
in die Vorhersage der Knochenst¨ arke mit einzubeziehen. Als Konsequenz davon
wurden viele morphometrische Parameter zur Charakterisierung der trabekul¨ aren
Knochenarchitektur entwickelt. Die meisten dieser Verfahren beschreiben dabei
Knochenproben als ganzes. Im Gegensatz dazu wurde in dieser Doktorarbeit ein Sys-
tem entwickelt und implementiert, um Knochenproben auf dem Niveau der Elemente
untersuchen zu k¨ onnen. Mit dieser Methode konnte trabekul¨ arer Knochen zum er-
sten Mal in seine volumetrischen St¨ abe und Platten zerlegt werden. Basierend auf
dieser r¨ aumlichen Zerlegung wurden zwei Anwendungen vorgestellt. In der ersten,
lokale Morphometrie genannt, wurden die Elemente mit herk¨ ommlichen morphome-
trischen Methoden analysiert. In der zweiten wurden die extrahierten Elemente zu
einem idealisierten Computermodell konvertiert, welches zur schnellen und exakten
Vorhersage der mechanischen Eigenschaften des Knochens verwendet werden kann.
Mit lokaler Morphometrie konnten grosse Unterschiede in der Zusammenset-
zung bez¨ uglich St¨ abe und Platten f¨ ur Strukturen verschiedener anatomischer Lagen
aufgedeckt werden. Es wurde vorgeschlagen, dass die St¨ arke von dichten, plat-
tenf¨ ormigen Strukturen haupts¨ achlich durch ein paar ,,Hauptelemente“, welche sich
quer durch die ganze Struktur erstrecken, bestimmt ist, wohingegen die St¨ arke in
lockeren, stabf¨ ormigen Strukturen durch die relative Anordnung, Qualit¨ at und Form
eines ganzen Sets von Elementen bestimmt ist. Diese Lage-Unterschiede konnten
auch in den unterschiedlichen altersabh¨ angigen Umbauprozessen beobachtet werden.
Wo sich in der Lendenwirbels¨ aule Knochenschwund haupts¨ achlich durch den Verlust
von St¨ aben manifestierte, dr¨ uckte sich Knochenverlust im Oberschenkelkopf durch
die Durchl¨ ocherung von grossen Platten, gefolgt von einer Transformation von Plat-
ten zu St¨ aben, sowie durch den Verlust von stabilisierenden Zwischenverstrebungen
aus. Es wurde auch gezeigt, dass solche Zwischenverstrebungen eine Schl¨ usselrolle
bei der Bruchbildung in inhomogenen Proben spielen k¨ onnten. Ausserdem konnten
mittels lokaler Morphometrie die mechanischen Eigenschaften von Knochen genau
vorhergesagt werden. Eine multiple lineare Regression mit dem mittleren Abstand
der Trabekel (Tb.Sp), der mittleren Schlankheit der St¨ abe (?Ro.Sl?) und dem rel-
ativen Stabvolumen (Ro.BV/BV) als erkl¨ arende Gr¨ ossen, konnte 90% der Varianz
des Elastizit¨ atsmoduls, welcher durch die konventionelle Methode der finiten Ele-
mente (FE) berechnet wurde, erkl¨ aren. Aus diesen Resultaten wurde gefolgert, dass
lokale Morphometrie eine n¨ utzliche Methode ist, um ein vertieftes Verst¨ andnis der
Knochenqualit¨ at zu erlangen und um die relative Wichtigkeit von lokalen struk-
turellen¨Anderungen im Alter sowie deren Effekt auf die Knochenst¨ arke besser ver-
stehen zu k¨ onnen.
Idealisierte Computermodelle konnten direkt aus den volumetrisch zerlegten
Bildern generiert werden, indem die trabekul¨ aren Elemente in einem entsprechen-
den Finiten-Elemente-Modell zu Balken konvertiert wurden. Obwohl diese Mod-
elle stark vereinfacht sind, konnte der scheinbare Elastizit¨ atsmodul in einem Satz
von humanen Knochenproben gleich gut vorhergesagt werden wie durch konven-
tionelle FE-Modelle (R2= 0.97). Der grosse Vorteil von Balken-FE-Modellen
gegen¨ uber herk¨ ommlichen voxelbasierten FE-Modellen ist die gewaltige Reduktion
in der Berechnungszeit (bis zu einem Faktor 10’000). Die starke Reduktion in der
CPU-Zeit ¨ offnet den Weg f¨ ur eine Forschung, die bisher nicht m¨ oglich war, wie
beispielsweise die routinem¨ assige Untersuchung von mechanischen Eigenschaften von
grossen Knochenproben oder sogar von ganzen Knochen.
Zusammenfassend wurde ein neues System zur elementbasierten Analyse von
trabekul¨ aren Knochenstrukturen eingef¨ uhrt. Lokale Morphometrie und idealisierte
Computermodelle k¨ onnten wichtige Werkzeuge werden, um alters- und krankheits-
bedingte¨Anderungen in der Knochenqualit¨ at und -st¨ arke zu erkl¨ aren.
Entwicklung von neuen, hochaufl¨ osenden in vivo Bildgebungsverfahren haben ide-
alisierte Computermodelle das Potential, in der routinem¨ assigen Knochenfraktur-
Vorhersage zu einem Standard zu werden.
Bone is a specialized connective tissue that makes up, together with cartilage, the
skeletal system. Bone serves three functions: (a) mechanical, support and site of
muscle attachment for locomotion, (b) protective, for vital organs and bone marrow,
and (c) metabolic, as a reservoir of ions, especially calcium and phosphate, for the
maintenance of serum homeostasis, which is essential to life.
The macroscopic bone architecture is classically described using long bones as a
model (Figure 1.1). A typical long bone shows two wider extremities called epiphy-
ses, a more or less cylindrical shaft in the middle called midshaft or diaphysis, and a
developmental zone between the two called metaphysis. In a growing long bone, the
epiphysis and the metaphysis originate from two independent ossification centers.
They are separated by a layer of cartilage, called the epiphyseal cartilage or growth
plate. This layer of proliferative cells and expanding cartilage matrix is responsible
for the longitudinal growth of bones. By the end of the growth period this layer is
remodelled, becomes entirely calcified and is finally replaced by bone. The external
part of the bones is formed by a thick and dense layer of calcified tissue, the cortex or
compact bone, which in the diaphysis encloses the medullary cavity where the bone
marrow is housed. Toward the metaphysis and the epiphysis, the cortex becomes
progressively thinner, and the internal space is filled with a network of thin, calcified
struts called trabeculae. These trabeculae are conventionally classified according to
their shape as rods and plates. This porous bone type is the cancellous bone, also
named spongy or trabecular bone (Figure 1.2). The spaces enclosed by these thin
trabeculae also are filled with bone marrow and are in continuity with the medullary
cavity of the diaphysis. The bone surfaces at the epiphyses that take part in the
joint are covered with a layer of articular cartilage that does not calcify.
2Chapter 1. Introduction
Growth plateGrowth plate
Cortical bone Cortical bone
Fused growth plate Fused growth plate
L. Weiss, ed., Histology, cell and tissue biology.
Elsevier Science Publishing, 1983:200-255.
Figure 1.1: Schematic drawing of a longitudinal section through a long bone (tibia).
There are consequently two bone surfaces at which the bone is in contact with the
soft tissues, an external surface, the periosteal surface and an internal surface, the
endosteal surface. These surfaces are lined with osteogenic cells organized in layers,
the periosteum and the endosteum. Cortical and trabecular bone are constituted of
the same cells and the same matrix elements, but there are structural and functional
differences. The primary structural difference is quantitative: 80% to 90% of the
volume of compact bone is bone tissue (as opposed to marrow tissue), whereas only
15% to 25% of the trabecular bone is bone tissue (the remainder being occupied by
bone marrow, blood vessels, and connective tissue). The result is that 70% to 85%
of the interface with soft tissues is at the endosteal bone surface, which leads to the
functional difference: the cortical bone fulfills mainly a mechanical and protective
function, and the trabecular bone, a metabolic function. However, this classification
is rather simplified and in certain bones such as the vertebral bodies, load is mainly
carried by trabecular bone (adapted from ).
plates plates plates
Cortical boneCortical boneCortical bone
Figure 1.2: Trabecular bone.
There are many metabolic bone diseases where one or both of the above mentioned
functions are not accomplished properly anymore. One of those diseases is osteo-
porosis, which is defined as a skeletal disorder characterized by compromised bone
strength predisposing to an increased risk of fracture. Bone strength reflects the
integration of two main features: bone volume fraction and bone quality referring to
bone architecture, turnover, damage accumulation, and mineralization . Before
the age of 50, it affects only a few, whereas in old age, few are left without fractures
due to age- or disease-related reduction of bone strength. It has been estimated
that 4 in 10 white women age 50 years and older will experience a hip, spine, or
wrist fracture sometime during the remainder of their lives (Figure 1.2) . The
risk for white men is lower due to their shorter life expectancy and lower fracture
incidence rates. Due to the large size of the affected population and because of
the devastating impact of osteoporotic fractures on morbidity, mortality, and on
social costs, osteoporosis is recognized to be an important public health problem .
Death rates in patients with a hip fracture is 12% to 20% higher than in persons
of similar age, race, and sex . The total medical expenditures of osteoporotic
hip-fractures, which is related with the highest medical costs, has been estimated
for many countries [6-14]. On a global scale, it was estimated that the number of
4Chapter 1. Introduction
126.96.36.1999.7Any of the three (%)Any of the three (%)
2.52.5 16.016.0 Wrist (%)Wrist (%)
5.05.0 16.5 16.5Vertebra (%)Vertebra (%)
6.06.017.517.5Hip (%)Hip (%)
menmen womenwomenType of fractureType of fracture
Lifetime risk at age 50 years for Caucasian.
L.J. Melton III, JBMR 15(12):2309-14; 2000.
Figure 1.3: Common fractures in osteoporosis.
hip fractures occurring in the world each year will rise from 1.66 million in 1990 to
6.26 million by 2050 . Thus, assuming a total annual cost of roughly US$ 30’000
per hip-fracture case , the global expenditures will rise from US$ 50 billion in
1990 to almost US$ 200 billion in 2050.
The increased use of densitometry throughout the last decades reflected a focus
on bone density as the most important predictor of osteoporotic bone fractures.
Significant correlations of apparent density and different mechanical properties of
cancellous bone have been demonstrated for large populations using power law re-
gressions [16-22]. Although many older persons may lose bone, as expressed by a
decrease in bone density, not all develop fractures. The reason is that bone density is
not the sole determinant of fracture risk. Neuromuscular function and environmental
hazards, influencing the risk of fall, the force of impact as well as bone strength are
equally important factors . Bone mineral density, geometry of bone, microarchi-
tecture of bone and quality of the bone material are all components that determine
bone strength as defined by the bone’s ability to withstand loading. On average,
seventy to eighty percent of the in vitro variability in bone strength is determined
by its density . On an individual basis, density alone accounts for 10% – 90% of
the variation in the strength of trabecular bone . This also means that 90% –
10% of the variation in strength cannot be explained by bone density. It has been
shown that changes in trabecular morphology lead to a disproportionate decrease in
bone strength . For this reason, microstructural information must be included in
the analysis to predict individual mechanical bone properties [27,28]. Preliminary
data have shown that predicting trabecular bone strength can be greatly improved
by including architectural parameters in the analysis [29-33]. However, the relative
importance of bone density, architecture and local tissue properties, in the etiology
of bone fractures, an issue referred to as bone “quality”, is poorly understood.
Quantification of bone microstructure
To assess the contribution of microstructure to bone mechanical competence, con-
ventionally relatively small bone samples have been extracted from the body, where
it was proposed to either use cubic or cylindrical specimen geometry . For the
assessment of bone mechanical properties, different experimental testing methods
have been proposed . For trabecular bone, compression [20,22,32,34,36,37], and
tensile testing [22,37] were used to assess apparent Young’s modulus and ultimate
strength. Additional to these experimental methods, the stiffness and strength of
trabecular bone samples was also computed by finite element model simulations [38-
40], which showed to yield qualitatively similar results as experimental approaches.
The assessment of bone strength has traditionally been used to invent inde-
pendent measures potentially predicting and explaining the variation in stiffness
and strength, where the main focus was set on the characterization of bone mi-
crostructure. Conventionally, histomorphometry has been used to investigate bone
structures. This method allows computing parameters such as bone volume den-
sity (BV/TV) and bone surface density (BS/TV). Model dependent algorithms
were developed to calculate mean trabecular thickness (Tb.Th), trabecular sepa-
ration (Tb.Sp), number of trabecular structures (Tb.N), and other indices needed
to better characterize bone architecture . However, histomorphometry is a two-
dimensional technique which does not allow to determine the real three-dimensional
structure of trabecular bone. New imaging methods such as micro-computed to-
mography are attractive for such a characterization.
Micro-computed tomography (µCT) is a non-destructive imaging technique
which allows to achieve images at a very high resolution (∼10 µm). From two-
dimensional projections a fully three-dimensional image can be reconstructed .
The advantage of three-dimensional images is that they enable assessment of truly
three-dimensional parameters which may not be correctly computed from two-
dimensional sections. Such indices can be used to estimate the influence of bone
architecture on the mechanical competence of bone. Additionally, µCT is an ideal
candidate to expand the classical mechanical protocols because it allows to monitor
bone architecture and its behavior under loading in a time lapsed fashion due to its
non-destructiveness. M¨ uller et al  developed a new technique for nondestructive