Mechanical and mechanobiological influences on bone
fracture repair : identifying important cellular
characteristics
Isaksson, H.E.
DOI:
10.6100/IR630668
Published: 01/01/2007
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Isaksson, H. E. (2007). Mechanical and mechanobiological influences on bone fracture repair : identifying
important cellular characteristics Eindhoven: Technische Universiteit Eindhoven DOI: 10.6100/IR630668
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Mechanical and mechanobiological influences
on bone fracture repair
- identifying important cellular characteristics
A catalogue record is available from the Eindhoven University of Technology Library
ISBN 978-90-386-1146-4
Copyright © 2007 by H. Isaksson
All rights reserved. No part of this book may be reproduced, stored in a database or retrieval
system, or published, in any form or in any way, electronically, mechanically, by print,
photoprint, microfilm or any other means without prior written permission of the author.
Cover design: Jorrit van Rijt, Oranje Vormgevers
Printed by Universiteitsdrukkerij TU Eindhoven, Eindhoven, The Netherlands.
Financial support from the AO Foundation, Switzerland is gratefully acknowledged.
AO Foundation
Research
Mechanical and mechanobiological influences
on bone fracture repair
- identifying important cellular characteristics
PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Technische Universiteit Eindhoven,
op gezag van de Rector Magnificus, prof.dr.ir. C.J. van Duijn,
voor een commissie aangewezen door het College
voor Promoties in het openbaar te verdedigen op
maandag 26 november 2007 om 16.00 uur
door
Hanna Elisabet Isaksson
geboren te Linköping, Zweden
Dit proefschrift is goedgekeurd door de promotoren:
prof.dr.ir. H.W.J. Huiskes
en
prof.dr.ir. K. Ito
Copromotor:
dr. C.C. van Donkelaar
To my family and friends for all their
support through these years
Contents
Contents ................................................................................................................................... vii
Summary................................................................................................................................... ix
List of original publications .................................................................................................... xi
1
Introduction......................................................................................................................... 1
2
Bone fracture healing and computational modeling of bone mechanobiology ............. 7
3
Comparison of biophysical stimuli for mechano-regulation of tissue differentiation
during fracture healing .................................................................................................... 27
4
Corroboration of mechano-regulatory algorithms: Comparison with in vivo results 41
5
Bone regeneration during distraction osteogenesis: Mechano-regulation by shear
strain and fluid velocity.................................................................................................... 55
6
A mechano-regulatory bone-healing model based on cell phenotype specific activity71
7
Determining the most important cellular characteristics for fracture healing, using
design of experiments methods ........................................................................................ 91
8
Remodeling of fracture callus in mice can be explained by mechanical loading ...... 107
9
Discussion and conclusions ............................................................................................ 123
Appendix A: Theoretical development of finite element formulation for modeling
cellular activity………… ...................................................................................................... 135
Appendix B:
Taguchi orthogonal arrays and design of experiments methods ........... 141
References.............................................................................................................................. 147
Samenvatting......................................................................................................................... 163
Acknowledgement................................................................................................................. 165
Curriculum Vitae .................................................................................................................. 167
vii
Mechanical and mechanobiological
influences on bone fracture repair
- identifying important cellular characteristics
Summary
Fracture repair is a complex and multifactorial process, which involves a well-programmed
series of cellular and molecular events that result in a combination of intramembranous and
endochondral bone formation. The vast majority of fractures is treated successfully. They heal
through ‘secondary healing’, a sequence of tissue differentiation processes, from initial
haematoma, to connective tissues, and via cartilage to bone. However, the process can fail and
this results in delayed healing or non-union, which occur in 5-10% of all cases. A better
understanding of this process would enable the development of more accurate and rational
strategies for fracture treatment and accelerating healing. Impaired healing has been associated
with a variety of factors, related to the biological and mechanical environments. The local
mechanical environment can induce fracture healing or alter its biological pathway by
directing the cell and tissue differentiation pathways. The mechanical environment is usually
described by global mechanical factors, such as gap size and interfragmentary movement. The
relationship between global mechanical factors and the local stresses and strains that influence
cell differentiation can be calculated using computational models.
In this thesis, mechano-regulation algorithms are used to predict the influence of mechanical
stimuli on tissue differentiation during bone healing. These models used can assist in
unraveling the basic principles of cell and tissue differentiation, optimization of implant
design, and investigation of treatments for non-union and other pathologies. However, this can
only be accomplished after the models have been suitably validated. The aim of this thesis is
to corroborate mechanoregulatory models, by comparing existing models with well
characterized experimental data, identify shortcomings and develop new computational
models of bone healing. The underlying hypothesis throughout this work is that the cells act as
sensors of mechanical stimuli during bone healing. This directs their differentiation
accordingly. Moreover, the cells respond to mechanical loading by proliferation,
differentiation or apoptosis, as well as by synthesis or removal of extracellular matrix.
In the first part of this work, both well-established and new potential mechano-regulation
algorithms were implemented into the same computational model and their capacities to
predict the general tissue distributions in normal fracture healing under cyclic axial load were
compared. Several algorithms, based on different biophysical stimuli, were equally well able
to predict normal fracture healing processes (Chapter 3). To corroborate the algorithms, they
were compared with extensive in vivo experimental bone healing data. Healing under two
distinctly different mechanical conditions was compared: axial compression or torsional
rotation. None of the established algorithms properly predicted the spatial and temporal tissue
distributions observed experimentally, for both loading modes and time points. Specific
ix
Summary
inadequacies with each model were identified. One algorithm, based on deviatoric strain and
fluid flow, predicted the experimental results the best (Chapter 4). This algorithm was then
employed in further studies of bone regeneration. By including volumetric growth of
individual tissue types, it was shown to correctly predict experimentally observed spatial and
temporal tissue distributions during distraction osteogenesis, as well as known perturbations
due to changes in distraction rate and frequency (Chapter 5).
In the second part of this work, a novel ‘mechanistic model’ of cellular activity in bone
healing was developed, in which the limitations of previous models were addressed. The
formulation included mechanical modulation of cell phenotype and skeletal tissue-type
specific activities and rates. This model was shown to correctly predict the normal fracture
healing processes, as well as delayed and non-union due to excessive loading, and also the
effects of some specific biological perturbations and pathological situations. For example,
alterations due to periosteal stripping or impaired cartilage remodeling (endochondral
ossification) compared well with experimental observations (Chapter 6). The model requires
extensive parametric data as input, which was gathered, as far as possible, from literature.
Since many of the parameter magnitudes are not well established, a factorial analysis was
conducted using ‘design of experiments’ methods and Taguchi orthogonal arrays. A few
cellular parameters were thereby identified as key factors in the process of bone healing.
These were related to bone formation, and cartilage production and degradation, which
corresponded to those processes that have been suggested to be crucial biological steps in
bone healing. Bone healing was found to be sensitive to parameters related to fibrous tissue
and cartilage formation. These parameters had optimum values, indicating that some amounts
of soft tissue production are beneficial, but too little or too much may be detrimental to the
healing process (Chapter 7).
The final part of this work focused on the remodeling phase of bone healing. Long bone postfracture remodeling in mice femora was characterized, including a new phenomenon
described as ‘dual cortex formation’. The effect of mechanical loading modes on fracturecallus remodeling was evaluated using a bone remodeling algorithm, and it was shown that the
distinct remodeling behavior observed in mice, compared to larger mammals, could be
explained by a difference in major mechanical loading mode (Chapter 8).
In summary, this work has further established the potential of mechanobiological
computational models in developing our knowledge of cell and tissue differentiation processes
during bone healing in general, and fracture healing and distraction osteogenesis in particular.
The studies presented in this thesis have led to the development of more mechanistic models
of cell and tissue differentiation and validation approaches have been described. These models
can further assist in screening for potential treatment protocols of pathophysiological bone
healing.
x
List of original publications
The work presented in this thesis was carried out at the AO Research Institute in Davos,
Switzerland, and within the Bone- and Orthopaedic Biomechanics section of the department of
Biomedical Engineering at Eindhoven University of Technology. It resulted in the following
peer-reviewed publications and manuscripts, referred to by their roman numerals. The thesis
also contains unpublished data.
I
Comparison of biophysical stimuli for mechano-regulation of tissue
differentiation during fracture healing.
H. Isaksson, W. Wilson, C.C. van Donkelaar, R. Huiskes, K. Ito
Journal of Biomechanics, 39(8):1363-1562, 2006
II
Corroboration of mechanoregulatory algorithms for tissue differentiation
during fracture healing: Comparison with in vivo results
H. Isaksson, C.C. van Donkelaar, R. Huiskes, K. Ito
Journal of Orthopaedic Research, 24(5):898-907, 2006
III
Bone regeneration during distraction osteogenesis:
Mechano-regulation by shear strain and fluid velocity
H. Isaksson, O. Comas, J. Mediavilla, W. Wilson,
C.C. van Donkelaar, R. Huiskes, K. Ito
Journal of Biomechanics, 40(9):2002-2011, 2007
IV
A mechano-regulatory bone-healing model based on cell phenotype
specific activity
H. Isaksson, C.C. van Donkelaar, R. Huiskes, K. Ito
Manuscript submitted for publication
V
Determining the most important cellular parameters for the characteristics
of proper fracture healing, using design of experiments methods
H. Isaksson, C.C. van Donkelaar, R. Huiskes, J. Yao, K. Ito
Manuscript submitted for publication
VI
Remodeling of fracture callus in mice can be explained by mechanical loading
H. Isaksson, I. Gröngröft, W. Wilson, B. van Rietbergen, A. Tami,
C.C. van Donkelaar, R. Huiskes, K. Ito
Manuscript submitted for publication
xi
xii
1
1
1
Introduction
This chapter includes a short introduction to the problems
concerning bone tissue regeneration, particularly with regard to
fracture healing. The role of the mechanical environment, both
globally and locally, is then introduced with a focus on how
computational models can be of assistance. The overall hypothesis
and an outline of the specific goals of the thesis are then described,
including the specific research questions investigated in each of the
constituent studies. The general methodology is briefly outlined.
1
Chapter 1
1.1 Problem
Bone healing is so common in life that it is easy to overlook how astonishing it is as a
biomechanical phenomenon. In contrast to other adult tissues, which heal with the production
of scar tissue, bone heals with bone. New bone is formed and continuously remodeled until the
original site of injury can hardly be recognized. During fracture repair, bone is formed by a
combination of processes, which are closely related to both embryonic development and adult
growth (Marks and Hermey, 1996).
Despite their natural healing capacity and the extensive amount of research conducted in this
area, delayed healing and non-union of bones are frequently encountered. For example in the
United States 5-10 % of the over 6 million fractures occurring annually develop into delayed
or non-unions (Praemer et al., 1992; 1999; Einhorn, 1995; 1998b). Bone fractures cost society
large amounts of money every year in primary treatment, follow-up operations due to delayed
or non-unions, and the cost of lost employment. Furthermore, ageing of the population is
expected to increase the prevalence of fractures due to osteoporosis. In the European Union, in
the year 2000, the number of osteoporotic related fractures was estimated at 3.8 million,
resulting in direct costs for osteoporotic fractures to the health care services of € 32 billion
(Reginster and Burlet, 2006). It has been predicted that 40% of all postmenopausal women
will suffer one or more fractures during their remaining lifetimes (Compston et al., 1998;
Reginster and Burlet, 2006). Hence, prevention and effective treatment of such complications
are desirable.
It is well recognized that mechanical stimulation can induce fracture healing or alter its
biological pathway (Rand et al., 1981; Brighton, 1984; Wu et al., 1984; Goodship and
Kenwright, 1985; Aro et al., 1991; Claes et al., 1997; Rubin et al., 2001). New bone formation
is also related to the direction and magnitude of loading, affecting the internal stress state in
the repairing tissue (Park et al., 1998; Augat et al., 2003; Bishop et al., 2006). However, the
mechanisms by which mechanical stimuli are transferred, via cellular mediators, into a
biological response remain unknown.
Mechanobiology describes the mechanisms by which biological processes are regulated by
signals to cells that are induced by mechanical loads (Roux, 1881; van der Meulen and
Huiskes, 2002). When the mechanisms of mechanically-regulated tissue formation are
understood and well defined at the cellular level, physiological conditions and
pharmacological agents may be developed and used to prevent non-unions and, furthermore,
to help accelerate fracture repair and restore optimal function. Computer modeling is having a
profound effect on scientific research (Sacks et al., 1989). Many biological processes,
including bone healing, are so complex that physical experimentation is either too time
consuming, too expensive, or impossible. As a result, mathematical models that simulate these
complex systems are more extensively used. In mechanobiology, these computational models
have been developed and used together with in vivo and in vitro experiments to quantitatively
determine the rules that govern the effects of mechanical loading on cells and tissue
differentiation, growth, and adaptation and maintenance of bone. Mechanical perturbations are
2
Introduction
applied to a model geometry, and the local mechanical environment is calculated, using the
finite element method. The biological aspects of the computations are based on different
premises for local mechanical variables stimulating certain cellular activities, for example cell
proliferation, or changes in bone structure. Computational models are gradually becoming
more sophisticated with increasing computational power and mechanobiological knowledge.
Both experimental and computational studies are critical to advance our knowledge in
mechanobiology. Integration of the fields is important, since models can help interpret
experiments and experiments can provide relationships and observations for model
development.
Using these principles, mechano-regulation algorithms were proposed to investigate the
influence of mechanical stimuli on tissue differentiation. These algorithms were extensively
applied to study bone healing (Chapter 2.8). They have used strain invariants and fluid
hydrostatic pressure or fluid velocity in different combinations as biofeedback variables.
These algorithms need to be validated against direct in vivo data, before further developments
can follow. Validation could help both the understanding of basic biology during bone
regeneration and in developing clinical treatment protocols for fracture healing. Additionally,
validated models can be useful in designing new experiments, and theoretical models and
animal experiments together can lead to new research questions and advances in
mechanobiology. However, to date validation attempts have not been carried out sufficiently.
This is partly due to the need for experimentally reliable and repeatable outcomes, and
controlled mechanical environments. These are rarely available, because the required
conditions are very difficult to meet in an experimental setting. Moreover, many experiments
that are used for validation are originally carried out with other scientific questions in mind.
The principles of bone healing are very similar to other bone forming processes. Bone healing
has great similarities to bone formation and growth during fetal development (Marks and
Hermey, 1996; Ferguson et al., 1999). Furthermore, it appears that the understanding of
principles in fracture repair may have implications beyond fracture treatment, with
applications in tissue regeneration in general, such as during distraction osteogenesis,
osseointegration of implants, and in tissue engineering. Therefore, a better understanding of all
the factors that influence the bone healing process in general, and mechanobiology in
particular, will have important applications in skeletal generation and regeneration.
1.2 Aims and outline of the thesis
The previous section identified the need for further research on mechanoregulatory
mechanisms of bone healing. The general objective of this work is to enhance the knowledge
of the role of mechanical factors in tissue differentiation during bone regeneration in general,
and fracture healing in particular, by corroborating mechanoregulatory algorithms. The
fundamental hypothesis in these studies is that the local level of mechanical stimulation, using
stress and strain invariants, determines the cell and tissue differentiation pathways. The cells
act as sensors, and they respond depending on their environment. Mechanical stimulation
influences where either fibrous tissue, cartilage or bone tissue forms by directing the
3
Chapter 1
differentiation of mesenchymal cells into fibroblasts, chondrocytes or osteoblasts. This thesis
develops methods for corroboration of computational models with direct in vivo experimental
data. The general objectives are divided into specific aims and hypotheses, which are
addressed in subsequent chapters. The specific objectives with each chapter are specified
below:
Chapter 2 – Literature review
•
To provide a comprehensive literature basis to describe the current knowledge and
previous research conducted in the area of bone healing and computational
mechanobiology.
Chapter 3 – Comparing existing models
•
To implement and compare several existing mechano-regulation algorithms with
regards to their abilities to predict the normal fracture healing processes.
•
To investigate whether individual parameters such as strain invariants, i.e. deviatoric
strain or volumetric deformation, i.e. pore pressure and fluid velocity can be used to
predict tissue differentiation during normal fracture healing.
Chapter 4 – Determining validation status and identify inadequacies
•
To corroborate the mechano-regulatory algorithms with extensive in vivo bone healing
data from animal experiments, including interfragmentary conditions, different from
those for which they were developed.
•
To reveal which of these algorithms reflect the actual mechanobiological processes the
best, by analyzing the corroborations at time points representing early and late healing.
Chapter 5 – Implementing volumetric growth
•
To investigate whether mechano-regulation by octahedral shear strain and fluid
velocity, the algorithm selected in Chapter 4, can predict the spatial and temporal
tissue distributions observed during experimental distraction osteogenesis.
•
To study variations in predicted tissue distributions due to alterations in distraction rate
and frequency.
Chapter 6 – Developing a mechanistic cell model
•
To develop a new model of tissue differentiation based on cell activity, including
matrix and cell phenotype-specific descriptions of migration, proliferation,
differentiation, apoptosis, matrix production and degradation, in order to overcome
discrepancies identified in Chapter 4.
• To determine the importance of including cell-specific activities when modeling tissue
differentiation and bone healing.
Chapter 7 – Establishing the relative importance of cellular characteristics
•
To determine the importance of each parameter in the mechanistic cell model, by
employing ‘design of experiments’ methods and Taguchi orthogonal arrays.
4
Introduction
Chapter 8 – Characterizing post fracture remodeling in mice
•
To experimentally describe the remodeling phase of fracture healing in mice.
•
To investigate the hypothesis that the differences during the remodeling phase of
fracture healing observed in mice compared to larger mammals and humans, can be
explained by a main difference in mechanical loading mode.
Chapter 9 – Discussion
• To summarize the results and conclusions and discuss the logic of the thesis as a whole
and to incorporate it with past research and future prospects.
1.3 General approach
Two- and three dimensional finite element models were developed as adaptive models for
tissue differentiation. Poroelastic finite element formulations were used to calculate the
biophysical stimuli and mass- or heat transfer finite element formulations were employed for
the calculations of cellular activities. Adapted tissue types (matrix production) regulated the
mechanical properties and could also alter the geometry of the tissue. Results from in vivo
animal experiments were employed for a range of qualitative and quantitative comparisons
between computational predictions and experimental data.
Verification was ensured by assessing the ability of the model to solve the mathematical
representations correctly and by performing convergence studies. Validation was performed
by assessing the models ability to represent the mechanical and biological behavior of specific
experimental outcomes (Figure 1-1). The software that was used to create the computational
models and the origin of the experimental data employed, is described below.
Figure 1-1: General scheme of the approach for validation of computational models. The
research within this thesis focused on the left hand side, and the experiments required (right
hand side) were adopted from other studies that were performed at the AO Research Institute.
5
Chapter 1
The numerical models developed in this thesis were implemented and solved with the
following software: The overall framework of the tissue differentiation model was
implemented and solved in Matlab (v 5.3-7.1 Mathworks). Depending on complexity, the
finite element meshes where created using either Marc Mentat (MSC Software), ABAQUS
CAE (v 6.3-6.5 Simulia, Dassault Systemés) or Matlab. All finite element models were solved
using ABAQUS (v 6.3-6.5 Simulia, Dassault Systemés). Parts of the codes were written in
external programs in FORTRAN 77 or C++. The remeshing algorithm (Chapter 5), was
modified based on existing code from Dr Jesus Mediavilla (2005). The biphasic swelling
model adapted to implement volumetric growth in Chapter 5 originated from Dr Wouter
Wilson (2005). The bone remodeling algorithm used in Chapter 8 was adopted from the theory
by Dr Ronald Ruimerman (2005). The mechanistic model in Chapter 6 and 7 was solved using
a special finite element formulation, developed for biological modeling of cell activity during
this work. The details are provided in Appendix A.
For validation purpose, results from several animal experiments, originally carried out to
answer other research questions were used (Figure 1-1, right side). The availability of
experimental results, such as radiographs, histology, histomorphometry, mechanical testing,
reaction force measurements and micro computed tomography were vital for the work
presented in this thesis. It allowed both quantitative and qualitative comparisons between
computational predictions and experimental results, a strategy which is a necessity for
validations of theoretical models. The in vivo ovine tibia fracture model employed in Chapter
4 was provided by Dr Nicholas Bishop, as part of his PhD studies (Bishop, 2007). The
experimental ovine distraction model used in Chapter 5 is from Dr. Ulrich Brunner’s MD
habilitation research (Brunner, 1992). The murine experimental fracture healing data, which is
part of Chapter 8, was conducted by Dr. Ina Gröngröft, DVM, as part of her dissertation
(Gröngröft, 2007).
6
2
2
Bone fracture healing and
computational modeling of
bone mechanobiology
This chapter provides a literature review of the topics addressed in
this thesis. It includes a brief description of bone morphology, the
mechanisms by which it is generated, regulated and repaired, and
the role that the cells play in these processes. This is followed by an
overview of skeletal disorders, in particular bone fractures and the
healing process, including different forms of healing, and possible
complications. Thereafter, the influence of mechanics on bone
healing and the current understanding of mechanobiology are
summarized. Finally, previous studies in the area of computational
mechano-regulation of tissue differentiation are reviewed and
theories and algorithms described.
7
2
Chapter 2
2.1 Bone and bone fracture
The adult human skeleton consists of 206 bones. They act as a support framework for the body
and protect the internal organs. Together with the muscles and joints, they facilitate movement
and participate in maintenance of the body’s mineral balance (Marks and Hermey, 1996).
2.1.1 Bone structure and composition
Morphologically, bones are classified as cortical or trabecular (cancellous) bone. Cortical bone
forms the outer shell of every bone. It is compact, stiff and strong and has a high resistance to
all loads: bending, axial and torsion, which are especially important in the shafts of long bones
(Buckwalter et al., 1996a). In contrast, trabecular bone is a less dense, less stiff, open pore
matrix, which acts as a mechanically efficient structure in supporting the thinner cortical shells
at the ends of long bones and in the vertebrae (Buckwalter et al., 1996a). The shafts of the
long bones are referred to as the diaphyses, and the expanded ends as the epiphyses. The ends
of the epiphyses are coated with articular cartilage and other bone surfaces are covered by a
well vascularized soft-tissue layer, known as the periosteum. The periosteum isolates and
protects the bone from surrounding tissues and provides cells for bone growth and repair.
Similarly, the inner surfaces of the long bones are lined by the endosteum. Bone marrow is the
soft tissue that fills the medullary cavity of the long bones and the spaces between the
trabeculae. It serves as storage for precursor cells, which are involved in repair.
Bone tissue can also be woven or lamellar. Woven bone is laid down rapidly and has
randomly oriented collagen fibers, and low strength. In adults it is observed mainly at sites of
repair, at tendon or ligament attachments and in pathological conditions. In contrast, the
collagen fibers in lamellar bone are aligned and are much stronger. Woven bone is mostly
replaced by lamellar bone during growth or repair (Buckwalter et al., 1996b).
Bone consists mainly of extracellular matrix (ECM), divided into organic and inorganic
components. The organic components consist primarily of type I collagen (Rossert and
Crombrugghe, 1996), and the inorganic component consists primarily of hydroxyapatite and
calcium carbonates (Marks and Hermey, 1996). The combination of organic fibers enclosed in
an inorganic matrix provides a stiff and strong composite structure, in which the mineral
component resists compression and the collagen fibers resist tension and shear (van der and
Garrone, 1991; Marks and Hermey, 1996). The remainder of the skeleton consists of cells and
blood vessels. There are four different cell types in human bones: osteoblasts, osteoclasts,
bone lining cells, and osteocytes. Osteoblasts are bone forming cells. They line the surfaces of
the bones and produce osteoid (Buckwalter et al., 1996a). Osteocytes are osteoblasts that
became surrounded by bone matrix growing around them, forming a cavity, or “lacunae”.
They remain active in the maintenance of bone and are believed to regulate bone remodeling
(Buckwalter et al., 1996b). Bone-lining cells, also called pre-osteoblasts, are found in the
periosteal and endosteal surfaces. Osteoclasts are multinucleated cells whose function is bone
resorption. They break down bone and release the minerals into the blood (Buckwalter et al.,
1996b). Osteoblasts, osteocytes and bone-lining cells differentiate from mesenchymal stem
cells, and osteoclasts from hemopoietic stem cells (Owen, 1970).
8
Bone fracture healing and computational modeling of bone mechanobiology
2.1.2 Bone formation and growth
Bone forms, grows and resorbs continuously, by remodeling processes. The formation of bone
occurs by two methods, intramembranous and endochondral ossification. These are discussed
in greater depth in the following chapters, since both are prominent during bone healing.
Briefly, intramembranous ossification occurs during formation of the ‘flat’ bones, for those in
the skull, for example (Buckwalter et al., 1996b). It forms directly from basic mesenchymal
tissue, by differentiation from pre-osteoblasts into osteoblasts, which lay down osteoid.
Intramembranous bone formation occurs as appositional growth on bone surfaces, thereby
increasing their width (Buckwalter et al., 1996b). Endochondral ossification occurs in long
bone formation and growth. The bone develops from a cartilage template, which calcifies
along a front and is replaced by bone as blood-capillaries tunnel through, providing bone
forming cells (Buckwalter et al., 1996b).
About 5% of the skeleton is undergoing remodeling, or renewal, at any time. Haversian
remodeling is a process of resorption followed by replacement of bone, with little change in
shape, and occurs throughout life (Marks and Hermey, 1996). A cluster of osteoclasts drill a
tunnel into the bone, creating a cone. Behind the tip, osteoblasts fill up the cone with new
bone with living cells, connected to the capillaries within the canal (Buckwalter et al., 1996b)
(Figure 2-1). Remodeling releases calcium and repairs micro damage. It is also responsible for
bone adaptation to the mechanical environment (Wolf, 1892), resulting in bone thickening in
regions of increased stress and bone thinning in regions of decreased stress.
Figure 2-1: Schematic diagram of haverisan remodeling (Reprinted from Rüedi et al. (2007),
Copyright by AO Publishing, Davos, Switzerland)
2.1.3 Bone fracture
Bone fractures when its strain limit is exceeded. A fracture disrupts the blood supply and
causes damage to the surrounding tissues, resulting in hemorrhage, anoxia, cell death and
aseptic inflammation (Simmons, 1985). Most fractures are caused by physical trauma. The
risk of fracture can increase when medical conditions, such as osteoporosis or cancer, weaken
the bones.
Bone fractures are classified by their appearance and the extent of damage to the surrounding
tissues (Rüedi et al., 2007). All fractures investigated in this thesis were simple transverse
fractures. Particular treatment strategies are used for each fracture type, including external
fixation, nailing and plating which are employed in Chapters 4, 5 and 8.
9
Chapter 2
2.2 Fracture healing
Fracture results in a series of tissue responses that remove tissue debris, re-establish the
vascular supply, and produce new skeletal matrix (Simmons, 1985). Unlike the healing
processes of other tissues, which produce scar tissue, bone has the ability to repair itself. Once
a fracture has healed and undergone remodeling, the structure will have returned to the preinjury state. There are two main types of fracture healing: Primary and secondary.
2.2.1 Primary healing
Primary fracture healing, also known as direct healing, involves intramembranous bone
formation and direct cortical remodeling without any external tissue (callus) formation (Rahn
et al., 1971; Perren, 1979). Primary healing only occurs when there is a combination of
anatomical reduction, small displacements of the bony ends, and either a small gap or direct
contact of the fractured cortical bone ends (Rüedi et al., 2007). Osteons traveling along the
length of the bone are able to cross the fracture site and bridge the gap, laying down cylinders
of bone (Figure 2-2). Gradually the fracture is healed by the formation of numerous osteons. It
is generally a slow process that can take months to years until healing is complete.
Figure 2-2: Primary healing. New osteons connecting the bone fragments across a fracture
line (Reprinted from Rüedi et al. (2007), Copyright by AO Publishing, Davos, Switzerland)
2.2.2 Secondary healing
In contrast to primary healing, secondary healing occurs in the presence of some
interfragmentary movement and is the process by which fractures heal naturally. It involves a
sequence of tissue differentiation processes by which the bone fragments are first stabilized by
an external callus (Rahn, 1987; Perren and Claes, 2000). Recovery of bone strength is
generally more rapid than in primary healing.
Stages of repair during secondary fracture healing
The process of bone repair by secondary healing can be divided into three overlapping stages
– the inflammatory, reparative and remodeling phases. Healing begins with inflammation
which is followed by the formation of soft and hard callus during the reparative phase. Finally
the callus is resorbed by remodeling (Cruess and Dumont, 1985; Frost, 1989). This thesis
10
Bone fracture healing and computational modeling of bone mechanobiology
focuses on the reparative (Chapter 3-7) and remodeling (Chapter 8) phases of fracture healing.
The relative duration of these phases is shown in Figure 2-3.
Figure 2-3: Phases of fracture healing and their relative length. The figure has been
recreated based on Cruess and Dumont, (1975).
a)
b)
c)
Figure 2-4: Schematic drawing of the three main stages of fracture repair. a) Inflammatory
phase, b) reparative phase and c) remodeling phase. The figure has been adapted from Cruess
and Dumont, (1975).
Inflammation
The inflammatory phase begins simultaneously with the occurrence of the fracture (Figure
2-4a). During the trauma, blood vessels, the periosteum and the surrounding soft tissues are
ruptured and a haematoma (blood cloth) forms. The haematoma serves as an important source
of haematopoeitic cells and platelets that initiate the inflammatory response (Buckwalter et al.,
1996b). Large numbers of signaling molecules, including cytokines and growth factors, are
released (Bolander, 1992). The disruption of the blood supply also causes bone necrosis at the
edges of the fracture ends. Many of the cytokines released have angiogenic functions to
restore the blood supply. Also, pluripotent mesenchymal stem cells invade the haematoma at
this time. Cell division is first observed along the periosteum, and within a few days the
activity is increased along the entire area next to the fracture, where it remains high for weeks
(McKibbin, 1978).
11
Chapter 2
Mesenchymal cells, originating from the periosteum, endosteum, bone marrow, and possibly
the vasculature of the muscle-tissue surrounding the haematoma (Postacchini et al., 1995;
Iwaki et al., 1997; Gerstenfeld et al., 2003b), migrate towards the fracture region. No cells
originate from the actual fracture gap. The mesenchymal cells and the inflammatory cells form
a loose granulation tissue. Mesenchymal cells proliferate, to later differentiate down specific
pathways to become fibroblasts, chondrocytes, or osteoblasts, which generate fibrous tissue,
cartilage and bone, respectively. These cells proliferate and generate a callus (Bostrom and
Asnis, 1998). The ends of the fractured bone themselves do not appear to participate in the
initial reaction, and become necrotic, indicated by the empty osteocyte lacunae at the fractured
ends (McKibbin, 1978).
Repair
The repair phase can be divided into the formation of hard callus (intramembranous
ossification) and the formation of soft callus (endochondral ossification). Once the blood
supply has started to be re-established and mesenchymal cells have invaded, callus formation
begins (Figure 2-4b).
Intramembranous ossification
The first bone to be formed is laid down beneath the periosteum. This rapid formation of
woven bone begins several millimeters away from the fracture gap (Einhorn, 1998b). This
bone is produced by committed osteoprogenitor cells that are already present in the cambium
layer of the periosteum (Owen, 1970). It occurs within the haematoma when a group of
mesenchymal or osteoprogenitor cells start producing osteoid at an ossification center.
Ossification extends progressively from the bony surface, pushing the surrounding soft tissue
away. Mineralized bone replaces the osteoid, and as the ossification centers expand, and
eventually fuse. Formation of these external bony cuffs proceeds in the direction of the
fracture gap (McKibbin, 1978; Brighton, 1984).
Endochondral ossification
Concurrently, callus formation through endochondral ossification occurs at and around the
fracture gap. The soft callus consists of fibrous and/or cartilaginous connective tissues, which
have differentiated from the mesenchymal stem cells. During this stage, chondrocytes within
the matrix proliferate and generate cartilaginous tissue. Eventually these chondrocytes
hypertrophy, and the cartilage calcifies. The calcified cartilage acts as a stimulus for the
ingrowth of new blood vessels (Webb and Tricker, 2000). The amount of cartilage present is
variable, and dependent on the amount of movement (McKibbin, 1978). The formation of
cartilage usually begins at the cortical bone ends and expands radially. The bone formation
occurs step by step toward the fracture plane. The formation of endochondral bone is
dependent on the existence of blood capillaries, which originate from the periosteal callus. The
process of endochondral bone formation strongly resembles the embryonic development of
long bones (Ferguson et al., 1999). Angiogenesis occurs in parallel with endochondral
ossification, eventually leading to erosion of mineralized cartilage and deposition of bone
(Mark et al., 2004).
12
Bone fracture healing and computational modeling of bone mechanobiology
Remodeling
Once bony bridging of the callus has occurred and reunited the fracture ends, the processes of
bone remodeling and resorption become the dominant activities in the callus (Figure 2-4c).
The woven bone is gradually replaced by lamellar bone (Marsh and Li, 1999). During this
process the medullary cavity is reconstituted. It is thought that fluid shear stresses in bone
modulate the remodeling activities, leading to osteocyte apoptosis and osteoclast recruitment
(Bakker et al., 2004). Eventually, osteonal remodeling of the newly formed bone tissue and of
the fracture ends restores the original shape and lamellar structure of the bone (Einhorn,
1998b). Resorption of the endosteal callus coincides with re-establishment of the original
blood supply.
2.3 Requirements for bone healing
Fracture healing is influenced by many variables including mechanical stability, electrical
environment, biochemical factors and vascular supply. Many of the basic influences of these
factors on connective tissue response during fracture healing are poorly understood. However,
biochemical and mechanical interactions are recognized as most important.
2.3.1 Mechanical stability
It has long been known that mechanical stimulation can induce fracture healing or alter its
biological pathway (Rand et al., 1981; Brighton, 1984; Wu et al., 1984; Aro et al., 1991; Claes
et al., 1997; 1998). However, fractures can heal successfully under both extremely rigid, as
well as relatively flexible fixation (Augat et al., 2005). In general, rigid fixation results in
primary healing and more flexible fixation results in indirect or secondary healing.
The most dominant mechanical factors identified are the fracture geometry and the magnitude,
direction and history of the interfragmentary motion. These factors determine the local strain
field in the callus. The distribution of local strain in the healing tissue is believed to provide
the mechanobiological signal for regulation of the fracture repair process that stimulates
cellular reactions. One of the most dominant mechanical factors is the fracture geometry,
described by fracture pattern and gap size. For example, even simple transverse fractures that
lack careful repositioning and adequate fixation, can result in delayed union or non-union
(Koch et al., 2002). Small gaps are beneficial for a fast and successful healing process, while
larger gaps result in delayed healing, with decreased size in the periosteal callus and reduced
bone formation in the fracture gap (Augat et al., 1998). The amount of interfragmentary
movement is dictated by external load and fixation stability. A stiff fixator limits the
stimulation of callus formation, while flexible fixation enhances callus formation. Unstable
fixation can lead to excessive motion and result in non-union (Kenwright and Goodship, 1989;
Claes et al., 1995). However, the effect of the interfragmentary movement depends on the size
of the fracture gap (Claes et al., 1998).
The direction of the interfragmentary movement influences the healing process. Moderate
axial interfragmentary movement is widely accepted to enhance fracture repair by stimulating
formation of periosteal callus and increasing the rate of healing (Kenwright et al., 1991;
13
Chapter 2
Larsson et al., 2001). Shear movements, however, have resulted in contradicting results.
Experimental studies have shown that shear movements at the fracture site result in healing
with decreased periosteal callus formation, delayed bone formation in the fracture gap, and
inferior mechanical stability, compared to healing with axial movement (Yamagishi and
Yoshimura , 1955; Augat et al., 2003). However, other experimental investigations have
demonstrated superior healing under shear, compared to axial interfragmentary motion (Park
et al., 1998; Bishop et al., 2006). Furthermore, clinical studies have shown shear movement to
be compatible with successful healing (Sarmiento et al., 1996). Hence, the effect of shear,
compared to axial motion, appears to be sensitive to timing, magnitude, and/or gap size (Augat
et al., 2005). These studies have all investigated shear at the level of a whole bone. However,
it is still uncertain how that translates to shear at the tissue and cell level. That translation can
be investigated with computational tools.
During the course of healing, the callus stabilizes the fracture by enlarging its cross sectional
area and increasing its stiffness through tissue differentiation. The interfragmentary movement
decreases with healing time, as the callus stiffens. Finally, the hard callus bridges the bony
fragments and reduces the interfragmentary movement to such a low level that bone formation
can occur in the gap. The rate of reduction of interfragmentary movement appears to be related
to the initial interfragmentary movement, with larger movements having a faster decline
(Claes et al., 1998).
2.3.2 Biochemical factors
Soft tissue coverage and blood supply
Over the last few decades, the importance of restoration of the soft tissues surrounding the
fracture has become emphasized (Rüedi et al., 2007). Restoration of blood supply is important
in providing the biological environment, necessary for fracture healing. The nutrient artery in
the intramedullary canal, the capillary-rich periosteum and metaphyseal vessels are all
important in providing cells with oxygen, nutrients and chemical factors, such as growth
factors and cytokines (Rüedi et al., 2007).
Growth factors
Several growth factors and cytokines are known to be involved in the process of skeletal tissue
repair and remodeling (Bostrom and Asnis, 1998; Lieberman et al., 2002). Many factors have
been studied in isolation. However, the interactions and feedback mechanisms are still far
from being understood. A short summary of known effects are provided below.
Members of the transforming growth factor beta (TGF-β) supergene family, which include the
bone morphogenic proteins (BMPs), have been shown to control a number of processes during
skeletal development and repair. Although these proteins are closely related, both structurally
and functionally, each has a distinct temporal expression pattern and a potentially unique role
in bone healing (Aspenberg, 2005; Einhorn, 2005). TGF-β factors promote proliferation and
differentiation of mesenchymal precursor cells into osteoblasts, osteoclasts and chondrocytes
(Linkhart et al., 1996). They also appear to stimulate both endochondral and intramembranous
14
Bone fracture healing and computational modeling of bone mechanobiology
bone formation. BMP signaling leads to activation of genes for proliferation and
differentiation along the chondrogenic and osteogenic pathways. BMP2 and BMP7 (OP1)
seem to have similar effects and were shown to induce bone locally and speed-up skeletal
defect repair. Fibroblast growth factors (FGFs) and insulin growth factors (IGFs) can increase
callus size and strength (Kawaguchi, 1994), by increasing proliferation of chondrocytes and
osteoblasts, and stimulation of angiogenesis (Schmidmaier et al., 2004). Platelet-derived
growth factors (PDGFs) stimulate osteoblast and osteoclast cell proliferation (Bourque et al.,
1993; Sandberg et al., 1993). Furthermore, several cytokines (interleukins and tumor necrosis
factors) affects processes during skeletal repair (Bolander, 1992; Sandberg et al., 1993).
Prostaglandins stimulate osteoblastic bone formation and inhibit remodeling by decreasing
osteoclast activity (Bakker et al., 2001), and hormones, such as estrogen and parathyroid
hormones, also affect the healing potential (Aspenberg, 2005). Elucidating these interactions
will be an important task for future research.
2.4 Delayed and non-unions
There is no universally accepted definition of delayed or non-union. A general definition of
delayed union is a more than average time lapse to achieve clinical healing (Biasibetti et al.,
2005). One commonly used description is that a delayed union occurs when periosteal callus
formation ceases prior to complete union, delaying union to the late endosteal healing phase
(Babhulkar et al., 2005). Non-union can be defined by the failure of both the endosteal and
periosteal callus formation (Babhulkar et al., 2005). Sclerosis, a stiffening and hardening of
the tissues in the medullary canal, occurs when the fracture remains open or becomes filled
with scar tissue, which is usually fibrous in nature. Occasionally a fibro-cartilaginous
pseudoarthrosis or ‘false’ joint forms (Figure 2-5). Some types of non-unions, such as
pseudoarthrosis and hypertrophic non-unions are usually treated mechanically. Other types of
non-unions related to infection, aseptic or septic tissues are treated biologically. Treatments
can also include the use of traditional bone grafts.
Figure 2-5: Non-union represented by a mid-diaphyseal femur fracture, which resulted in a
pseudoarthrosis formation (Reprinted from Rüedi and Murphy (2000), Copyright by AO
Publishing, Davos, Switzerland)
15
Chapter 2
Non-union can be associated with patient factors, characteristics of the fracture, type of
treatment, and pharmacological factors. Patient factors can include smoking, diabetes and
vascular insufficiency, and muscle quality, as well as nutritional status, anemia, and growth
hormone deficiency. Smoking is related to an increased rate of delayed union because of the
vascoconstrictive effects of nicotine (Raikin et al., 1998; Hollinger et al., 1999). Moreover,
injury sites and high-energy injuries that lead to extensive soft tissue damage are associated
with higher rates of non-union. Treatment techniques can also impede fracture healing, with
inadequate immobilization or mobilization, fracture distraction, periosteal stripping and
repeated manipulations being common examples. Furthermore, cytostatics work by killing
cells that proliferate quickly. Hence, it has a negative effect on bone regeneration (Sauer et al.,
1982). Corticosteroids and non-steroidal anti-inflammatory drugs impair the inflammatory
response, and therefore impair healing (Einhorn, 2003; Gerstenfeld et al., 2003a).
2.5 Distraction osteogenesis
Distraction osteogenesis (DO) is a bone regeneration process, which was first performed by
the Russian physician Ilizarov (Ilizarov, 1989a; 1989b). Ilizarov created an osteotomy on a
patient to correct a severe deformity of the lower limb. To avoid stretching nerves and blood
vessels, his strategy was to use percutaneous wires to transfix the bone proximal and distal to
the osteotomy site, and to use them to gradually distract the ends of the bone at a steady rate.
He assumed that this treatment would create a large gap in the osteotomy site, which would
require a subsequent bone graft procedure. However, when he attempted to perform the bone
graft operation, he found that the gap was completely filled with new bone (Ilizarov, 1989a).
This method has become adopted world wide as the primary procedure for limb lengthening,
correcting deformities and treating non-unions due to trauma, infection or tumor (Richards et
al., 1998; Einhorn, 1998a).
This biological phenomenon seems to contradict some of the earlier basic assumptions about
the formation of bone and the way in which mechanical forces affect osteogenesis (Einhorn,
1998a). Researchers have believed that compression, weight bearing and stress-generated
potentials in bone lead to osteogenesis. On the contrary, distraction (tension) has never been
thought to be a stimulus for osteogenesis and in fact, most surgeons consider a fracture that is
subjected to tension to be at risk of becoming a non-union (Einhorn, 1998a). However, the
outcome has been shown predictable and reproducible. DO is usually separated into three
phases: the latency phase, immediately following osteotomy, the distraction phase, during
which the active distraction of the bony segments take place, and the consolidation phase,
which finally leads to bony union. The rate of bone formation during DO is directly related to
distraction rate (Ilizarov, 1989b; Li et al., 1999; 2000), frequency (Ilizarov, 1989b; Aarnes et
al., 2002; Mizuta et al., 2003) and the local strain/stress generated in the distraction gap (Li et
al., 1997; 1999). In distraction osteogenesis bone forms just as rapidly as during fracture
healing, and as long as distraction force is applied, bone regeneration can be sustained almost
indefinitely (Einhorn, 1998a). Hence, it is a suitable model for studying the potential
mechanisms that stimulate bone formation and examination of the role of mechanical forces.
16
Bone fracture healing and computational modeling of bone mechanobiology
2.6 Stem cell and tissue differentiation
In general, stem cells may be characterized as cells which have the capacity for extended selfrenewal, as well as the ability to produce differentiated cells to maintain tissue structure and
renew tissue after damage (Bianco et al., 2001; Triffitt, 2002). Mesenchymal stem cells (MSC)
in adult organisms are known to exist in several locations, including the marrow, periosteum
and muscle-connective tissues, all which are potentially important during bone healing
(Postacchini et al., 1995; Iwaki et al., 1997; Gerstenfeld et al., 2003b). The MSCs have so far
been difficult to recognize and there are still no markers to exclusively identify mesenchymal
stem cells. In a healthy tissue there is little stem-cell activity with stem cells resting in a stable
non-proliferating state and this state being maintained until more cells are required for tissue
regeneration or repair (MacArthur et al., 2004). However, upon large disturbances, or where
additional tissue is required, stem cells are capable of producing tissue progenitor cells, which
then differentiate, whilst at the same time maintaining a stem cell pool.
MSCs can differentiate towards several highly different cell phenotypes, including fibroblasts,
chondrocytes or osteoblasts (Figure 2-6). These differentiated cells begin to synthesize the
extracellular matrix of their corresponding tissues. Several factors influence which lineage
pathways the cell and tissue differentiation will take. These factors include biochemical
signaling molecules (Chapter 2.3.2) and mechanical conditions (Chapter 2.3.1) (Ashhurst,
1986; Sandberg et al., 1993). Bone marrow-derived MSCs play an important role as
progenitors of skeletal tissue components, in skeletal morphogenesis and healing (Baron,
1999; MacArthur et al., 2004). Consequently, the understanding of MSC-derived cell
proliferation and differentiation is of great interest in tissue regeneration, as well as from
clinical and tissue-engineering perspectives (Yang et al., 2001; Rose and Oreffo, 2002;
Cancedda et al., 2003).
Figure 2-6: Mesenchymal lineage pathways displayed by the end-stage phenotypes and the
possible differentiation pathways a MSC can take. The differentiation of a pluripotent MSC
goes through a multi-step series of changes in response to environmental stimuli before the
end-state cell phenotype is reached. The figure was adapted from Caplan and Bruder (2001).
17
Chapter 2
2.7 Mechanobiology
The principle of mechanobiology is that biological cellular processes are regulated by signals,
generated by mechanical loading, a concept dating back to Roux (1881). Mechanobiology
aims to determine how loads are transferred to the tissues, how the cells sense these loads, and
how the signals are translated into the cascade of biochemical reactions that stimulate cell
expression and cell- or tissue differentiation (van der Meulen and Huiskes, 2002).
Computational mechanobiology attempts to determine the quantitative rules that govern the
effects of mechanical loading on tissue differentiation, growth, adaptation and maintenance.
The biological side of the computation is based on the premise that local mechanical variables
stimulate cell expression to regulate matrix composition, density or structure. Modeling
considerations include force application at the boundary, force transmission through the tissue
matrix, mechanosensation and transduction by cells, cell gene expression, and transformation
of extracellular matrix characteristics. All these parts are combined in a computer simulation
model. These processes must be represented by variables, parameters and mathematical
relationships. Some of these are known, or can be measured (e.g. morphology, mechanical
tissue properties, external loading characteristics), whereas others have to be estimated.
2.8 Computational mechanobiological models
Many biological processes, including bone healing, are so complex that physical
experimentation is either too time consuming, too expensive, or impossible. As a result,
mathematical models became increasingly important. Finite element modeling was first
introduced into orthopaedic biomechanics in 1972 to evaluate stresses in human bones
(Brekelmans et al., 1972; Huiskes and Chao, 1983). Since then finite element models have
been used, for example, to design and analyze implants, to obtain fundamental mechanical
knowledge about musculoskeletal structures, and to investigate time-dependent adaptation
processes in tissues (Huiskes and Hollister, 1993; Prendergast, 1997). By combining the
power of computers with the knowledge of mechanobiology, theories have been proposed in
terms of computer algorithms, to explain how the mechanical environment influences tissue
growth, maintenance, remodeling and degeneration. The theories have then been tested using
finite element models. Some of the proposed algorithms regarding tissue differentiation and
bone healing are described below.
2.8.1 Early theories
Pauwels’ theory
In 1960, Pauwels proposed the first rigorous theoretical framework by which the effects of
mechanical forces on tissue differentiation pathways occur through mechanical deformation of
the tissues (Pauwels, 1960). Building on initial work by Roux (1881), Pauwels suggested that
tissues were suited to sustain distinct mechanical stressing. Fibrous tissue forms in regions of
tension, since collagen fibres are highly resistant exclusively to tensile stressing. Cartilaginous
tissue forms fluid-filled spherical structures around chondrocytes, which swell osmotically,
and are suited to support hydrostatic pressure only. Hence, he identified strain and pressure, as
18
Bone fracture healing and computational modeling of bone mechanobiology
two distinct stimuli, stimulating or allowing fibrous tissue and cartilage, respectively. Primary
bone formation requires a stable, low-strain mechanical environment and endochondral bone
formation will proceed only after the soft tissues have stabilized the environment sufficiently
to create this low strain environment (Pauwels, 1960) (Figure 2-7).
Figure 2-7: Pauwels scheme for differentiation of mesenchymal cells into musculoskeletal
tissues, depending on the combination of volumetric and deviatoric deformation components
(Pauwels, 1960). This figure is created based on Pauwels (1960)
The fundamental concept in Pauwels’ theory was that in the case of a healing fracture, it is
impossible for direct bone formation to bridge an unstable gap without being destroyed.
Therefore the purpose of the intermediate tissues is to stabilise and stiffen the fracture callus
and to create a mechanically undisturbed environment where bone can form. Pauwels’ theory
was based on clinical observation and logic, but he did not have the means of measuring or
calculating the tissue strains or stresses in detail.
Interfragmentary strain theory
Perren and Cordey proposed that tissue differentiation is controlled by the resilience of the
callus tissues to strain (Perren, 1979; Perren and Cordey, 1980). Their main idea was that a
tissue that ruptures or fails at a certain strain level cannot be formed in a region experiencing
strains greater than this (Figure 2-8).
Figure 2-8: Perren and Cordey’s ideas were based on how much elongation each tissue type
can tolerate. This figure is created based on Perren and Cordey (1980).
19
Chapter 2
The interfragmentary strain is determined by taking the longitudinal fracture-gap movement
and dividing it by the size of the gap. As a tissue in the fracture gap stiffens, the
interfragmentary strain is reduced allowing healing by progressive tissue-differentiation from
the initial granulation tissue, to fibrous tissue, cartilaginous tissue and finally bony tissue.
However, the hypothesis only considered longitudinal or axial strains; important strain
contributions from radial and circumferential strains were neglected.
2.8.2 Single phase models
Carter’s mechanobiological hypothesis
From the ideas of Pauwels, Carter et al (1988) proposed a model in which local stress or strain
history explained tissue differentiation over time. Later, Carter and colleagues developed their
ideas and proposed a more generalised mechano-transduction model (Carter et al., 1998)
(Figure 2-9). When the tissue is subjected to high tensile strains (above the tension line)
fibrous matrix is produced. Production of cartilaginous matrix is predicted to occur under high
pressure, i.e. to the left of the pressure line, since this tissue can support and resist hydrostatic
pressure. When the hydrostatic pressure is very low, i.e. to the right of this line, formation of
bone occurs. No specific threshold values were specified for tension or pressure lines.
Figure 2-9: Mechanobiological model as proposed by Carter et al., (1998). Two lines
separate the different predicted tissue types, one line based on tensile strain and one line
based on hydrostatic pressure. This figure is created based on Carter et al. (1998).
The studies of Carter et al. were the first to employ finite element analysis to explore
relationships between local stress/strain levels and differentiated tissue types. They modeled
the tissue in the callus as a single solid (linear elastic) phase. The first tissue differentiation
scheme was tested with a model of a developing joint to determine whether this approach
could predict the emergence of secondary ossification centres (Carter and Wong, 1988). Using
the refined model, predictions for endochondral ossification during fracture healing, and
healing around orthopaedic implants were investigated (Giori et al., 1995; Carter et al., 1998;
Carter and Beaupre, 2001). Carter’s studies stressed that a good blood supply is necessary for
bone formation, while a compromised blood supply favours cartilaginous tissue formation.
20
Bone fracture healing and computational modeling of bone mechanobiology
Carter’s mechanobiological model has also been used in other studies. For example,
computational studies of oblique fractures (Blenman et al., 1989), pseudoarthrosis formation
(Loboa et al., 2001), asymmetric clinical fractures (Gardner et al., 2004) and distraction
osteogenesis (Morgan et al., 2006) have been performed based on Carters mechanobiological
model. However, none of the studies predicted tissue differentiation adaptively over time.
Claes and Heigele’s fracture healing model
Claes and associates performed an interdisciplinary study comparing data from animal
experiments, finite element analysis and cell cultures to assess the influence of gap size and
interfragmentary strain on bone healing (Claes et al., 1995; 1997; 1998). Based on histological
observation, Claes and Heigele (1999) formulated a mechano-regulation algorithm, similar to
that of Carter. For the first time, they quantified thresholds for when the various tissues were
to form (Figure 2-10). The finite element analysis performed, as a basis for the threshold
determination, was a solid hyperelastic analysis, performed at a few specific time points
during fracture healing. The comparison of histology with mathematical analyses of stress and
strain allowed attribution of intramembranous bone formations to local strains of less than 5%.
The 5% limit for bone formation was also supported by cell-culture experiments involving
stretching of osteoblasts (Claes et al., 1998). Compressive hydrostatic pressures greater than
-0.15 MPa and strains smaller than 15% appeared to stimulate endochondral ossification, with
all other conditions corresponding to areas of connective fibrous tissue or fibrocartilage. Their
theory was based on observations that bone formation occurs mainly near calcified surfaces.
Figure 2-10: The fracture healing model proposed by Claes and Heigele (1999), including
threshold values for when each tissue type will form. This figure is created based on Claes and
Heigele (1999).
The fracture healing algorithm from Claes and Heigele has also been used by others. Gardner
and Mishra (2003) studied a clinical fracture and found favourable correlations with the
algorithm. Moreover the model has been combined with other rules of bone healing, using an
21
Chapter 2
iterative finite element analysis controlled by a ‘fuzzy logic’ algorithm (Ament and Hofer,
2000). Rules were based on cell culture experiments and histological investigations,
specifically incorporating vascularity to successfully simulate the main patterns of fracture
healing. The combination of Claes and Heigele’s algorithm and ‘fuzzy logic’ rules was also
used by Simon et al. (2004) to investigate differences between shear and axial stimulation, and
by Shefelbine et al. (2005) to study healing of trabecular bone fracture.
2.8.3 Biphasic adaptive models
In recent years biphasic and poroelastic finite element formulations became available for
modelling fluid-saturated solid materials. External loading is resisted by the linear
combination of stress in the solid matrix and pressure of the fluid. The solid matrix deforms
according to its elastic modulus and fluid flows at a rate proportional to the pressure gradient
and the permeability, according to Darcy’s Law. Depending on the theoretical
implementation, the solid and fluid components can be assumed as incompressible, such that
the rate of change of solid volume and fluid volume are equal, or the solid can be
compressible while the fluid is incompressible. This type of formulation leads to timedependent behaviour of the material, as fluid is extruded from and redistributed within the
solid matrix. The poroelastic formulation was originally proposed to model soil mechanics
(Biot, 1941) and the biphasic formulation was proposed by Mow to model cartilage behaviour
(Mow et al., 1980). The theories are slightly different but have been shown similar in
outcomes (Prendergast et al., 1996).
Prendergast and Huiskes
In a biphasic analysis of a tissue differentiation experiment around a spring piston implanted
in the femoral condyles of dogs, it was found that local tissue fluid pressure does not change
as the tissues differentiate (Soballe et al., 1992a; 1992b; Huiskes et al., 1997; Prendergast et
al., 1997). It was also found that the stresses on tissues are not only generated by the tissue
matrix, but also to a large extent by the drag forces from interstitial fluid flow (Huiskes et al.,
1997; Prendergast et al., 1997). This indicated the need for dynamically loaded, biphasic
models, because these effects could not be examined with static or linear elastic
representations. It was concluded that interstitial fluid flow and pressure need to be
investigated as potential signalling variables. Prendergast et al. introduced a model of tissue
differentiation based on a biphasic poroelastic finite element model of the tissues, found
experimentally at a loaded implant interface (1997). They proposed two biophysical stimuli:
solid shear (deviatoric) strain in the solid phase and fluid velocity in the interstitial fluid phase,
where high magnitudes of either, favors fibrous tissue, and only when both stimuli are low
enough, can ossification occur. The spatial and temporal comparison of the fluid velocity and
solid shear strain, with tissue type indicated a pattern of increasing tissue stiffness (maturity)
as these mechanical variables decreased in magnitude (Figure 2-11).
22
Bone fracture healing and computational modeling of bone mechanobiology
Figure 2-11: The tissue differentiation scheme proposed by Prendergast et al. (1997) and
Huiskes et al. (1997). Mesenchymal stem cells differentiate depending on the magnitudes of
fluid velocity and tissue shear strain. Reprinted from Lacroix and Prendergast (2002),
Copyright (2002), with permission from Elsevier.
Based on work by Prendergast et al. (1997), Lacroix et al. applied the same algorithm to
investigate tissue differentiation during fracture healing (Lacroix and Prendergast, 2002;
Lacroix et al., 2002). They used a 2D axisymmetric finite element model with a poroelastic
material description. The dynamic model created by Lacroix was able to simulate direct
periosteal bone formation, endochondral ossification in the external callus, stabilisation when
bridging of the external callus occurs, and resorption of the external callus (Lacroix et al.,
2002). The model was able to predict slower healing with increasing gap size and increased
connective tissue production with increased interfragmentary strain. These studies introduced
some biological representations by prescribing stem cell concentrations initially at the external
boundaries and using a diffusive mechanism to collectively simulate migration, proliferation
and differentiation of cells. Actual tissue differentiation depended on resulting cell
concentration and stimulus.
This model has later been used for successful predictions of tissue differentiation in a rabbit
bone chamber (Geris et al., 2003; 2004), and during osteochondral defect healing (Kelly and
Prendergast, 2005).
2.8.4 Models based on biochemical factors
The mechanoregulatory algorithms discussed in previous sections incorporates the effects of
vascularity and growth factors implicitly. However, since the isolation and clinical use of
growth factors such as TGF-β and BMPs, it has become necessary to incorporate them into
models of bone healing for some research questions.
Framework by Bailon Plaza and van der Meulen
Bailon-Plaza and van der Meulen (2001) developed a mathematical framework to study the
effects of growth factors during fracture healing. They used finite difference methods to
simulate sequential tissue regulation and cellular events, studying the evolution of
chondrocytes and osteoblasts existing in the callus. In their model, cell differentiation was
23
Chapter 2
controlled by the presence of osteogenic and chondrogenic growth factors. The rate of change
of cell density, matrix density and growth factor concentrations, as well as matrix synthesis
and degradation and growth factor diffusion, were included into their model. The model was
further refined in an attempt to include the influences of the mechanical environment (BailonPlaza and van der Meulen, 2003).
This model was recently adopted by Geris et al. (2006a). They developed the model further by
including key aspects of healing, such as angiogenesis, and comparing the results with
experimental data of normal fracture healing (Geris et al., 2006b; Geris, 2007). This study also
evaluated the models’ ability to predict certain pathological cases of fracture healing, and took
a first step towards attempts to test therapeutic strategies.
2.8.5 Other computational models of tissue differentiation
Recently, the focus has been shifted towards incorporating a more accurate description of
cellular processes. Two models with different approaches are described below.
Models of callus growth
Garcia et al. (2006) developed a continuum mathematical model that simulated the process of
tissue regulation and callus growth, taking different cellular events into account. The model
attempts to mimic events such as mesenchymal-cell migration, and mesenchymal stem cell,
chondrocyte, fibroblast, and osteoblast proliferation, differentiation and cell death, and matrix
synthesis, degradation, tissue damage, calcification and remodeling over time. They aimed to
analyze the main components that form the matrix of the different tissues, such as collagen
types, proteoglycans, mineral and water, and used that composition to determine mechanical
properties and permeability of the tissue. They chose the second invariant of the deviatoric
strain tensor as the stimulus guiding the tissue differentiation process.
This model was the first to include tissue growth in adaptive simulation of fracture healing
(Garcia-Aznar et al., 2006). Even though the predicted callus geometries in their growth
model are not completely physiological, it is able to predict increased callus size for increased
interfragmentary movements (Garcia-Aznar et al., 2006), as well as realistic variations when
gap size, and fixator stiffness were varied (Gomez-Benito et al., 2005; 2006).
Stochastic cell modelling
Recently, a study by Perez and Prendergast (2006) developed a new model for cell dispersal in
the callus. A ’random walk’ model was included to represent cell migration both with and
without a preferred direction. The study simulated an implant-bone interface, using the
stochastic cell model and the mechano-regulatory model by Prendergast et al (1997), and
compared the results with those using the diffusion model for cell migration (Lacroix et al.,
2002). The predictions of both models are similar, although the ‘random walk’ model was able
to predict a more irregular tissue distribution than the diffusion model.
24
Bone fracture healing and computational modeling of bone mechanobiology
2.8.6 Summary of computational models of tissue differentiation
The models proposed and the mechanobiological processes included, discussed in the previous
sections, are summarized in Table 2-1.
Mechanical
Bone
regeneration
process
Time point
evaluation
Material
description
Carter et al., 1988
fracture healing
single
linear elastic
Carter and Wong,
1988
joint
development
single
linear elastic
octahedral shear
stress and
dilatational stress
Blenman et al,. 1989 fracture healing
single
linear elastic
octahedral shear
stress and
dilatational stress
Carter et al., 1998
fracture healing
single
linear elastic
principal tensile
strain and
hydrostatic stress
Loboa et al., 2001
oblique fracture
healing
single
linear elastic
principal tensile
strain and
hydrostatic stress
Claes and Heigele,
1999
fracture healing
single
hyper elastic
principal strain and
hydrostatic
pressure
Prendergast et al.,
and Huiskes et al.,
1997
implant
adaptively
osseointegration
poroelastic
shear strain and
fluid flow
Bailon Plaza and
van der Meulen,
2001
fracture healing
adaptively
Bailon Plaza and
van der Meulen,
2003
fracture healing
adaptively
linear elastic
Lacroix and
Prendergast, 2002
fracture healing
adaptively
poroelastic
Geris et al., 2003
bone chamber
adaptively
poroelastic
bone chamber
adaptively
linear elastic
osteochondral
defects
adaptively
Shefelbine et al.,
2005
trabecular bone
healing
adaptively
Geris et al., 2006
fracture healing
adaptively
Garcia et al., 2006
fracture healing
adaptively
Kelly and
Prendergast, 2005
Biophysical
stimuli
octahedral shear
stress and
dilatational stress
Biological
Cell
modeling
Growth
factors
MSC, CC,
OB
osteogenenic,
chondrogenic
deviatoric strain
and dilatational
strain
MSC, CC,
OB
osteogenenic,
chondrogenic
shear strain and
fluid flow
MSC
diffusion
shear strain and
fluid flow
principal strain and
hydrostatic
pressure
MSC
diffusion
poroelastic
shear strain and
fluid flow
MSC
diffusion
linear elastic
ostahedral shear
strain, hydrostatic
strain, fuzzy logic
MSC, CC,
OB
poroelastic
shear strain
invariant
MSC, FB,
CC, OB
Table 2-1: Summary of computational models of tissue differentiation.
25
Tissue
growth
osteogenenic,
chondrogenic
volume
growth
Chapter 2
Over the last two decades, the computational models employed for studies of bone healing
have progressed a great deal. As described in previous sections, it has gone from single phase
linear elastic models which were evaluated at only one time point (Carter et al., 1988; 1998)
via hyperelastic (Claes and Heigele, 1999) to poroelastic material descriptions implemented in
models that adapt tissue distributions over time (Huiskes et al., 1997; Prendergast et al., 1997).
Poroelastic material description is especially important when describing the soft tissues
involved in the early stages of healing, and has become the standard. Unfortunately, the
material properties of these soft tissues are not yet well characterized.
Over the last couple of years, the focus has shifted from pure mechanical analyses, towards
implementing more mechanobiological aspects, initially only including stem cell
concentrations (Lacroix and Prendergast, 2002), as well as solely biological models (BailonPlaza and van der Meulen, 2001) including effects of growth factors and directed cell
movement. The models are becoming more complex as the knowledge about the detailed
processes during bone healing increases. The work in this thesis contributes to the
development in this area. Many of the above discussed models and methods are recent
developments, and were not available when this thesis work was initiated. This development is
still progressing focusing on describing cells and their activities (Garcia et al., 2002; Geris,
2007). In this work, a slightly different approach is taken, which is described in Chapter 6-7 of
this thesis.
26
3
3
Comparison of biophysical
stimuli for mechano-regulation
of tissue differentiation during
fracture healing
The aim of this chapter is to compare the ability of various
mechano-regulation algorithms to predict normal fracture healing in
a computational model. Additionally, the question whether tissue
differentiation during normal fracture healing can be equally well
regulated by individual mechanical stimuli, e.g. deviatoric strain,
pore pressure or fluid velocity is assessed.
It was concluded that all the previously published mechanoregulation algorithms simulated the course of normal fracture
healing correctly, including intramembranous bone formation along
the periosteum and callus tip, endochondral ossification within the
external callus and cortical gap, and creeping substitution of bone
towards the gap from the initial lateral osseous bridge. Furthermore,
simulation as a function of only deviatoric strain accurately
predicted the course of normal fracture healing.
The content of this chapter is based on publication I
Journal of Biomechanics, 2006
27
3
Chapter 3
3.1 Introduction
Most fractures heal through indirect or secondary healing. Indirect fracture healing starts from
the injury-induced haematoma, and involves the sequential differentiation from one
connective tissue type to another (Chapter 2.2). Cartilage forms within the fracture gap, which
is calcified and replaced by immature bone, which is later remodeled into mature bone. This
sequence of tissue differentiation is known to be sensitive to the local mechanical environment
within the tissue. However, the mechano-transduction mechanisms are not well understood.
Many scientists have tried to determine the mechanical and biological parameters influencing
the process of tissue differentiation, either by using experimental or computational models.
Experimentally, the ovine tibia model is a common well characterized model often used in
fracture healing studies. It was shown that the amount of callus formed is related to the
interfragmentary movement in the fracture gap (Goodship and Kenwright, 1985; Claes et al.,
1995). If the interfragmentary movement is too high, the healing process might be delayed or
lead to nonunion (Kenwright and Goodship, 1989). Several mechano-regulation algorithms for
investigating the influence of mechanical stimuli on tissue differentiation during fracture
healing with finite element analysis (FEA) were proposed. They are described in Chapter 2.8.
Three mechano-regulation proposals, although different in theory, have shown consistent with
the actual tissues formed during fracture healing (Carter et al., 1988; Claes and Heigele, 1999;
Prendergast et al., 1997). It has so far been difficult to compare these theories since they were
investigated in FE models with different geometrical and material parameters. The algorithms
of Carter and Claes both predicted changes in tissue phenotype at specific time points, but
their simulations were not continued over the complete healing period. Their material
properties were linear elastic, whereas those used by Lacroix et al. (2002), with the algorithm
by Prendergast et al. (1997), were poroelastic. Until recently, no studies were performed to
compare these mechano-regulation algorithms or to investigate the individual contributions of
the stimuli in the mechano-regulation algorithms. Two of the algorithms’ ability to predict
bone formation inside a rabbit bone chamber was compared (Geris et al., 2003). Although this
study introduced both algorithms in one geometrical model, different material descriptions for
each algorithm were used. The aim of this study was to compare the existing mechanoregulation algorithms with regards to their ability to predict the normal fracture healing
processes. For this purpose they were implemented in the same computational FEA model.
Additionally, we studied the hypothesis that tissue differentiation could equally well be
regulated by the individual mechanical stimuli, e.g. deviatoric strain, pore pressure or fluid
velocity alone.
3.2 Methods
3.2.1 Finite element model
For the computational model, a mechano-regulatory adaptive, axisymmetric finite element
model of an ovine tibia was created. The geometry involved a 3 mm transverse fracture gap
and an external callus (Figure 3-1). The external surface of the callus, the ends of the cortical
28
Comparison of biophysical stimuli for mechano-regulation of tissue differentiation during fracture healing
bone and the intramedullary canal were assumed to be covered by fascia and impermeable
(Einhorn, 1998b). The loads were applied to the cortical bone at the top of the model. The
callus, marrow and cortical bone consisted of 779, 1060 and 540 elements respectively, which
were all 8 noded biquadratic displacement, bilinear pore-pressure elements. The finite element
solver used was ABAQUS (v 6.3).
Figure 3-1: Axisymmetric finite element model of an ovine tibia with a 3 mm fracture gap, and
external callus. The cortical bone was modeled with diameters of 14 mm (inner) and 20 mm
(outer). The callus extended 15 mm along the periosteum with a maximal diameter of 28 mm.
3.2.2 Adaptive tissue differentiation model
The adaptive process of fracture healing was implemented in MATLAB (v 6.5) (Figure 3-2).
The model was implemented as described by Lacroix and Prendergast (2002).
Figure 3-2: Fracture healing in a mechano-regulated, adaptive model simulated in MATLAB.
The iterative procedure starts with a stress analysis in ABAQUS, where the biophysical
stimuli for each theory are calculated. Tissue phenotype was determined for each element, and
tissue properties were smoothened to account for slower changes in phenotype. A rule-ofmixture was used and the material properties were updated, before the next iteration starts.
29
Chapter 3
Initially, the entire callus was assumed to consist of granulation tissue, into which precursor
cells could migrate. The precursor cells originated at the periosteal surface of the bone, from
the soft tissue external to the callus, and from the marrow (Lacroix and Prendergast, 2002;
Gerstenfeld et al., 2003b). Migration and proliferation of the cells were modeled as a
combined diffusive process, un-coupled to the tissue-deformation stress analysis:
d [cells]
= D∇ 2 [cells] ,
dt
(Eq 3-1)
where the change in cell concentration over time (d[cells]/dt) was determined from the
diffusion constant D [m2/day] and the current cell density. The diffusion constant was
optimized so that the entire callus had reached maximal cell density after 16 weeks (Frost,
1989). Neovascularization was assumed to follow cell density patterns such that osteogenic
bone-cell activity was not inhibited. Hence, it was not explicitly modeled.
The cells within an element of callus tissue were enabled to differentiate into fibroblasts,
chondrocytes or osteoblasts and to produce their respective matrices, dependent on the average
mechanical environment of that element for that day. Production of matrix was assumed to be
dependent on cell density and to occur over time. To account for changes in cell phenotype
and matrix production, a rule of mixtures was used to calculate the element material
properties:
En+1 =
1 n
∑ Ei
10 i =n−9
(Eq 3-2)
where Ei is the elastic modulus at iteration i, and En+1 is the temporary, new elastic modulus
before considering cell distribution. The mixture was further based on the cell types stimulated
by the mechanical environment in the previous 10 days, and on the cell density to maximumcell density ratio:
En+1 =
[cell ]max − [cell ]
[cell ]
En+1
EGran +
max
[cell ]
[cell ]max
(Eq 3-3)
where En+1 was the final new elastic modulus, [cell] was the cell density and Egran was the
elastic modulus for granulation tissue. Once the new material properties were determined, the
next iteration began. The simulation ran for 120 iterations, where an iteration represented one
day of healing. Based on the mechanical environment, resorption of bone was also simulated,
by deactivation of the element. Marrow and original cortical bone were not allowed to change.
All tissues were described as linear poroelastic. Both the solid and the fluid constituents were
modeled as compressible with the material properties shown in Table 3-1. Material properties
for granulation tissue are not very well established; the values used were similar to those of
the marrow, but with a lower Young’s modulus of 1 MPa.
30
Comparison of biophysical stimuli for mechano-regulation of tissue differentiation during fracture healing
Cortical
bone
Marrow
Gran.
Tissue
Fibrous
Tissue
Cartilage
Immature
Bone
Mature
Bone
Young’s
modulus (MPa)
15750 a
2
1
2f
10 i
1000
6000 k
Permeability
(m4/Ns)
1E-17 d
1.00E-14
1.00E-14
1E-14 f
5E-15 g
1.00E-13
3.7E-13 l
Poisson’s ratio
0.325 b
0.167
0.167
0.167
0.167 j
0.325
0.325
Solid Bulk
Modulus
17660 a
2300 e
2300
2300
3400 h
17660 a
17660 a
Porosity
0.04 c
0.8
0.8
0.8
0.8 m
0.8
0.8
Table 3-1: Material properties, a Smit et al. (2002); b Cowin (1999); c Schaffler and Burr
(1988); d Johnson et al. (1982); e Anderson (1967); f Hori and Lewis (1982); g Armstrong and
Mow (1982); h Tepic et al. (1983); i Lacroix and Prendergast (2002); j Jurvelin et al. (1997);
k
Claes and Heigele, (1999); l Ochoa and Hillberry (1992); m Mow et al. (1980).
3.2.3 Feedback regulated loading
Two different loading regimes were applied to the model, both as axial ramps (1 Hz)
simulating the loading for one day. The biophysical stimuli were calculated at the peak load.
The first loading pattern peaked at 300 N, which was kept the same over the healing period
and used for validation of the model. The second loading pattern was created to simulate, in
general, in vivo experimental loading regimes with increasing peak loads as healing
progressed (Aranzulla et al., 1998; Duda et al., 1998). The initial magnitude of the load was
100 N, but was then adapted in a biofeedback-loop, as a function of the interfragmentary
movement in the previous iteration. If the interfragmentary movement had decreased, this was
considered a sign of healing and the force was increased. If the interfragmentary movement
had increased since the previous iteration, the tissue was considered unfit for the force applied
and the peak force was decreased. Once the loading peak reached 600 N, the maximal load
applied on a normal sheep tibia (Duda et al., 1998), it was kept constant. The generated peak
loads applied with two algorithms are presented in Figure 3-3.
31
Chapter 3
Figure 3-3: Peak loads applied, determined from the interfragmentary movement according
to the algorithm regulated by deviatoric strain and fluid velocity (Prendergast et al., 1997)
and the algorithm regulated by deviatoric strain alone.
3.2.4 Investigated mechano-regulatory algorithms
Several mechano-regulation theories were explored with this model. First was the algorithm
proposed by Carter et al. (1998) according to which principal tensile strains and hydrostatic
pore pressures were assessed (Figure 2-9), using optimized threshold values. The threshold
values found by Claes and Heigele (1999) were chosen as a starting point. Approximately 10
different combinations were evaluated. The threshold values that predicted the temporal and
spatial tissue distributions best corresponding with normal fracture healing were chosen.
Normal fracture healing was characterized as: 1) initial intramembranous bone formation from
the periosteum and the callus tip; 2) followed by endochondral ossification of the external
callus and 3) bridging across the gap first at the external callus with creeping substitution of
bone towards the cortical gap. The final limits were 0.2 MPa hydrostatic pore pressure and 5%
maximal principal tensile strain, as shown in Table 3-2. The second algorithm explored was
proposed by Claes and Heigele (1999), where intramembranous bone formation was assumed
for strains lower than ±5% and pore pressures smaller than ±0.15 MPa (Figure 2-10).
Endochondral ossification was associated with compressive pressures larger than 0.15 MPa
and strain lower than 15%. All other conditions stimulated fibrous tissue formation. Both the
algorithms of Carter et al. (1998) and Claes and Heigele (1999) were originally explored with
hydrostatic stress or pressure of the solid in a linear elastic analysis. The poroelastic approach
taken in this study instead assumed the solid hydrostatic stress as equivalent to the pore
pressure of the fluid. The hypothesis of Prendergast et al. (1997), assumed the combined
effects of deviatoric strain and fluid velocity to describe the differentiation processes was
implemented as described by Lacroix and Prendergast (2002) (Figure 2-11). Finally, to
determine the contribution of each of the constituents within the mechano-regulation theories,
the effects of deviatoric strain, pore pressure and fluid velocity alone were simulated after first
optimizing the threshold values for these mechano-regulation algorithms, similarly as done for
32
Comparison of biophysical stimuli for mechano-regulation of tissue differentiation during fracture healing
the model of Carter (Table 3-2). Deviatoric strain was calculated based on the principal strains
(ε1, ε2, ε3) according to Eq 3-4. The limits used for the algorithm are displayed in Figure 3-4.
y0 =
1
(ε 1 − ε 2 ) 2 + (ε 2 − ε 3 ) 2 + (ε 3 − ε 1 ) 2
3
(Eq 3-4)
Figure 3-4: The threshold values in the simulations with only deviatoric strain (y0) as the
biofeedback variable.
Carter
Biophysical stimuli
Predicted tissue type
Fibrous tissue
Principal
Hydrostatic Principal
Tensile
stress
Strain
strain
>5%
Cartilage / Endochondral
ossification
Bone / Intramembranous
ossification
Claes and Heigele
≤5%
Hydrostatic
Pore
Pressure
Prendergast
Single parameter studies
Deviatoric
Fluid Flow
Octahedral
Shear Strain
μm/s (FF)
shear strain
% (SS)
i = SS / 3.75 + FF / 3
Fluid
Velocity
Pore
Pressure
≤ 0.2 MPa
> 15 %
>5%
> 0.15 MPa
> -0.15 MPa
< -0.15 MPa
i>3
> 5%
> 5μm/s
> 0.6MPa
> 0.2 MPa
≤ 15 %
> 0.15 MPa
i>1
> 2.5%
> 2.5μm/s
> 0.4MPa
≤ 0.2 MPa
≤5%
< ± 0.15 MPa
> 0.1MPa
Immature Bone
i > 0.267
> 0.05%
> 0.6μm/s
Mature Bone
i > 0.010
> 0.005%
> 0.03μm/s > 0.03MPa
i ≤ 0.010
≤ 0.005%
≤ 0.03μm/s ≤ 0.03MPa
Resorption
Table 3-2: The boundaries of the biophysical stimuli for tissue formation according to the
mechano-regulation algorithms investigated, including the three algorithms with only one
stimulus (Carter et al., 1988; Claes and Heigele, 1999; Prendergast et al., 1997).
33
Chapter 3
3.3 Results
3.3.1 Predicted tissue distributions
Published algorithms
Using the same mechano-regulation algorithm and the load case where the peak load was
300N, the FEA model itself was validated by direct comparison of tissue types and material
properties obtained to those found by Lacroix and Prendergast (2002). All further results
presented were predicted by our adapted load pattern determined from the interfragmentary
movement. The previously published mechano-regulation algorithms (Carter et al., 1988;
Claes and Heigele, 1999; Prendergast et al., 1997) simulated the course of normal fracture
healing. Intramembranous bone formation first occurred along the periosteum and callus tip,
followed by endochondral ossification within the external callus and cortical gap, and finally,
as the external callus stabilized the fracture, bone growth occurred towards the gap from the
initial lateral osseous bridge (Figure 3-5a-c). Resorption of the external callus was only
simulated with the algorithm of Prendergast, as it was suggested by Lacroix and Prendergast
(2002), since the other two algorithms did not specify this aspect of fracture healing.
Single stimuli regulators
Tissue differentiation as a function of only deviatoric strain also correctly simulated the same
normal fracture healing patterns (Figure 3-5d). As with the other theories, bone growth also
occurred in the intramedullary canal, and unlike the others, the bone in the intramedullary
canal was resorbed. Tissue differentiation as a function of only fluid velocity or pore pressure
did not correctly simulate the temporal and spatial distributions of tissue types in the callus.
With fluid velocity as the signal, islands of ossification formed in the external callus, bridging
occurred initially within the intracortical gap. Creeping substitution did not develop from
external to internal, and the final prediction displayed only small amounts of mature bone in
the gap. Using pore pressure as the differentiation signal, initially the entire callus was
stimulated to ossify, but as tissue matured in the callus, higher pressures were observed and
softening of the callus occurred.
34
Comparison of biophysical stimuli for mechano-regulation of tissue differentiation during fracture healing
Figure 3-5: Predicted fracture healing pattern with the biofeedback loading regime, with the
algorithms of a) Carter et al. (1998), b) Claes and Heigele (1999), c) Prendergast et al.
(1997) and d) Deviatoric strain. Tissue type was based on the average element moduli as
determined by mixture theory.
3.3.2 Interfragmentary movement and stiffness
To monitor healing, the decrease in interfragmentary movement was computed, as shown in
Figure 3-6a. Additionally, the interfragmentary stiffness was calculated (Figure 3-6b). The
time to heal was determined as the time after which 90 % of final stiffness was obtained. The
final stiffness for the various algorithms was not the same (Figure 3-6b), which was due to the
pattern and the varying extent of resorption of the external callus. Prior to resorption, all
obtained stiffness curves followed the characteristic S-shape (Richardson et al., 1994). For
each algorithm, healing time was most sensitive to the cell diffusion speed. With the same cell
diffusion rate, healing with the algorithms of Carter and Claes was three times faster than with
the regulation algorithm using deviatoric strain and fluid velocity (Prendergast), and the one
using deviatoric strain alone. The algorithms were all sensitive to the rate of load increase.
When the load increased too high at a too early stage of healing, e.g. before bony bridging, a
non union was predicted. The algorithms by Carter et al. (1998) and Claes and Heigele (1999)
were noticeably more sensitive to fast load increases than the algorithms by Prendergast et al.
(1997) and the algorithm regulated by deviatoric strain. Some instability was detected in two
of the models. The mechano-regulation algorithm of Claes and Heigele produced temporary
softening of the callus at about 13 days, but thereafter went on to normal healing. The model
of Lacroix produced isolated bone bridging across the gap between iterations 20 and 30, prior
to creeping of the bone front from the external callus towards the gap (Figure 3-5c).
35
Chapter 3
a)
b)
Figure 3-6: a) Interfragmentary movement in the fracture gap over time, and b)
interfragmentary stiffness in the fracture gap over time. These parameters were used to
analyze healing time for the various mechano-regulation algorithms.
3.4 Discussion
This study compared previously proposed mechano-regulation algorithms for tissue
differentiation during bone healing (Carter et al., 1998; Claes and Heigele, 1999; Prendergast
et al., 1997) in one and the same computational FEA model. Additionally, we investigated the
individual contributions of deviatoric strain, fluid velocity and pore pressure as stimuli for
tissue differentiation. The model developed was versatile and allowed direct comparison
between the different mechano-regulation algorithms. However, cellular mechanisms were not
included implicitly. It was assumed that the mesenchymal cells would spread homogeneously
throughout the callus during a period of 16 weeks. Cell mitosis, cell removal, cell death and
36
Comparison of biophysical stimuli for mechano-regulation of tissue differentiation during fracture healing
the mediating effects of cytokine and nutrition concentrations were not considered (BailonPlaza and van der Meulen, 2001; 2003). It is known that during the tissue differentiation
process the callus not only changes in stiffness and cell density, but it also tends to change
shape. Such tissue growth was neglected (Garcia et al., 2002). Furthermore,
neovascularization was only implicitly considered (Simon et al., 2002). A natural next step
would be to couple these cellular processes not only to tissue deformation but also to each
other. Finally, it was observed that healing speed was most sensitive to cell diffusion rate. This
diffusion was intended to model combined cell migration and proliferation as well as matrix
production when combined with the smoothing and mixture of material properties over an
element. Due to this combined diffusion model, real synthesis rates, as e.g. in the model of
Bailon-Plaza & van der Meulen were not modeled. Therefore, the results related to the time
course of fracture should not be over-interpreted and “model time” has less physical meaning.
Nevertheless, given that these conditions would be consistent with tissue differentiation
regulated by the mechanical environment and would not change with the regulation algorithm,
comparison between the various biophysical stimuli for fracture healing in this study would
still be valid.
3.4.1 Finite element model validation
The FE model itself was validated by comparing its results to those reported by Lacroix and
Prendergast (2002) with the same peak load and fracture gap size. The temporal and spatial
distribution of the tissue differentiation process was very similar, but not identical. Some
differences that might have lead to small variations were the material properties. In this study
the most recently reported bulk and elastic moduli for bone were used (Smit et al., 2002) and
the Young’s modulus used for the granulation tissue was set higher in order to avoid
numerical problems with the very soft tissue during the first iterations. Furthermore, mesh
refinement was similar, but the element shape was different; where we used 8 node elements
with 9 integration points, Lacroix and Prendergast (2002) used 4 node elements with 9
integration points. The impermeability of the external callus boundary was mainly taken from
experimental observations. Histological analysis of fracture calluses has shown a thin fascia
separating the external parts of the callus and the surrounding tissue (Einhorn, 1998b). The
fascia is believed to be relatively impermeable, but no quantification exists to our knowledge.
To asses the importance of this boundary we performed simulations with a fully permeable
external callus. Generally, increased permeability at the boundary did not affect the temporal
and spatial tissue distributions significantly. Increased permeability decreased the pore
pressure externally and promoted slightly earlier ossification with the algorithms by Carter
(1998) and Claes and Heigele (1999). Further, it increased fluid velocity which impeded
ossification externally with the algorithm by Prendergast et al. (1997). The differences were
minor and mostly observed at the two element lines closest to the external boundary. It did not
change the overall healing pattern considerably.
37
Chapter 3
3.4.2 Healing outcome
Established algorithms
Generally speaking, all of the previously proposed mechano-regulation theories correctly
predicted the spatial and temporal tissue differentiation patterns in normal fracture healing.
The similarities produced by the theories were not surprising, since they are all composed of
one volumetric and one deviatoric component of tissue loading and deformation. The
algorithms proposed by Carter et al. (1998) and Claes and Heigele (1999) are virtually
identical. This was confirmed by the similarity of the threshold values obtained for Carter’s
proposal, and by the similarity in predicted tissue distributions and healing times determined.
The threshold values for the algorithms by Carter et al. (1998) was determined by a ‘best fit’
of the temporal and spatial tissue distributions to normal fracture healing. The sensitivity of
the threshold values was relatively high. If the significant digit was altered, for example by
changing the pressure limit for cartilage to form 0.2 MPa to 0.3 MPa, the temporal tissue
distribution was altered and less endochondral ossification was observed. The tissue
distributions when the pore pressure threshold was varied slightly was similar, as seen by the
comparison between the algorithm by Carter et al. (1998) and the algorithm by Claes and
Heigele (1999), where the pore pressure threshold was 0.15MPa.
Individual stimuli
The simulation of fracture healing solely as a function of deviatoric strain magnitude, i.e.
tissue deformation, however, was surprisingly accurate. Those results were best comparable to
the regulation based on fluid velocity and deviatoric strain (Prendergast et al., 1997) in terms
of tissue distribution and healing times. Their algorithm predicted resorption of the external
callus but not of the mature bone in the intramedullary canal, because fluid velocity was too
high. When only deviatoric strain regulated the tissue differentiation process, resorption of the
internal callus was indeed predicted.
The algorithms regulated by fluid velocity or pore pressure only did not predict the healing
process correctly. Some of their inconsistencies were also seen, to a lesser extent, in the
algorithms regulated partly by these two stimuli. The exclusively fluid velocity regulated
algorithm experienced isolated intracortical bony bridging prior to creeping from the external
bone front, which was also partly seen with the algorithm of Prendergast et al. (1997). The
pore pressure algorithm built up pressure in the external callus, which led to softening of the
tissue, which was also observed to a certain extent with the algorithm by Claes and Heigele
(1999).
3.4.3 Feedback regulated loading
The results presented were based on predictions using the biofeedback regulated loading
regime, because of its better consistency with physiological loading during fracture healing.
However, when these simulation results were compared to those from the 300 N loading
regime some differences were observed. Initially, when the tissue was soft, the biofeedback
load was small as were the calculated stimuli, which led to a faster differentiation from fibrous
tissue to cartilage. When the tissue became stiffer and the load increased the differentiation
38
Comparison of biophysical stimuli for mechano-regulation of tissue differentiation during fracture healing
process was slowed down, until the two loading regimes provoked similar tissue distributions.
When the biofeedback regulated load reached its maximum, the stimuli were higher and the
time for the bone to heal became longer. For instance, the model only regulated by deviatoric
strain displayed external callus resorption for the constant load (300 N), but not for the
biofeedback regulated load (maxima of 600 N) (Figure 3-7). Additionally, all of the
algorithms were sensitive to how the biofeedback loop was created, i.e. how fast the load was
increased. When the load-increase rate was too high, the tissue differentiation process was
interrupted and a non-union predicted. This highlights the need for initial reduced loading over
the fracture gap to ensure complete healing.
Figure 3-7: The two different loading regimes produced similar overall healing patterns, but
in a different time frame. For the case of deviatoric strain, with a) the 300 N load, resorption
of both the external and the internal callus occurred, but with b) the biofeedback load regime
and a maximal peak load of 600 N, only the internal callus was resorbed. Tissue type was
based on the averaged element moduli as determined by mixture theory.
3.5 Conclusions
The previously proposed algorithms were all able to predict the most important aspects of
normal fracture healing. Differences were seen, but the diversities were not extensive and no
algorithm could be rejected or determined as the superior one. Furthermore, deviatoric strain
alone was able to simulate the tissue differentiation process during normal fracture healing
equally well as the previously proposed algorithms. The deviatoric component predicted
proper tissue differentiation while the volumetric components did not, suggesting that the
deviatoric component is the more significant mechanical parameter in the guidance of tissue
differentiation during indirect fracture healing.
39
4
4
Corroboration of mechanoregulatory algorithms:
Comparison with in vivo results
4
In the previous chapter it was determined that several mechanoregulation algorithms can accurately predict temporal and spatial
tissue distributions during normal fracture healing. In an attempt to
separate these algorithms, the study in this chapter aimed to
corroborate the algorithms with more extensive bone healing data
from animal experiments at two time points. An in vivo study where
axial compression or torsional rotation was used as two distinct
mechanical stimulations was adopted.
By applying torsional rotation, the predictions of the algorithms
were distinguished successfully. In torsion, the algorithm regulated
by deviatoric strain and fluid velocity was the only one that
predicted bridging and healing, as observed in vivo. However, none
of the algorithms predicted patterns of healing entirely similar to
those observed experimentally for both loading modes and time
points.
The content of this chapter is based on publication II
Journal of Orthopaedic Research, 2006
41
Chapter 4
4.1 Introduction
The local mechanical environment in a fracture callus, characterized by interfragmentary
movement, modulates the progress of healing (Goodship and Kenwright, 1985; Kenwright et
al., 1991; Goodship et al., 1993; Claes et al., 1995). Despite this knowledge, the mechanism
by which mechanical stimuli regulate differentiation of the tissue is not understood. Several
mechano-regulation algorithms have been proposed to regulate this process during normal
secondary fracture healing (Chapter 3). The theories behind these algorithms are different and
they use distinct mechanical stimuli as regulators. These algorithms, although dissimilar, have
been shown to be consistent with some aspects of normal fracture healing by others, and in the
previous chapter of this thesis. Can all these different stimuli and theories be correct? This
issue needs to be resolved to determine which aspects of the algorithms truly reflect mechanoregulation of tissue differentiation during bone healing.
The close agreements between the ability of the algorithms to predict normal fracture healing,
which was presented in the previous chapter, may have been due to the particular loading
scenario simulated, i.e. axial load controlled compression. Some of the algorithms and their
threshold values were developed from experimental data for healing as an effect of this
loading mode. Carter et al. (1998) developed their semi-quantitative mechano-biological
relationships based on general patterns of fracture healing in humans. The threshold values
(boundaries between tissue types) in the tissue-regulation scheme proposed by Claes and
Heigele (1999) were determined to resemble an ovine fracture healing experiment only
allowing axial stimulation (Claes et al., 1995). In contrast, the algorithm used by Lacroix and
Prendergast (2002) was initially proposed by Prendergast et al. (1997) and calibrated from
tissue differentiation patterns observed around loaded bone implants (Huiskes et al., 1997).
They then used the same algorithm and identical threshold values to successfully predict tissue
differentiation during fracture healing under axial load-control (Lacroix and Prendergast,
2002). To further validate such mechano-regulation algorithms, their ability to predict tissue
differentiation under mechanical conditions other than axial load-controlled stimulation needs
to be evaluated by direct comparison with well-controlled experimental data.
The purpose of the study in this chapter was to corroborate each of the models by comparing
their predictions with in vivo data for interfragmentary conditions, different from those for
which they were developed, i.e. both axial compression and torsional rotation, as two separate
load cases. By analyzing the corroborations we further aimed to determine which mechanoregulation algorithm best resembles the experimental data. For that purpose, a threedimensional finite element model was required. The mechano-regulation algorithms
investigated were those by Carter et al. (1998), Claes and Heigele (1999), Prendergast et al.
(1997) and an algorithm regulated by deviatoric strain alone (Chapter 3). To study mechanoregulation of tissue differentiation, the mechanical environment needs to be well controlled. In
a recent study, Bishop et al. (2006) characterized in vivo bone healing with well-defined,
contrasting mechanical stimulation. They applied pure interfragmentary torsional shear across
a transverse osteotomy in sheep tibiae, and examined its effect on tissue differentiation during
fracture healing in comparison with axial compression. These two conditions develop
42
Corroboration of mechano-regulatory algorithms: Comparison with in vivo results
contrasting local mechanical conditions, whereby torsional shear isolates deviatoric
deformation by elimination of the local volumetric component, while axial compression
results in both volumetric and deviatoric deformation components. Interfragmentary strain was
applied daily with a load limit using an external fixator (Bishop et al., 2003) and external
loading was minimized by Achilles tenotomy (Bishop et al., 2006). Observational time points
were chosen to investigate both early (4 weeks) and late (8 weeks) stages of healing.
Histological assessment and weekly intermediate radiographs were available for the
comparison described in this article.
4.2 Methods
4.2.1 Finite element model
A three-dimensional mechano-regulated adaptive finite element model of an ovine tibia with a
healing transverse fracture gap of 2.4 mm and an external callus was developed based on the
two-dimensional model described in Chapter 3, with geometry and boundary conditions
according to those described experimentally (Bishop et al., 2006) (Figure 4-1). To increase
computational efficiency, a 22.5º wedge was modeled, with proper constraints to impose
rotational symmetry. The external surface of the callus, the ends of the cortical bone and the
intramedullary canal were assumed to be impermeable. The displacements applied were 0.6
mm (0.5 Hz) of axial compression, with a 360 N load limit, or 7.2º (0.5 Hz) of torsional
rotation, with a 1670 Nmm load limit, all equal to experimentally applied stimulation. The
reaction force was monitored and when the load limit was reached, the displacement was
truncated to allow the peak strain to decrease as healing progressed, resembling the
experimental set up. Poroelastic elements were used, with 20 nodes, triquadratic displacement
interpolation and trilinear pore pressure (ABAQUS, v6.4).
Figure 4-1: Three-dimensional finite element model of an ovine tibia with a 2.4 mm fracture
gap, and external callus. Cortical bone: inner diameter 14 mm, outer diameter 20 mm.
External callus: 15 mm along the periosteum, maximal diameter at fracture site 28 mm.
43
Chapter 4
4.2.2 Adaptive tissue differentiation model
The adaptive process of fracture healing was implemented in custom subroutines (MATLAB v
6.5) (Figure 4-2). Fracture healing was simulated by using the biophysical stimuli calculated
from the finite element analysis, at maximum displacement or when the load limit was
reached, to predict new element material properties, according to the mechano-regulation
rules. As in previous chapter, the callus initially consisted of granulation tissue into which
precursor cells could migrate/proliferate according to a diffusive process from the soft tissue
external to the callus, the periosteum and the marrow (Lacroix et al., 2002; Gerstenfeld et al.,
2003b). The cells within an element of callus were able to differentiate into fibroblasts,
chondrocytes or osteoblasts, based on threshold values in each regulation scheme, and to
produce their respective matrices. A rule of mixtures was used to calculate element material
properties based on the stimulated cell phenotype in the previous ten days, and on cell density
(Lacroix et al., 2002). Once the new material properties were determined, the next iteration
began. The simulations ran until a steady state tissue distribution was reached. Resorption of
bone was also simulated, based on the mechanical environment, by deactivation of the
element. Material properties for marrow and original cortical bone elements were not varied.
All materials were described as linear poroelastic, with properties taken from literature (Table
3-1, page 31).
Figure 4-2: Fracture healing in a mechano-regulated adaptive model in MATLAB. The
iterative procedure starts with a stress analysis in ABAQUS, where the biophysical stimuli for
each theory are calculated. The tissue phenotype is determined for each element, and the
tissue properties are smoothed to account for slower changes of phenotype. A rule of mixtures
is used and the material properties are updated before the next iteration begins.
44
Corroboration of mechano-regulatory algorithms: Comparison with in vivo results
4.2.3 Simulations
First of all it was confirmed that the applied mechanical conditions resulted in the preferred
contrasting local mechanical conditions. Maximal stimulation under both axial compression
and torsional rotation was applied, and the deviatoric and volumetric components observed.
Thereafter, all the mechano-regulation algorithms that were successful in the study presented
in Chapter 3 were investigated. The algorithms were those proposed by Carter et al. (1998)
(Figure 2-9), Claes and Heigele (1999) (Figure 2-10), Prendergast et al. (1997) (Figure 2-11),
as well as the algorithm regulated by deviatoric strain alone (Figure 3-4) which was presented
in previous chapter. All algorithms were implemented using identical threshold values as in
the previous study (Table 4-1). The progressive tissue patterns from the computational
predictions were compared to experimental tissue distributions from histological analyses at 4
and 8 weeks and by comparison to weekly radiographs. The comparison was made by
identifying similarities in the sequential development of tissues instead of focusing on specific
iteration numbers (i.e., time points).
Carter
Biophysical stimuli
Predicted tissue type
Fibrous tissue
Cartilage or Endochondral
ossification
Bone or Intramembranous
ossification
Immature Bone
Claes and Heigele
Principal
Hydrostatic
Tensile strain
stress
>5%
≤5%
Principal
Strain
Hydrostatic
Pore Pressure
≤ 0.2 MPa
> 15 %
>5%
> 0.2 MPa
≤ 15 %
> 0.15 MPa
> -0.15 MPa
< -0.15 MPa
> 0.15 MPa
≤ 0.2 MPa
≤5%
< ± 0.15 MPa
Mature Bone
Resorption
Prendergast
Deviatoric Shear Fluid Flow
Strain % (SS)
μm/s (FF)
i = SS / 3.75 + FF / 3
Octahedral
shear strain
i>3
>5%
i>1
> 2.5 %
i > 0.267
> 0.05 %
i > 0.010
> 0.005 %
i ≤ 0.010
≤ 0.005 %
Table 4-1: The threshold values of the biophysical stimuli for tissue formation according to
the various mechano-regulation algorithms investigated. Including the algorithm regulated
only by deviatoric strain.
4.2.4 In vivo fracture healing model
The complete results from the experimental study were presented by Bishop et al. (2006) and
are briefly summarized here. The histological analysis at 4 weeks (Figure 4-3a-b), showed no
substantial general differences between axial and torsional loading. There was new woven
bone formation in the external callus at some distance from the gap. The gap was filled mainly
with fibrous connective tissue and small islands of cartilaginous tissue. There was no bridging
of the external callus or within the gap. However, the data showed a trend towards more bone
formation closer to and inside the intercortical gap with torsional stimulation. At 8 weeks
(Figure 4-3c-d) the differences between the effects of the two loading modes were more
distinct. For axial loading, the range between biological responses was large. There was some
bony bridging (2/5 animals), limited to the periphery of the external callus and no creeping
substitution towards the gap had yet occurred. The intercortical gap was still filled with mainly
45
Chapter 4
soft tissue, rich in proteoglycans. Two animals also showed signs of delayed union, with only
little bone formation after 4 weeks. In torsional stimulation, bridging (4/5 animals) was more
advanced and newly formed high-density bone was found closer to the gap. There was also
more bone formation within the intercortical gap, although bony bridging was mainly found
externally. Additional examination of weekly radiographs confirmed contrasting healing
sequences between the groups. In the axial group, initial periosteal bone formation was seen
further away from the gap, followed by growth of bony cuffs, which bridged externally. In
torsion, periosteal bone growth started outside the gap, very near to the cortical corners next to
the gap. The periosteal callus developed slightly before bone formation started to creep around
the corners into the intercortical gap. There was a significant amount of bone formed within
the gap, before the bone bridged externally, followed by intercortical gap bridging.
a)
b)
c)
d)
Figure 4-3: Histological slides used for comparison (Bishop et al., 2006). After 4 weeks
specimens were stained with light green and toluidine blue; a) axial compression and b)
torsional stimulation. After 8 weeks specimens were stained with eosin and toluidine blue; c)
axial compression and d) torsional stimulation.
4.3 Results
The three-dimensional computational model was validated against our earlier two-dimensional
axisymmetric model (Chapter 3) by comparing the computed mechanical stimuli values and
the calculated reaction forces in axial compression. The mechanical stimuli agreed entirely,
while the reaction forces were slightly lower (< 1%) for the three-dimensional model.
The computational analysis of the initial stage revealed that the volumetric deformation was at
least two orders of magnitude lower for torsional displacement than for axial compression
over the entire callus. With torsional rotation the pressure was minimal throughout the callus
(Figure 4-4a). Under axial compression, the pressure distribution peaked within the gap area,
and decreased towards the medullary cavity and periosteal boundaries (Figure 4-4c). The
deviatoric deformation also displayed differences between axial compression and torsional
rotation. Under torsion, maximum principal strain increased radially to the outer cortical
radius and was constant over the gap height (Figure 4-4b). With axial compression, the strain
was found to peak intracortically and was especially high at the outer corners of the cut cortex
and was greater in these regions than for torsional rotation (Figure 4-4d).
46
Corroboration of mechano-regulatory algorithms: Comparison with in vivo results
Figure 4-4: a) Volumetric deformation under torsional displacement represented by the pore
pressure. b) Deviatoric deformation under torsional rotation represented by the maximum
principal strain. c) Volumetric deformation under axial compression represented by the pore
pressure. d) The deviatoric deformation under axial compression represented by the maximal
principal strain. Note that the scaling is different in figures a) and c), but not in b) and d).
4.3.1 Axial compression
Both algorithms regulated by strain and pressure (Carter et al., 1998; Claes and Heigele, 1999)
produced largely similar tissue distributions. In both cases early bone formation along the
entire external callus boundary resulted in immediate bone bridging (Figure 4-5a-b, iteration
(it) 5). This was not in agreement with histology after 4 weeks, where bony cuffs were
observed prior to bridging. In the computational simulations with Carter’s algorithm (Figure
4-5a) endochondral ossification of the external callus and maturation of the bone followed
(Figure 4-5a, it 10-20), which resulted in tissue distributions similar to those observed in some
of the animals after 8 weeks (Figure 4-5a, it 30). In contrast, Claes’s scheme led to unstable
tissue predictions and temporarily isolated bone formation, prior to final bridging (Figure
4-5b, it 20-50). With both algorithms, the tissue within the gap differentiated from fibrous
tissue to cartilage. Creeping substitution of bone and ossification of the gap was interrupted,
and cartilaginous tissue remained in the gap (Figure 4-5b, it 120), stimulated by high
pressures; no final healing was predicted. Figure 4-6a displays the magnitude of the
hydrostatic pressure in the callus. Although complete healing was not seen in vivo either, it is
believed that once a fracture has bridged with bone, it becomes stable enough for complete
healing (Perren and Claes, 2000).
47
Chapter 4
Figure 4-5: Predicted fracture healing pattern with axial compression, with the algorithms of
a) Carter et al. (1998), b) Claes and Heigele (1999), c) Prendergast et al. (1997) and d)
Deviatoric strain. Tissue type is based on the element moduli determined by mixture theory.
The algorithm regulated by deviatoric strain and fluid velocity (Prendergast et al., 1997)
predicted initial intramembranous bone formation at the tip of the external callus and along the
periosteal surface, growing into a bony external callus by endochondral ossification (Figure
4-5c, it 5-20). This agreed well with observations made at the 4th week of histology. No
further bone growth or bridging was predicted. Soft tissue remained in the gap, which was
related to high fluid velocities, and the prediction resulted in a steady state non-union (Figure
4-5c, it 30-80). The variability of experimentally observed healing under axial compression
was large; some animals experienced delayed union. The predictions by this algorithm could
therefore be in partial agreement with the biological response observed. However, a steady
state non-union as predicted by this algorithm was not generally observed experimentally.
Therefore, there was at best only partial agreement with the eventual experimental
development.
48
Corroboration of mechano-regulatory algorithms: Comparison with in vivo results
Figure 4-6: The algorithms proposed by Carter et al. (1998) and Claes and Heigele (1999)
did not predict complete healing. The figure shows the magnitude of the regulating stimuli in
the two loading cases; a) with axial compression, high volumetric deformation (hydrostatic
pressure) impeded creeping substitution of bone and complete healing, b) with torsional
rotation, the magnitudes of principal strain were too high for further bone formation and
healing to occur.
The algorithm based on deviatoric strain alone predicted full bony healing (Figure 4-5d). In
comparison with histological tissue distributions there were disagreements. The algorithm
predicted external bridging immediately (Figure 4-5d, it 10), which disagreed with
experimental findings after 4 weeks, where an external bony callus was found prior to
bridging. This was followed quickly by endochondral ossification of the external callus with
creeping substitution towards the gap (it 20-30), and healing (it 50). This corresponded better
to histological findings at 8 weeks (it 20), but was still accelerated compared to experiments.
The gap was predicted to ossify completely; bone formed in the intramedullary canal, and
some of the external callus was resorbed (it 120).
4.3.2 Torsional rotation
The mechano-regulation algorithms by Carter et al. (1998) and Claes and Heigele (1999)
simulated identical tissue distributions with torsional loading (Figure 4-7a). With minimal
local volumetric deformation, these algorithms were unable to fully predict the in vivo tissue
distributions observed at neither 4 nor 8 weeks. They predicted intramembranous bone
formation externally (Figure 4-7a, it 5), which resulted in a bony cuff (it 10-20). Also, isolated
bone formation in the canal (it 20), which originated from the marrow, was predicted, but no
bone formation within the intercortical gap was observed (Figure 4-7a). According to these
algorithms local volumetric deformation is necessary to predict tissue differentiation into
cartilage. Thus, no endochondral ossification could have subsequently occurred. Since the
volumetric deformation component is eliminated under torsional rotation, all bone formation
predicted was intramembranous. The rest of the callus tissue, where strains exceeded the
49
Chapter 4
thresholds to predict bone formation, remained as fibrous tissue, without providing any
stabilization of the gap, and the load limit was never attained. Figure 4-6b displays the
magnitudes of the principal strain in the gap.
Figure 4-7: Predicted fracture healing pattern with torsional rotation, with the algorithms of
a) Carter et al. (1998) and Claes and Heigele (1999), b) Prendergast et al. (1997) and c)
Deviatoric strain. Tissue type is based on the element moduli determined by mixture theory.
The algorithm regulated by deviatoric strain and fluid velocity (Prendergast et al., 1997) was
generally very successful in predicting the experimental results (Figure 4-7b). Initially,
predicted healing was accelerated compared to the experiments, and immature bony bridging
occurred earlier (Figure 4-7b, it 5-10). Thereafter, maturation of bone, and further bone
growth, progressed similar to the 4 week histological and radiographic observations (it 20-30).
Bone formation was predicted adjacent to the original cortical bone and some mature bone
formation was observed in the cortical gap, while the rest of the tissue in the gap remained soft
(it 30-50). The first bony bridges occurred externally, with similar tissue distributions as
observed in the 8 week data (it 70). Thereafter, creeping substitution of bone was predicted
and the entire callus filled with mature bone (it 100). After final healing, resorption of the
internal callus was observed in the model. This event was not assessed in the in vivo study,
due to the experimental time line, but would likely have occurred at a later stage.
50
Corroboration of mechano-regulatory algorithms: Comparison with in vivo results
The algorithm regulated by deviatoric strain only was unable to simulate healing (Figure
4-7c). Initially, some bone formation was observed similar to early time points in radiographs
(Figure 4-7c, it 5-10). The tissue distributions that followed led to extreme local strain
magnitudes in the remaining soft tissue, which would have prevented healing. It also led to
numerical failure of the finite element simulation. A summary of the comparisons between
experimental and modeling data is provided in Table 4-2.
Carter et al.
Principal strain &
hydrostatic stress
Claes and Heigele
Strain & hydrostatic
pressure
Prendergast et al.
Deviatoric strain & fluid
velocity
Deviatoric strain
Axial 4 weeks
Axial 8 weeks
Torsion 4 weeks
Torsion 8 weeks
No,
external bridging too early
Reasonable,
but inhibited final healing
due to high pressure
No,
strain limits too low
No,
failed to bridge
No,
bridging too early, unstable
tissue predictions
Reasonable,
but inhibited final healing
due to high pressure
No,
strain limits too low
No,
failed to bridge
Yes,
bony external callus prior to
bridging
No,
failed to bridge due to high
fluid velocity
Yes,
mature bone formation
in gap prior to bridging
Yes,
including bone
formation in gap area
No,
external bridging too early
Reasonable,
but too quick
No,
threshold values not
transferable to torsion
No,
extreme strain
magnitudes
Table 4-2: Summary of comparison between modeling and experimental data. “Yes”
corresponds with agreement and “No” when deviations were found.
4.4 Discussion
Several algorithms have been proposed to describe mechano-regulation of tissue
differentiation during secondary fracture healing. In the study presented in Chapter 3, they
showed similar abilities in predicting normal fracture healing under axial load. However, in
this study their predictions were separated by comparing them to in vivo healing under more
diverse mechanical conditions, i.e. axial compression and torsional rotation.
Tissue differentiation during fracture healing, as predicted by mechano-regulation based on
deviatoric strain alone, was not confirmed by experimental observations. Under axial loading,
healing occurred too fast. The algorithms regulated by both a deviatoric and a volumetric
deformation component in axial compression correctly simulated some features of early
(Prendergast et al., 1997) or intermediate healing (Carter et al., 1998; Claes and Heigele,
1999), but none of them predicted final healing (Figure 4-5). The threshold values for the
volumetric deformation stimulus previously found to predict normal fracture healing were not
transferable to new mechanical conditions. These inconsistencies might be resolved by
establishing new volumetric threshold values. This was not done, since the goal was to test the
algorithms’ abilities in predicting healing as observed in vivo in their established formats,
thereby avoiding subsequent adaptation of threshold values to each mechanically different
situation.
51
Chapter 4
The application of contrasting axial and torsion interfragmentary loading conditions resulted
in greater contrast between the results obtained from the different algorithms than simulating
axial compression only. With torsional rotation providing the mechanical stimuli, only the
algorithm by Prendergast et al. (1997) was able to correctly simulate full bridging and healing
as seen experimentally. In retrospect, the fact that this algorithm performed better under
torsional rotation was not very surprising. The algorithm was originally developed to describe
tissue differentiation at implant interfaces (Huiskes et al., 1997; Prendergast et al., 1997). It
has also been shown to successfully simulate various aspects of fracture healing (Lacroix and
Prendergast, 2002), tissue differentiation in osteochondral defects (Kelly and Prendergast,
2005) and bone formation in bone chambers (Geris et al., 2004) with the same threshold
values. Although this algorithm was successful for torsion, it did not fully predict bridging and
final healing for axial compression.
4.4.1 Experimental model
A combination of a well-controlled mechanical environment and physiologically comparable
conditions is difficult to achieve experimentally. Although the in vivo data used in this study
was the result of well-defined mechanical conditions, the manner in which the loads were
applied makes direct correlation to clinical fracture healing inappropriate, i.e. pure
interfragmentary loading mode such as only axial compression or only torsion are not
generally expected clinically. However, for studies of mechano-regulation, precise stimulation
was considered more relevant, and the specified experimental data was used. Still, there were
limitations associated with the in vivo study. Because of limited significant differences in hard
callus morphometry and mechanical characteristics, due to unexpected high intra-group
variability, no histomorphometric parameters were quantified. Although this made
comparisons in the healing response between loading groups difficult, there were still
substantial mean differences or similarities between groups. These results are those to which
the mechano-regulation simulations were corroborated, e.g.. intra-gap and periosteal callus
bone formation under torsion, and which were able to distinguish between the investigated
regulation algorithms.
4.4.2 Assumptions
With respect to the computational model, several steps were taken to verify its
appropriateness. Experimentally, 120 cycles per day were applied, which were mimicked with
one cycle per day, sampling stimuli at maximal displacement or when the load limit was
reached. The stress relaxation effect of applying consecutive cycles to the model was
negligible. The reaction force was monitored when the displacement was applied, and the load
limit was assumed to be achieved when the reaction force was within 1% of the defined limit.
Cell migration/proliferation and matrix production were modeled mechanistically, without
incorporating real matrix synthesis rates. Hence, modeled time was only approximate, and
comparisons with experimental results were made by identifying similar sequential
transformations of tissues instead of focusing on specific iteration numbers.
52
Corroboration of mechano-regulatory algorithms: Comparison with in vivo results
The external callus boundary was assumed as impermeable (Figure 4-1), based on
experimental observations. Histological analyses of fracture calluses showed a thin fascia
separating the external parts of the callus and the surrounding tissue (Einhorn, 1998b). The
fascia is believed to be relatively impermeable, although no confirmation of this exists, to our
knowledge. The significance of assuming an impermeable boundary was addressed by
investigating the effect of a fully permeable external boundary. This affected the local
volumetric deformation under axial compression, but not in torsion. With the algorithms of
Carter et al. (1998) and Claes and Heigele (1999), decreased pore pressure was found close to
the boundary where it was already sufficiently low, stimulating ossification. Hence, the
predicted tissue distributions did not change. With the algorithm of Prendergast et al. (1997)
increasing fluid velocity was observed mostly in the external callus close to the boundary,
where the fluid velocity was already high. The bony cuff (Figure 4-5c) became slightly
smaller and developed close to the cortical bone. Thus, external callus permeability did not
significantly affect the simulated tissue distributions.
The callus size in the model was kept constant, while in reality the callus develops over time
to increase stability and allow bony bridging. By increasing the callus diameter it was found
that, in axial compression, increased callus volumes decreased pore pressure and fluid
velocity, which promoted differentiation and enhanced the possibility of bony bridging and
complete healing. In torsion, increased callus sizes decreased local strains, which enhanced
differentiation and further bone formation. However, with the algorithms of Carter et al.
(1998) and Claes and Heigele (1999), in torsion, the callus diameter had to increase by 200%
for the predicted tissue distributions to change significantly and for bony bridging to be
correctly predicted. Thus, an increase in callus size, necessary to modify the outcome of the
study, was unrealistic.
4.5 Conclusions
This study examined the validity of four mechano-regulation algorithms. Their capacities to
predict tissue differentiation as observed in vivo, for both axial displacement and torsional
rotation, were evaluated. The algorithms had earlier produced similar results for axial loads,
consistent with normal fracture healing as shown in previous chapter. For both axial
compression and torsional rotation, however, none of the algorithms were completely correct;
therefore no full corroboration was possible. Nevertheless, the algorithm regulated by
deviatoric strain and fluid velocity was the most accurate in this study, and alone able to
predict healing as observed in vivo for torsional rotation.
53
5
5
Bone regeneration during
distraction osteogenesis:
Mechano-regulation by shear
strain and fluid velocity
5
In this study, the most promising algorithm from the study in Chapter
4 was applied to distraction osteogenesis. In vivo data, with a well
controlled mechanical environment and repeatable outcomes were
used for validation. It was hypothesized that mechano-regulation by
octahedral shear strain and fluid velocity could predict spatial and
temporal tissue distributions observed during experimental
distraction osteogenesis. Variations in predicted tissue distributions
due to alterations in distraction rate and frequency were studied.
The predicted temporal and spatial tissue distributions agreed well
with experimental observations. It was observed that decreased
distraction rate increased the overall time necessary for complete
bone regeneration, while increased distraction frequency stimulated
faster bone regeneration, similar to experimental findings by others.
The content of this chapter is based on publication III
Journal of Biomechanics, 2007
55
Chapter 5
5.1 Introduction
In osteogenesis, the differentiation of precursor cells is sensitive to the local mechanical
environment. There have been several propositions of how this relationship is regulated.
However, as was shown in the previous chapter, corroboration of these algorithms is difficult
(Chapter 4), particularly because repeatable experimental outcomes under controlled
mechanical environments are required, but rarely available in laboratory or clinical studies.
In distraction osteogenesis (DO) (Chapter 2.5), a controlled displacement of a bone fragment
is used to generate new bone (Ilizarov, 1989a; Richards et al., 1998). The outcome is
predictable and reproducible. Therefore, it is a suitable model for studying the potential
mechanisms underlying the stimulation of bone formation and investigation of the role of
mechanical loading. DO is usually separated into three phases. The first is the latency phase,
immediately prior to distraction. The second is the distraction phase in which the bony
segments are actively separated at a particular rate (total distance per day), over a particular
time period and at a particular frequency (number of distractions per day). During this phase,
tissue differentiation is initiated, with some sparse bone formation. In the third ‘consolidation’
phase there is no further distraction, and bony union is achieved. Within limits, the rate of
bone formation during DO has been directly related to the distraction rate (Ilizarov, 1989b; Li
et al., 1999; 2000), and frequency (Aarnes et al., 2002). The bone formation rate has been
directly related to (Ilizarov, 1989b; Mizuta et al., 2003) the strain/stress generated in the gap
tissue (Li et al., 1997; 1999) and the phenotypic differentiation of the cells within the
distraction gap has been related to the interfragmentary tension (Meyer et al., 2001a).
Although DO provides an attractive paradigm for the study of mechanical effects on bone
regeneration, very little computational evaluation has been performed. Morgan et al. (2006)
investigated the local physical environment within an osteotomy gap during long bone DO and
correlated tissue dilatation (volumetric strain) with differentiation of mesenchymal tissue.
They successfully evaluated distraction and tissue relaxation during one single day of the
distraction period. Loboa et al. (2005) used FEA to correlate bone formation with magnitudes
of tensile strain and hydrostatic pressure (Carter et al., 1998) during mandibular DO at four
time points. So far no studies have described the process of tissue differentiation during DO
both spatially and temporally during the complete distraction process. This type of
computational evaluation of DO could indicate local stress and strain magnitudes necessary
for optimal bone formation.
A mechano-regulation algorithm based on octahedral shear strain and fluid velocity was
proposed by Prendergast et al. (1997) to stimulate the generation of specific mesenchymal
tissues. The threshold values for this algorithm were initially determined from experimental
models of bone formation around implants (Huiskes et al., 1997). The same scheme has been
used to predict tissue differentiation during secondary fracture healing (Chapter 3) (Lacroix
and Prendergast, 2002; Lacroix et al., 2002), in bone chambers (Geris et al., 2003; 2004) and
during osteochondral defect repair (Kelly and Prendergast, 2005). In the preceding chapter, it
was demonstrated that the scheme is nearly consistent with bone healing under both shear and
56
Bone regeneration during distraction osteogenesis: Mechano-regulation by shear strain and fluid velocity
compressive deformations (Chapter 4) and that other algorithms are less consistent (Carter et
al., 1998; Claes and Heigele, 1999).
During distraction osteogenesis the tissue is subjected to tension, a mechanical mode for
which this algorithm has not yet been tested. This study tests the hypothesis that mechanoregulation by octahedral shear strain and fluid velocity, using the same thresholds as previous
studies (Huiskes et al., 1997), can predict spatial and temporal tissue distributions observed
experimentally during DO, including variations due to alteration in distraction rate and
frequency.
5.2 Method
5.2.1 Experimental model
Data from an ovine in vivo experiment for evaluation of bone segment transport over an
intramedullary nail, previously conducted in our institution, was used for comparison with a
computational model (Brunner et al., 1993; 1994). A distal diaphyseal defect, of either 20 or
45 mm, was created in the left tibia of 6 sheep in each group. The tibia was then stabilized
with a 7mm diameter unreamed static interlocking nail. After corticotomy, bone segments
were transported (distracted) using subcutaneous traction wires over the nail (Figure 5-1a).
Each full rotation of the screw represented a lengthening of 1 mm, and pulling was achieved
using small external devices (Brunner et al., 1993; 1994). Distraction started on post-operative
day 1 at a rate of 1 mm/day, and was conducted at one time point, until the defect was closed,
followed by consolidation. Animals were sacrificed after 12 weeks for the short defects and 16
weeks for the long defects. Daily distraction forces were measured before (resting force),
during (peak force) and 5 min after distraction. The resting force was the tension between the
distracted segment and the fixator before distraction. The peak force was the force between the
fixator and distracted segment after 1 mm of distraction. The relaxation behavior was
characterized by the difference between resting force and peak force, divided by the peak
force, and was used to represent the viscoelasticity of the tissue. Weekly standardized
radiographs and undecalcified histology at the time of completed transport were available.
5.2.2 Finite element model
A 2D axisymmetric FE mesh was created based on the geometry of the tibia, the nail and the
callus from the experimental data (Brunner et al., 1993; 1994) (Figure 5-1b). The initial
corticotomy gap was set to 1 mm. Boundary conditions were applied according to the
experimental model. The ends of bone and marrow and the external callus boundary were
assumed to be impermeable. Distraction (1 mm/day for 20 or 45 days) was applied as
displacement to the end of the cortical bone and started on post-operative day 1. Distraction
was followed by consolidation, where no active mechanical stimulation was applied,
according to the experimental protocol. One iteration simulated one day and included
distraction performed over 1 sec followed by 24 hours of relaxation, during which reaction
forces were monitored. All tissues were assumed to follow linear poroelasticity theory with
properties taken from literature (Table 3-1, page 31). The intramedullary nail was assumed to
57
Chapter 5
be rigid compared to the biological tissues and the interfaces between the nail and the tissues
were modeled using finite sliding and zero friction (ABAQUS, v 6.5).
Figure 5-1: Experimental and computational model. a) The experimental model from Brunner
et al (Brunner et al., 1993) including initial defect and corticotomy, followed by a distraction
phase (bone segment transport) and final consolidation period. b) The initial two-dimensional
axisymmetric finite element model was created from the experimental measurements. The
initial gap size was 1 mm. The nail diameter was 7 mm, the cortical bone’s inner diameter 14
mm, and outer diameter 20 mm.
5.2.3 Adaptive tissue differentiation model
The adaptive tissue differentiation process was implemented using custom-written subroutines
(MATLAB, v 7.1) (Figure 5-2). The meshing of the callus was performed by automatic
meshing into triangular elements, which were transformed into quadrilateral elements with a
maximum area of 0.1mm2 for the FE analysis (Brokken, 1999). The initial corticotomy
material consisted of granulation tissue, without any precursor cells. The precursor cells
migrated into the callus from the boundaries between the callus, marrow and periosteum, with
unlimited supply. A diffusive process was implemented to model migration and proliferation
of cells (Lacroix et al., 2002). Distraction was applied and the biophysical stimuli were
calculated in the FE analysis at maximal distraction. The new tissue phenotypes were
predicted according to the local magnitudes of octahedral shear strain and fluid velocity
(Prendergast et al., 1997). The cells within an element of callus tissue were able to
differentiate into fibroblasts, chondrocytes or osteoblasts and to produce their respective
matrices. Cell differentiation was only restricted by the mechanical environment, as well as
the type of matrix produced by the residing cells. The differentiation of the cells between one
phenotype and another was not explicitly modeled, but by having the type of matrix
modulated by the mechanical environment, tissue transformation over time and space was
therefore modeled. There was one additional requisite that bone could only form on already
calcified surfaces (Claes and Heigele, 1999).
58
Bone regeneration during distraction osteogenesis: Mechano-regulation by shear strain and fluid velocity
Figure 5-2: Bone regeneration as simulated in a mechano-regulated adaptive model in
MATLAB. The iterative procedure starts with a mass transport analysis to determine cell
concentrations followed by a stress analysis where the biophysical stimuli, i.e. octahedral
shear strain and fluid velocity are calculated (Figure 2-11). The tissue phenotype is
determined for each element followed by matrix production simulated with a biphasic swelling
model. The callus geometry is re-meshed and the tissue properties re-mapped, before the
material properties and cell concentrations are updated and the next iteration begins.
5.2.4 Matrix production and growth
A particular matrix production rate was modeled for each tissue type, depending on cell type
and density. Matrix production and growth were simulated by applying a swelling pressure to
the element and considering the subsequent volume expansion as being an increase in matrix.
A biphasic swelling model was adopted for this growth simulation (Wilson et al., 2005), in
which the swelling pressure is given by
2 ⎞
Δπ = RT ⎛⎜ c F2 + 4cext
⎟ − 2 RTcext
⎝
⎠
(Eq 5-1)
where R is the gas constant, T the absolute temperature, cext is the external salt concentration
and cF the fixed charged density which can be expressed as a function of the tissue
deformation as
59
Chapter 5
⎛
⎞
n f ,0
⎟
c F = c F ,0 ⎜
⎜ n f ,0 − 1 + J ⎟
⎝
⎠
(Eq 5-2)
where nf,0 is the initial fluid fraction of the tissue, cF,0 the initial negative charge in the tissue
and J the determinant of the deformation tensor. Before a simulation was carried out, all
negative charges were set to zero and the displaced geometry after the previous distraction
served as input. Identical geometrical boundary conditions were applied. Growth was induced
by introducing a non-zero fixed charge to the growing element, dependent on the cell type
stimulated in the element (Table 5-1). The fixed-charge density changes were chosen such that
they resulted in growth/volume changes within the range of those found experimentally for
each tissue type (Table 5-1). For bone, the fixed charge introduced was also dependent on
whether bone formation was intramembranous or endochondral in nature, i.e. which tissue
type was previously located in that element. The tissue was allowed to swell for 24 hours. The
resulting tissue was assumed stress free and its geometry used as input for the next increment.
Hence, all stresses induced by growth were assumed to fully relax within 1 day. The new
tissue material properties were then calculated as the result of matrix production and
degradation over the past 5 iterations, using a rule of mixtures:
n
∗
E n +1 =
∑E
i=n−4
n
i
⋅ vg i
∑ vg
i =n−4
(Eq 5-3)
i
where Ei was the elastic modulus at iteration i, vgi was the volume growth fraction, calculated
as the elemental volume after swelling divided by the elemental volume before swelling
pressure was induced. En+1* was the temporary new elastic modulus before considering cell
distribution. The cell concentrations were adjusted to the new tissue volumes (Eq 5-4), such
that the total number of cells remained the same after matrix production. The new modulus
was calculated assuming a linear relation between the modulus of the tissue and the number of
cells with the corresponding phenotype (Lacroix et al., 2002) (Eq 5-5),
[cells]n +1 =
[cells]n
vg n
(Eq 5-4)
⎛ [cells ]max − [cells ]n +1 ⎞
⎛ [cells ]n +1 ⎞
⎟⎟ EGran
⎟⎟ ⋅ En +1∗ + ⎜⎜
En +1 = ⎜⎜
[cells ]max
⎠
⎝
⎝ [cells ]max ⎠
(Eq 5-5)
where [cells]n and [cells]n+1 were the cell densities in the elements before and after
considering growth, [cells]max was the maximal cell density (assumed to be 100%), En+1 was
the new elastic modulus and Egran was the elastic modulus for granulation tissue.
60
Bone regeneration during distraction osteogenesis: Mechano-regulation by shear strain and fluid velocity
Model parameters
cF,0
nf,0
Resulting growth
(meq/mm )
Volume
growth
Fibrous tissue
0.8
7×10-2
~ 15-20 %
Cartilage
0.8
3.5×10-2
~ 5-7 %
Appositional
bone growth
0.8
3.5×10-2
~ 5 µm/day a
Endochondral
bone growth
0.8
5.25×10-2
~ 25µm/day b
T = 298 K
R =8.3145 Nmm/mmolK
3
Bone growth
rate
-4
c-ext = 1.5×10 mmol/mm
3
Table 5-1: Properties and constants used in the osmotic swelling model to predict tissue
growth. The assumed fluid fraction (nf,0) and the fixed charge density (cF,0) are inputs and the
volume growth and bone growth rate are the calculated growth of the various tissue types.
5.2.5 Remeshing and remapping
To avoid highly deformed elements the callus was remeshed prior to every new increment
(Brokken, 1999). The remeshing was applied to the deformed geometry defined by the nodal
positions after the last converged increment (Mediavilla 2005). After remeshing all tissue
properties were mapped from the integration points of the old mesh to the integration points of
the new mesh, by interpolation (Peric et al., 1996; Brokken, 1999) (Figure 5-3).
a)
b)
c)
d)
Figure 5-3: Transport: a) old mesh integration points; b) old mesh nodes; c) new mesh nodes;
d) new mesh integration points. Old elements are drawn with dashed lines and new elements
are drawn with solid lines (Adapted from Mediavilla 2005).
5.2.6 Model implementation
Marrow and cortical bone were prevented from changing their material properties or
producing matrix. The nail-callus interface did not influence matrix production or
differentiation. The reaction forces of the transported bone segment were monitored during the
24 hours of matrix production to calculate relaxation behavior. Temporal and spatial tissue
distributions, reaction forces and force relaxation data were evaluated and compared with the
experimental results. Additionally, simulations with altered distraction rate (0.5 mm/day and
0.25 mm/day) and frequencies (0.5 mm/12 hours, and 0.25 mm/6 hours) were conducted. For
frequencies of 2 or 4 distractions per day, each iteration simulated 12 or 6 hours, respectively.
61
Chapter 5
5.3 Results
5.3.1 Experimental results
The experimental results, published in detail elsewhere, showed good reproducibility (Brunner
et al., 1993; 1994). Post operatively there was a narrow corticotomy gap. During the first week
of distraction ‘graining’ appeared radiographically, i.e. small slightly radio-opaque areas
appeared throughout the distraction gap without any organized pattern. From the second week
of distraction, strips of increased radio-opacity were observed, running from the cortical bone
ends and growing towards each other. A small overlapping callus was observed on the
periosteal side. Furthermore, greater bone growth was observed periosteally than endosteally
at the nail interface. During distraction of the segment, bone formation was clearly observed in
the longitudinal direction of distraction, particularly for the larger defect, with increasing
density over time, and with the greatest density close to the cortical ends, where initial bone
formation had been observed. During continued distraction of the bone fragment, an area of
soft tissue was located in the middle of the regenerate. During consolidation, reorganization
and maturation of the regenerated bone occurred. Over time, the soft tissue in the gap
differentiated and bony bridging presided. The same general patterns were observed in the
shorter and longer regenerates, but in the longer gaps, the different stages of healing were
more clearly distinguished.
5.3.2 Computational predictions
Overall, the predicted tissue distributions agreed well with those seen experimentally (Figure
5-4). During the first week, the mesenchymal stem cells that migrated into the callus and
proliferated mainly differentiated into fibroblasts. Thus, the predicted tissue distributions
consisted primarily of fibrous tissue. After 7 days, differentiation into osteoblasts was first
observed along the periosteum and in the gap area (Figure 5-4b, iteration (it) 10). After 15
days bone tissue could be distinguished close to the periosteum. During distraction of the
segment, bone continued to develop. Slow creeping substitution by bone was seen in the
longitudinal direction of distraction, with a higher density at the periosteal side (Figure 5-4a, it
35). Throughout distraction of the bone segment, soft tissue was observed between the bone
ends, which reached a steady length after day 30. The predicted areas of bone mainly
remained immature until the end of the distraction phase. During consolidation, maturation of
the bone occurred followed by final bony bridging. Similar temporal and spatial tissue
distributions were predicted for both defect sizes (similarly to experimental observations), but
continuous bone growth during distraction was mainly predicted in the longer defects.
62
Bone regeneration during distraction osteogenesis: Mechano-regulation by shear strain and fluid velocity
Figure 5-4: Tissue differentiation during distraction of the long defects followed by
consolidation. Distraction rate and frequency are identical to the experimental study, i.e.
1mm/day distracted once. a) Predicted bone regeneration pattern. The tissue type was based
on the average element moduli as determined by the mixture theory (Eq 3-5). b) Stimulated
cell types.
63
Chapter 5
5.3.3 Reaction forces
The rate of increase of reaction forces was almost constant during the first weeks of the
experiment with a temporary decrease during the third week observed in four out of five sheep
(Figure 5-5a) (Brunner et al., 1994). Computationally, the peak force was initially higher than
that measured experimentally and over time it decreased slightly due to the increased soft
tissue regenerate (Figure 5-5a). Predicted relaxation forces compared well with the
experiments (Figure 5-5b). The stress relaxation curves of the tissues during transport were
initially between 60-70% in the experiment, compared to 65% computationally. During
distraction the relaxation increased to about 80% for both experimental results and the
computational prediction (Figure 5-5b).
a)
b)
Figure 5-5: a) Peak reaction forces after 1 mm distraction and b) relaxation behavior
calculated by the computational model and measured experimentally (Brunner et al., 1994).
64
Bone regeneration during distraction osteogenesis: Mechano-regulation by shear strain and fluid velocity
5.3.4 Distraction rate and frequency
When the distraction rate was decreased to 0.5 mm/day or 0.25 mm/day the total time for bone
regeneration increased, even though the amount of bone formation at the same magnitude of
total distraction increased (Figure 5-6). When the distraction frequency was increased to 0.5
mm two times per day, or 0.25 mm four times per day, the overall rate of bone formation
increased (Figure 5-7). During the first week of distraction the tissue distributions were similar
and mainly fibrous for all frequencies, but as distraction preceded into the second and third
weeks the amount of bone formation increased with frequency. Also the consolidation period
necessary to achieve complete bridging became shorter with increased distraction frequency.
Figure 5-6: Bone regeneration patterns with various distraction rates. The distraction rates
simulated were a) 1 mm/day, b) 0.5 mm/day and c) 0.25 mm/day, all with a frequency of 1.
The tissue type was based on the average element moduli as determined by the mixture theory.
65
Chapter 5
Figure 5-7: Bone regeneration patterns with various distraction frequencies. The total
distraction rate was 1 mm/day divided into a) 1 (1 mm/24 hours), 2 (0.5 mm/12 hours), or 4
(0.25 mm/6 hours) distractions per day. The tissue type was based on the average element
moduli as determined by the mixture theory.
5.4 Discussion
The bone formation pattern predicted using a mechano-regulation algorithm based on
octahedral shear strain and fluid velocity was consistent with experimental observations
during DO, from initial corticotomy to final consolidation. Initial bone formation was
observed in the cortical gap at the end of week two in both the experiments and model. These
events were followed by progressive bone growth in the direction of distraction, with
increased bone density at the periosteal side. Areas of soft tissue remaining in the gap
throughout distraction of the segment, and bone maturation during consolidation, were similar
in the experiment and computational predictions. The mechano-regulation algorithm has
previously been used to predict fracture healing, as well as other bone regeneration processes.
Its application to bone formation patterns during distraction osteogenesis in this study
broadens its field of application further.
66
Bone regeneration during distraction osteogenesis: Mechano-regulation by shear strain and fluid velocity
5.4.1 Tissue relaxation
During DO the tissues are subjected to tension, in contrast to fracture healing where
interfragmentary compression predominates. With tensile loads, for example, the fluid
velocity is directionally reversed, when compared to compressive loading. However, in terms
of the mechano-regulation algorithm, the magnitudes of the mechanical stimuli in the callus
are similar. The magnitudes of the two mechanical stimuli after distraction are displayed in
Figure 5-8, and the relaxation behavior over a period of 24 hours is also shown. This confirms
that the peak values of the stimuli occur around the time of maximal distraction. The
magnitudes of fluid velocity decreased rapidly as soon as distraction was completed (Figure
5-8c), while the deviatoric strain remained high during the beginning of the relaxation period
(Figure 5-8d). Depending on the location in the callus, the strain values even slightly increased
initially during relaxation. In those cases, the increases were minimal and did not affect the
predicted phenotype.
Figure 5-8: Spatial distributions and relaxation behavior of fluid velocity and deviatoric
strain at day 5. a) Model geometry at the beginning of day 5. Spatial distributions of b)
deviatoric strain and c) fluid velocity after 1 mm distraction. Two elements are highlighted
and their relaxation behavior is displayed in d) and e). Note that the time scale is logarithmic.
The relaxation behavior of the tissue in the model corresponded well with that measured
experimentally. This occurs because relaxation is dominated by the modulus/permeability
ratio of the callus tissue, which did not change much during the distraction period. In contrast,
the reaction forces from the model became progressively lower than those measured
67
Chapter 5
experimentally. This is probably not due to the mechano-regulation algorithm itself and the
predicted pattern of tissue differentiation, but to additional factors in the experimental model
that were not included in the computational model. In limb lengthening, or simple DO, there
are progressively higher tensions on adjacent fascia, tendons and muscles (Simpson et al.,
1995; Williams et al., 1999). This is because muscles are often attached across the distraction
gap. It can increase reaction forces substantially (Aronson and Harp, 1994), and also cause
considerable pain for patients undergoing DO. However, in the current experimental model of
bone segment transport (Brunner et al., 1994), these effects were reduced because the total
length of the tibia was kept constant, and most muscles were only attached to the proximal and
distal main fragments. Still, most likely there were some contributions from the adjacent soft
tissue (including muscles) on the measured forces. Another possible cause for the
disagreement in forces is that, unlike simple DO, the segment transport model required the
creation of a large gap distal to the transported segment, which would have been filled with
soft tissues. With distraction of the segment, these tissues would have been compressed,
eventually completely, resulting in increased reaction forces. Finally, throughout the
distraction period, collagen fibers are known to align longitudinally (Meyer et al., 2001b),
implying that the axial modulus of the soft tissue in the gap would have increased, similar to
other collagen-oriented soft tissues, e.g. fascia, vessels, etc (Hudetz et al., 1981; Birk and
Silver, 1984; Billiar and Sacks, 2000a; 2000b). None of these effects were included in the
computational model and, in combination, may be the source of the lower reaction forces in
comparison with experimentally measured magnitudes.
Even though DO is mechanically well-defined, some assumptions were necessary. Only the
peak magnitudes of the mechanical parameters immediately after distraction were considered
for the mechano-regulation, and the subsequent relaxation was assumed to have minimal
mechano-biological effects, and was therefore neglected. Loading during consolidation was
also neglected, because assessing the performance of the algorithm during distraction (tensile
displacements) was the main focus of this study. Thus, the resorption criteria initially
suggested for this mechano-regulation algorithm were excluded in this study. The nail was
modeled with finite sliding and was assumed to have no influence on tissue development. This
assumption was chosen since experimentally no bone formation on the nail was observed
during transport. Additionally, to overcome computational difficulties with high relative
strains, the initial experimental corticotomy (~0.5 mm) was modeled as a gap of 1 mm, and
the distraction rate was initially chosen 0.5 mm/day, increasing to 1 mm/day at day 3. This did
not influence the tissue differentiation process since even with the lower distraction
magnitudes, fibroblasts were stimulated during this period and fibrous tissue was produced.
5.4.2 Tissue growth model
Tissue growth and matrix production were modeled using a new approach. The effect of local
matrix production on tissue morphology was simulated by inducing local tissue swelling in
response to simulated osmotic pressure. The parameters were chosen such that fibrous tissue
would grow faster than cartilage and bone. More specifically, volume increases of up to 20 %
occurred in the regions where fibroblasts saturated the tissue producing fibrous matrix, while
the growth rate for cartilage was lower (5-10 %). These relative growth rates are compatible
68
Bone regeneration during distraction osteogenesis: Mechano-regulation by shear strain and fluid velocity
with experimental findings. The growth during bone formation corresponded to a bone
apposition rate of 5-10µm/day (Vedi et al., 2005) and during cartilage calcification growth has
been found to be higher prior to mineralization due to cell hypertrophy (20 µm/day) (1996a;
Wilsman et al., 1996b). Figure 5-9 displays an example of this model, where the stimulated
cell types after distraction are compared with the resulting matrix production and tissue
growth generated with the biphasic swelling model after 10 days of distraction. The osmotic
swelling model, originally developed to describe cartilaginous tissues, was applied to compute
matrix production. Hence, the concentrations of fixed charges included during matrix
production are only used to achieve a new geometrical shape of the callus. The fixed charge
densities and their effects on solid/fluid content in the tissue have no physical meaning and are
not used in subsequent iterations. The assumption of a stress free geometrical shape after
swelling/growth was made to avoid incremental stress increases in the tissue and to allow the
use of the same set of parameters to achieve the same amount of volume increase throughout
the simulation. The measured relaxation times for the tissues are on the order of hours (Weiss
et al., 2002; Huang et al., 2003; Bonifasi-Lista et al., 2005; Park and Ateshian, 2006). Hence,
by modeling full relaxation after 24 hours this assumption should be valid. The matrix
constitution in each iteration is based only on the differentiation algorithm and the rule of
mixtures (Eq 5-3).
Figure 5-9: The osmotic swelling model used simulates matrix production in an element
specific manner. a) The stimulated cell phenotypes and b) the resulting volume growth after
10 iterations with distraction rate of 1mm/day and frequency of 1 distraction per day.
5.4.3 Distraction rate and frequency
Experimental findings by others have shown that the rate of bone formation is directly related
to the local strain/stress generated in the distraction gap (Li et al., 1997; 1999), and that the
amount of mechanical tension directly influences the phenotypic differentiation of the cells
within the distraction gap (Meyer et al., 2001a). Our simulations with variations of the
distraction rates were consistent with those findings. When the tension in the gap was lowered
by a reduction in distraction rate, the bone formation per day increased. Still, the most
favorable distraction rate was 1 mm/day, because the total time necessary for regeneration of
the bone in the defect was shorter than for lower distraction rates. This also agrees with the
findings of Ilizarov which showed 1 mm/day to be the most favorable rate. Additionally,
69
Chapter 5
experimental studies have shown that further increases in distraction rate can be detrimental to
healing (Ilizarov, 1989b; Choi et al., 2004) and lead to a distraction gap filled with mostly
fibrous tissue (Choi et al., 2004). In the current study, distraction rates above 1mm/day were
not examined. Hence, we cannot compare those experimental observations with our
computational model. Ilizarov’s studies further showed that the greater the distraction
frequency, the better the outcome (Ilizarov, 1989b). Our predictions demonstrated the same
pattern, where the rate of bone regeneration increased with distraction frequency (Figure 5-7).
In our model, the best possible bone regeneration was achieved with a total distraction of 1
mm/day divided into 4 sub distractions of 0.25 mm/6 hours. Experimental studies have also
suggested that the division into endochondral and intramembranous bone formation during
DO is related to the distraction rate (Li et al., 1999; Mizuta et al., 2003; Kessler et al., 2005).
With this model the stimulated cell phenotypes and tissue types produced were similar. With a
lower distraction rate the proportion of the cells that differentiated into osteoblasts without
first going through a cartilage intermediate was increased.
5.5 Conclusions
Tissue differentiation during DO, by a mechano-regulation algorithm based on octahedral
shear strain and fluid velocity, was successfully simulated from distraction to consolidation
and was confirmed by experimental observations in a model of bone segment transport. The
rate of bone formation increased with distraction rate and frequency, similarly to experimental
observations, advocating that this algorithm could potentially be used to optimize treatment
protocols.
70
6
6
A mechano-regulatory bonehealing model based on cell
phenotype specific activity
6
A more mechanistic model of cell and tissue differentiation is
presented, which directly couples cell phenotype-specific
mechanisms to mechanical stimulation during bone healing, based
on the belief that the cells act as transducers during tissue
regeneration. The model is assembled from coupled, partial
differentiation equations, which are solved using a newly developed
finite element formulation.
The additional value of the new model and the importance of
including cell phenotype-specific activities, when modeling tissue
differentiation and bone healing, are demonstrated by comparing the
predictions with previously used models. The model’s capacity is
established by showing that it can correctly predict several aspects
of bone healing. These aspects include cell and tissue distributions
during normal fracture healing and experimentally established
alterations due to excessive mechanical stimulation, periosteal
stripping and impaired effects of cartilage remodeling.
The content of this chapter is based on manuscript IV
71
Chapter 6
6.1 Introduction
Mechano-regulation algorithms have been proposed to investigate possible relationships
between mechanical stimulation and cell and tissue differentiation, particularly in bone
healing (Chapter 2.8). They have been incorporated into computational models of tissue
regeneration and provide enormous predictive potential. However, to realize their clinical
potential, they first require robust validation. To date, they have been able to simulate many
key aspects of bone regeneration as seen experimentally (Loboa et al., 2001; Lacroix and
Prendergast, 2002; Geris et al., 2004; Duda et al., 2005; Kelly and Prendergast, 2005; Loboa et
al., 2005; Shefelbine et al., 2005) (Chapter 5), but when compared to more diverse healing
conditions (Chapter 4), they have only been partially corroborated.
Part of the problem is that in many computational mechano-regulation models of tissue
differentiation the simulation is not only dependent on the mechanoregulatory algorithm, but
also on other aspects of healing. Among these aspects, particularly the description of cellular
processes has generally been simplified to their most basic levels. Modeling of cells via a
diffusive process was introduced by Lacroix et al. (2002), to indirectly account for combined
migration, proliferation and differentiation of cells. Concentrations of progenitor cells,
originating from the periosteum, the bone marrow and the soft tissue external to the callus
were modeled using a diffusion equation (Lacroix et al., 2002). In this phenomenological
model, which has been used by many groups, including Chapter 3-4 in this thesis, the cells
were subsequently assumed to differentiate into fibroblasts, chondrocytes or osteoblasts,
according to a mechano-regulation scheme. Presence and concentration of cells influenced the
amount of tissue differentiation that occurred. All cell types were included in one ‘cell pool’,
and represented by one single diffusion constant. These models can be of assistance when
trying to find mechanical conditions under which bone healing would be promoted. However,
they are purely phenomenological, in terms of converting mechanical stimulation to biological
results. Hence, they cannot be used to understand cellular or molecular mechanisms which
would be necessary to develop not only mechanical methods to promote bone healing, but also
to enhance bone repair in combination with cellular and molecular therapy. For this purpose, it
is necessary to consider differences between cell phenotypes. The cell phenotypes involved
have different rates of proliferation, capacity of migration and capacities of differentiation and
de-differentiation from one cell phenotype to another. Furthermore, matrix production rates
are different. Such cell phenotype-specific actions need to be accounted for to enable more
accurate predictions of cell-mediated processes, such as fracture healing.
Some tissue differentiation models have already been developed to focus on cell and
molecular mechanisms of bone healing (Bailon-Plaza and van der Meulen, 2001), capturing
important aspects of bone healing, such as angiogenesis (Geris et al., 2006b). However, they
have not been coupled to the mechanobiology of bone healing. To overcome this deficit, we
have developed a more mechanistic model of tissue differentiation, which describes the
mechanobiology by directly coupling cellular mechanisms to mechanical stimuli during bone
healing. Our underlying hypothesis is that, during stimulated tissue regeneration, the cells act
as sensors. The cells within the matrix proliferate, differentiate, migrate, and produce
72
A mechano-regulatory bone-healing model based on cell phenotype specific activity
extracellular matrix based on the mechanical stimulation they experience. This response is
cell-phenotype specific. In this study, the computational model is presented, and its predictive
capacity is evaluated by showing that it can correctly predict several aspects of bone healing.
The study further aimed to determine the additional value of including cell-phenotype specific
activities and rates, when modeling tissue differentiation during bone healing, by comparing
the results with the phenomenological model described above.
6.2 Methods
A computational model was developed to describe the temporal and spatial distributions of
fibrous tissue, cartilage and bone, regulated through cellular activity. The activities of the four
cell types, mesenchymal stem cells, fibroblasts, chondrocytes and osteoblasts, are dependent
on the mechanical stimulation. At each time point and location, each cell type can migrate,
proliferate, differentiate and/or apoptose, depending on their mechanical stimulation and the
activity of other cell types in the environment. They can also produce matrix, or stimulate
matrix degradation. The active cellular processes, the developed computational model to
describe the cellular processes, the already excisting mechanical model, as well as how the
models interact are described in detail in the following sections.
6.2.1 Cellular processes
The main cellular processes active during tissue repair were identified as original location and
concentration of cells, rates of proliferation, migration, differentiation and maturation,
apoptosis, matrix production and matrix degradation. The cells are also involved in a complex
feedback control system in which numerous growth factors, cytokines and signaling proteins
are active (Bolander, 1992; Bostrom and Asnis, 1998). These factors were not explicitly
included as parameters, but were implicit in the cell activities modeled. An extensive literature
review was conducted to find experimental data for all parameters necessary in the cellular
model. The literature data that were found to support the events are described below and
summarized in Table 6-1, and the calculated normalized parameters that were used as input in
the model are given in Table 6-2.
Cell origin
Bone lining cells and pre-osteoblasts reside in the cambial layer of the periosteum, overlying
the cortical bone (Nakahara et al., 1990; Gerstenfeld et al., 2003b). These cells can be
stimulated to rapidly differentiate into osteoblasts and deposit new bone. The periosteum is
also rich in mesenchymal stem cells (Nakahara et al., 1990; Gerstenfeld et al., 2003b). The
importance of the periosteum in bone healing has been established by demonstrating a lower
capacity for fracture-callus development when it was removed (Aro et al., 1985; Buckwalter et
al., 1996b). This indicates that the periosteum may be the greatest source of stem cells during
fracture healing. The bone marrow is also a potent source (Gerstenfeld et al., 2003b). Human
adult long bones mainly contain yellow bone marrow, with a lower stem cell concentration
than the red bone marrow (Postacchini et al., 1995). However, many laboratory animals have
bones that contain a large proportion of red marrow (Postacchini et al., 1995). Furthermore,
studies using radiolabeling and bone marrow transplantation have shown that cells originating
73
Chapter 6
from the marrow play a role during fracture repair (Taguchi et al., 2005b; Colnot et al., 2006).
There is also clear evidence that there are stem cells in the muscle tissue surrounding the
callus (Urist, 1965; Iwata et al., 2002). However, there is no confirmation that they actively
participate in the bone regeneration process, and their distance to the fracture callus is large
compared to the stem cells from the periosteum and bone marrow. Therefore, the external
musculature is not assumed to be a major source of stem cells. Stem cells are most probably
released into the haematoma, which forms immediately following the fracture trauma
(Buckwalter et al., 1996b; Gerstenfeld et al., 2003b). Hence, when the reparative phase of
bone healing begins, there should be a random distribution of mesenchymal stem cells in the
callus tissue (Gerstenfeld et al., 2003b). Unfortunately, little quantitative data is available on
cell concentrations.
Max cell density
Initial cell
Cell
density
Periosteum
MSC
5 (1)
5 •10
cells/mm
6 (3)
10
8 (4)
FB
10
CC
2•10
5 (5,1-2)
2.7•10
OB
3
mm /cell
C
B
4 (6)
rate
Doubling
Mitotic
µm
µm/min
time hours
fraction
10
(4)
18
(7)
25
(5,8-9)
20
(10)
Production
Matrix
FT
3
-6 (27,28)
5•10
5•10
-6 (27,28)
3•10
-6 (27,28)
pg/cell h
0.3
3
Proliferation rate
Migration
Cell size
µm/day
(29)
60
(3,11)
40
(11,12)
12-24
(14,15-17)
12-16
low
(13)
10
(18)
35
(1-2)
20
(14)
0.50
Differentiation
Apoptosis
Maturation time
cells/mm
(14,15,19)
0.45
0.2
0.35
(18,20)
(16)
(14,20)
14-21 days
(21-22)
8-10 doubling
(23,24)
4.5*norm
35
(25)
(6,26)
Degradation
3
mm /cell
-6 (27,28)
5•10
(30)
-6 (27,28)
5•10
15
(1,31,32)
-6 (27,28)
3•10
Table 6-1: Summary of the literature data that was used to identify cell processes and
calculate normalized cell parameter rates. 1 Wilsman et al. (1996b), 2 Wilsman et al. (1996a),
3
Bailon-Plaza and van der Meulen (2001), 4 Stephan Miltz, AO Research Institute, Personal
communication, 5 Hunziker et al. (1987), 6 Olmedo et al. (1999), 7 McGarry and Prendergast
(2004), 8 Fazzalari et al. (1997), 9 Kember and Sissons (1976), 10 Lian and Stein (2001),
11
Friedl et al. (1998), 12 Shreiber et al. (2003), 13 Fiedler et al. (2002), 14 Manabe et al.
(1975), 15 Huang et al. (1999), 16 Ekholm et al. (2002), 17 Colter et al. (2000), 18 Spyrou et al.
(1998), 19 Deasy et al. (2003), 20 Fedarko et al. (1995), 21 Bosnakovski et al. (2004),
22
Bosnakovski et al. (2005), 23 Malaval et al. (1999), 24 Aubin et al. (1995), 25 Li et al. (2002),
26
Olmedo et al. (2000), 27 Martin et al. (1998), 28 Gomez-Benito et al. (2005), 29 Howard et al.
(1998), 30 Sengers et al. (2004), 31 Eriksen and Kassem (1992), 32 Vedi et al. (2005)
Cell proliferation
Stem cells are characterized by their dual ability to self-renew and to differentiate into a range
of progenitor cell phenotypes. They can undergo asymmetric division, i.e. produce a daughter
cell identical to the mother cell and another cell committed to differentiation (Huang et al.,
1999; Punzel et al., 2003). Several mathematical models of cell proliferation exists (Sherley et
al., 1995; Murray, 2002; Deasy et al., 2003; MacArthur et al., 2004). We adopted a model
based on the logistic growth equation, where the rate of cell division decreases linearly with
74
A mechano-regulatory bone-healing model based on cell phenotype specific activity
cell density, due to limitations in space and nutritional resources (Murray, 2002). Cell
proliferation rates depend both on phenotypes and magnitudes of mechanical stimuli, and are
generally very sensitive. Cell phenotype-specific proliferation rates were adopted, where
proliferation was assumed to be ‘on’ or ‘off’, depending on the mechanical stimulation, and
did not vary in magnitude within one cell-phenotype group. Proliferation rates for
mesenchymal stem cells agree relatively well between studies (Manabe et al., 1975; Huang et
al., 1999; Javazon et al., 2001; Punzel et al., 2003), even though they are highly dependent on
plating density, mechanical stimulation and culture conditions. Proliferation of osteoblasts and
chondrocytes has also been frequently studied (Manabe et al., 1975; Aubin et al., 1995;
Malaval et al., 1999; Chowdhury et al., 2004; Sengers et al., 2006). Fibroblasts were assumed
to have proliferation rates in the range of those for mesenchymal stem cells (Spyrou et al.,
1998).
Cell maturation and differentiation
Once a mesenchymal stem cell has been stimulated down the osteogenic pathway, it
proliferates 8-10 times before it becomes a mature osteoblast and produces bone matrix
(Aubin et al., 1995; Malaval et al., 1999). Chondrocytes are believed to require between 14-21
days to mature (2004; Bosnakovski et al., 2005). Maturation time was not modeled explicitly,
but it was taken into account when calculating differentiation rates. The rates were calculated
as time to achieve complete transformation of one cell group into another, after the full
maturation time. The differentiation and de-differentiation of cell phenotypes is complex. The
following relationships are supported by various studies: Mesenchymal stem cells can
differentiate into any cell phenotype, i.e. fibroblasts, chondrocytes or osteoblasts. Fibroblasts
can differentiate into either chondrocytes or osteoblasts (Mizuno and Glowacki, 1996; Yates,
2004; Zhou et al., 2004). Chondrocytes cannot differentiate into osteoblasts. Instead, they
undergo hypertrophy, followed by apoptosis under certain conditions (Lee et al., 1998; Bland
et al., 1999; Ford et al., 2003). With the apoptosis of chondrocytes and degradation of the
cartilage matrix, osteoblasts are able to migrate into the tissue and mineralize it (Ford et al.,
2004). Osteoblasts cannot de-differentiate into chondrocytes, but there is some evidence that
they can de-differentiate into fibroblasts (Jones et al., 1991).
Cell migration
Literature advocates that migration of stem cells is partly occurring through chemotaxis
(Fiedler et al., 2002; 2004; 2005; Makhijani et al., 2005). Fibroblast migration has been
suggested to occur randomly at a high rate (Radomsky et al., 1998) and has been studied in a
variety of ways, in models of wound healing for example (Radomsky et al., 1998; Spyrou et
al., 1998; Horobin et al., 2006). Chondrocytes, on the other hand, are believed to have a very
low potential for migration (Morales, 2007). Osteoblasts are known to migrate, to some extent,
by crawling along calcified surfaces (Stains and Civitelli, 2005; Kaplan et al., 2007). Our
model describes migration as a diffusive process, where transport is limited by increasing cell
concentration. The diffusion constants were assumed to decrease linearly with concentration.
75
Chapter 6
Matrix production and degradation
Matrix production rates of fibrous tissue, cartilage and bone have all been extensively studied.
Production of fibrous tissue is known to occur relatively rapidly, and has been quantified in
studies of wound healing and cell culture experiments (Tranquillo and Murray, 1993;
Midwood et al., 2004; Ahlfors and Billiar, 2007). Regeneration of cartilage has been widely
studied in the field of cartilage regeneration and tissue engineering (Wilkins et al., 2000;
Williamson et al., 2001; Mauck et al., 2003). However, the relevance of those results during in
vivo bone-healing, where cartilage is produced during endochondral bone formation, is likely
to be limited. Therefore, we adopted matrix production rates from studies of growth plates
(Hunziker et al., 1987; Wilsman et al., 1996a; 1996b). The matrix production rate of cartilage
was lower than for other tissues. Rates of bone formation, either through bone apposition or as
endochondral replacement, were taken both from studies of growth plates and from cellculture experiments (Wilsman et al., 1996a; 1996b; Vedi et al., 2005). Experimental
measurements of matrix degradation and removal were not found in the literature. Therefore,
assumptions were made based on other computational models in the literature (Bailon-Plaza
and van der Meulen, 2001; Gomez-Benito et al., 2005; Garcia-Aznar et al., 2006), and the
degradation rates were assumed to be identical for all tissue types.
6.2.2 Theoretical development
The computational cell model consists of seven coupled non-linear partial differential
equations. The first four of the seven variables describe the concentrations of mesenchymal
stem cells (MSC), fibroblasts (FB), chondrocytes (CC), and osteoblasts (OB), as
⎛
∂c(x, t ) i
c ⎞
= ∇D(ci ) i ∇ci + f PR (Ψ ) i ci ⎜1 − i ⎟ − FD (Ψ , c1−4 ) − f AP (Ψ ) i ci
⎜ c
⎟
∂t
space ⎠
⎝
(Eq 6-1)
where t represents time, x is the two-dimensional position in space, and ci is the concentration
of cell type i, where i = 1 (MSC), 2 (FB), 3 (CC) or 4 (OB). The four parts of the equation
describe transport/migration, proliferation, differentiation and apoptosis of each of the cell
types i. Migration is described as diffusion, and Di is the concentration-dependent diffusivity
for cell type i. fPR, FD and fAP are functions which regulate rates of proliferation, differentiation
and apoptosis, respectively. fPR and fAP are either turned ‘on’ or ‘off’ depending on the
mechanical stimulation ( Ψ ). Ψ is calculated based on the magnitudes of deviatoric shear
strain and fluid velocity (Prendergast et al., 1997), and can have values between 1 and 4 for
stimulation of MSC, FB, CC, or OB. This is described in more detail below. FD is dependent
on the mechanical stimulation, as
If
If
Ψ = i,
Ψ ≠ i,
FD =
∑ (− f
k =1− 4 ,k ≠i
c
Dk k
)
FD = f D i ci
76
(Eq 6-2)
A mechano-regulatory bone-healing model based on cell phenotype specific activity
Additionally, FD is also dependent on the individual differentiation potential of each cell type.
Hence, if differentiation is not cell biologically possible, it will not occur. The maximum rates
of fPR, fD and fAP for each cell type are shown in Table 6-2. cspace is the ‘available space’ in the
element, and is calculated as the maximum cell concentration minus the sum of the current
cell concentrations. All parameters are normalized and the normalized maximal cell
concentrations are calculated based on literature data of cell size and occupied space, or cell
concentrations (Table 6-2). These cells can produce the skeletal tissue types fibrous tissue
(FT), cartilage (C) and bone (B), which are the remaining three variables, described as
∂m(x, t ) j
∂t
⎛
mj ⎞
⎟ − f DM (Ψ ) j ci m j
= f PM (Ψ ) j ci ⎜1 −
⎟
⎜ m
space ⎠
⎝
(Eq 6-3)
where mj represents the concentration of matrix type j, and i is the corresponding cell type to
the matrix type j, e.g. fibroblasts to fibrous tissue. The equation is divided into production and
degradation of matrix, where fPM and fDM are functions which regulate the rates of production
and degradation of matrix, respectively. Similar as for the cell processes, matrix production
and degradation are turned ‘on’ or ‘off’ depending on Ψ . For example, if Ψ is 2, stimulating
fibroblast cell activity, that also results in maximal matrix production of fibrous tissue, and
minimal degradation of fibrous tissue. The maximal values of fPM and fDM are shown in Table
6-2. mspace represents the ‘available space’ in the element, and is calculated as the maximal
matrix concentration minus the sum of the current matrix concentrations. The parameters were
normalized based on literature data on cell concentrations as matrix production-rates per cell
(Table 6-2).
Cell
MSC
FB
CC
OB
Matrix
FT
C
B
Initial cell density
Periost
0.5
0,0
0,0
0,0
Initial
conc
0,0
0,0
0,0
Marrow
0,30
0,0
0,0
0,0
External
0,05
0,0
0,0
0,0
Callus
0,005
0,0
0,0
0,0
Transport
D
mm2/it
0.65
0.50
0.0
0.20
Proliferation Differentiation Apoptosis
fD
f AP
f PR
day -1
day -1
day -1
0,60
0,30
0,05
0,55
0,20
0,05
0,20
0,10
0,10
0,30
0,15
0,15
Production Degradation
f PM
f DM
day -1
day -1
0,20
0,05
0,05
0,05
0,10
0,05
Table 6-2: Normalized cell parameter data that was used as input in the model. Initial cell
densities and initial concentrations indicate the initial conditions at day 0. D, fPR, fD, fAP, fPM
and fDM are maximal values. D decreased linearly with cell concentration, and the f’s were
turned on or off depending on the mechanical stimulation.
77
Chapter 6
Implementation of cell model
To solve the equations and determine the evolution of the skeletal tissue types, a new finite
element formulation was written, including all seven degrees of freedom (MSC, FB, CC, OB,
FT, C, B), and their coupling. The 4-noded linear element was implemented into ABAQUS (v.
6.5) as a user-defined element. The development of the element formulation is discussed in
Appendix A. The system of equations was solved through a transient heat transfer analysis,
using backward difference principals for time integration. Coupling of the freedom degrees
was done individually, to permit cell phenotype-specific differentiation pathways and cell
concentrations to affect matrix production and degradation. The cell type-specific
differentiation rules and rates were added as extra conditions to the function FD, so that
mesenchymal stem cells could differentiate into any other celltypes. Fibroblasts could
differentiate into chondrocytes and osteoblasts. Chondrocytes could not de-differentiate and
osteoblasts could de-differentiate into fibroblasts. The mechanical stimulation was assumed to
stimulate each process, either at maximal rate, or not at all, depending on the mechanoregulation algorithm (Prendergast et al., 1997). Mechanical stimulation of cell type i resulted
in: 1) maximal proliferation of cell type i, 2) no proliferation of other cell types, 3) minimal
apoptosis of cell type i and 4) maximal differentiation of other cell types into cell type i when
differentiation was permitted. Furthermore, mechanical stimulation of cell type i resulted in:
1) maximal matrix production of the corresponding tissue type j, 2) no matrix production of
other tissue types, and 3) minimal matrix degradation for tissue type j.
The initial conditions in the cell model include concentrations of mesenchymal stem cells at
the periosteum, the marrow, the outer boundary interface and randomly in the callus tissue at
locations shown in Figure 6-1 and wth magnitudes shown in Table 6-2. All other cell and
tissue types have a zero initial concentration. Additional boundary conditions throughout the
simulations were that no cell or tissue type can have a negative concentration, and that the sum
of all cell concentrations and all matrix concentrations cannot exceed 100 %.
Figure 6-1: Geometrical finite element model that were used (left). The mechanical model
was axisymmetric. Additional geometrical details are described in Figure 3-1. The cell model
(right) was only solved for the callus areas and included initial MSC concentrations at
periosteum, marrow and outer boundaries as well as randomly distributed within the callus.
78
A mechano-regulatory bone-healing model based on cell phenotype specific activity
6.2.3 Adaptive tissue differentiation model
Within the overall tissue differentiation model, two separate finite element analyses are
conducted, using the same geometrical model (Figure 6-1); 1) an analysis of cellular processes
as described above and 2) a mechanical poroelastic analysis. The axisymmetric finite element
model, which was presented in Chapter 3, including an ovine tibia with a 3 mm healing
transverse fracture gap and external callus was used for both analyses (Figure 6-1). A 1 Hz
cyclic load of 300 N was applied. A poroelastic analysis was conducted, and the biophysical
stimuli were calculated at maximal load. A mechano-regulation algorithm, assuming the
combined effects of deviatoric strain and fluid velocity to regulate cell and tissue
differentiation was adopted (Prendergast et al., 1997) since it has previously been shown to be
versatile in predicting fracture healing under different loading conditions (Lacroix and
Prendergast, 2002; Geris et al., 2003; Kelly and Prendergast, 2005) (Chapters 4-5). The
magnitudes of deviatoric shear strain (SS) and fluid velocity (FV) was used to determine the
value of Ψ .
stim =
SS
FV
+
3.75 3.0
(Eq 6-4)
According to the mechanoregulation algorithm by Prendergast et al., (1997), stim > 3 results in
stimulation of fibroblasts and fibrous tissue ( Ψ = 2). stim > 1 results in stimulation of
chondrocytes and cartilage ( Ψ = 3). stim > 0.267 results in stimulation of osteoblasts and bone
( Ψ = 4) and stim > 0.01 also resulted in mature bone tissue formation. The mechanical stimuli
for each element and integration point ( Ψ ) were transferred to the cell model using
subroutines (Figure 6-2b), and all the active cell processes were calculated (Figure 6-2a). The
normalized amounts of matrix, predicted by the cell model, were used to calculate the new
mechanical properties as,
⎛
⎞
E = ⎜⎜ mmax − ∑ m j ⎟⎟ ⋅ EGT +
j = FT ,C , B ⎠
⎝
⎛ mj
⎞
⎜
⎟
⋅
E
∑ ⎜
j ⎟
j = FT ,C , B ⎝ mmax
⎠
(Eq 6-5)
where E is the Young’s modulus for the element, mj the current amount of matrix type j, and
mmax is the maximum amount of matrix in an element, normalized to 1. EGT is the Young’s
modulus for granulation tissue, and Ej is the Young’s modulus for tissue type j. The material
properties were then transferred back into the mechanical analysis, using subroutines, and the
next iteration began. Material properties for cortical bone and marrow remained constant and
all material properties were taken from the literature (Table 6-3).
79
Chapter 6
a)
b)
Figure 6-2: Sketch of the adaptive tissue differentiation model. a) the cellular processes,
which all occurs over time. In addition, migration occurs in space. b) the adaptive
computational models including the coupling between mechanical and cell analysis.
Cortical
bone
Marrow
Gran.
Tissue
Young’s modulus
(MPa)
15750 a
2
1
Permeability
(m4/Ns)
1E-17 d
Poisson’s ratio
0.325 b
0.167
Porosity
0.04 c
0.8
Fibrous
Immature
Cartilage
Tissue
Bone
Mature
Bone
2f
10 i
1000
6000 k
1E-14 f
5E-15 g
1.00E-13
3.7E-13 l
0.167
0.167
0.167 j
0.325
0.325
0.8
0.8
0.8 m
0.8
0.8
1.00E-14 1.00E-14
Table 6-3: Tissue material properties used. a Smit et al. (2002); b Cowin (1999); c Schaffler
and Burr (1988); d Johnson et al. (1982); e Anderson (1967); f Hori and Lewis (1982);
g
Armstrong and Mow (1982); hTepic et al. (1983); iLacroix and Prendergast (2002); jJurvelin
et al. (1997); kClaes and Heigele (1999); lOchoa and Hillberry (1992); m Mow et al. (1980).
80
A mechano-regulatory bone-healing model based on cell phenotype specific activity
6.2.4 Phenomenological tissue level model
A phenomenological model with cell and matrix concentrations based on diffusion only
(Lacroix et al., 2002), was used for comparison. It was described in detail in Chapter 3-4, and
was implemented as in previous studies (Lacroix et al., 2002). Briefly, one pool of cells was
included and regulated by a diffusion equation, separate from the stress analysis, to
collectively account for migration, proliferation and differentiation of cells. The cells
originated at the periosteum, the bone marrow and the muscle tissue external to the callus, all
with 100% concentration (Lacroix et al., 2002). The diffusion constant was adjusted to fit the
overall rate of healing, observed with the new model, to simplify comparison. All cells within
an element were assumed to differentiate into either fibroblasts, chondrocytes or osteoblasts,
dependent on the mechanical stimulation characteristics of that element for that day.
Production of matrix was dependent on cell density. The new material properties were
calculated based on a ‘rule of mixtures’ of the stimulated cell-types over the last 10 days
(Lacroix et al., 2002).
6.2.5 Simulations
To assess the potential of the developed model, the ability to predict normal fracture healing
was determined and evaluated. Thereafter, the importance of including cell phenotype-specific
parameters was assessed by comparison of the predicted tissue distributions between the new
more mechanistic model and the phenomenological model. Furthermore, effects of alterations
in both the mechanical and two distinct biological conditions were evaluated separately. First,
the capacity of the new model to predict spatial and temporal tissue patterns, due to excessive
mechanical stimulation was evaluated by increasing the load from 300N to 400N, or 500N
respectively. The magnitudes were chosen based on experimental measurements (Claes et al.,
1998; Duda et al., 1998). Secondly, the biological environment was varied by evaluating the
effect of initial periosteal stripping. Removal of parts of the periosteum has been used
experimentally, to create models of delayed healing and non-unions (Aro et al., 1985). This is
known to delay initial bone formation and results in reduced periosteal callus formation. It was
simulated by removing the initial condition cell-source along the periosteum (Figure 6-1).
Third, decreased cartilage turnover during endochondral ossification was simulated. It is
known that cartilage turnover plays an important role during endochondral ossification. There
are several factors influencing this process. For example, matrix metalloproteinases (MMP’s)
are known to be important during angiogenesis and cartilage removal, and MMP-deficient
mice have shown delayed healing, characterized by retarded cartilage resorption (Colnot et al.,
2003b; Kosaki et al., 2007). MMP’s are not modeled explicitly. Instead impaired
endochondral ossification and cartilage remodeling was simulated by reducing the apoptosis
rate of chondrocytes to 50%, and the degradation of cartilage to 10% of normal.
81
Chapter 6
6.3 Results
6.3.1 Normal fracture healing
The new cell model successfully captured various characteristic events of normal fracture
healing (Figure 6-3). There was initial intramembranous bone formation along the periosteum,
starting at some distance from the gap (Figure 6-3, day 10). Concurrently, there was soft tissue
formation in the gap area, where fibrous tissue formed prior to cartilage. The bony callus grew
mainly through intramembranous ossification (day 20), followed by endochondral
replacement of the cartilage in the lower callus and cortical gap areas (day 20-30). Initial bony
bridging occurred externally, and was followed by creeping substitution of bone until
complete healing was achieved. Figure 6-4 shows the temporal variations of cell- and matrix
concentrations in the gap area and in the external lower callus area. In the gap, mesenchymal
stem cells initially differentiated into fibroblasts, which were stimulated both to proliferate and
produce matrix (Figure 6-4a). This was followed by a period of chondrocyte stimulation.
Mesenchymal stem cells and fibroblasts differentiated into chondrocytes, which produced
extracellular matrix and cartilage. The cartilage was later replaced by bone. The evolution of
cell phenotypes and skeletal tissue types in the external callus (Figure 6-4b) at some distance
from the gap, showed no initial fibroblast stimulation. The mesenchymal stem cells
differentiated directly, although more slowly, into chondrocytes. The matrix-production rate of
cartilage was lower than that for fibrous connective tissue. The peak concentration of cartilage
occurred around the same time as in the gap region. Thereafter, the cartilage was replaced
through endochondral ossification.
Figure 6-3: Predicted distributions of skeletal tissue types calculated with the new cell model
during normal fracture healing at an axial load of 300N. The top row shows the normalized
concentration of fibrous tissue, the mid and bottom rows show concentrations of cartilage and
bone, respectively.
82
A mechano-regulatory bone-healing model based on cell phenotype specific activity
Figure 6-4: Predicted evolution of normalized cell (left) and matrix (right) concentrations in
one element in a) the gap area and b) the periosteal callus area.
6.3.2 Comparison with diffusion model
With the phenomenological diffusion model, normal fracture healing was simulated as
described previously in Chapter 3. However, when comparing the predictions for the models,
differences were observed. As discussed in Chapter 3, the phenomenological model produced
some instabilities in the tissue predictions. Also a non-physiological, isolated bony bridge
across the gap was predicted between days 20 and 30, prior to creeping of the bone front from
the external callus towards the gap (Figure 6-5). With the new cell model, bony bridging
occurred externally, with no isolated islands of new bone. It was followed by successive
creeping-substitution of bone until the gap was completely filled with bone matrix (Figure
6-5), similar to experimental observations of fracture repair (Perren and Rahn, 1980; Cruess
and Dumont, 1985), in which new bone grows only at bony fronts (Claes et al., 1997; Claes
and Heigele, 1999). Furthermore, the relative rates of individual events or phases of healing
were different. With the diffusion model the duration of fibrous tissue in the gap was
relatively short and cartilage was formed earlier and more intensive, as compared to the cell
model, where cartilage stimulation occurred later and more sparsely, similar to most larger
animal models (Sarmiento et al., 1996; Claes et al., 1998). The reason was that with the
diffusion model the tissue phenotype was determined based on cell partitions over a number of
iterations, instead of actual extracellular matrix-production rates, as in the cell model. In
addition to these improved predictions of tissue differentiation during fracture healing there
are some clear advantages of the new model. It had the ability to evaluate isolated parameters,
which was not possible with the diffusion model. For example, effects of individual
parameters on cell distributions (Figure 6-4) could be analyzed. This allowed direct
comparison with, for instance, experimentally assessed cell distributions, if these become
available.
83
Chapter 6
Figure 6-5: Comparison of predicted tissue types as determined by modulus between a) the
cell model (top) and b) the diffusion model (bottom).
Figure 6-6: The effect of load on the predicted tissue distributions displayed by the varying
tissue distributions and cell concentrations when the load was a) 300N, b) 400N or c) 500N.
6.3.3 Predictive capacity of the model
The new model’s potential to predict tissue distributions during fracture healing was assessed
by simulating a number of situations which have also been studied experimentally. The effect
on the predicted tissue distributions due to an increase in load from 300N to 400N and 500N,
respectively, are shown in Figure 6-6. No clear differences could be observed in the initial
stages of healing: Bone formed along the periosteum and fibrous soft tissue in the gap area,
independently of loading. The effect of the load became more prominent during endochondral
ossification and the phase of bony bridging. An increased load (400N) almost doubled the
time to bony bridging (Figure 6-6a-b), and a further load increase (500N) resulted in a steadystate non-union (Figure 6-6c). The loading magnitude also had an effect on the predicted cell
84
A mechano-regulatory bone-healing model based on cell phenotype specific activity
concentrations. Under a 300N load there was a short peak in fibroblast concentration in the
immediate gap area, followed by chondrocyte proliferation, chondrocyte hypertrophy and
finally osteoblast invasion (Figure 6-7a). Increasing the load to 400N increased the length of
the fibrous phase, with a higher concentration of fibroblasts. The chondral phase and the time
to hypertrophy and osteoblast formation were longer than under the 300N load (Figure 6-7b).
Under 500N, the gap-area resulted in a steady-state combination of fibroblasts and
chondrocytes (Figure 6-7c). Hence, the extracellular matrix that formed was fibrocartilaginous.
Figure 6-7: The effect of load on the predicted tissue distributions displayed by the varying
cell concentrations in the immediate gap (shown by ‘.’) with the loads a) 300N, b) 400N or c)
500N.
The model’s potential was also evaluated by simulating initial periosteal stripping and
deficiency in cartilage turnover and remodeling. When the initial cell-source of the periosteum
was removed, the spatial and temporal bone-formation patterns changed (Figure 6-8a). Initial
periosteal reaction and bone formation observed during normal fracture healing was
diminished (Day 10-30). There was almost no periosteal callus formation until after the timepoint at which external bony bridging would normally occur (Day 30-40). Bone healing was
delayed and the time until complete healing was two times that for normal healing.
Furthermore, the amount of cartilage formed was greater than in normal fracture healing, all in
agreement with actual observations in the literature.
The importance of cartilage turnover for the process of endochondral ossification is well
established (Lee et al., 1998; Einhorn, 1998b; Ford et al., 2004). A decreased cartilage
remodeling capacity resulted in distorted tissue distributions. The initial processes of
intramembranous bone formation and soft tissue production were similar to the normal healing
case (Figure 6-8b), and a periosteal bony callus formed (Day 30). Thereafter, cartilage
replacement and external bony bridging were interrupted. Cartilage still remained in the gap
and the adjacent periosteal callus, even after the ‘normal’ case had achieved complete healing.
Eventually, there was external bony bridging (Day 60) and the fracture healed completely
around day 80, after more than two times that of normal healing (Figure 6-3).
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Chapter 6
Figure 6-8: When a load of 300N was combined with; a) Initial stripping of the periosteum, it
resulted in delayed periosteal callus formation and delayed union, b) reduced ability for
endochondral ossification, it resulted in a delayed union, with a cartilaginous callus in the
gap area.
6.4 Discussion
In this study a mechanistic cell model of tissue differentiation was developed, and used to
model various aspects of bone healing. The model was based on the way in which cellular
activities control the evolution in concentrations of seven variables: Mesenchymal stem cells,
fibroblasts, chondrocytes and osteoblasts, as well as fibrous tissue, cartilage and bone. It was
combined with a pre-existing mechano-regulation algorithm, in which cell and tissue
differentiation is regulated by the magnitudes of deviatoric strain and fluid velocity
(Prendergast et al., 1997). This algorithm was adopted because it had shown to predict more
versatile experimental observations than alternative algorithms. To evaluate the new cell
model, and demonstrate the importance of including mechanistic descriptions of cell activities
and individual rates, the outcome was compared with a tissue-phenomenological level model
in which diffusion was used as a collective mechanism for migration, proliferation,
differentiation and matrix generation. The new model predicted events observed during
normal fracture healing, and captured phenomena which the tissue-level model did not. Other
benefits of our mechanistic model are that relative time, individual cell activity and
concentrations, as well as the effects of each parameter on the process of tissue regeneration
can be evaluated and compared to experimental results. Moreover, the model was able to
predict alterations in healing patterns due to periosteal stripping and impaired cartilage
remodeling. This has not been possible with former computational models, and therefore
illustrates the additive value of the presented approach.
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A mechano-regulatory bone-healing model based on cell phenotype specific activity
6.4.1 Predictive capacity
The predictive capacity and potential of the new model was assessed by showing that delayedand non-unions can result from excessive load magnitudes, periosteal stripping and impaired
cartilage turnover during endochondral ossification. The effects of increased mechanical
stimulation, predicted by the model, agree well with literature findings. High interfragmentary
movements (low fixator stiffness) were shown, both experimentally (Sarmiento et al., 1996;
Claes et al., 1997; Duda et al., 1998) and clinically (Kenwright et al., 1986; 1991), to result in
delayed healing or non-union, particularly by persistence of cartilage through inhibition of
endochondral replacement of the tissue (Kenwright and Goodship, 1989; Sarmiento et al.,
1996; Claes et al., 1997; Choi et al., 2004). This was also shown in our current model, where
an increase in load delayed healing in this manner, and a further increase in load predicted a
steady-state non-union, with fibrocartilage in the gap and cartilage in the adjacent periosteal
callus (Figure 6-6, Figure 6-7). Similar alterations in tissue predictions due to increased load
have also been observed with the phenomenological tissue level model (Lacroix and
Prendergast, 2002).
Varying the extent of periosteal stripping has been employed as a reliable experimental
method of achieving delayed or non-union (Aro et al., 1985; Buckwalter et al., 1996a;
Einhorn, 1998b). Also in the present model, periosteal stripping resulted in a delayed
periosteal response, decreased periosteal callus formation and delayed healing (Figure 6-8a).
These observations concur with findings in the literature, where Aro et al. (1985) reported that
removal of the periosteum resulted in delayed periosteal bone formation, smaller callus size
and delayed bony bridging. The effect of periosteal stripping was also evaluated with the
phenomenological model (data not shown). However, although the predictions were altered by
removal of the periosteal cell source, tissue distributions were not comparable with
experimental observations. This was partly due to the differences in initial cell concentrations
between the models, where the phenomenological model started with maximal concentration
also at the external muscle tissue. Hence, removal of the periosteal cell source did not alter the
tissue distributions similarly as to the cell model or as in experimental observations.
Endochondral ossification is a complex process involving several steps of chondrocyte
hypertrophy and apoptosis, cartilage mineralization, angiogenesis, and replacement by bone
(Lee et al., 1998; Ford et al., 2004). Experimentally it was shown that absence or disruption of
any one of these processes can result in impaired healing (Lee et al., 1998; Colnot and Helms,
2001). Using the ability of the present model to change particular parts of this process, we
performed a simulation in which the ability of cartilage degradation and remodeling was
impaired. Cartilage remained at the fracture site, whereas with normal healing it was replaced
earlier, and bony bridging and complete healing was significantly delayed. The findings relate
well to experimental studies using MMP knockout mice (Colnot et al., 2003b; Kosaki et al.,
2007). MMP 9 and MMP 13 in deficient mice were shown to result in delayed fracture
healing, caused by persistent cartilage at the fracture site. MMPs are known to mediate both
vascular invasion into the hypertrophic cartilage and cartilage resorption (Colnot et al.,
2003b). This effect was represented in the model by decreasing the cartilage-remodeling
capacity. This would not be possible with the phenomenological model. These simulations
87
Chapter 6
demonstrate the potential of the current model of predicting bone healing, where the effects of
any factor can be investigated, once its effect on cellular processes are known.
6.4.2 Cell model
The cell model is based on a mechanistic description of cell proliferation, differentiation and
tissue synthesis, whereas the diffusion model uses a phenomenological description by which
these processes are captured implicitly by a single parameter, which does not distinguish
between cell types and cellular processes. This increases the new models’ applicability and
enhances possibilities for validation with experimental data from the literature. Two processes
known to be important during bone healing are not included here. These are biochemical
signaling and vascularization. The complex interactions between growth factors and cytokines
have not yet been described sufficiently; including them would require many more
assumptions. However, with our approach, their effects on cell proliferation and
differentiation are indirectly captured in the present equations. Angiogenesis and revascularization have received increased attention recently (Colnot and Helms, 2001; Carano
and Filvaroff, 2003; Carvalho et al., 2004; Lienau et al., 2005). Currently, the model includes
angiogenesis implicitly in function-regulating cartilage degradation and endochondral
replacement. The present model extensions focused on important and well known cellular
processes, where the necessary assumptions are well based in literature. This has opened
ample opportunities for further tissue- differentiation studies that yet need to be explored.
With the finite element formulation created for our model it will be possible to implement
other aspects of bone healing, once it becomes pertinent for solving specific research
questions.
Several recent studies have also focused on the incorporation of biological phenomena at the
cell level in models of tissue differentiation. Perez and Prendergast have included a stochastic
model of cell dispersal, together with a mechano-regulation algorithm (Perez and Prendergast,
2006). This model included anisotropic aspects and directed movements, and was applied to
an implant-bone interface. It predicted similar results as the phenomenological model, with the
exception that predicted tissue distributions were rather discontinuous (Perez and Prendergast,
2006). However, identical rates were applied to describe proliferation, maturation and
differentiation for all cell types, but the possibility of de-differentiation of cells was not
included. Geris et al. (2006b) adopted the biological model of Bailon-Plaza and van der
Meulen (2001), and included angiogenesis. The influence of osteogenic, chondrogenic and
angiogenic growth factors on the development of tissue distributions was described. However,
their model did not include the influence of mechanical stimulation. Garcia-Aznar et al. (2006)
and Gomez-Benito et al. (2005) have developed a model of tissue differentiation which
includes some cell processes, as well as volumetric tissue growth. This model has been shown
to predict some physiological aspects of tissue regeneration, but includes many assumptions
that are currently difficult to validate.
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A mechano-regulatory bone-healing model based on cell phenotype specific activity
6.4.3 Model parameters
Some model parameters are not well quantified in the literature and were estimated. For
example, experimental quantifications of matrix degradation were not found. Other
assumptions include the choice to turn each cellular process, for example proliferation, ‘on’ or
‘off’, instead of varying the amount of activity within each cell group, dependent on either
mechanical stimulation or cellular environment. The influences of these assumptions are
difficult to evaluate, but with the model we developed they can easily be implemented when
essential for the scientific question and/or when reliable quantifications are found. However,
the aim of this study was to create a mechanistic model and demonstrate its applicability in
describing cell and tissue-differentiation during bone healing. An extensive parametric study
is necessary to evaluate the importance of each assumption and its influence on the overall
healing process. This was beyond the scope of the present study, but will be presented in the
following chapter of this thesis.
6.5 Conclusions
In summary, a new mechanoregulation model based on cell activity and cell-phenotype
specific processes was developed. This study shows that the cell-phenotype specific processes
are very important to take into account, as they largely determine the outcome of the
simulations. This suggests that computational models should describe cellular processes
during tissue differentiation and bone healing accurately. The mechanistic cell model
presented here was shown to correctly predict general aspects of normal fracture healing.
Furthermore, it captured the effects of excessive loading, periosteal stripping, and impairedcartilage turnover, as described in literature. These are events which are known to disrupt the
healing process and the two latter are events that former computational models were unable to
capture. The present model therefore seems promising in its ability to predict pathological
conditions and could be used in the future to evaluate potential treatments.
89
7
7
Determining the most important
cellular characteristics for
fracture healing, using design of
experiments methods
Mechano-regulation models of tissue differentiation do not account
for uncertainties in input parameters, and often include assumptions
about parameter values that are not yet established. The objective of
this study was to determine the most important cellular
characteristics during bone healing. The mechanistic model
described in Chapter 6 was used in combination with a statistical
‘design of experiments’ approach, including fractional factorial
analysis and Taguchi orthogonal arrays.
The assessed bone healing parameters were predominantly
influenced by matrix production rate of bone and the rate of
production and degradation of cartilage. Parameters related to bone
were linear, while parameters related to soft tissues were nonlinear.
Hence, optimum values were found to achieve most successful bone
healing. The most important parameters and processes identified
were similar to what is known from in vivo animal experiments. The
study suggests that experiments should preferably focus at
establishing values of parameters related to endochondral bone
formation.
The content of this chapter is based on manuscript V
91
7
Chapter 7
7.1 Introduction
Mechano-regulation of cell and tissue differentiation during bone healing involves a sequential
cascade of highly coordinated cellular events. Most events are known to be sensitive to the
local mechanical environment (Einhorn, 1998b; Gerstenfeld et al., 2003b). However, the
complete sequence of activated cellular and molecular events is not yet known, and the
regulatory mechanisms are still under investigation. Computational modeling of bone healing
has been used to study potential mechano-regulation pathways. Although such models of
bone healing has become powerful tools, they remain phenomenological (Prendergast et al.,
1997; Carter et al., 1998; Claes and Heigele, 1999) (Chapter 2-8).
Mechanobiological models have the potential to help develop biological and mechanical
interventions for treatment of skeletal pathologies. However, the models need to be
mechanistic in order to understand the processes involved. Developments of mechanistic
tissue differentiation models have followed increasing biological knowledge and
computational power. The development of a novel mechanistic mechanobiological model of
tissue differentiation during bone healing was described in Chapter 6. This model combines a
detailed description of cell phenotype-specific activities and rates in fracture repair with
mechanical stimulation, based on finite element analysis. Initial cell concentrations, rates of
proliferation, differentiation, migration, apoptosis, as well as production and degradation of
cell-associated matrix are implemented in combination with a mechano-regulation algorithm,
based on tissue shear strain and fluid velocity (Prendergast et al., 1997). This mechanistic
model was shown to be applicable to bone healing. It was also shown able to predict known
alterations in spatial and temporal tissue formation patterns due to both altered mechanical and
biological environments (Chapter 6).
More sophisticated mechanistic mechanobiological models require more parameter values to
be quantified. However, many of the ongoing processes and interactions occurring during
bone healing are only partly resolved. Moreover, current models do not always account for the
uncertainty in model input parameters. Hence, more complex models require more
assumptions about parameter data. The importance of these assumptions has not always been
investigated properly. What are the most important parameters for modeling tissue
differentiation and bone healing? Identifying these parameters and establishing how well they
are currently known, could lead towards designing specific experiments to measure unknown
parameters of significance.
A number of statistical methods are available to conduct parametric analyses to account for
variations in parameter data and quantify the importance of each parameter. Full factorial
designs are useful for studying systems with a low number of parameters and levels, and can
provide more information concerning interactions between parameters. However, when the
number of parameters and levels are large, the total number of required simulations often
becomes impractical or impossible. A statistical method developed by Taguchi (Taguchi and
Wu, 1980; Taguchi, 1987) utilizes an orthogonal array, which is a form of fractional factorial
design containing a well-chosen subset of all possible combinations of test conditions. The
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Determining the most important cellular characteristics for fracture healing, using design of experiments methods
Taguchi method lends itself well to finite element analysis (Dar et al., 2002) and can be used
to identify the input parameters that have the largest influence on the outcome variables
(Phadke, 1989; Taguchi et al., 2005a). The value of such an experimental approach, in which
the effects of multiple parameters are tested concurrently, has long been recognized in
industrial development and manufacturing operations (Phadke, 1989; Dar et al., 2002). One
important advantage, compared to the traditional approach of varying one parameter at a time,
is that the potential problem associated with selection of a single base line condition is
avoided. ‘Design of experiments’ methods have recently been increasingly employed in areas
such as orthopaedic biomechanics (Mburu et al., 1999; Dar and Aspden, 2003; Lee and Zhang,
2005; Yao et al., 2006), and has the potential to reveal the underlying mechanisms of
mechanobiological models.
This study aimed to reveal which cellular parameters are of the greatest influence to each of
the major processes during tissue differentiation and to the bone healing capacity. This was
carried out by using ‘design of experiments’ methods and Taguchi orthogonal arrays to
investigate the effects of each of the cell parameters in the aforementioned mechanistic model
of bone healing on the outcome. The outcome was assessed as sequential spatial and temporal
tissue differentiation, bone formation rate and time until complete healing. The parameters
investigated were related to original cell distributions, rates of proliferation, migration,
differentiation and apoptosis, and rates of production and degradation of the extracellular
matrix.
7.2 Methods
To study individual cellular parameter values on bone healing, we used a theoretical
description of cell processes as part of a mechanobiological tissue-differentiation model. A
two step parametric analysis was conducted, in which all the parameters in the cell model
were investigated and their influences on the outcome were evaluated using a statistical
approach. The first step was a screening experiment and the second step was a higher level
examination of critical variables identified in the first step. All these parts are described
separately in detail in the following sections.
7.2.1 Cell model
The cell model was described in the previous chapter (Chapter 6.2.1-6.2.3). Briefly, cells
responded to mechanical stimulation by conducting one or several of the following processes:
Proliferation, differentiation into fibroblasts, chondrocytes or osteoblasts, migration or
apoptosis. Additionally, the cells could produce or remove extracellular matrix for its
respective tissue type. The computational model consists of seven coupled non-linear partial
differential equations. The first four of the seven variables describe concentrations of
mesenchymal stem cells (MSC), fibroblasts (FB), chondrocytes (CC), and osteoblasts (OB), as
described by Eq 6-1 (Chapter 6.2.2). These cells can produce the skeletal tissue types fibrous
tissue (FT), cartilage (C) and bone (B), respectively, which are the remaining three variables,
described by Eq 6-2 (Chapter 6.2.2). The system of equations was implemented as user-
93
Chapter 7
defined finite elements into ABAQUS (v 6.5), and solved as a transient heat transfer problem
(Chapter 6, Appendix A).
7.2.2 Finite element model and tissue differentiation
The axisymmetric finite element model of an ovine tibia was also described in the previous
chapter (Chapter 6.2.3). It was used to both calculate the biophysical stimuli and for the cell
analysis. The geometry involved a 3 mm transverse fracture gap and an external callus (Figure
7-1a). For the mechanical analysis, a 1 Hz cyclic load of 300 N was chosen based on
experimental measurements (Claes et al., 1998), and applied. The biophysical stimuli were
calculated at the peak load using ABAQUS (v 6.5) and transferred into the cell model via
subroutines (Figure 7-1b).
Figure 7-1: a) FE model and b) description of the adaptive tissue differentiation model
including the cell processes involved.
Mechano-regulation of cell processes was adopted from the model of Prendergast et al.
(1997), assuming the combined effects of deviatoric strain and fluid velocity to predict
activation of processes for each cell phenotype. The normalized amounts of matrix predicted
by the cell model were used to calculate the new mechanical properties as:
⎛
⎞
E = ⎜⎜ mmax − ∑ m j ⎟⎟ ⋅ EGT +
j = FT ,C , B ⎠
⎝
⎛ mj
⎞
⎜⎜
⋅ E j ⎟⎟
j = FT ,C , B ⎝ mmax
⎠
∑
(Eq 7-1)
where E is the total Young’s modulus for the element, mj is the current amount of matrix type
j, and mmax is the maximum amount of matrix in an element, normalized to 1. EGT is the
94
Determining the most important cellular characteristics for fracture healing, using design of experiments methods
Young’s modulus for granulation tissue, and Ej is the Young’s modulus for tissue type j.
Cortical bone and marrow were not allowed to evolve, and did not contain any cells. All
tissues were described as linear poroelastic, with the material properties shown in Table 6-3,
(see page 80).
7.2.3 Normal fracture healing simulation
A simulation of normal fracture healing with averaged normalized literature values was
conducted, to determine reference predictions for normal fracture healing. The parameter
values used in the cell model were based on literature as described in Chapter 6.2.1 and their
normalized values implemented as shown in Table 7-1.
Cell
MSC
FB
CC
OB
Matrix
FT
C
B
Initial cell density
Periost
0.5
0.0
0.0
0.0
Initial
conc
0.0
0.0
0.0
Marrow
0.30
0.0
0.0
0.0
External
0.05
0.0
0.0
0.0
Callus
0.005
0.0
0.0
0.0
Transport
D
2
mm /it
0.65
0.50
0.0
0.20
Proliferation Differentiation Apoptosis
f PR
fD
f AP
-1
-1
-1
day
day
day
0.60
0.30
0.05
0.55
0.20
0.05
0.20
0.10
0.10
0.30
0.15
0.15
Production Degradation
f PM
f DM
-1
-1
day
day
0.20
0.05
0.05
0.05
0.10
0.05
Table 7-1: Normalized cell parameter data that was used as input in the model for normal
fracture healing. Parameter values were calculated based on the literature review, which was
presented in Chapter 6.
7.2.4 Parametric study
Control factors and levels
Within the parametric study, all properties included in the cell activity model were examined.
The investigated cell properties are initial origin and concentrations of mesenchymal stem
cells, rates of proliferation, differentiation, migration, and apoptosis for each cell type (MSC,
FB, CC, OB). Furthermore, the rate of matrix production and degradation of fibrous tissue,
cartilage and bone were investigated. Within the parametric study each of the properties varied
in this way is referred to as a control factor, the values it is set to be its levels, and each
combination of control factor levels that is evaluated is called a treatment condition (Phadke,
1989). The orthogonal arrays employed are denoted as LN(SM), where M is the number of
control factors, S is the number of levels of each control factor, and N is the total number of
treatment conditions in the evaluation. The chosen parameter space for each of the
investigated cell parameters was based on the literature summary in Chapter 6.2.1. Generally,
the space between the highest and lowest values found in the literature was investigated. When
little or no support was found in literature, the parameter space was estimated.
95
Chapter 7
Design of the matrix experiments
Design for this problem was constrained by the large number of factors and the potential for
significant interactions among the factors. In the first stage a two-level screening experiment
was used to identify the most important factors. In the second stage a three-level factorial
design was carried out on the smaller subset of the factors that showed important effects, to
also study non-linear factor effects.
For the screening experiment, all 26 factors were investigated at two levels, high (-1) and low
(+1). The lowest possible resolution design is III (Phadke, 1989; Montgomery, 2005).
However, since in a resolution III design, at least one factor is confounded with two-factor
interactions (Phadke, 1989; Montgomery, 2005), a resolution IV design was employed.
Resolution IV designs are used extensively as screening experiments (Montgomery, 2005),
and a resolution IV L64(231) orthogonal array was chosen, with a total of 64 treatment
conditions, and leaving factor 27-31 vacant (Phadke, 1989). The configurations of the factors
and the levels in each simulation are shown in Table 7-2, and the orthogonal array is available
in Appendix B.
x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Levels
High
Low
Factors
Initial concentration
Proliferation
Differentiation
Migration
Apoptosis
Matrix production
Matrix degradation
26
Periost
Marrow
Outer
callus
MSC
FB
CC
OB
MSC
FB
CC
OB
MSC
FB
CC
OB
MSC
FB
CC
OB
FT
CC
B
FT
CC
0.8
0.5
0.1
0.05
0.8
0.7
0.4
0.5
0.4
0.3
0.2
0.3
0.8
0.7
0.1
0.4
0.2
0.2
0.2
0.2
0.3
0.1
0.2
0.1
0.1
0.2
0.1
0
0
0.5
0.3
0.1
0.15
0.1
0.05
0.025
0.05
0.4
0.3
0
0.05
0.05
0.05
0.05
0.05
0.05
0.01
0.02
0.015
0.015
B
0.1
0.015
Table 7-2: Cell model variables used during the screening experiment (L64) for each of the
two levels and 26 factors. The L64 array was prepared to match that in (Phadke, 1989).
In the next step, a three-level factorial design was used to study the curvature of the factors
identified as most important in the screening experiment. A resolution III L27(313), Taguchi
orthogonal array was chosen (Phadke, 1989) (Appendix B) to investigate the effects of the 10
most influential factors, and leaving factor 11-13 vacant. Each factor was assigned 3 levels,
high (-1), normal (0) and low (1) (Table 7-3), and the experiment included a total of 27
treatment conditions.
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Determining the most important cellular characteristics for fracture healing, using design of experiments methods
x
Factors
1
2
3
4
5
6
7
8
9
10
Initial MSC concentration
P
M
FB
CC
OB
FT
CC
B
FT
C
Proliferation
Matrix production
Matrix degradation
High
Mid
Low
0.8
0.5
0.7
0.4
0.5
0.5
0.1
0.15
0.1
0.1
0.5
0.3
0.5
0.25
0.325
0.3
0.05
0.08
0.05
0.05
0.2
0.1
0.3
0.1
0.15
0.1
0.01
0.05
0.01
0.01
Table 7-3: Cell model variables used during the higher level experiment (L27) for each of the
three levels and 10 factors. The L27 Taguchi array was prepared based on (Phadke, 1989).
Response
To assess the results obtained from the parametric study, parameters that characterize the
performance of the system for each treatment condition were determined. Clinically and
experimentally employed methods for evaluating the progress of healing include several
radiographic and histological scoring systems (Bos et al., 1983; Heiple et al., 1987; Lane and
Sandhu, 1987; Yang et al., 1994; Johnson et al., 1996). These are based on subjective scores
of several parameters: periosteal reaction or callus formation, bone union, marrow changes
and fracture remodeling (An et al., 1999). Each parameter is assessed as ‘no reaction’, ‘mild’,
‘moderate’ or ‘full reaction’, and given the corresponding score between 0-3. The total score
is then calculated and used to assess the progress of healing.
Based on these tables, a scoring system was formed for this study. Three different outcome
analyses were performed. The first one was an overall performance measure, to assess the
ability of the model parameters to predict sequential spatial events observed during normal
fracture healing, independent of time. Each of the following events received a score of 0 or 1,
where 0 was non-physiological and 1 a normal event: 1) Fibrous tissue formation in the
fracture gap, 2) initial periosteal-bone formation, 3) growing periosteal callus including
endochondral bone formation, 4) fibrous/cartilage formation within the gap area, 5) external
bony bridging, 6) bone creeping substitution, 7) complete callus filled with bone. The second
analysis was performed to measure the progression of bone healing, based on the amount of
bone formation in different areas at specified time points. Three time points were chosen,
representing early (day 10), mid (day 25), and late (day 50) phases of healing. Regions of
interest (ROI) were chosen (Figure 7-2) to represent the events to be evaluated: periosteal
reaction, callus formation, intramedullary canal reaction/endosteal callus formation, bony
bridging and complete healing. In each ROI, the total amount of bone matrix was calculated.
During early phase of healing, the periosteal reaction and callus formation were seen as
measurements of progression. During the mid-phase of healing the amount of bone formation
in the bridging and endosteal callus regions were selected to measure healing progression, and
for the late phase, the bridging and the gap regions were chosen. The third analysis was a
measure of total time required until complete fracture healing. It was based on an absolute
97
Chapter 7
number of days until the whole callus was predicted to be filled with bone, i.e. when all the
elements in the callus were filled with more than 75 % bone matrix.
Figure 7-2: Regions of interests (ROI) that were used during outcome analysis. For early
stage of bone formation, the periosteal reaction and callus formation were measured. For the
mid-phase of healing, the endosteal (intramedullary canal) reaction and the bridging regions
were assessed. To assess the amount of bone formation during the late stage, the bridging and
gap 9complete healing) regions were evaluated.
Data analysis
The loss function which the Taguchi method seeks to minimize is generally taken to be a
quadratic function (Phadke, 1989). Analysis of variance (ANOVA) was used to investigate the
significance and contribution of each factor. The ANOVA process, including calculation of
the total sum of squares of the deviation about the mean, gave
n
SST = ∑ ( yi − y ) 2
(Eq 7-2)
i =1
where n was the number of experiments, yi the outcome parameter for the ith treatment
condition, y was the overall mean of y. For each factor, the sum of the squares of deviation
about the mean was
n
SS F = ∑ N Fi ( y Fi − y ) 2
(Eq 7-3)
i =1
where Fi are the factors from 1-26, or 1-10 respectively, and n is the number of levels, i.e. 2 in
the screening experiment and 3 in the three-level experiment. NFi is the number of treatment
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Determining the most important cellular characteristics for fracture healing, using design of experiments methods
conditions at each level of each factor, and y Fi is the mean outcome parameter at each level of
each factor. The sum of the squares of the error was
Fi
SS E = SST − ∑ SS i
(Eq 7-4)
i =1
The fraction of the variance explained by each factor was calculated from the F-value of the
ANOVA for each parameter as the mean square of deviation for each factor over the mean
square of the error. The percentage of the total sum of squares represented the contribution of
each factor to the variance, and was calculated as (SS F SST ) ⋅ 100% . It was considered a
measure of ‘importance’ of the factors (Dar et al., 2002).
7.3 Results
7.3.1 Normal fracture healing simulation
The model predicted the characteristic events of normal fracture healing successfully (Figure
7-3) (Chapter 6). The predictions were used to determine the baseline for evaluation of normal
fracture healing in the parametric study and included initial bone formation along the
periosteum, starting at some distance from the gap (Figure 7-3, day 10) with concurrent soft
tissue formation in the gap area and fibrous tissue prior to cartilage. The bony callus grew
mainly through intramembranous ossification (day 25), followed by endochondral
replacement of the cartilage in the lower callus and cortical-gap area (day 40). Initial bony
bridging occurred externally, and was followed by creeping substitution of bone into the gap
until complete healing was achieved.
Figure 7-3: Predicted distributions of skeletal tissue types with the mechanistic cell model
during normal fracture healing and an axial load of 300N. The top row shows the normalized
concentrations of fibrous tissue, the mid and bottom rows show concentrations of cartilage
and bone, respectively.
99
Chapter 7
7.3.2 Screening experiment
The 64 treatment conditions required for the L64 array were simulated and the outcome
performance calculated. The percentage of the total sum of squares for each of the outcome
parameters, 1) sequential normal healing, 2) bone formation rate assessed at early, mid and
late stages, and 3) time to complete healing, represented the approximate contribution of each
factor to the variance (Table 7-4). The parameters that were generally most influential were
related to formation and degradation of cartilage and matrix production of bone. The ability of
the model to predict the expected sequences of normal fracture healing was most influenced
by the rate of matrix production of bone (42%), followed by rate of cartilage replacement
(degradation) and proliferation rate of fibroblasts (Table 7-4). When the healing sequence was
separated into early and late parts, it was observed that the bone formation rate increased its
contribution during the early part, whereas cartilage degradation during endochondral
replacement and fibroblast proliferation was most important during the last parts.
The analysis of the amount of bone formation at early, mid and late stages of bone healing was
solely dependent on parameters related to chondrocytes and osteoblasts (Table 7-4). During
early healing, when the external callus formation was analyzed, matrix production of bone was
accountable for 78% of the contributions, followed by proliferation rate of osteoblasts.
However, the analysis of the mid and late phase revealed the matrix production of cartilage to
be the most influential parameter with 43% and 16%, respectively. For the mid phase, which
was assessed by the amount of bone in endosteal and periosteal regions of interest, matrix
production of bone was the second most influential factor. For the late stage, determined as the
amount of bone in the bridging and gap region of interest (Figure 7-2), proliferation rate for
chondrocytes and osteoblasts were second most influential parameters, followed by the
apoptosis rate of fibroblasts.
The time to complete healing was mostly dependent on the rate of matrix degradation of
cartilage (20%). This was followed by matrix production rate of cartilage (20%) and matrix
production rate of bone, and some lower influences related to proliferation of fibroblasts and
osteoblasts (Table 7-4).
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Determining the most important cellular characteristics for fracture healing, using design of experiments methods
Sequential
normal healing
SSF
x Source
% TSS
1.0
1 Origin conc
Periost 4.48E-01
2
Marrow 1.57E-01
0.3
1.57E-01
0.3
3
Outer
4.48E-01
1.0
4
callus
2.14E+00
4.6
5 Proliferation MSC
4.67E+00 10.0
6
FB
7
CC
1.12E-01
0.2
1.12E-01
0.2
8
OB
5.34E-01
1.1
9 Differentiation MSC
1.96E+00
4.2
10
FB
5.34E-01
1.1
11
CC
12
OB
3.74E-03
0.0
1.12E-01
0.2
13 Migration
MSC
14
FB
1.01E+00
2.2
1.12E-01
0.2
15
CC
1.96E+00
4.2
16
OB
17 Apoptosis
MSC
5.34E-01
1.1
1.12E-01
0.2
18
FB
19
CC
1.12E-01
0.2
5.34E-01
1.1
20
OB
3.74E-03
0.0
21 Matrix prod.
FT
8.08E-28
0.0
22
CC
1.95E+01 41.8
23
B
24 Matrix deg.
FT
1.57E-01
0.3
6.21E+00 13.3
25
CC
26
B
3.02E-01
0.6
ANOVA
Time to complete
healing
SSF
% TSS
1.89E+00
0.2
2.29E+00
0.3
9.88E+00
1.2
1.21E+01
1.5
7.38E-01
0.1
3.57E+01
4.4
1.71E+01
2.1
3.38E+01
4.1
2.08E+01
2.5
4.26E+00
0.5
1.11E+01
1.4
9.50E+00
1.2
2.80E+00
0.3
3.70E+00
0.5
4.10E+00
0.5
5.88E+00
0.7
1.11E+00
0.1
8.57E+00
1.0
1.14E+00
0.1
3.00E+00
0.4
9.15E-02
0.0
1.61E+02
19.7
1.41E+02
17.2
6.19E+00
0.8
1.65E+02
20.1
3.28E-02
0.0
_______________ Amount of bone formation_______________
Early phase
Mid phase
Late phase
SSF
SSF
SSF
% TSS
% TSS
% TSS
8.02E+00
0.6
8.57E+00
0.3
1.40E+02
1.1
1.29E-01
0.0
3.91E+00
0.1
1.52E+01
0.1
2.55E+00
0.2
2.63E+01
1.0
6.55E+01
0.5
1.28E-01
0.0
6.73E+01
2.5
4.51E+01
0.4
2.94E+00
0.2
3.61E+00
0.1
7.90E+01
0.6
1.53E+01
1.2
6.60E+00
0.2
4.20E+01
0.3
2.54E+00
0.2
1.09E+02
4.0
1.07E+03
8.7
1.19E+02
1.08E+02
4.0
1.01E+03
9.2
8.2
2.38E+01
1.8
5.62E+01
2.1
1.58E+02
1.3
2.28E+01
1.8
3.11E+00
0.1
1.92E+02
1.6
1.07E+00
0.1
6.21E-01
0.0
4.78E+02
3.9
6.39E-01
0.0
8.54E+00
0.3
5.13E+01
0.4
4.70E+00
0.4
4.32E+00
0.2
1.62E+02
1.3
8.21E-02
0.0
1.29E+01
0.5
1.97E+02
1.6
1.34E+00
0.1
4.57E+00
0.2
2.21E+02
1.8
4.53E+00
0.3
1.78E+01
0.7
2.27E+02
1.8
2.03E+00
0.2
5.73E+00
0.2
5.86E+01
0.5
2.40E+00
0.2
5.97E+01
2.2
5.09E+02
4.1
5.06E-01
0.0
7.06E+00
0.3
1.96E+02
1.6
1.18E-02
0.0
3.75E+01
1.4
8.65E+01
0.7
3.89E+00
0.3
1.61E+01
0.6
7.76E+00
0.1
2.14E+00
0.2
1.16E+03 42.9
1.96E+03 15.9
1.01E+03 77.6
2.86E+02 10.6
3.75E+02
3.0
1.78E+00
0.1
3.97E+01
1.5
4.03E-01
0.0
1.61E+00
0.1
1.81E-01
0.0
2.40E+02
1.9
9.83E+00
0.8
7.67E+00
0.3
2.44E+01
0.2
Table 7-4: ANOVA of each of the outcome variables for the L64 screening experiment. The
sum of squares for each factor (SSF) and the percentage of the total sum of squares (%TSS)
are listed. The most influential (>5%) parameters are highlighted.
From the results of the screening experiment (Table 7-4), the overall most contributing factors
were collected to design the three-level experiment. The selected factors were proliferation
rates of fibroblasts, chondrocytes and osteoblasts, matrix production rates of all tissue types,
as well as matrix degradation of fibrous tissue and cartilage. Moreover, initial concentrations
of cells at the periosteum and marrow were included. The chosen levels and factors are shown
in Table 7-3.
7.3.3 Higher level experiment
Throughout the outcome analyses of the L27 array, the factor that was most important in the
screening experiment also had the highest influence in the three-level experiment (Table 7-5).
However, parameters of lower influence shifted in some cases, due to non-linearity. The
sequence of normal fracture healing was still most influenced by the matrix production rate of
bone followed by the proliferation rate of fibroblasts. Relative to the screening experiment, the
influence of degradation of fibrous tissue increased. Separating this sequence into two parts,
early and late events, revealed that for the early sequences of healing, only matrix production
rate of bone was important, whereas the fibroblast and fibrous tissue related parameters were
most important during the late events, as well as matrix degradation of cartilage.
The amount of bone formation at early, mid and late stages of healing was again most
sensitive to parameters related to cartilage and bone (Table 7-5). During the early phase of
healing the rate of matrix formation of bone was responsible for 71% of the variation. Also,
the proliferation rate of osteoblasts had a high influence (14%). During the mid and late
101
Chapter 7
phases, the matrix production rate of cartilage was most important, followed by the
proliferation rate of chondrocytes during the mid phase, and the matrix production rate of
fibrous tissue and proliferation of osteoblasts during the late phase of healing (Table 7-5).
The time until complete healing was most sensitive to matrix degradation of cartilage (31%)
followed by matrix production of cartilage and proliferation of osteoblasts, where both a high
rate of cartilage production and degradation corresponded to a shorter time until complete
healing (Table 7-5).
ANOVA
x
1
2
3
4
5
6
7
8
9
10
Source
Origin conc
Periost
Marrow
Proliferation FB
CC
OB
Matrix prod. FT
CC
B
Matrix deg.
FT
CC
Sequential
normal healing
SSF
% TSS
1.99E-01
0.8
1.99E-01
0.8
4.22E+00 16.9
1.86E-01
0.7
1.06E+00
4.2
4.39E-01
1.8
2.35E+00
9.4
1.22E+01 48.9
2.08E+00
8.4
1.85E+00
7.4
Time to complete
healing
SSF
% TSS
8.42E+01
1.0
2.75E+02
3.4
5.49E+01
0.7
7.51E+02
9.2
1.37E+03
16.8
3.29E+01
0.4
18.9
1.54E+03
2.65E+02
3.3
1.58E+02
1.9
2.90E+03
35.7
______________ Amount of bone formation_______________
Early phase
Mid phase
Late phase
SSF
SSF
SSF
% TSS
% TSS
% TSS
3.92E-03 0.9
1.75E+01
2.3
1.79E-01
5.8
1.27E-02 3.0
2.27E+01
3.0
5.65E-02
1.8
4.95E-03 1.2
1.23E+01
1.6
5.20E-02
1.7
6.17E-04 0.1
5.32E+01
7.1
2.69E-01
8.6
5.33E-02 12.7
2.38E+01
3.2
4.07E-01 13.1
3.79E-03 0.9
2.64E+00
0.4
6.09E-01 19.6
1.17E-03 0.3
5.61E+02 75.3
1.06E+00 34.0
3.18E-01 75.8
3.99E+01
5.3
3.68E-02
1.2
5.98E-03 1.4
1.34E+00
0.2
5.55E-02
1.8
2.41E-03 0.6
4.65E+00
0.6
1.36E-01
4.4
Table 7-5: ANOVA of each of the outcome variables for the L27 higher level experiment. The
sum of squares for each factor (SSF) and the percentage of the total sum of squares (%TSS)
are listed. The most influential (>5%) parameters are highlighted.
The non-linearity provides an explanation for how the relative importance of some parameters
could shift between the screening experiment and the three-level experiment. For example, the
time to complete healing was most influenced by cartilage parameters (Figure 7-4a).
Relatively speaking, both mid- and high levels of cartilage production and degradation
resulted in similar time to healing, whereas low production and degradation rates resulted in
longer times to complete healing (Figure 7-4a). The dependency of osteoblast proliferation
was linear, where higher rate of proliferation resulted in shorter time until complete healing. In
general, the parameters related to bone and osteoblasts were more linear in appearance than
those related to cartilage and fibrous tissue, where higher rates solely resulted in more bone
formation. This is displayed by the amount of bone formation at the early stage (Figure 7-4b)
and parameter behavior for fibrous tissue and cartilage during the late phase (Figure 7-4c). In
this case, the mid level (normal value) of both production of cartilage and fibrous tissue
resulted in more bone formation relative to both the high and the low production rates.
Finally, the results of the experiments were confirmed by running single simulations with the
most beneficial parameter values for each outcome analysis. This is an important part for
validating the design of experiments approach (Phadke, 1989; Montgomery, 2005) and it was
confirmed that those simulations resulted in the highest fracture healing sequential score, the
most amount of bone formation at both early, mid and late stages, as well as the shortest time
until complete healing.
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Determining the most important cellular characteristics for fracture healing, using design of experiments methods
Time to complete healing
Outcome parameter (days)
75
65
Prol OB
MatProd C
MatDeg C
Average
55
45
35
High
a)
Mid
Low
Levels
Outcome parameters (% bone)
Amount of bone formation - Early phase
0.8
Prol OB
MatProd B
Average
0.7
0.6
0.5
High
Mid
Low
Levels
b)
Amount of bone formation Late phase
Outcome parameter (% bone)
1
0.8
Prol OB
MatProd FT
MatProd C
Average
0.6
0.4
High
c)
Mid
Low
Levels
Figure 7-4: Nonlinear behavior of highly influential parameters determined from the outcome
analysis of a) time to complete healing b) amount of bone formation at early stage c) amount
of bone formation at late stage.
103
Chapter 7
7.4 Discussion
Computational modeling of fracture healing is a very challenging problem because of its
complexity both mechanically and biologically. The uncertainty in reaction pathways and the
quantification of responses (property values) requires further investigation. This study aimed
to use a design of experiments approach to screen for factors that are most influential on the
outcome of bone healing. A technique was employed using computational modeling and a
statistically based method to evaluate the importance of several cell parameters on the process
of fracture healing. The parameters that were found to be most important for the bone
regeneration process were further studied in a multilevel approach. Although only a small
number of all the possible combinations of various levels of the parameters were studied, we
believe that the critical parameters were successfully identified (Table 7-5). The most
important parameters identified overall were matrix production rates of bone and cartilage,
and cartilage replacement (degradation). The fact that they were not the same for all outcome
analyses conducted further establishes the complexity of the mechanobiological processes
during bone healing. The outcome analyses were chosen specifically to capture different
aspects of this process.
7.4.1 Relevance of results
Throughout all outcome analyses, formation rates of bone and cartilage were most important,
followed by the degradation rate of cartilage. Worth noting is that parameters related to
cartilage were often more influential than parameters related to bone, even for the measure of
the amount of bone formation (Table 7-4 and Table 7-5). This highlights the importance of
cartilage formation and replacement (endochondral ossification) during bone healing. It relates
well to many in vivo experimental studies that have identified this step as crucial for
successful bone healing (Lee et al., 1998; Colnot and Helms, 2001; Ford et al., 2003).
Additionally, animal studies have shown that the amount of cartilage and the duration of the
cartilaginous phase is a key factor for the required time for a long bone to heal (Claes et al.,
1998; Choi et al., 2004; Ortega et al., 2004). This relates well to our outcome parameter based
on time to complete healing, in which replacement of cartilage, i.e. endochondral ossification
was the most influential parameter. The amount of bone formation was regulated by different
parameters during early, mid and late phases. The early phase was evaluated as the amount of
periosteal reaction and callus formation (Figure 7-2). These are regions where it is well
established clinically and experimentally, that very little soft tissue is formed (Einhorn, 1998b;
Rüedi et al., 2007). This is consistent with the result that callus formation was solely
influenced by the rates of osteoblastic proliferation and bone matrix formation (Table 7-5).
The mid and late phases were evaluated as the amount of bone formed in the intramedullary
canal and bridging regions, and the bridging and gap regions, respectively (Figure 7-2). These
are regions where in vivo animal studies have shown large amounts of soft tissues initially
(Claes et al., 1997; Einhorn, 1998b; Choi et al., 2004). In agreement with these observations,
the amount of fibrous tissue and cartilage that formed, also affected the amount of bone
formation. The matrix production of both fibrous tissue and cartilage had an optimum level
(Figure 7-4c). Some fibrous connective tissue and cartilage formation was beneficial to the
amount of bone formed during the late phase, but too much of it delayed bone formation. This
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Determining the most important cellular characteristics for fracture healing, using design of experiments methods
is also known from experimental observations, where formation of these tissues is known to
stabilize the gap and subsequently allow for external bony bridging to occur, followed by
creeping substitution and complete healing (Buckwalter et al., 1996b; Einhorn, 1998b;
Gerstenfeld et al., 2003b). In vivo bone healing studies have also shown that too much
cartilage formation delays healing (Claes et al., 1998), which is also suggested by our results.
Angiogenesis and re-vascularization are processes that are known to be of great importance
during bone healing, especially for the replacement of cartilage (Colnot and Helms, 2001;
Carano and Filvaroff, 2003; Carvalho et al., 2004). The model includes angiogenesis
implicitly in the function that regulates cartilage degradation and endochondral replacement.
Thus, the model predictions of cartilage degradation during endochondral ossification as a
critical parameter, strengthens the correspondence with biological knowledge.
7.4.2 Taguchi factorial analysis
The Taguchi fractional analysis is a powerful statistical tool to fairly assess the relative
importance of parameters while reducing experimental effort (Dar et al., 2002). In this study it
was successfully used to determine relative importance of parameters in a computational,
mechanistic cell model. Full factorial designs can provide more information about interactions
between factors than the Taguchi method. However, when the numbers of factors and levels
are large, the total number of simulations required becomes impossible. Although we chose a
more computationally expensive higher resolution design (Resolution IV, L64 array), only 64
analyses were required to retrieve information about the importance of 26 parameters, rather
than 226 simulations for a full factorial analysis. A factorial experiment is able to identify the
most important factors and determine the response of a system within the parameter space
chosen, as well as predicting that response for a given set of input parameters. It is sensitive to
the range of values used as well as to the underlying mechanics (Dar et al., 2002). The range
of each value was chosen according to literature (Chapter 6). When the values were well
defined in literature, a smaller parameter space was chosen than for less established values.
For all parameters that used purely estimated values, the relative space was identical.
7.4.3 Outcome analysis
The results from the parametric study are highly dependent on the outcome variables chosen.
Most studies employing this technique first determine an ‘ideal stage’ as a baseline
comparison. There are also other studies using this approach to solely focus on determining
the factors that are of most importance to the system (Meakin et al., 2003), Similar approaches
were taken in this study. However, during fracture healing it is difficult to find one parameter
which can be used to characterize the performance of the system. We explored possibilities of
using existing clinical and experimentally used scoring systems (An et al., 1999), and
proposed a similar scoring system more appropriate for our model. By evaluating three
outcome parameters, we were able to study the performance in terms of 1) expected sequential
events, 2) evaluate the progression of healing at specified time points and 3) calculate the
absolute time required until complete healing. After evaluating these three outcome
parameters individually and comparing the results, we believe to have characterized the
system well, in terms of which parameters are of the greatest influence.
105
Chapter 7
When studying tissue differentiation and bone healing computationally, there are also other
factors that affect the results, such as material parameters, geometry and volumetric growth.
However, due to the already large number of factors in this experiment, we decided to focus
only on the parameters in the newly developed mechanistic cell model. The results of the
parametric study are based on this particular cell model (Chapter 6), and at this stage, care
must be taken when extrapolating the results to bone healing in general. However, the
outcome of this study does indicate parameters of importance for computational modeling of
this phenomenon, and indicate where most concern should be taken when describing the
processes.
Some variables, which our approach suggests are of importance to the outcome, have not been
established well experimentally in the literature. Therefore we suggest that more attention is
paid to these variables. One such example is the degradation and replacement of cartilage,
which was identified as one of the key factors, but for which no experimental quantification or
direct dependencies could be found. In general, the cell processes differentiation, migration
and apoptosis, were found to have less impact on the outcome variables than the cell
proliferation rates and tissue formation and degradation. This does not mean that they are not
crucial for successful bone healing, but that their relative rates of importance within the
chosen parameter space were not as influential.
7.5 Conclusions
For the first time, this study employed ‘design of experiments’ methods to evaluate relative
importance of cellular parameters in mechanobiological models. It was able to identify the
most critical parameters in a computational model of bone healing based on cellular activities.
The parameters and processes that were found to be most important are similar to those that
have been suggested crucial steps, from a biological standpoint. To experimentally prove such
suggestions is extremely challenging, and computational analyses have shown to be a valuable
tool to strengthen these ideas. Our study further suggested that future experimental efforts
should be undertaken to understand the processes and rates of cartilage production and
degradation or replacement during endochondral ossification, as well as further computational
studies of interactions between parameters. Establishing experimental values for the parameter
of greatest importance will be a necessary part of the validation process of future
computational mechanobiological models.
106
8
8
Remodeling of fracture callus in
mice can be explained by
mechanical loading
Small animals have similar patterns of healing as larger animals
during inflammatory and reparative phases. However, in the final
part of bone healing, remodeling is different. In mice the callus
gradually transforms into a double cortex, which thereafter merges
into one cortex. These differences could be due to biological
differences in species, or to differences in mechanical loading. The
study presented in this chapter investigates remodeling of the
fracture callus in mice and aims to establish whether the patterns of
remodeling can be explained by mechanical loading.
This study demonstrates how a difference in major loading
directions can explain the differences between the remodeling
phases in small rodents and larger mammals, including humans.
Although biological differences between species may also be
involved in this process, the contrasting behavior in post fracture
remodeling can be explained by differences in loading direction.
The content of this chapter is based on manuscript VI
107
8
Chapter 8
8.1 Introduction
Bone fracture repair is usually divided into an inflammatory phase, a soft and hard callus
reparative phase, and a final remodeling phase (Chapter 2.2) (Cruess and Dumont, 1975;
Einhorn, 1998b). Most current research focuses on the reparative phase, during which the
stiffness of the bone is restored (Einhorn, 1995; Marsh and Li, 1999). Only few studies have
characterized the remodeling phase of healing, although behavior in this phase is important,
since in this phase the full strength of the bone returns and the chance of re-fracture decreases
(Rüedi et al., 2007).
Remodeling is generally described as the replacement of woven bone, that is rapidly laid down
during the reparative phase, by highly organized lamellar bone with a well organized structure
(Marsh and Li, 1999). In this process, the periosteal and endosteal callus are slowly resorbed
until the original shape of the bone is restored (Owen, 1970; Willenegger et al., 1971).
Resorption of the endosteal callus also coincides with reestablishment of the intramedullary
canal and thüüüe original blood supply (Rhinelander, 1968). It is thought that fluid shear
stresses in bone modulate the remodeling activities (Bakker et al., 2004).
Small animal models are becoming more frequently used in studies of fracture repair
(Hiltunen et al., 1993; Thompson et al., 2002; Cheung et al., 2003; Manigrasso and O'Connor,
2004; Holstein et al., 2007). In particular, mice have been used extensively in basic research of
developmental biology (Hiltunen et al., 1993; Tay et al., 1998). They have several benefits,
including cost effectiveness and ease of experimentation. The availability of knockout mice
and the elucidation of the entire murine genome has further extended the scientific importance
of murine models of human physiology (Mundlos and Olsen, 1997a; 1997b; Einhorn, 1998b).
The inflammatory and reparative phases of fracture healing in mice are similar to those in
larger mammals, including humans. However, the remodeling phase has not been thoroughly
investigated.
During analysis of a murine model of fracture repair, a phenomenon was noted that differs
from that in larger mammals or humans (Gröngröft et al., 2007). At the end of the reparative
phase, an external large callus had developed, similar to that in large mammals, and periosteal
and endosteal bony bridging was achieved. However, during remodeling the callus gradually
transformed into double, concentric cortices. Later, the two cortices, equally thick and dense,
merged together into one cortex. This behavior has not been described in literature, but seems
to occur in fracture remodeling in rodents with no fixation or flexibly fixed fractures
(Gerstenfeld et al., 2006). Why do mice respond differently than larger mammals during bone
fracture remodeling? The contrasting response between mice and larger mammals could be
due either to biological differences between species, or to distinct mechanical loading patterns.
This study describes the remodeling phase in mice and aims to provide an explanation for the
differences seen between rodents and larger mammals. During murine gait, the knee and ankle
are always flexed, whereas in human gait, the knee and ankle are more extended. We
hypothesize that the differences in bone geometry during the remodeling phase of fracture
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Remodeling of fracture callus in mice can be explained by mechanical loading
healing observed in mice, compared to large mammals, can be explained by the difference in
main mechanical loading mode. To investigate this hypothesis, a bone remodeling theory was
used (Huiskes et al., 2000; Ruimerman et al., 2005), in which adaptation of bone mass and
geometry over time were modulated through osteocyte mechano-sensing and signaling. The
predicted bone distributions for various loading cases were then compared to the remodeling
behavior observed in an in vivo murine fracture healing model.
8.2 Method
8.2.1 Murine fracture healing model
The in vivo experiment described in this study was part of a larger animal study, which
characterized fracture healing in a mouse model with plating fixation of two different
stabilities (Gröngröft et al., 2007). The study employed one rigid and one flexible plate, and
focused on the altering healing pathways during the reparative phase of healing caused by
these different plate stiffnesses. Part of that experiment (late time points, groups with flexible
plates) was re-analyzed for the purpose of this study and used to determine the morphology of
the remodeling phase of healing fracture calluses. In short, a flexible bridging plate was
attached to the anterior aspect of the mid-femur with four angle-stable screws through a lateral
approach (Matthys-Mark, 2006) (Figure 8-1a). It was used to stabilize a 0.45mm middiaphyseal gap osteotomy in the femora of female C57BL/6 mice, 20-25 weeks of age (RCC
Ltd, Füllinsdorf, Switzerland), (Figure 8-1b). All procedures were approved by the Animal
Experimentation Commission and followed the guidelines of the Swiss Federal Veterinary
Office for use and care of laboratory animals.
Figure 8-1: a) the flexible locked bridging plate used to fixate the osteotomies in the
experimental study (Matthys-Mark, 2006; Gröngröft et al., 2007) and b) postoperative
radiograph showing the osteotomy and bridging plate in place.
The mice were able to freely weight-bear postoperatively. They were euthanized after 21, 28
or 42 days of healing (n=10 per healing duration). After excision, four point bending stiffness
of the osteotomized femur was determined as a percentage of the stiffness of the contralateral
femur by non-destructive mechanical testing. Femora were bent around the former plate
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Chapter 8
position on the compression side (deformation rate of 2.1 deg/min), with a load limit of 4.5
Nmm. The femora were then fixed in methanol for 10 days. Thereafter, µCT imaging of the
healing bones was performed (µCT 40, Scanco Medical, Bassersdorf, Switzerland), with an
isotropic voxel resolution of 12 µm. Three-dimensional segmented reconstructions were used
for qualitative evaluation of sub-volumes of the bone. A thresholding algorithm protocol, was
used to segment tissue into three attenuation types, i.e. soft tissue (< 145); woven bone (low
mineralization, 145 to 360); and lamellar bone (high mineralization, > 360, in per mille of
maximal image gray value) (Gabet et al., 2004).
For histological analysis, femora of 8 mice per time point were embedded in
methylmethacrylate, and serially sectioned on a circular saw. The mid-longitudinal section of
each femur was stained with 1% Toluidin blue (Fluka) for 15 minutes and blot dried after
washing in deionized water. Sections in the region between the outer screw holes were
examined with a macrofluoroscope at x64 magnification (MacroFluo™, Leica Microsystems,
Heerbrugg, Switzerland). A custom macro (AxioVision, KS400, Zeiss) was used to measure
the area of woven bone, lamellar bone, cartilage and total callus area. Each processed image
was visually checked for proper segmentation.
8.2.2 Bone remodelling
Bone remodeling was simulated based on the established theory by Huiskes and co-workers
(Huiskes et al., 2000; Ruimerman et al., 2005). It assumes that osteocytes sense the local
strain-energy-density (SED) and send a corresponding signal to the bone surface, which either
activates osteoblasts or inhibits osteoclasts. It is assumed that the osteocyte signals decay
exponentially with distance. Therefore, only osteocytes within a distance smaller than the
decay distance D, are taken into account when calculating the total stimulus (Ruimerman et
al., 2005). The total stimulus P is given by the sum of all osteocyte signals within the
influence distance of location x (Ruimerman et al., 2005), as
n
P( x, t ) = ∑ f ( x, x k ) μU k ,
(Eq 8-1)
k =1
where n is the number of osteocytes within the influence volume, μ the osteocyte
mechanosensitivity and Uk the strain-energy-density sensed by osteocyte k. Depending on the
total stimulus bone can be either formed or resorbed. Resorption is activated when the
stimulus is lower than the resorption threshold Ctr, or by random microcracks formed in the
bone (Huiskes et al., 2000). The probability of osteoclastic resorption is given by
p resorb = Rmax −
Rmax
P ( x, t ) ,
Ctr
(Eq 8-2)
where Rmax is the maximal chance of resorption. It is assumed that osteoclasts resorb a fixed
amount of bone. Hence, the change in bone volume due to osteoclastic resorption at any time
point (dVr / dt) is given by
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Remodeling of fracture callus in mice can be explained by mechanical loading
dVr ( x, t )
= Vcl ,
dt
(Eq 8-3)
where Vcl is the amount of bone resorbed by one osteoclast. When the local stimulus is higher
than the formation threshold ktr, osteoblastic bone formation will occur. The amount of bone
formed (dVf / dt) is given by
dV f ( x, t )
dt
= τ ( P( x, t − k tr )) ,
(Eq 8-4)
where τ is a material constant called the proportionality tensor. The change in bone volume is
calculated as the sum of the resorbed and deposited bone volume, which then allows the total
amount of bone in each integration point to be updated. Based on the local amount of bone,
local bone volume density ρ is calculated, which allows the homogenized stiffness of the
volume to be updated as
E = E max ρ ( x, t ) γ ,
(Eq 8-5)
where Emax is the maximal bone stiffness and γ a material constant (Carter and Hayes, 1977;
Currey, 1988).
The bone remodeling theory was implemented for calculations into ABAQUS (v. 6.5). The
osteocytes were positioned at random locations within the integration-point volume. The
number of osteocytes per integration point volume Vip was defined as
nosteo = Vip ρ osteo ,
(Eq 8-6)
where ρosteo is the osteocyte density. When more bone was formed than what was present
within the integration point volume, the rest of the bone volume formed was distributed over
the surrounding integration point volumes. The same was done for over-resorption. All the
parameter values that were used in the bone remodeling theory are displayed in Table 8-1.
111
Chapter 8
Variable
Symbol
Unit
Value
Osteocyte density
n
mm--3
96 000 a
Osteocyte
mechanosensitivity
µ
mol mm J-1 s-1 day-1
1
Osteocyte influence
distance
D
µm
50 b
Formation threshold
ktr
mol mm-2 day-1
2.0 · 10-4
Proportionality factor
τ
mm5 mol-1
5.0 · 10-4
Resorption amount per
cavity
R
mm3
Emax
GPa
Maximum elastic
modulus Bone
1.5 · 10-3
c
5.0 d
Poisons ratio
ν
0.3
Exponent gamma
γ
3.0 e
Table 8-1: Parameters and constants used for the bone remodeling theory. a Mullender et al.
(1996), b Mullender and Huiskes (1995) , c Eriksen and Kassem (1992), d Schriefer et al.
(2005), e Currey (1988) and Carter and Hayes (1977) .
8.2.3 Micro-CT imaging and conversion
Fracture calluses after 21 days of healing were assumed to be starting points for the
remodeling phase. At this time point all experimental calluses showed comparable
morphology, and five representative bones were chosen for the simulation. Using the µCT
images, bone tissue was segmented from soft tissue as described above, and the full range of
output density data was converted to mineral content and bone density. Two-dimensional
midsagittal sections were converted to two-dimensional finite element meshes, where each
voxel provided initial bone density for one integration point (Figure 8-2). Hence, each linear
4-node plain strain element was 24μm x 24μm. Bone was modeled as a homogenous linear
isotropic material with a Poisson’s ratio of 0.3 and a maximum Young’s modulus of 5GPa
(Schriefer et al., 2005). Elements with no bone were assumed to be filled with marrow, and
were modeled with a Young’s modulus of 1MPa and a Poisson’s ratio of 0.3. Bone volumes
and densities were recorded for each location and time point. The areas were classified as
woven or lamellar, according to the same threshold values as employed experimentally, and
were used for quantitative comparison between experimental and computational results.
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Remodeling of fracture callus in mice can be explained by mechanical loading
Figure 8-2: Finite element meshes created from the five animals mid-sagittal section of the
μCT reconstruction after 21 days of healing. The meshes were used as the initial
computational models and assumed to be the starting point for the remodeling phase.
8.2.4 Loading conditions
The main purpose of this study was to assess the effect of loading mode on the bone
remodelling. In large mammals, loading is well defined both in direction and magnitude,
whereas in mice the loading is mainly unknown. To investigate if direction of loading could
explain the bone remodelling differences observed, we assumed two different loading
directions, either an axial force, or a bending moment. The force was tied to all nodes on the
proximal cortical ends, while the distal cortical ends were rigidly fixed. It was applied either in
the y-direction (axial) or around the z-axis producing a bending moment (Figure 8-3a).
Figure 8-3: a) The load was determined by applying a force with varying magnitudes axially
or around the bending axis (z) and recording the steady state cortex diameter. b) The load of
0,75N was chosen for the simulations because it resulted in steady state thicknesses, both with
axial and bending loads, which were close to those measured experimentally. c-d) The steady
state diameters recorded c) axially and d) under bending due to loads of 0.25N, 1.0N and 5N.
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Chapter 8
The physiological magnitudes of the loads in mice are also unknown, and were therefore
estimated by first running the simulation with the bone remodeling algorithm with loads
applied axially or as a moment of varying magnitudes on an unfractured cortex. The
magnitudes analyzed were 0.20N, 0.50N 0.75N, 1N, 2N, 3N and 5N. The load magnitudes
that resulted in steady-state cortical geometry, similar to intact femora, were selected. They
were thereafter used for simulations of the experimentally collected fracture calluses after 21
days of healing. The computational predictions from day 21 onward were compared with the
experimental results after 28 and 42 days.
8.3 Results
8.3.1 Murine fracture callus remodelling
During the experiment, 1/10 animals in the 21 and 28 days groups and 4/10 in the 42 days
group were excluded, due to failure of the flexible plate because of bending of the wires
bridging the gap, or due to technical problems while removing the plate. At 21 days,
radiographs exhibited abundant amounts of callus around the fractures. The callus size reached
its peak around 21 days post fracture, which then decreased to day 42. Mechanical testing
provided further indication of remodeling activity. Between 21 and 42 days of healing, the
stiffnesses determined by mechanical testing of the osteotomized femora had increased from
50-74 % of those measured in the contralateral intact femora (Table 8-2) (Gröngröft et al.,
2007).
The quantitative μCT evaluation showed that the amount of woven bone was greater at day 21,
compared to later time points, while the amount of lamellar bone was greater at days 28 and
42 (Table 8-2). Qualitatively, remodeling of the fracture callus began around day 21, when
woven bone was evenly distributed throughout the callus (Figure 8-4 a-b) (8/9 mice). After 28
days, most of the woven bone within the periosteal callus was resorbed, while the periosteal
part of the callus and the direct cortical gap continued to become denser (Figure 8-4 c-d). A
dual cortex had clearly developed in 6/9 mice, and was evidently under development in the
remaining 3/9 animals. There was still some endosteal callus remaining, with mainly new
woven bone in the gap area and in the outer cortex. By 42 days, most of the endeosteal callus
had resorbed (Figure 8-4 e-f). The double cortices, now visible in all mice (6/6 mice), were
equally thick and the bone directly bridging the fracture gap and the bone in the outer double
cortex was predominantly highly mineralized lamellar bone. The diameters of the callus and
the distances between the two cortices were smaller after 42 days of healing as compared to 28
days.
The histological findings were consistent with the results of the μCT analysis. Quantitatively,
the amount of woven bone decreased between day 21 and day 42, while the amount of
lamellar bone increased (Table 8-2). The double cortex formation was also visible in the
histological sections (Figure 8-5). It generally developed between 21 and 28 days and
remained, although smaller at 42 days post fracture.
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Remodeling of fracture callus in mice can be explained by mechanical loading
Figure 8-4: Representative examples of the μCT dataset after a-b) 21, c-d) 28 and e-f) 42 days
of healing. The top part of the figure, a), c) and e) shows mid-sagittal cross sections and in the
lower part of the figure b), d) and f), transverse cross-sections through the fracture gap are
displayed.
Figure 8-5: Histology of the mid-sagittal section through the osteotomy gap after a) 28 days
and b) 42 days post-op, stained with toluidine blue (plate position, above). The upper
periosteal area is partly blocked by the plate and bridging in the external periosteal callus is
only possible in the area where the wires were. The double cortex formation is visible after 28
days. It remained until 42 days, although the callus size was reduced after 42 days as woven
bone was progressively remodeled into lamellar bone.
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Chapter 8
21 days
Experimental data
28 days
42 days
Mechnical testing
Stiffness % 50% (± 15 %)
61% (± 16%) 74% (± 20%)
μCT (mm3)
woven bone
lamellar bone
4.5 (± 0.7)
2.6 (± 0.4)
2.7 (± 0.8)
3.3 (± 0.5)
1.0 (± 0.3)
3.1 (± 0.3)
μCT dual cortex (DC)
DC / total mice
0/9
6/9
6/6
Histomorphometry (mm2)
woven bone
lamellar bone
cartilage
1.8 (± 0.4)
0.3 (± 0.1)
0.06 (± 0.09)
0.5 (± 0.3)
1.6 (± 0.5)
0.08 (± 0.2)
0.4 (± 0.2)
1.7 (± 0.4)
0.06 (± 0.1)
Table 8-2: Quantitative experimentally results from mechanical testing, μCT and
histomorphometry. The experimental reported numbers are the mean values from all analyzed
mice, and the numbers in prentices are standard deviations about the mean. The
histomorphometric data is compared to those calculated with the computational model in
Figure 8-8.
8.3.2 Computational predictions
Loading
To estimate the loads for a mouse femur, a range of loading magnitudes were applied on an
intact cortex (Figure 8-3). The load chosen, resulting in steady-state cortex diameters, was a
force of 0.75N, either applied axially or about the bending axis (Figure 8-3a). With this force,
steady state cortical thicknesses closest to those measured experimentally were recorded
(Figure 8-3b). Figure 8-3c-d shows the variations in steady state geometries with some of the
load magnitudes that were investigated.
Axial simulation
All five samples predicted similar evolution of bone tissue distributions. With an axial force of
0.75N, the callus began the remodeling process by relatively fast subsequent resorption of the
periosteal callus (Figure 8-6a, iteration 1-10, or day 21-31 of healing). At the same time, the
density in the area close to the cortex and immediately surrounding the fracture gap increased.
Thereafter, most of the endosteal callus was resorbed, along with a further increase in density
in the intercortical gap (Figure 8-6a, iteration 10-25). By iteration 30, only very little endosteal
callus remained. After 50 iterations the cortex was almost completely restored, and after 100
iterations, there was no evidence of the fracture (Figure 8-6a).
Bending simulation
All five samples predicted similar evolution of bone tissue distributions. When a bending
moment was applied, the remodeling process and the spatial and temporal bone distributions
were different (Figure 8-6b). Initially, the external periosteal-callus shell and the immediate
116
Remodeling of fracture callus in mice can be explained by mechanical loading
cortical gap became gradually denser, while the trabecular bone inside the periosteal callus
was resorbed (Figure 8-6b, iteration 1-10, or day 21-31 of healing). Thereafter the resorption
of the endeosteal callus was initiated, while the cortical gap and the periosteal shell became
even denser (iteration 10-25). After iteration 25 a dual cortex had developed. At that point
most of the newly developed additional cortex had a lower bone density, compared to the
original cortex. Over time, both cortices were of equally high bone density (Figure 8-6b,
iteration 50). From iteration 50 onwards, the dual cortex slowly migrated together again, and
over time, the original cortex was restored (iteration 150).
Figure 8-6: The predicted bone distributions and densities during fracture callus remodeling,
when the callus after 21 days of healing was stimulated with a) an axial load, and b) a
bending moment. a) axial: The periosteal callus gradually resorbed, followed by resorption of
the endosteal callus. Lamellar bone bridged the direct fracture gap between iteration 10-25,
and at iteration 100 the original cortex was restored. b) bending: A dual cortex developed
where the external periosteal callus and the direct fracture gap remodeled into high density
lamellar bone (iteration 25). Thereafter the dual cortex merged together slowly and one single
cortex was restored after 150 iterations
8.3.3 Comparison between experimental and simulated results
There was almost no resemblance between experimental data at 28 and 42 days of healing and
the simulations under axial load. In the computational model, the entire callus was resorbed
and the original cortex restored, which was also evident from the strain-energy-density (SED)
distributions (Figure 8-7). The SED after 5 iterations, i.e. in-between experimental days 21
and 28, showed high stimulation at the cortex and the immediate fracture gap surroundings.
This compares well with normal fracture healing in larger mammals. In contrast, the
simulations in which the callus was loaded in bending corresponded well with the progression
of bone remodeling observed experimentally. A dual cortex was formed, which became
gradually denser and thicker, and thereafter merged. Experimentally the merging of the dual
cortex was not observed, but the distance between the cortices at day 42 was less than after 28
117
Chapter 8
days, indicating that they were slowly merging together. The SED distributions after iteration
5 of the bending stimulation (Figure 8-7) showed the same progression, where both external
periosteal callus line and direct cortex gap was stimulated to form bone. Examination of the
SED distributions (Figure 8-7a-b) clearly showed the differences between mechanical
stimulation in axial and bending direction. Already during the first couple of iterations, the
dark areas of SED predicted that a double cortex would form from bending stimulation, while
the load transmission from axial stimulation predicts that the callus will be resorbed.
Furthermore, Figure 8-7 shows the SED distribution after the cortex was restored: 100 (axial)
or 150 (bending) iterations still left a small ‘bony bump’ on the cortex, at the former location
of the fracture gap. The SED distributions around those areas indicated that this area would be
restored over time.
The computational results from the bending simulations were also compared to quantitative
data from the experimental results (Figure 8-8). The total callus area, and areas of woven and
lamellar bone that were calculated by the computational model, were similar to the
histomorphometrically determined areas (Figure 8-8). The time sequence in the computational
model was slower, but still all computationally quantified areas, except the area of woven
bone after 28 days, were within one standard deviation of those determined experimentally.
Figure 8-7: The distributions of strain-energy density explain the differences in remodeling
behavior observed with the two mechanical loading conditions. Dark gray display areas
where bone will form, and light gray areas where resorption might be activated. With an axial
load, already after 5 iterations, it is shown that the strain energy density is almost only
stimulating bone formation in the direct cortical gap and will resorb the external callus
rapidly. Hence it also restores the cortex faster. With a bending moment both the external
periosteal callus and the direct cortical gap experiences high strain energy densities.
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Remodeling of fracture callus in mice can be explained by mechanical loading
Figure 8-8: Quantitative comparison of computationally calculated tissue areas and
equivalent experimentally quantified data. The calculated tissue areas of total callus, woven
bone and lamellar bone were determined from the computational model and compared with
experimentally quantified tissue areas of total callus, woven bone and lamellar bone at 21, 28
and 42 days post fracture are displayed as mean and standard deviation.
8.4 Discussion
This study demonstrates how a difference in major loading modes can explain the distinct
remodeling characteristics of fracture healing in small rodents compared to larger mammals,
including humans. For the first time, to our knowledge, this behavior has been characterized,
which includes temporary dual cortex formations during post fracture remodeling in mice. The
progression of healing and remodeling were monitored experimentally, and qualitatively and
quantitatively compared to simulations with a bone remodeling theory under different loading
modes. Five animals were simulated and the outcomes of all animals were similar. The
simulated patterns for the callus loaded in bending were very similar to the experimentally
observed remodeled bone morphology in mice, both after 28 and 42 days. When axial load
was applied, there were no similarities between our predictions and the actual mice fracture
healing patterns. Instead, the bone density distributions resembled the general remodeling
pattern during fracture healing as observed in larger mammals. Hence, the murine remodeling
behaviour observed experimentally can be explained by a difference in main mechanical
loading mode. This difference in loading mode can be due to the construction of their
skeletons.
8.4.1 Experimental observations
The experimental model, for which a flexible plate was used to fix a mid-diaphyseal femur
gap osteotomy, clearly demonstrated the formation of a double cortex during the remodeling
phase of healing. It developed around 28 days post fracture (5/9 mice) and was evident in all
animals after 42 days of healing. Many murine models of fracture repair have been developed,
but very little description of the remodeling phase or of this behavior can be found in the
119
Chapter 8
literature. Most investigators focused on other aspects of healing (Meyer, Jr. et al., 2003;
Manigrasso and O'Connor, 2004; Holstein et al., 2007; Meyer and Meyer, Jr., 2007), or ended
their experiments prior to remodeling (Colnot et al., 2003a; Lu et al., 2007; Meyer and Meyer,
Jr., 2007). However, three studies have shown radiographs after 4 weeks of healing in a ratfemur fracture healing-model in which a double cortex can be recognized (Desai et al., 2003;
Meyer, Jr. et al., 2003; Ashraf et al., 2007). Furthermore, a study by Gerstenfeld et al
(Gerstenfeld et al., 2006) provides a three dimensional reconstruction of fracture callus
morphogenesis. This study also showed similar double-cortex formation in a reconstructed rat
femur fracture callus from µCT data after 35 days of healing. Hence, this appears to be a
general phenomenon when fractures in rodents undergo secondary fracture healing.
Mouse models are used in studies of bone healing, with the desire to extract the results to
physiological and clinical problems encountered in humans. Therefore, it needs to be ensured
that the biological behavior in mice is comparable to that in humans. The current and
previously mentioned studies displays that the remodeling phase of healing is different.
However, this study provides a possible explanation for these differences, i.e. the mechanical
loading mode. Hence, the formation of dual cortex during the remodeling phase of healing
does not have to be seen as a biological difference between species and is not a reason to
choose other animal models.
8.4.2 Mechanical loading
The loading conditions in this model have been simplified. The real loading conditions in mice
are unkown. Therefore, the complex loading in mice femora was replaced by only the two
extreems, i.e. either axial load or bending moment. All additional muscle forces were
neglected. The hypothesis in this study is derived from the assumption that mechanical
loading in long bones in mice is significantly different to that in larger mammals. The actual
loading conditions in humans (Bergmann et al., 1993; 2001) and larger animals such as sheep
(Duda et al., 1998; Taylor et al., 2006) are well described. Large quadrupeds, such as sheep,
are commonly used for fracture studies because of their similarity in long bone loading to
humans. All long bones experience some bending moments, likely to give them their tubular
shape. However, the amount of bending moments relative to the amount of axial load is low.
Unfortunately, the gait and contact force mechanics in rats and mice are not well characterized
(Howard et al., 2000; Clarke et al., 2001). However, the construction of their skeleton implies
that the relative amount of bending moments is most likely higher than in humans. This was
also suggested by a characterization of joint mechanics in a rabbit model (Gushue et al.,
2005).
The constructed finite element models originated from 2D slices of the 3D µCT
reconstructions that were available experimentally. A 2D model was chosen since a 3D model
would have been computationally expensive. The proximal and distal ends of the cortex in the
computational model were tied to ensure that their relative distance remained, and that the full
3D structure of the cortex was mimicked. By creating a 3D model, additional, more complex
and realistic loading conditions could be investigated. However, since the loading in mice is
not characterized, it would be very difficult to accurately determine which loading conditions
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Remodeling of fracture callus in mice can be explained by mechanical loading
to apply. Hence, at this stage, simplified loading as used in this study can serve to provide a
possible explanation for why the remodeling behavior, i.e. the temporal bone distributions, in
mice is different from those seen in larger mammals. In addition, the bone remodeling theory
does not account for the contribution from cartilage and mineralized cartilage completely.
However, since the amount of cartilage remaining after 21 days of healing was very small
(Table 8-2), the contribution from cartilage to the stiffness of the callus was assumed low.
8.4.3 Bone remodelling theory
The model assumes a constant osteocyte density both over different bone types and densities
and over time. However, it is known that areas of high bone turnover, such as a fracture callus,
are characterized by a higher number of bone cells (Miyamoto and Suda, 2003). The osteocyte
populations in woven bone is believed to be larger than in lamellar bone (Buckwalter et al.,
1996a; 1996b) and to decrease with remodeling until normal levels are restored. Recent
studies that have quantified osteocyte density have concluded that the densities in the center
region of a fracture callus are about 100% larger than in normal lamellar bone (Hernandez et
al., 2004). The same study also showed that the osteocyte density in woven bone formed
during fracture healing through intramembranous or endochondral pathways, can be different
(Hernandez et al., 2004). This could be implemented in our model by starting the simulation
with variations in osteocyte densities, depending on the initial bone density, and then updating
those during the bone remodeling process. However, it was shown by Mullender and Huiskes
(1995), that the osteocyte density in the model only affected the remodeling rate. Hence, it
would not affect the remodeling pattern or the conclusions of this study.
Computational models of fracture repair have previously centered their attention on the
reparative phase (Carter et al., 1998; Claes and Heigele, 1999; Lacroix and Prendergast, 2002;
Doblare et al., 2004). Some mechano-regulation models have included simplified conditions
for when remodeling or resorption of the fracture callus would occur (Lacroix and
Prendergast, 2002) (Chapter 3). However, no former computational models of fracture healing
have predicted the remodeling behavior during post fracture remodeling in a mechanistic
manner. The complexity shown in murine fracture healing models during the remodeling
phase, including the double cortex formation described in this study, encourages use of a more
detailed model such as a bio-physical bone remodeling theory for computational investigations
of the remodeling phase. The fact that these contrasting differences in post-fracture
remodeling between species can be induced by different loading patterns further supports the
hypotheses underlying current load-based bone-remodeling theories (Huiskes et al., 2000).
The remodeling algorithm has previously been shown to apply to cortical and trabecular bone
remodeling (Ruimerman et al., 2005; Ruimerman, 2005), and is able to describe osteoporotic
changes in bone (Ruimerman, 2005). In this study the unifying theory of bone remodeling was
applied to small animals for the first time. However, it has not previously been used or
validated on whole bones or in fracture repair. In this study the unifying theory of bone
remodeling was applied to small animals for the first time. Although mice bones are different
in many aspects (lack of osteons, rather thin cortical bone and less developed trabecular
structure), this study was successful. Equally important, this study is also novel in showing
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Chapter 8
that the unifying theory for bone remodeling (Huiskes et al., 2000) applies to an abnormal
bone remodeling process, i.e. post fracture remodeling.
8.5 Conclusion
In conclusion, a computational bone remodeling theory, where activities of osteoblasts and
osteoclasts are modulated by external loads through osteocyte signaling, can produce distinct
remodeling patterns, depending on the loading regime applied. The bending load, which gives
rise to the dual cortex type remodeling, is likely to mimic the loading of mouse femurs (knee
and ankles always flexed). Hence, the contrasting behavior during the remodeling phase
observed in mice compared to humans, could be explained by differences in mechanical
loading and does not necessarily arise from biological differences.
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9
9
Discussion and conclusions
The last chapter summarizes the current thesis and its conclusions,
discusses the logic and how it incorporates with past research and
future prospects. It also discusses the potential drawbacks
associated with the development of more detailed models to clarify
and further investigate the relationship between mechanical
stimulation, biological relations and bone fracture healing. It
predicts how knowledge of mechanical factors can potentially be
used to enhance fracture treatment. It also elucidates the role that
computational models can play in the development of those
treatments.
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9
Chapter 9
9.1 Overview
In the first chapter, the need for further research in the field of bone mechanobiology and its
regulatory pathways were established. Fracture healing remains a costly and inconvenient
problem, mainly due to the incidence of delayed or non-unions. Over recent decades, the
incidence has been rising due to aging of the population. Increased duration of hospital stay
due to failure of bone healing has a major impact on medical and hospital resources and
adequate recovery of the patient is essential. Understanding the basic biology and the
mechanoregulatory mechanisms of bone regeneration should lead to cheaper and more
effective treatments.
This work has employed computational models to investigate the role of mechanical factors in
tissue differentiation during bone healing. The underlying hypothesis is that the magnitude of
mechanical stimuli at a local level (tissue or cell) influences the temporal and spatial tissue
distributions which determine the biological repair process. The cells are proposed to act as
sensors, which respond actively to their mechanical environment. To investigate this
hypothesis such stimuli must be determined at numerous local positions within the healing
tissue, which would be impossible to measure experimentally. Computer modeling allows
such scales of sampling and is consequently having a profound effect on all fields of scientific
research. However, such models require experimental validation and the aim of this thesis has
been to do so by comparing computational models with data from well-characterized
experiments. Chapters 3-7 focus on the reparative phase of bone healing, while Chapter 8
focuses on the remodeling phase of healing. The objectives for each of the chapters are
summarized below, followed by the most important conclusions (Chapter 9.2).
In Chapters 3-5, established mechanoregulation algorithms were implemented and compared
with each other in terms of their capacities to predict healing patterns observed in vivo. The
more diverse healing conditions that can be correctly predicted, the more evidence there is for
the universal application of the model. Therefore, biological data was sought for healing under
a variety of interfragmentary motion modes. The models were applied to ‘normal’ clinical
fracture healing under axial compression (Chapter 3), experimental healing under carefully
controlled axial compression and torsional rotation (Chapter 4), as well as to bone
regeneration during distraction osteogenesis (Chapter 5), which places the tissue under
tension. The computational predictions for the different algorithms were compared with the
experimental results both qualitatively and quantitatively. Shortcomings of the algorithms
were identified and strategies to overcome them were shaped.
One such strategy was to describe the cellular behavior in detail. This involved the
development of a more mechanistic cell model (Chapters 6-7). Many key aspects of bone
healing were incorporated by direct coupling of mechanical stimuli with cell phenotype
specific actions. The model was applied to bone healing and successfully predicted well
established aspects of normal and pathological bone regeneration (Chapter 6). Design-ofexperiments statistical methods were employed to perform an extensive sensitivity analysis of
healing patterns to the parameters introduced in the cell model (Chapter 7). In Chapter 8 post-
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Discussion and conclusions
fracture remodeling in mice was characterized and the phenomenon of temporary dual cortex
formation was presented for the first time. An explanation for this phenomenon was developed
by utilizing an established bone remodeling algorithm and comparison of predictions for
different mechanical loading modes with experimental data, both quantitatively and
qualitatively.
9.2 Conclusions
•
All of the established algorithms investigated, satisfactorily predicted the most
important aspects of normal fracture healing. Differences were observed, but these
were not extensive and no algorithm could be rejected or determined to be superior to
the others. Furthermore, local deviatoric strain alone was able to simulate the tissue
differentiation during normal fracture healing equally well to the established
algorithms, suggesting that the deviatoric deformation component might be the more
significant mechanical parameter in the control of tissue differentiation during
secondary bone healing.
•
To distinguish the predictive capacities of algorithms in vivo data for both axial
displacement and torsional rotation was employed for comparison of the four
algorithms that satisfactorily simulated normal fracture healing. Torsional rotation,
which eliminated local volumetric deformation, elicited differential responses between
algorithms. None was entirely satisfactory in predicting both axial compression and
torsional rotation, but the algorithm regulated by deviatoric strain and fluid velocity
resulted in the closest predictions to the experimental results (Prendergast et al., 1997).
It was speculated that the most profound reasons for the shortcomings with this
algorithm were lack of tissue volumetric growth and an insufficiently mechanistic
description of cellular behavior.
•
Volumetric growth was implemented in a soft-tissue model using a biphasic swelling
approach. Tissue differentiation during distraction osteogenesis was successfully
simulated using the mechano-regulation algorithm based on deviatoric shear strain and
fluid velocity. It correctly predicted the course of tissue differentiation from distraction
to consolidation in an experimental model of bone segment transport. Predicted
reaction forces, including tissue relaxation, were not completely correct, which was
attributed to the use of fixed material properties. The rate of bone formation increased
with distraction rate and frequency, similarly to experimental observations, suggesting
that this algorithm could be used to optimize treatment protocols.
•
A new mechano-regulation model based on cell activity was developed and the
importance of describing cell-phenotype-specific processes was determined. The
additional value of the more mechanistic model was demonstrated by improved
predictions in comparison with previous models. As well as correctly predicting
several aspects of normal bone healing, this model also simulated experimentally
established impediments to fracture healing, including excessive mechanical
125
Chapter 9
stimulation, periosteal stripping and impaired cartilage turnover. The improved
mechanistic nature of this model allows the fracture healing process and its pathologies
to be understood in terms of cellular processes, which can be further investigated to
evaluate modern treatments related directly to cell behavior. Many necessary
parameters were not well established in literature.
•
Statistical Design-of-Experiments methods were employed to evaluate the relative
importance of the cell parameters in the mechanobiological model developed. It was
possible to identify the most critical parameters and processes to successful healing in
the previous described model of bone healing. They were found to correspond to
established biological facts. In particular, cartilage production and the replacement of
cartilage during endochondral ossification were found to be critical to proper healing.
Parameters related to cartilage were found to have optimum values. Thus, moderate
cartilage turnover (production and degradation) was beneficial over both low and high
rates, which delayed healing. To experimentally confirm such suggestions is extremely
challenging. However, computational analyses applied in the way proposed in this
study have the power to generate and develop hypotheses to the point at which they
may be tested experimentally. It was suggested that future experimental efforts should
be undertaken to understand the processes identified as important in this study.
•
The remodeling phase of fracture repair in mice was characterized, and the
phenomenon of temporary formation of dual cortices was documented. An established
bone remodeling theory, in which osteoblastic and osteoclastic activity is modulated
by local mechanical stimuli, was applied to subject-specific models to show that a
difference in major loading directions could explain the differences between
remodeling patterns in different species. In contrast to axial loading, a bending load
was found to give rise to dual cortex remodeling in all samples. It is hypothesized that
this is due a greater proportion of bending acting in the mouse femur than in larger
mammals, since the knee and ankles are more flexed in rodents. Although biological
differences between species may also be involved in this process, the contrasting
behavior during post fracture remodeling could be explained by differences in loading
direction.
In summary, the conclusions derived in this thesis confirm that computational models can be
used as an important tool in studies of bone mechanobiology. The combined studies have
demonstrated both the possibilities and limitations of computational models of tissue
differentiation and bone healing. The studies presented in this thesis were compared
extensively with a wide variety of experimental data, which led to the development of more
mechanistic models of cell and tissue differentiation. This model was created with further
development in mind, and depending on the research questions, it can easily be extended to
include more key aspects of bone healing. Future prospects for such approaches will be
discussed in the following sections as well as further possibilities for their validation.
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Discussion and conclusions
9.3 Aspects of interest
Is bone healing still a problem?
The incidence of hospital admission in the Netherlands due to a fracture is 50,000 per year, of
which about 28,000 are operated upon (den Boer et al., 2002). During the last decades, this
number has been rising due to increased aging of the population. Improved treatment and
prevention is therefore necessary in order to limit the considerable impact on medical and
hospital resources.
The reported frequency of non-union varies depending on the operative technique and the
patient group. It is generally estimated as being between 5-10% (Praemer et al., 1992;
Einhorn, 1995), although other reports range from 11% for open tibial fractures (Siebenrock et
al., 1993), to 23% for high-energy femur fractures treated with intramedullary osteosynthesis
(Harris et al., 2003). Such complications are compounded by the fact that patients with nonunions require repeated surgical procedures and frequently develop complex outcomes.
Functional outcome of fracture healing in patients with non-unions is often chronically
compromised. Furthermore, the burden of osteoporosis related fractures is increasing. In 2005,
2 million fractures occurred in the United States due to osteoporosis, resulting in direct
medical costs of $17 billion. A recent study predicts the annual incidence of fractures and
costs to grow by 50% by 2025, exceeding 3 million cases and costing more than $25 billion
(Burge et al., 2007). Despite considerable developments in fracture treatment there remains
enormous scope for further improvement. Understanding basic bone regenerative biology, its
regulatory mechanisms, and the alterations with age should be of highest priority.
How can the healing process be influenced?
Biological, physiological, and mechanical factors are the major influences on the repair
process. During recent years, bone healing research has begun to shift its focus towards
biological factors such as angiogenesis and re-vascularization. However, when identified and
well characterized, many biological factors could be controlled, replaced or supplemented
(Aro and Chao, 1993a). For example, growth factors are likely to be part of future treatment
protocols for problematic fractures (Einhorn, 1995). The competing factors are various Bone
Morphogenetic Proteins (BMP), Parathyroid hormone and selective Prostaglandin agonists, all
of which have been shown capable of stimulating and improving the quality of fracture repair
(Aspenberg, 2005). To date, clinical evidence only exists for BMPs.
The fundamental aspect of mechanical stability undoubtedly influences the outcome of
fracture healing and can reverse a delayed union. It may be the only irreplaceable element in
governing complete healing following successful initiation of the fracture repair process (Aro
and Chao, 1993a). Most studies investigating the effects of mechanical loading have focused
on avoiding delayed healing and non-union. One of the future challenges will be to determine
whether healing can also be accelerated by mechanical stimulation. So far there has been no
definite answer to this question, because the design of experimental studies is often too
imprecise to determine whether the comparison is made to a normal or slow healing fracture
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Chapter 9
(Aro and Chao, 1993b). Effective and reliable mechanical methods to maintain or enhance
bone regeneration may be established for the treatment of both healthy fractures as well as
difficult fractures in patients with deficient osteogenic potential (Aro and Chao, 1993b).
Do we need computational models?
Computational models have been shown to be very useful for research on bone healing. The
complexity of the biological problem results in difficulty of performing in vivo experiments
and interpretation of the results, which may vary across species, ages, geometries, loading
conditions et cetera. These factors, as well as the influence of other isolated factors can be
investigated with numerical models (Sacks et al., 1989). Moreover, in comparison with
experimental studies, in which only a few time points are evaluated, computational models
have the advantage that continuous evolution in time can be calculated. Computational models
also provide the possibility to interpolate and extrapolate from known experimental time
points. Furthermore, computational simulations can be used for parametric examination of
factors that are difficult or impossible to examine experimentally (Prendergast, 1997; Doblare
et al., 2004). The work presented in this thesis demonstrates these advantages.
Advances in computational power have allowed problems of greater complexity to be studied
with broader applications. This has for example resulted in the development of more
mechanistic computational models (Chapter 6-7). They have the potential to help develop
biological and mechanical interventions for treatment of skeletal pathologies. For example,
they can be used to understand cellular or molecular mechanisms which would be necessary to
develop not only mechanical methods to promote bone healing, but also to enhance bone
repair in combination with cellular and molecular therapy. Moreover, development of patient
specific models and incorporation of genetic variability are in progress.
Can computational models be validated?
Mathematical and numerical models must be validated with experimental data. However,
scientists have argued that “validation of numerical models of natural systems is impossible”
(Oreskes et al., 1994). This line of thinking is comparable to the argument made by Popper
(1992) that, like scientific theories, correctness of model predictions cannot be proven, only
disproven. Therefore, tolerance levels must be defined and hypotheses tested in terms of
whether sufficient validation can be achieved. According to Anderson et al. (2007),
corroboration of a computational model is achieved when enough evidence is generated and
credibility established that a computer model yields results with sufficient accuracy for its
intended use. That begs the question of what is sufficient accuracy. There is no universal
answer to that question, but for some models achieving predictions within the experimental
variability might be considered sufficient (Chapter 8, Figure 8-8).
In line of research of this thesis, computational models are often developed to simulate
biological processes that cannot be measured directly by experimentation. They also
necessitate inputs that can only be estimated. Interpretation of predictions from such biophysical models may appear to contradict the validation requirements described above. Still,
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Discussion and conclusions
indirect validations can be performed by comparing outcomes of computer simulations with
parameters that can be measured experimentally (for example the boundary conditions on load
and displacement, or qualitative and quantitative histology) or with clinical outcome. Then
credibility of the model can be established by showing that it can predict much versatile
experimental data. Hence, integration of theoretical models, experiments and clinical outcome
is necessary. Moreover, sensitivity studies such as the one presented in Chapter 7 can be used
as a step in the validation process to interpret the mechanobiological response of the model to
both assumed and known inputs. The limitations of any study that incorporates computational
modeling of biological systems must be assessed relative to the degree of validation to ensure
that the interpretations are reasonable (Anderson et al., 2007). Hence, as this thesis has
demonstrated, corroboration of mechanobiological models is possible to some extent.
However, care must be taken when extrapolating results to other situations.
Why are fluid velocity and shear strain the most probably stimuli?
In the first part of this thesis, several mechano-regulation schemes were investigated and their
predictions compared. The conclusion of the study in Chapter 4 was that the algorithm
regulated by fluid velocity and tissue shear stain was, although not completely correct, more
consistent with experimental data than the other algorithms. However, the mechanisms that
underlie these mechanotransduction systems have not been established. One reason for this is
that many cell phenotypes are active during tissue regeneration. These cells have been studied
extensively in isolation, and have been shown to react to various mechanical stimuli in their
original tissue environment. However, current theoretical and experimental evidence suggest
that many cell types are sensitive to shear strain and/or shear stress generated by fluid flow
(Jin et al., 2001; Cowin, 2002). For bone, it is currently thought that mechanotransduction is
governed by the bone cells, which respond to a load-induced flow of interstitial fluid through
the canalicular network which connects them (Cowin, 2002; McGarry et al., 2005). In most of
the early literature addressing mechanical stimulation of cells, cartilage was proposed to be
stimulated by hydrostatic compression. However, more recent studies have shown that fluid
flow during dynamic compression of cartilage explants can stimulate proteoglycans and
protein synthesis (Buschmann et al., 1999). Other studies have shown that isolated deviatoric
deformation (shear strain) also stimulates chondrocyte activity and cartilage synthesis (Jin et
al., 2001). Fibroblasts originating in tendons have also been shown to be responsive to fluid
flow. Strain and shear combinations of these stimuli have been shown to activate
mechanotransduction pathways that modulate tissue maintenance, repair and pathology (Wall
and Banes, 2005).
Further support exists for interstitial fluid flow as a stimulus of cellular differentiation, as
opposed to hydrostatic pressure. It has been demonstrated using computer models, that the
stresses acting on tissues are generated predominantly by the drag forces acting due to the
flow of the interstitial fluid (Huiskes et al., 1997; Prendergast et al., 1997), while the local
fluid pressure in the tissue does not change as the tissues differentiate (Soballe et al., 1992a;
1992b). Also, fluid flow is known to stimulate anabolic cell expressions in in vitro studies
(Jacobs et al., 1998), and to transport signalling molecules and nutrients.
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Chapter 9
Tissue level models
Most computational studies of cell and tissue differentiation presented so far (Chapter 3-7)
have been conducted at the tissue level. A basic assumption is that one tissue type transforms
into another tissue type. Is this description sufficient? Bone remodeling theories (Chapter 8)
demonstrate how computational models can be used to study variations within the tissues
themselves. This has been shown to be crucial when studying whole bone structures and
trabecular bone alignment. Perhaps the same approach could contribute to studies of other
tissue types? For example, cartilage that undergoes endochondral ossification passes through
several stages that involve specific cell actions and tissue transformations, such as
hypertrophy, mineralization, degradation and angiogenesis. Still, today it is assumed to be one
single type of tissue, when computationally modeling tissue differentiation. However, adding
complexity to a model is not always better, per se. The necessary level of complexity will
depend on the research question, and one should keep a model as simple as possible, as long
as it can answer the particular research question, within the boundaries of the current
knowledge.
Cell model hypothesis
The underlying hypothesis in this thesis has been that cells sense mechanical stimuli and
respond to it. Basic cellular responses can be proliferation, migration, differentiation and
matrix synthesis. These concepts were developed into the mechanistic model presented in
Chapter 6-7. In distraction osteogenesis bone forms just as rapidly as during fracture healing,
and as long as distraction force is applied, bone regeneration can be sustained almost
indefinitely. Studies have shown that the rate and frequency of distraction do not influence any
of the morphometric parameters (Einhorn, 1998a). The enhanced bone formation appears to
result from increased recruitment and activation of bone cells, rather than from an increase in
individual cellular activity (Welch et al., 1998). This supports our underlying hypothesis and
the reasoning behind the cell model described in Chapter 6, which assumes that more cells
will be recruited as long as the simulation is sustained, instead of altered activity of each cell.
9.4 Limitations
Comparison of computational predictions with experimental data allows for further
development of the theoretical models, as has been shown both in this thesis and by others
(Claes and Heigele, 1999; Geris et al., 2006b). However, the predictions must be interpreted in
the correct context. Developing a theoretical model of a biological phenomenon includes
simplifications of the biological processes. These simplifications involve, for example, not
specifying certain biological or mechanical processes that might be involved, or describing
them in a phenomenological sense. The accuracy with which material behavior is modeled and
the way in which boundary conditions are imposed will be discussed in more detail below, as
well as limitations in modeling the time scale. Models should be designed with particular
research questions in mind, and adjusted as the questions change.
130
Discussion and conclusions
Finite element modeling
In most computational models simplifications must be made in terms of the material
characterization and application of boundary conditions. Throughout this work, a single set of
mechanical properties was used. Some of these properties are well established while others are
unclear. In particular, the mechanical properties of the soft tissues are not well established.
However, it was shown by Lacroix (2001) that when modeling tissue differentiation during
fracture healing, the relative material properties do not have any profound effect on the
sequence of tissue differentiation. However, the rate and duration of individual phases of the
process were altered. These observations justify the use of the single set of properties in
Chapters 3-5. With progress towards more mechanistic models (Chapter 6-7), including rates
of individual cell processes, it will be necessary to investigate the mechanical properties more
closely and assess their affects on the computational predictions. To do so, ‘design of
experiments methods’ similar to those employed in Chapter 7 are suggested. Since material
behavior is not well established, particularly for the soft tissues, one critical task for
biomechanics community is to determine constitutive laws for these tissues.
Equally important when performing finite element studies are the descriptions of the boundary
conditions, including the load application. This is especially difficult when simulating in vivo
experimental data, since the precise loading in animals is very difficult to control or measure.
In Chapter 4, in vivo data was used in which the mechanical loading was carefully controlled.
Such experiments are very important in achieving model validation. On the other hand, such
controlled loading regimes are not completely physiological in respect to the normal fracture
healing process in the animal.
Several authors have discussed the use of computer models to evaluate mechanical stimuli at a
macroscopic (homogenized) continuum level. With our hypothesis that the cells are the
sensors that react to local stimuli, it is not clear whether the continuum approach is completely
valid (Humphrey, 2001; van der Meulen and Huiskes, 2002). This assumption may become
more critical with the development of mechanistic models and the use of denser meshes. With
the rapid development of imaging tools, such as nano-CT, more realistic geometrical meshes
will be possible. When the mesh-size is on the level of only a few cells (with their surrounding
matrix) it will not be sufficient to treat materials as homogenous. Therefore, in the future,
microscopic scale models (homogenized at a micro-scale instead of macro-scale) that can be
used to investigate the actual stimuli acting at a cellular level will probably become important.
This development is already evident in the use of multi-level modeling where both a
continuum and a micro-scale modeling approach are combined. Furthermore, anisotropic
material formulations should be included in future computational models. This may be
particularly important in the late remodeling phase of healing (Chapter 8) where the tissues
become more organized, for example to distinguish between woven (more isotropic) and
lamellar bone (more anisotropic).
Despite the limitations mentioned above, which in many cases are also concerns in both in
vivo and in vitro experimental studies, theoretical models have been shown to be very useful
in examining possible mechano-regulatory pathways. This has been further established by the
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Chapter 9
work presented in this thesis. In the future such modeling techniques might be used to study
more problematic cases of bone healing, and also to identify new future research questions.
9.5 Future directions
This research
The computational models and the methodology for validation that were developed in this
thesis were applied to a variety of examples of bone regeneration, including fracture healing
and distraction osteogenesis. The models describe the cell and tissue differentiation processes,
and can therefore also be applied to other examples of tissue regeneration. Current prospects
are to apply the models to study cell differentiation and metabolism in tissue engineering
constructs, which are stimulated mechanically through a bio-reactor. The main advantage is
that these setups allow controlled variations of mechanical stimulation and biological agents.
Hence, it is a good experimental model to obtain quantitative data and relationships between
certain parameters. However, care must be taken since it also introduces disadvantages due to
its non-physiological in vitro environment. Further prospects are to study pathological
conditions of bone healing, such as to optimize protocols during distraction osteogenesis.
The mechanistic model presented in Chapters 6-7, demonstrated the potential and benefits
over solely phenomenological models. A next step would be to combine the cell model with
the tissue volumetric growth model, which was applied in Chapter 5, to determine whether
these additional modeling aspects can help in achieving both quantitative and qualitative
corroboration with experimental data. The underlying hypothesis was that cells sense
mechanical stimuli and respond to it. With that in mind, the model in Chapter 6-7 was
developed. The model was created with further developments in mind. There are other aspects
of tissue differentiation and bone healing that may alter the outcome. These aspects were
anticipated to be less important. In this aspect, ‘less important’ could be defined as parameters
that do not alter the response outside the experimental variability. However, the importance of
many of these variables (for example biochemical factors, explicit description of angiogenesis,
or aspects of the inflammatory phase of healing) is difficult to quantify without adding them to
the model. Therefore, the model was created with this in mind, and adding other aspects of
bone healing is possible, and can be incorporated whenever necessary for the particular
research questions.
Computational mechanobiology in general
The goal of many investigators is to search for relationships between mechanobiological
factors and cellular responses under normal and deficient bone healing conditions. However,
to establish the interdependence of biophysical stimulation and bone repair and remodeling at
the material and structural level, experiments must be carefully designed (van der Meulen and
Huiskes, 2002). These studies include a) appropriate animal models to investigate the cellular
and tissue responses under different forms of mechanical stimulation; b) in vitro cell and
tissue culture studies with well-controlled biophysical stimuli to eliminate other confounding
132
Discussion and conclusions
factors at the systemic level and c) computational models to investigate mechanobiological
relationships and possible signaling pathways (van der Meulen and Huiskes, 2002).
Most experimental and computational models of bone healing investigate tissue differentiation
under a known interfragmentary movement, which is assumed to be the main stimulus.
However, the challenges of the future will involve combining mechanical aspects with
essential biological factors to predict treatments for more pathological cases and explore
potential signaling pathways. Despite the steady progress in the use of computer models to
study bone mechanobiology, it is still very difficult to obtain quantitative conclusions because
of the differences between individual patients and animal species. This is also the case for in
vivo and in vitro experimental models. In particular, identification and inclusion of the affects
of patient variability and patient ageing will become important aspects. The burden of
osteoporosis-related fractures is increasing. A recent study estimates the annual incidence and
costs of such fractures to grow by 50% between 2005-2025 (Burge et al., 2007). Therefore,
efforts to quantify the response and variability of physiological parameters between
individuals and animal species will be a task for the future. In future research in bone
biomechanics, more complex and realistic computer simulations will be employed to in order
to reduce animal experimentation and clinical trials, with related economic benefits. With the
progress made in this field in recent years, and the work conducted within this thesis, the tools
are now available to be able to distinguish between the mechanical and biological effects of
healing. This will enable studies of specific cases in which the biology is altered in a known
way. As examples, the effect of altered osteoclast/osteoblast activity, as observed in
osteoporotic patients (Jilka, 2003), or the lower cell differentiation and proliferation rates
which are observed with age (Lu et al., 2005), etc., may be investigated
In the near future it will be important to focus research on the integration of simulations,
experiments and theoretical aspects (van der Meulen and Huiskes, 2002). Not only should
there be greater interaction between experimental studies and computational modeling, but
experiments should ultimately be designed and carried out with the associated computational
investigation in mind, in order to improve the value of numerical modeling. Use of
computational techniques for parametric examination of factors that are difficult or impossible
to examine experimentally will contribute to the advance of biomechanics, as indicated in this
thesis and by others (Prendergast, 1997; Doblare et al., 2004).
In the far future, once the influence of mechanical stimulation on transcription factors,
signaling pathways and genomic elements have been elucidated, it might be possible to
eliminate today’s reliability on mechanical forces for stimulation and to induce these signals
by other means. This would result in simpler and more efficient fracture treatment and
prevention. Perhaps, such technology could be used to restore bone mass systemically?
Answers to questions such as these would have direct bearing on many of the clinical
problems that now confront the orthopaedic community. Until then, however, the influence of
mechanical factors on the bone regeneration process and the entire musculoskeletal system
remains.
133
Appendix A: Theoretical development of finite element
formulation for modeling cellular activity
This appendix describes the theoretical development of the element formulation used to
predict cellular behavior in Chapter 6-7.
A1
Global equations
The global equation that is used to describe all the activities in the two dimensional space over
time for each degree of freedom is shown in Eq A-1.
⎛
∂φ ( x, t ) i
φ
= ∇D (φi ) i ∇φ i + f PR ( Ψ ) i φ i ⎜1 − i
⎜
∂t
⎝ φ space
⎛
φ
f PM ( Ψ ) i φ j ≠i ⎜1 − i
⎜ φ
space
⎝
⎞
⎟ − FD ( Ψ , φ i −1− 4 ) − f AP ( Ψ ) i φi +
⎟
⎠
(Eq A-1)
⎞
⎟ − f PM ( Ψ ) i φ j ≠iφ i
⎟
⎠
where φi represents the variable of degree of freedom i. φ j indicates coupling between degree
of freedom i and j. Each variable and part of the equation is described in detail below. The
implementation was regulated by only prescribing non-zero values for the constants associated
with each degree of freedom (Appendix A5).
The global equation can be divided into one part regulating cell processes (Eq A-2) and one
part determining extracellular matrix production and degradation (Eq A-3). The cell equation
can be divided into parts representing transport/migration of cells, proliferation,
differentiation, and apoptosis, as
⎛
∂c(x, t ) i
c ⎞
= ∇D (ci ) i ∇ci + f PR ( Ψ ) i ci ⎜1 − i ⎟ − FD (Ψ , c1− 4 ) − f AP (Ψ ) i ci
⎜ c
⎟
∂t
space ⎠
{
~
~ }
~
~y
{
~
~
~
~
~
~
~
~ ⎝}
~
~
~
~
~
~
~
~y
{
~
~ }
~
~y
{ }y
Transport
Proliferation
Differentiation
(Eq A-2)
Apoptosis
where t represents time, x corresponds to the two-dimensional space, and ci the normalized
concentration of cell type i. Di is the concentration dependent diffusivity for cell type i, and
fPR, FD and fAP are functions which regulates proliferation, differentiation and apoptosis,
respectively. Ψ represents the mechanical stimulation, and is used to turn on or off cell
activities accordingly. cspace represents the ‘available space’ in the element, and is calculated as
the maximum cell concentration minus the sum of the current cell concentrations. FD is
dependent on the mechanical stimulation, as well as the concentrations of other cell types, as
If
If
Ψ = i,
Ψ ≠ i,
FD =
∑ (− f
j =1− 4 , j ≠i
Dj
cj
)
FD = f D i ci
135
(Eq A-3)
A
Appendix A
Similarly, the matrix equation can be divided into parts specifying matrix production and
degradation, as
∂m(x, t ) j
∂t
⎛
mj ⎞
⎟ − f DM (Ψ ) j ci m j
= f PM (Ψ ) j ci ⎜1 −
⎜ m
⎟
space ⎠
{
~
~
~
~
~
~
~
~
~
~
~
~⎝ }
~
~
~
~
~
~
~
~
~
~
~
~y {
~
~
~
~
~ }
~
~
~
~
~y
Production
(Eq A-4)
Degradation
where mj represents the normalized concentration of matrix type j. fPM and fDM are functions
which regulate production and degradation of matrix, respectively. mspace represents the
‘available space’ in the element, and is calculated as the maximum matrix concentration minus
the sum of the current matrix concentrations, and i is the corresponding cell type to the matrix
type j, for example fibroblasts to fibrous tissue.
A2
User defined element formulation
A2.1 Transport/Migration
Constitutive equations
The finite element formulation was based on the diffusion equation. Hence, it was defined
from the requirement of mass conservation of the diffusing phase. The phenomenological
Fick’s 1st law assumes that the flux of the diffusing material in any part of the system is
proportional to the local density gradient. Combining the mass conservation equation with
Fick’s 1st law, results in the diffusion equation:
b
c
` a
∂c
f
f
f
f
f
f
f
= 5 D c 5 ci x , t
∂t
b
(Eq A-5)
where c is the concentration, t represents time, x is the two dimensional space x = x x,y , and
b b
D is the concentration dependent diffusivity.
`
a
Deriving the finite element solution
The strong form of the model equation is:
∂c
f
f
f
f
f
f
Z f
dV + Z n A J dS = 0
V
∂t
S
(Eq A-6)
b c
where V is any volume whose surface is S, J = @ D c A 5 c x , t
c
` a
b
b
c
is the flux of
concentration, n is the normal to S, and n · J is the flux leaving S. Using the divergence
theorem, and assuming arbitrary volume gives the weak form:
Z δξ
V
f
∂c
∂f
f
f
f
f
f
f
f f
f
f
f
f
f
f
+
A J dV = 0
∂t ∂x c
g
(Eq A-7)
136
Appendix A
where δξ is an arbitrary, suitable continuous scalar field. After re-writing Eq A-7, and using
the divergence theorem again:
h
i
∂c
f
f
f
f
f
f ∂δξ
f
f
f
f
f
f
f
f
f
f
f
f
f k
Z jδξ f
@
A J dV + Z δξ n A J dS = 0
∂t
∂x c
b c
f
V
g
(Eq A-8)
S
In the element formulation, the flux J was described as:
b cf ∂φ g
f
f
f
f
f
f
f
J =@sD φ f
∂x
c
(Eq A-9)
where φ is the activity, or the normalized concentration at the nodes φ = c/s, where s is the
solubility of the diffusing phase into the base material and D( φ ) is the diffusivity. The
solubility was assumed constant.
Discretization and time integration
The shape functions (NN) for each node are associated with the coordinate system on the
master element, r (r1,r2). The boundaries between these nodes are governed by isoparametric
mapping:
x=N
NT
b
(Eq A-10)
r xN
` a
b b
where the vector x contains the position vectors of the element nodes, and r contains the
coordinates of the shape functions on the master element. The normalized concentration field
is interpolated by:
N
δφ = N A δφ
N
(Eq A-11)
where NN(r) represents the interpolation functions. The discretized equation can then be
written as:
H
Z JN
N
V
f
N
I
g
∂φ
∂φ
∂N
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
fK
s f
+ f
AsD f
dV =Z N
∂x
∂t
∂x
g
f
N
q dS
(Eq A-12)
S
where q a @ n A J . The backward Euler method (modified crank-Nicholson operator) was
b c
used for time integration, with time points t and t+ Δt:
u&t + Δt =
1
(ut +Δt − ut )
Δt
(Eq A-13)
Where u are the nodal variables, resulting in:
H
Z JN
V
N
f
g
f g
N
φf
@φ
∂φ
∂N
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
fK
tf
s
+
AsD
dV =Z N
Δt
∂x
∂x
I
S
137
N
q dS
(Eq A-14)
Appendix A
The Jacobian
The Jacobian contribution to the conservation equation is obtained from the variation of
Eq A-14 with respect to φ at time t+Δt:
H
Z JN
N
V
d
N
N
I
g
δ∂φ
∂N
∂N
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f ∂D
f
f
f
f
f
f
f
f
f
f ∂φ
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
fK
∂φ +
As
∂φ +
AsD
dV
∂φ ∂x
Δt
∂x
∂x
∂x
sf
f
f
f
f
f
f
e
f
g
f
(Eq A-15)
Re-arranging and using the interpolation in Eq. A-10 gives:
H
d e
sf
f
f
f
f
f
ZJ f
N
N
Δt
V
A3
N
M
N
M
N
I
∂N
∂N
∂N
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f∂D
f
f
f
f
f
f
f
f
f
f ∂φ
f
f
f
f
f
f
f
f MK
+ f
AsD f
+ s f
N
dV
∂x
∂x
∂x ∂φ ∂x
f
g
(Eq A-16)
Implementation
N
To solve user defined elements in ABAQUS, the elemental contribution to the residual ( F )
N
at degree of freedom N, to the overall residual R has to be defined. Additionally, the
elemental contribution to the Jacobian KNM must be defined as:
K
NM
N
∂F
f
f
f
f
f
f
f
f
f
f
f
f
f
=@ f
∂u M
(Eq A-17)
The elements were 4 noded with 4 integration points. The shape functions were defined as:
h
N
N
ci
+
l1 4 1 @ r 1 1 @ r 2 m
l
m
b
cb
b
cb
cm
l
l
m
b c l 1 + 4 1 + r1 1 @ r 2 m
l
m
r =l
cb
cm
l 1+ 4b
m
b
l
1 + r1 1 + r 2 m
l
m
l
b
cb
cm
j
k
1+ 4 1 @ r 1 + r
1
(Eq A-18)
2
where r1 and r2 are the local nodal coordinates. To calculate the derivative of the shape
functions with respect to the global coordinate system, intermediate steps were taken,
including calculating the derivatives to the interpolation functions with respect to the
coordinates r1 and r2. The determinant of the Jacobian vector J was then calculated as:
dx
f
f
f
f
f
f
f
f dy
f
f
f
f
f
f
f
f
f dx
f
f
f
f
f
f
f
f
f dy
f
f
f
f
f
f
f
f
det J = f
A f
@ f
A f
dr1 dr 2 dr 2 dr1
` a
f
g
(Eq A-19)
The total derivatives with respect to x in the global coordinate system were calculated for each
degree of freedom as:
N
N
N
N
N
N
` a
` a
dN
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f dy
f
f
f
f
f
f
f
f
f
f dN
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f dy
f
f
f
f
f
f
f
f
f
f(
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f dx
f
f
f
f
f
f
f
f
f
f dN
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f dx
f
f
f
f
f
f
f
f
f
f(
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f dN
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f dN
det J and dN
det J
= dr A dr @ dr A dr
= dr A dr @ dr A dr
1
2
2
1
2
1
1
2
dx
dy
f
g
f
g
(Eq A-20)
The contribution to the residual (F) and the overall system matrix (K) were calculated for each
degree of freedom as:
138
Appendix A
F i = F i old @ X W int
`a
`a
i
h
` al
det J jN
i
As
f
g
dφ
f
f
f
f
f
f
f
f
dt
b
c
i
j
i
i
dx
dy
dx
(Eq A-21)
dy
i i
sf
dN
l i
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
fdN
f
f
f
f
f
f
f
f
f
f
f
f
f dN
f
f
f
f
f
f
f
f
f
f
f
fdN
f
f
f
f
f
f
f
f
f
f
f
f
f
k+ m
m
lN N j A f
+ DAj f
A f
+ f
A f
m
l
Δt
dx dx
dy dy
m
` al
m
det J l
m
l
m
l
i
i
m
l dD
f
f
f
f
f
f
f
f
f j dN
f
f
f
f
f
f
f
f
f
f
f
f
fdφ
f
f
f
f
f
f
f
f dN
f
f
f
f
f
f
f
f
f
f
f
f
fdφ
f
f
f
f
f
f
f
f
k
j f
h
K i , j = X X W int
+
ii
dN
f
f
f
f
f
f
f
f
f
f
f
fdφ
f
f
f
f
f
f
f dN
f
f
f
f
f
f
f
f
f
f
f
fdφ
f
f
f
f
f
f
f
km
k
DAj f
A f
+ f
A f
h
h
dt
AN A
dx
A
dx
i
j
+
A
dy
i
j
(Eq A-22)
dy
where Wint is the gauss weights for each integration point. Wint was equal to 1 for all 4
integration points. That concludes the part of the equation that regulates cell transport.
A4
Proliferation, differentiation and apoptosis of cells
The parts of the equation regulating other cell activities than transport (Eq A-2) were
implemented as boundary conditions, which could be turned on and off, for each element and
node. The corresponding parts were added to the residual F (Eq A-23, line 2) and the matrix K
(Eq A-24, line 3).
h
F i =F i
` a
` a
old
h
h
ii
i
i
f g
` al
dφ
dφ
dφ
dN
dN
i
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
ff
f
f
f
f
f
f
f f
f
f
f
f
f
f
f
f
f
f
ff
f
f
f
f
f
f
fm
det J jA N A s
+ D Aj
+
A
A kk
l
lW
l int
l
l
l
l
l
X
@ l
l +W int
l
l
l
l
l
l
j +W int
h
m
m
m
dt
dx dx dy dy
m
m
m
h
i
m
m
φ
i
i
i
f
f
f
f
f
f
f
f
f
f
f
k @ W Ν A B A φ @ W Ν A E A φm
m
Ν A A A φ j1 @
int
int
m
φ lim
m
m
h
i
m
m
φ
m
i
i
f
f
f
f
f
f
f
f
f
f
f
f
kφ @ W Ν A DmA φ A φ
k
Ν A Pmj1 @
int
h
` a
i
jN
det J l
φ max
X
ii
dN
sf
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
fdN
f
f
f
f
f
f
f
f
f
f
f
f
f dN
f
f
f
f
f
f
f
f
f
f
f
fdN
f
f
f
f
f
f
f
f
f
f
f
f
fm
k
k+
+ D Aj
+
A
A
h
A5
φ lim
(Eq A-23)
X
i
l
j
lW
N A
l int
l
dx
Δt
l
l
l
i
i
l
l dD
f
f
f
f
f
f
f
f
f
f j dN
f
f
f
f
f
f
f
f
f
f
f
fdφ
f
f
f
f
f
f
f
f dN
f
f
f
f
f
f
f
f
f
f
f
fdφ
f
f
f
f
f
f
f
f
l
+
+
N
A
A
A
A
b
c
l dt
dx
dy
dx
dy
K i , j = X Xl
l
h
i
l
l
φ
i
j
f
f
f
f
f
f
f
f
f
f
f
l @W
k+ W
N N A Aj1 @ 2A
l
int
int
l
φ
lim
l
l
l
φf
l
i
j
i
f
f
f
f
f
f
f
f
f
f
j @W
N N A PmA X + W N N
int
i
j
i
dx
int
i
dy
j
j
dy
i
N N A B + W int N N
j
A DmA φ X
i
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
j
AE m
m
m
m
m
m
m
k
(Eq A-24)
Matrix production and degradation
To have one governing equation to solve, matrix production and degradation were
implemented to the same global equation as the cell activities (Eq A-1). A similar approach as
for proliferation, differentiation and apoptosis of cells was used. The corresponding parts were
added to the residual F (Eq A-24, line 3) and the matrix K (Eq A-25, line 4). In these
139
Appendix A
equations, φ is the concentration of the current variable, and φ X indicates that a different
variable is influencing φ , i.e. fibroblast concentration influencing the matrix production of
fibrous tissue. The implementation was regulated by only prescribing non-zero values for the
constants associated with each degree of freedom:
Solub
Diffus
Prolif
Differ
Cm
ax
M
atP
M
atD
M
m
ax
z
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~Apop|
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~x
i
h
1
l
l
l1
l
l
l1
l
l
l1
l
l
l1
l
l1
l
j
1
D M SC
D FB
DCC
DOB
0
0
0
A M SC
AFB
ACC
AOB
0
0
0
B M SC
B FB
BCC
BOB
0
0
0
E M SC
E FB
ECC
EO B
0
0
0
1
0
0
1 0
0
1 0
0
1 0
0
0 PmFT DmFT
0 PmC DmC
0 PmB DmB
0m
0m
m
m
0m
m
m
0m
m
m
1m
m
m
1m
m
k
1
MSC
FB
CC
OB
FT
C
B
(Eq A-25)
Then the overall equation reduces to the cell equation (Eq A-2) for degree of freedom 1-4 and
to the matrix equation (Eq A-4) for degree of freedom 5-7.
A6
Coupling
Coupling of the degrees of freedom was necessary to implement for example differentiation of
one cell phenotype into another and to allow cell concentrations to affect matrix production
rates. The differentiation rules were individual for each cell type, and were included as
additional conditions, which could alter the function FD.
Coupling
Mesenchymal stem cell concentration was coupled with all other cell phenotypes, through
possible differentiation, and coupled to the total cell concentration.
Fibroblast concentration was coupled with CC and OB concentrations through possible
differentiation, and to the total cell concentration, and matrix production of FT.
Chondrocyte concentration was not coupled with any other cell phenotype through possible
differentiation. However, CC apoptosis was coupled with OB stimulation. It was coupled to
the total cell concentration, and to matrix production of C.
Osteoblast concentration was coupled with FB concentration through possible differentiation
and with the total cell concentration, and matrix production of B.
Mechanical stimulation
The cell activities were turned on or off. Stimulation of cell type i resulted in maximal
proliferation of cell type i, and no proliferation of other cell types. It also resulted in minimal
apoptosis of cell type i, and maximal differentiation of other cell types into cell type i when
differentiation is possible. Moreover, it resulted in maximal matrix production of the
corresponding tissue type, j, while there was no matrix production of other tissue types, as
well as minimal matrix degradation of tissue type j.
140
Appendix B: Taguchi orthogonal arrays and design of
experiments methods
This appendix provides tables and details about the design of experiments method employed
in Chapter 7. Taguchi technique uses multifactor experimental plans which are called
orthogonal arrays. The arrays are denoted as LN(SM), where M is the number of test factors, S
is the number of levels, and N is the total number of runs in the experiment. For more
information, see Taguchi (1987), Montgomery (2005), or Phadke (1989).
B1
Statistical calculations
Outcome parameters are chosen and determined during the experiment. The loss function
which the Taguchi method seeks to minimize is generally taken to be a quadratic function.
The outcome parameters are transformed into a the-higher-the-better (Eq B-1) or a the-lowerthe-better signal to noise ratio, (S/N), as
S / N = −10 log(
1
)
yi2
(Eq B-1)
where yi is the score from the outcome analysis of the ith treatment condition. Analysis of
variance was used to investigate the significance and contribution of each factor. It includes
calculating the total sum of squares of deviation about the mean, as
n
SST = ∑ ( S / N i − S / N ) 2
(Eq B-2)
i =1
where n was the number of experiments, S/Ni the signal-to-noise ratio for the ith treatment
condition, S / N was the overall mean of S/N. For each factor, the sum of the squares of
deviation about the mean was
n
SS F = ∑ N Fi ( S / N Fi − S / N ) 2
(Eq B-3)
i =1
Where F is the factors, and n is the number of levels, NFi is the number of experiments at each
level of each factor, S / N Fi is the mean of signal-to-noise ratio at each level of each factor.
The mean square of deviation (MSF), the sum of the squares of the error (SSE), and the mean
square of the error (MSE) were calculated according to Eq B-4-6.
MS F = SS F factorDOF
(Eq B-4)
F
SS E = SST − ∑ SS i
(Eq B-5)
MS E = SS E factorDOF
(Eq B-6)
i =1
The fraction of the variance explained by each factor is calculated from the F value
as F = MS F MS E . The percentage of the total sum of squares represents the approximate
contribution of each factor to the variance and was calculated as TSS % = (SS F SST ) ⋅100% .
141
Appendix B
B2
Orthogonal arrays
Screening experiment
For the screening experiment, an L64(231) orthogonal array was used, with a total of 64
treatment conditions, leaving factor 27-31 unused. The configuration of the L64 array and the
factors and levels in each simulation is shown in Table B-1.
L64
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
Initial MSC conc.
Proliferation
P
M
O
C
MSC FB
x1
x2
x3
x4
x5
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
Differentiation
CC
OB MSC FB
x6
x7
x8
x9
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
CC
Migration
OB MSC FB
CC
Apoptosis
OB MSC FB
CC
Matrix prod.
OB
FT
C
B
Matrix deg.
FT
C
B
x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25 x26
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
Table B-1: The L64 orthogonal array was found in Taguchi et al., (2005a). -1 corresponds to
high level and 1 corresponds to low level of each factor.
142
Appendix B
Higher level experiment
For the three level experiment, an L27(213) orthogonal array was used, with a total of 27
treatment conditions, and leaving factor 11-13 unused. The configuration of the L27 array and
the factors and levels in each simulation was
L27
Exp
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Initial MSC
concentration
Proliferation
Matrix production
Matrix
degradation
P
M
FB
CC
OB
FT
C
B
FT
C
x1
-1
-1
-1
-1
-1
-1
-1
-1
-1
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
x2
-1
-1
-1
0
0
0
1
1
1
-1
-1
-1
0
0
0
1
1
1
-1
-1
-1
0
0
0
1
1
1
x3
-1
-1
-1
0
0
0
1
1
1
0
0
0
1
1
1
-1
-1
-1
1
1
1
-1
-1
-1
0
0
0
x4
-1
-1
-1
0
0
0
1
1
1
1
1
1
-1
-1
-1
0
0
0
0
0
0
1
1
1
-1
-1
-1
x5
-1
0
1
-1
0
1
-1
0
1
-1
0
1
-1
0
1
-1
0
1
-1
0
1
-1
0
1
-1
0
1
x6
-1
0
1
-1
0
1
-1
0
1
0
1
-1
0
1
-1
0
1
-1
1
-1
0
1
-1
0
1
-1
0
x7
-1
0
1
-1
0
1
-1
0
1
1
-1
0
1
-1
0
1
-1
0
0
1
-1
0
1
1
0
1
-1
x8
-1
0
1
0
1
-1
1
-1
0
-1
0
1
0
1
-1
1
-1
0
-1
0
1
0
1
-1
1
-1
0
x9
-1
0
1
0
1
-1
1
-1
0
0
1
1
1
-1
0
-1
0
1
1
-1
0
-1
0
1
0
1
-1
x10
-1
0
1
0
1
-1
1
-1
0
1
-1
0
-1
0
1
0
1
-1
0
1
-1
1
-1
0
-1
0
1
Table B-2: The L27 orthogonal array was found in Taguchi et al., (2005a). -1 corresponds to
high level, 0 to mid level and 1 to low level of each factor.
B3
Factors and levels
Screening experiment
The normalized values implemented for each factor as high (-1) and low (1) in the L64 array
screening experiment was shown in Table 7-2, page 108.
Higher level experiment
The normalized values that were implemented for each factor as high (-1), mid (0) and low
(+1) in the L27 three level experiment was shown in Table 7-3, page 109.
143
Appendix B
B4
Outcome analysis
Screening experiment
The results from each of the outcome analyses and the calculated signal-to-noise ratio for the
screening experiment was
Time to
Exp. complete
No. healing
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
mean
SST
SSE
MSE
20
24
82
41
92
73
24
24
54
70
41
56
71
63
64
83
49
56
67
72
59
66
36
41
33
34
118
46
59
49
49
81
43
91
37
34
98
63
34
82
82
35
41
62
60
59
43
53
25
46
69
83
73
70
64
35
84
90
30
43
31
30
88
81
S/N
-26.021
-27.604
-38.276
-32.256
-39.276
-37.266
-27.604
-27.604
-34.648
-36.902
-32.256
-34.964
-37.025
-35.987
-36.124
-38.382
-33.804
-34.964
-36.521
-37.147
-35.417
-36.391
-31.126
-32.256
-30.370
-30.630
-41.438
-33.255
-35.417
-33.804
-33.804
-38.170
-32.669
-39.181
-31.364
-30.630
-39.825
-35.987
-30.630
-38.276
-38.276
-30.881
-32.256
-35.848
-35.563
-35.417
-32.669
-34.486
-27.959
-33.255
-36.777
-38.382
-37.266
-36.902
-36.124
-30.881
-38.486
-39.085
-29.542
-32.669
-29.827
-29.542
-38.890
-38.170
-34.44
818.20
817.21
408.61
General
normal
healing
6
7
6
6
5
5
7
7
6
6
7
7
7
7
6
7
7
7
6
7
6
6
7
7
6
6
7
7
7
7
6
7
7
7
6
6
6
7
7
7
6
5
7
7
6
7
6
6
5
7
6
6
6
6
7
7
6
6
7
6
7
7
6
6
S/N
15.563
16.902
15.563
15.563
13.979
13.979
16.902
16.902
15.563
15.563
16.902
16.902
16.902
16.902
15.563
16.902
16.902
16.902
15.563
16.902
15.563
15.563
16.902
16.902
15.563
15.563
16.902
16.902
16.902
16.902
15.563
16.902
16.902
16.902
15.563
15.563
15.563
16.902
16.902
16.902
15.563
13.979
16.902
16.902
15.563
16.902
15.563
15.563
13.979
16.902
15.563
15.563
15.563
15.563
16.902
16.902
15.563
15.563
16.902
15.563
16.902
16.902
15.563
15.563
16.13
46.57
45.67
22.83
EARLY
STAGE
Bone form
0.819
0.810
0.289
0.326
0.323
0.354
0.785
0.823
0.213
0.352
0.826
0.782
0.832
0.669
0.298
0.310
0.778
0.778
0.350
0.277
0.348
0.331
0.661
0.788
0.435
0.216
0.624
0.851
0.648
0.835
0.388
0.196
0.729
0.615
0.188
0.487
0.162
0.737
0.806
0.479
0.222
0.353
0.815
0.700
0.815
0.598
0.160
0.351
0.784
0.758
0.346
0.181
0.352
0.266
0.539
0.815
0.297
0.260
0.725
0.785
0.680
0.741
0.306
0.255
S/N
-1.736
-1.829
-10.769
-9.747
-9.818
-9.009
-2.098
-1.693
-13.449
-9.059
-1.666
-2.132
-1.602
-3.492
-10.507
-10.178
-2.178
-2.185
-9.115
-11.163
-9.173
-9.599
-3.602
-2.068
-7.223
-13.298
-4.095
-1.399
-3.769
-1.561
-8.233
-14.148
-2.743
-4.216
-14.531
-6.241
-15.835
-2.652
-1.870
-6.401
-13.091
-9.049
-1.778
-3.101
-1.780
-4.460
-15.907
-9.100
-2.119
-2.407
-9.222
-14.853
-9.063
-11.509
-5.362
-1.776
-10.556
-11.710
-2.797
-2.097
-3.352
-2.603
-10.287
-11.861
-6.66
1300.03
1300.03
650.02
MID
STAGE
Bone form
0.991
0.988
0.101
0.599
0.138
0.250
0.972
0.981
0.275
0.132
0.838
0.822
0.559
0.658
0.171
0.160
0.237
0.199
0.512
0.398
0.553
0.380
0.318
0.240
0.808
0.873
0.165
0.287
0.195
0.308
0.536
0.094
0.765
0.201
0.708
0.851
0.077
0.250
0.580
0.123
0.317
0.656
0.282
0.222
0.194
0.180
0.399
0.439
0.971
0.392
0.390
0.099
0.140
0.428
0.693
0.906
0.142
0.115
0.822
0.768
0.647
0.849
0.120
0.136
S/N
-0.081
-0.105
-19.895
-4.455
-17.173
-12.029
-0.245
-0.166
-11.211
-17.613
-1.537
-1.697
-5.057
-3.640
-15.358
-15.893
-12.513
-14.009
-5.808
-8.000
-5.147
-8.414
-9.949
-12.404
-1.850
-1.178
-15.628
-10.845
-14.191
-10.221
-5.418
-20.505
-2.330
-13.947
-2.995
-1.405
-22.294
-12.032
-4.738
-18.227
-9.968
-3.656
-10.993
-13.073
-14.259
-14.880
-7.982
-7.155
-0.251
-8.144
-8.173
-20.060
-17.076
-7.369
-3.183
-0.858
-16.951
-18.807
-1.705
-2.294
-3.777
-1.421
-18.450
-17.356
-9.19
2694.67
2694.67
1347.33
LATE
STAGE
Bone form
1.000
1.000
0.081
0.990
0.006
0.325
1.000
1.000
0.895
0.036
0.980
0.924
0.856
0.900
0.683
0.663
0.945
0.355
0.865
0.810
0.894
0.782
0.993
0.979
0.998
0.992
0.003
0.942
0.661
0.926
0.948
0.338
0.983
0.004
0.995
0.995
0.007
0.877
0.988
0.040
0.720
0.999
0.985
0.493
0.322
0.626
0.993
0.931
1.000
0.941
0.850
0.006
0.163
0.861
0.902
0.987
0.332
0.009
1.000
0.986
1.000
1.000
0.078
0.286
S/N
-0.003
0.000
-21.834
-0.088
-44.620
-9.760
0.000
-0.002
-0.963
-28.912
-0.177
-0.683
-1.349
-0.919
-3.312
-3.568
-0.492
-8.993
-1.261
-1.828
-0.975
-2.131
-0.065
-0.185
-0.019
-0.069
-51.858
-0.520
-3.599
-0.667
-0.463
-9.423
-0.151
-48.983
-0.043
-0.046
-43.182
-1.143
-0.108
-28.048
-2.858
-0.011
-0.129
-6.136
-9.834
-4.067
-0.065
-0.621
-0.001
-0.524
-1.417
-44.482
-15.757
-1.297
-0.896
-0.117
-9.588
-40.729
0.000
-0.127
-0.004
0.000
-22.135
-10.887
-7.69
12328.48
12328.48
6164.24
AVERAGE
Bone
form
0.936
0.933
0.157
0.638
0.156
0.310
0.919
0.935
0.461
0.173
0.881
0.843
0.749
0.742
0.384
0.378
0.653
0.444
0.576
0.495
0.598
0.498
0.657
0.669
0.747
0.694
0.264
0.693
0.501
0.690
0.624
0.209
0.826
0.273
0.630
0.778
0.082
0.621
0.791
0.214
0.420
0.669
0.694
0.472
0.444
0.468
0.517
0.574
0.918
0.697
0.529
0.095
0.218
0.518
0.712
0.903
0.257
0.128
0.849
0.846
0.776
0.863
0.168
0.225
S/N
-0.571
-0.605
-16.071
-3.903
-16.151
-10.174
-0.732
-0.587
-6.728
-15.224
-1.100
-1.483
-2.513
-2.591
-8.314
-8.455
-3.698
-7.052
-4.795
-6.109
-4.463
-6.060
-3.648
-3.492
-2.532
-3.174
-11.567
-3.181
-5.998
-3.224
-4.098
-13.577
-1.665
-11.269
-4.008
-2.185
-21.750
-4.134
-2.035
-13.408
-7.545
-3.487
-3.172
-6.527
-7.061
-6.590
-5.726
-4.829
-0.740
-3.135
-5.538
-20.411
-13.214
-5.707
-2.956
-0.891
-11.810
-17.864
-1.424
-1.450
-2.208
-1.276
-15.499
-12.939
-6.32
1780.00
1780.00
890.00
Table B-5: Results from each of the outcome analyses performed for the screening
experiment. Signal to noise ratio for each treatment condition is calculated.
144
Appendix B
Higher level experiment
The results from each of the outcome analyses and the calculated signal-to-noise ratio for the
higher level experiment was
Time to
Exp. complete
No.
healing
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
mean
SST
20
31
30
82
19
32
40
70
32
61
23
47
32
55
42
35
50
84
33
25
54
55
49
70
61
40
60
S/N
-26.021
-29.827
-29.542
-38.276
-25.575
-30.103
-32.041
-36.902
-30.103
-35.707
-27.235
-33.442
-30.103
-34.807
-32.465
-30.881
-33.979
-38.486
-30.370
-27.959
-34.648
-34.807
-33.804
-36.902
-35.707
-32.041
-35.563
-32.492
336.92
General
normal
healing
6
6
6
7
5
7
5
7
7
7
6
7
5
7
7
6
7
7
7
5
6
6
7
7
7
6
4
S/N
15.563
15.563
15.563
16.902
13.979
16.902
13.979
16.902
16.902
16.902
15.563
16.902
13.979
16.902
16.902
15.563
16.902
16.902
16.902
13.979
15.563
15.563
16.902
16.902
16.902
15.563
12.041
15.892
44.61
EARLY
STAGE
Bone form
8.42E-01
5.02E-01
4.83E-01
7.46E-01
5.95E-01
7.57E-01
5.91E-01
8.29E-01
6.10E-01
8.16E-01
6.90E-01
4.65E-01
7.30E-01
5.65E-01
7.59E-01
5.90E-01
8.22E-01
6.23E-01
8.03E-01
6.74E-01
3.87E-01
6.79E-01
5.10E-01
7.33E-01
5.99E-01
8.09E-01
6.04E-01
S/N
-1.494
-5.988
-6.321
-2.550
-4.505
-2.417
-4.561
-1.625
-4.296
-1.761
-3.224
-6.652
-2.737
-4.959
-2.392
-4.587
-1.699
-4.114
-1.910
-3.433
-8.256
-3.368
-5.856
-2.693
-4.453
-1.842
-4.377
-3.78
81.98
MID STAGE
Bone form
0.999
0.796
0.215
0.978
0.670
0.228
0.694
0.807
0.200
0.229
0.247
0.319
0.264
0.865
0.741
0.234
0.784
0.737
0.962
0.232
0.500
0.627
0.191
0.794
0.871
0.268
0.776
S/N
-0.007
-1.976
-13.357
-0.189
-3.473
-12.843
-3.177
-1.865
-13.998
-12.791
-12.159
-9.918
-11.563
-1.264
-2.609
-12.610
-2.111
-2.647
-0.335
-12.700
-6.024
-4.060
-14.358
-1.998
-1.197
-11.441
-2.202
-6.403
745.95
LATE
STAGE
Bone form
1.000
1.000
0.021
1.000
0.851
0.064
0.889
1.000
0.014
0.712
0.185
0.973
0.931
1.000
0.877
0.950
0.883
0.955
0.992
0.736
0.990
0.861
0.069
0.904
1.000
0.897
0.866
S/N
0.000
-0.002
-33.446
0.000
-1.398
-23.895
-1.019
0.000
-36.915
-2.950
-14.659
-0.240
-0.620
-0.004
-1.142
-0.443
-1.083
-0.396
-0.066
-2.665
-0.085
-1.300
-23.204
-0.874
0.000
-0.941
-1.254
-5.504
3014.79
AVERAGE
Bone form
0.947
0.766
0.240
0.908
0.706
0.350
0.725
0.879
0.275
0.586
0.374
0.586
0.642
0.810
0.792
0.591
0.830
0.772
0.919
0.547
0.626
0.722
0.257
0.811
0.823
0.658
0.749
S/N
-0.472
-2.315
-12.406
-0.838
-3.027
-9.128
-2.796
-1.123
-11.228
-4.643
-8.546
-4.647
-3.854
-1.833
-2.023
-4.563
-1.621
-2.249
-0.733
-5.240
-4.075
-2.829
-11.810
-1.822
-1.688
-3.635
-2.515
-4.136
310.84
Table B-6: Results from each of the outcome analyses performed for the higher level
experiment. Signal to noise ratio for each treatment condition is calculated.
B5
Statistical outcome
Screening experiment
The final processed results were presented as calculated sum of squares for each factor and the
percentage of the total sum of squares. These were presented in Table 7-4, page 113 for the
L64 screening experiment.
Higher level experiment
The final processed results were presented as calculated sum of squares for each factor and the
percentage of the total sum of squares. These were presented in Table 7-4, page 113 for the
L27 higher level experiment.
145
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Samenvatting
Mechanische en mechanobiologische effecten bij fractuurheling
- identificatie van belangrijke cellulaire eigenschappen
Het herstel van botfracturen verloopt volgens een complex proces waarbij cellulaire en
moleculaire activiteiten een rol spelen. Normaal volgt dit een proces van weefseldifferentiatie
waarbij het hematoom dat direct na de breuk ontstaat geleidelijk verandert in bindweefsel,
kraakbeen, en tenslotte volledig functioneel botweefsel. Echter, in 5-10% van de gevallen
verloopt de fractuurheling zeer traag, onvolledig of geheel niet. Om zulke probleemgevallen
adequaat te kunnen behandelen is een beter begrip nodig van de processen tijdens botheling.
Biologische of mechanische omstandigheden tijdens het helingsproces beïnvloeden botheling.
Zo heeft mechanische belasting een direct effect op de cel- en weefseldifferentiatie. Dit proces
heet mechanoregulatie. Mechanische belasting wordt tijdens fractuurheling meestal niet op
cel- maar op orgaanniveau beschreven, bijvoorbeeld als afstand en frictie tussen de botdelen.
De vertaalslag van deze globale factoren naar lokale spanningen en rekken in het weefsel die
de differentiatie van cellen beïnvloedt, vereist numerieke modellen. Diverse hypothetische
algoritmen zijn gebruikt om te verklaren hoe deze lokale spanningen en rekken de
weefseldifferentiatie beïnvloeden. Het toepassingsgebied van dergelijke algoritmen is zeer
groot. Ze kunnen helpen bij het ontrafelen van basisprincipes van cel- en weefseldifferentiatie,
het optimaliseren van implantaten en het zoeken naar potentiële behandelingen voor niethelende botbreuken of andere aandoeningen waarbij botvorming een rol speelt. Echter, een
correcte voorspelling van het botvormingsproces vereist een degelijke validatie van deze
computermodellen.
In dit proefschrift zijn verschillende mechanoregulatie algoritmen onderzocht en gevalideerd.
Daartoe zijn voorspellingen op basis van in de literatuur beschreven algoritmen vergeleken
met experimentele resultaten, onvolkomenheden in de voorspellingen geïdentificeerd, en
verbeteringen voorgesteld. De onderliggende hypothese van al deze algoritmen is dat cellen
als sensoren functioneren tijdens de fractuurheling. Ze registreren de mechanische belasting en
gebruiken die informatie om de weefseldifferentiatie te sturen door te prolifereren,
differentiëren of apoptotisch te worden en door hun extracellulaire matrix aan te passen.
In het eerste deel van deze studie werden bestaande en nieuwe mechanoregulatie algoritmen
geïmplementeerd in een numeriek model om het normale fractuurhelingsproces onder axiale
belasting te voorspellen. Hoewel de algoritmen verschillende biofysische stimuli als
uitgangspunt hadden, bleken alle in staat om normale botheling te voorspellen (Hoofdstuk 3).
Als validatie werden ze vervolgens gebruikt om een specifiek dierexperiment te simuleren
waarin een botfractuur belast werd met axiale compressie of torsie. Geen van de algoritmen
bleek in staat om de experimenteel waargenomen verdeling van weefseltypen correct te
voorspellen. De beste resultaten werden verkregen met een algoritme waarin
weefseldifferentiatie gestuurd werd door deviatorische rek en vloeistofstroming (Hoofdstuk
4). Verder onderzoek naar dit algoritme bracht aan het licht dat dit algoritme ook de verdeling
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Samenvatting
van weefsels tijdens diverse stadia van distractie osteogenese kon voorspellen als rekening
werd gehouden met de groei van individuele weefseltypen. Ook veranderingen in het distractie
osteogenese proces bij andere distractiesnelheden of -frequenties werden met dit model goed
voorspeld (Hoofdstuk 5).
In het tweede deel van deze studie werd een nieuw mechanistisch model voor cel activiteit
ontwikkeld waarmee veel tekortkomingen van de eerdere modellen konden worden opgelost.
In dit model werd het celfenotype gestuurd door mechanische factoren, en was de activiteit en
snelheid waarmee een weefsel zich ontwikkelt weefselspecifiek. Dit model kon, net als de
voorgaande modellen, normale fractuurheling simuleren. Daarnaast kon het ook vertraagde of
onvolledige botheling als gevolg van excessieve belasting of biologische verstoringen en
pathologische condities voorspellen. Ook veranderingen in het helingsproces na verwijdering
van het periosteum of bij verstoring van het mineralisatie proces van kraakbeen kwamen
overeen met experimentele data (Hoofdstuk 6). Dit nieuwe model bevat een groot aantal
parameters waarvan de meeste uit de literatuur zijn gehaald. Voor een aantal parameters kon
echter geen betrouwbare waarde bepaald worden. Met de zogenaamde ‘design of
experiments’-methode in combinatie met een Taguchi orthogonale matrix analyse was
mogelijk om die modelparameters te identificeren die de meeste invloed hebben op het
bothelingsproces. Deze parameters bleken direct gerelateerd aan botformatie en aan
kraakbeensynthese en -degradatie, hetgeen goed overeenkomt met in de biologie gehanteerde
concepten. Parameters die invloed hebben op bindweefsel en kraakbeenvorming hadden een
sterk niet-lineair effect op de resultaten met begrensde optima. Deze resultaten geven aan dat
afwijkingen van de optimale waarden de botheling sterk en negatief kunnen beïnvloeden
(Hoofdstuk 7).
Het laatste deel van deze studie was gericht op de remodelleringfase van botheling. Een
experiment toonde aan dat bij muizen een dubbele cortex ontstaat in een laat stadium van
fractuurheling. Dit tot nu toe onbekende karakteristieke fenomeen werd succesvol
gereproduceerd met een simulatiemodel voor botremodellering. De simulaties toonden aan dat
dit remodelleringsgedrag een gevolg is van een afwijkende gewrichtsbelasting bij kleine
zoogdieren zoals muizen, veroorzaakt door de specifieke stand van gewrichten (Hoofdstuk 8).
Samenvattend heeft dit onderzoeksproject bevestigd dat mechanobiologische numerieke
modellen leiden tot een beter begrip van cel- en weefseldifferentiatie tijdens fractuurheling en
distractie osteogenese. De studies in dit proefschrift zetten belangrijke stappen naar de
ontwikkeling van meer mechanistische modellen van cel- en weefseldifferentiatie en de
validatie daarvan. Het ligt in de verwachting dat deze modellen kunnen helpen bij het screenen
van potentiële behandelprotocollen voor verschillende vormen van pathofysiologische
fractuurheling.
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Acknowledgement
I would like to express my sincere gratitude to those who have contributed to this thesis and
without whom the thesis would never have been completed. First of all, I would like to thank
the AO Foundation, Davos, Switzerland, for financing this research.
I would like to thank my promotors Prof Keita Ito, Dr. René van Donkelaar, and Prof Rik
Huiskes. Keita, for giving me an excellent predoctoral training, combining the medically
oriented environment in Davos with the engineering scenery at the university in Eindhoven.
Also for being a great advisor in science and life, for trusting my scientific judgment, and for
being incredibly supporting and patient during the year when things went a little slow. René,
for being interested and enthusiastic and for always taking the time to discuss both progress
and setbacks whenever they occurred. Rik, for sharing his broad interest in and knowledge on
orthopaedic research, and for teaching me valuable things about scientific writing.
I would like to thank all the former and current group members of the Bone- and Orthopaedic
Biomechanics group, as well as the Mechanobiology group in Davos. Especially, my thanks
go to Wouter for always finding time to discuss solutions to computational problems.
Moreover, all my office mates in Eindhoven and Davos deserve thanks for amoung other
things pleasant dinner parties, yummy birthday cakes and interesting discussions regardless
topic during weekends and late nights.
I would also like to thank Nick and Damien, for being great friends and for taking the time to
read, correct and improve my work considerably by sharing their English writing skills.
My thanks also go to my previous supervisor, Prof Amy Lerner, Rochester, for passing on her
enthusiasm for orthopaedic research, and for supporting me in the pursuit of a PhD.
I am grateful for all the grand new friends I have made over the years: Roz, Regula, Nick,
Marije, Ang and Ina for great times in and around Davos. My housemates in Eindhoven;
Yvonne, Nollaig and Kathi, for always having someone to come home to. Machiel and Leda
for welcoming me to Eindhoven. Katy, Jakob, Pieter and Matej for good friendship, rewarding
discussions, and endless cookie supply. Maarten for taking care of me when I needed. I would
also like to thank my close old friends from Sweden for coming to visit me frequently: Erika,
Eva-Maria, Jimmie, Anna, Anna-Karin, Frida and Nina.
Finally, I wish to express my deepest gratitude to my family; my parents for their endless
support and encouragement throughout the years, and Petro, whose love and patience helped
to make this work possible and final.
Hanna Isaksson
Davos, August 30th, 2007
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Curriculum Vitae
Hanna Isaksson was born on April 12, 1979 in Linköping, Sweden. In 1998 she finished her
secondary education at Katedralskolan in Linköping. Thereafter she studied Chemical
Engineering and Material Science at Uppsala University, Sweden. During her studies she
worked for the student union in Uppsala to improve student’s influence and rights during their
education and she also taught mathematics. Part of her studies was carried out as an exchange
student at the University of Rochester, NY, USA. During this stay she shifted focus towards
Biomedical Engineering, and her degree project was accomplished at the Department of
Biomedical Engineering, University of Rochester. After receiving her Master’s degree, she
started her doctoral studies. Since October 2003, she has been working in the research group
for Mechanobiology at the AO Research Institute, Davos, Switzerland, and in the Bone- and
Orthopaedic Biomechanics section of the Department of Biomedical Engineering at
Eindhoven University of Technology.
2003-2007
Ph.D. Biomedical Engineering
Eindhoven University of Technology, Eindhoven, NETHERLANDS
1998-2003
M.Sc. Chemical Engineering, Material Science
Uppsala University, Uppsala, SWEDEN
2002-2003
Biomedical Engineering, Material Science, Chemical Engineering
University of Rochester, Rochester, NY, USA
1995-1998
Secondary School, Natural Science Program
Katedralskolan, Linköping, SWEDEN
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