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Mechanical and mechanobiological influences on bonefracture repair : identifying important cellular characteristicsCitation for published version (APA):Isaksson, H. E. (2007). Mechanical and mechanobiological influences on bone fracture repair : identifyingimportant cellular characteristics. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR630668

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

vii

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

ix

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

Summary

x

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 post-fracture remodeling in mice femora was characterized, including a new phenomenon described as ‘dual cortex formation’. The effect of mechanical loading modes on fracture-callus 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.

xi

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

xii

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 1

Chapter 1

2

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

Introduction

3

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

Chapter 1

4

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.

Introduction

5

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.

Chapter 1

6

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).

7

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.

2 2

Chapter 2

8

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).

Bone fracture healing and computational modeling of bone mechanobiology

9

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.

Chapter 2

10

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 pre-injury 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


11

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).

Chapter 2

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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).


13

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;

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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


15

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)

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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.


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2.6 Stem cell and tissue differentiation In general, stem cells may be characterized as cells which have the capacity for extended self-renewal, 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).

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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


19

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).

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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.


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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

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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 time-dependent 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).


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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

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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 (Bailon-Plaza 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.


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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.

Material description

Biophysical stimuli

Cell modeling

Growth factors

Tissue growth

Carter et al., 1988 fracture healing single linear elasticoctahedral shear

stress and dilatational stress

Carter and Wong, 1988

joint development single linear elastic

octahedral shear stress and

dilatational stress

Blenman et al,. 1989 fracture healing single linear elasticoctahedral shear

stress and dilatational stress

Carter et al., 1998 fracture healing single linear elasticprincipal 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 osseointegration adaptively poroelastic shear strain and

fluid flow

Bailon Plaza and van der Meulen, 2001

fracture healing adaptively MSC, CC, OB

osteogenenic, chondrogenic

Bailon Plaza and van der Meulen, 2003

fracture healing adaptively linear elasticdeviatoric strain and dilatational

strain

MSC, CC, OB


Lacroix and Prendergast, 2002 fracture healing adaptively poroelastic shear strain and

fluid flowMSC

diffusion

Geris et al., 2003 bone chamber adaptively poroelastic shear strain and fluid flow

MSC diffusion

bone chamber adaptively linear elasticprincipal strain and

hydrostatic pressure

Kelly and Prendergast, 2005

osteochondral defects adaptively poroelastic shear strain and

fluid flowMSC

diffusion

Shefelbine et al., 2005

trabecular bone healing adaptively linear elastic

ostahedral shear strain, hydrostatic strain, fuzzy logic

Geris et al., 2006 fracture healing adaptively MSC, CC, OB


Garcia et al., 2006 fracture healing adaptively poroelastic shear strain invariant

MSC, FB, CC, OB

volume growth

Mechanical BiologicalBone

regeneration process

Time point evaluation

Table 2-1: Summary of computational models of tissue differentiation.

Chapter 2

26

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 (Bailon-Plaza 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.

27

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 mechano-regulation 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

3 3

Chapter 3

28

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 mechano-regulation 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

Comparison of biophysical stimuli for mechano-regulation of tissue differentiation during fracture healing

29

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-of-mixture was used and the material properties were updated, before the next iteration starts.

Chapter 3

30

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:

][][ 2 cellsDdtcellsd

∇= , (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:

∑−=

+ =n

niin EE

91 10

1 (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 maximum-cell density ratio:

1maxmax

max

1 ][][

][][][

++ +−

= nGrann Ecell

cellEcell

cellcellE (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.


31

Cortical

bone Marrow Gran. Tissue

Fibrous Tissue Cartilage Immature

BoneMature Bone

Young’s modulus (MPa) 15750 a 2 1 2 f 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.

Chapter 3

32

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


33

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.

213

232

2210 )()()(

31 εεεεεε −+−+−=y (Eq 3-4)

Figure 3-4: The threshold values in the simulations with only deviatoric strain (y0) as the biofeedback variable.

Biophysical stimuli

> 15 % > 0.15 MPa Fibrous tissue > 5 % ≤ 0.2 MPa > 5 % > -0.15 MPa > 5% > 5μm/s > 0.6MPa

< -0.15 MPa

Cartilage / Endochondral > 0.2 MPa ≤ 15 % > 0.15 MPa > 2.5% > 2.5μm/s > 0.4MPa ossification

Bone / Intramembranous ≤ 5 % ≤ 0.2 MPa ≤ 5 % < ± 0.15 MPa ossification

Immature Bone > 0.05% > 0.6μm/s > 0.1MPa

Mature Bone > 0.005% > 0.03μm/s > 0.03MPa

Resorption ≤ 0.005% ≤ 0.03μm/s ≤ 0.03MPai ≤ 0.010

i = SS / 3.75 + FF / 3

Prendergast

i > 3

i > 1

i > 0.267

Deviatoric Shear Strain

% (SS)

Fluid Flow μm/s (FF)

Predicted tissue type

Single parameter studies

Octahedral shear strain

Fluid Velocity

Pore Pressure

Carter

Principal Tensile strain

Hydrostatic stress

Principal Strain

Claes and Heigele

Hydrostatic Pore

Pressure

i > 0.010

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).

Chapter 3

34

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.


35

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).

Chapter 3

36

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


37

the mediating effects of cytokine and nutrition concentrations were not considered (Bailon-Plaza 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.

Chapter 3

38

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


39

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.

41

4 Corroboration of mechano-regulatory algorithms:

Comparison with in vivo results In the previous chapter it was determined that several mechano-regulation 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

4

4

Chapter 4

42

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 mechano-regulation 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 mechano-regulation algorithm best resembles the experimental data. For that purpose, a three-dimensional 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 mechano-regulation 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

Corroboration of mechano-regulatory algorithms: Comparison with in vivo results

43

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.

Chapter 4

44

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.


45

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).

Biophysical stimuli

> 15 % > 0.15 MPa Fibrous tissue > 5 % ≤ 0.2 MPa > 5 % > -0.15 MPa > 5 %

< -0.15 MPaCartilage or Endochondral > 0.2 MPa ≤ 15 % > 0.15 MPa > 2.5 %

ossificationBone or Intramembranous ≤ 5 % ≤ 0.2 MPa ≤ 5 % < ± 0.15 MPa ossification

Immature Bone > 0.05 %

Mature Bone > 0.005 %

Resorption ≤ 0.005 %

Carter Claes and Heigele Prendergast

Octahedral shear strain

Principal Tensile strain

Hydrostatic stress

Principal Strain

Hydrostatic Pore PressurePredicted tissue type i = SS / 3.75 + FF / 3

i > 3

Deviatoric Shear Strain % (SS)

Fluid Flow μm/s (FF)

i > 1

i > 0.267

i > 0.010

i ≤ 0.010

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

Chapter 4

46

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).


47

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).

Chapter 4

48

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.


49

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

Chapter 4

50

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.


51

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.

Axial 4 weeks Axial 8 weeks Torsion 4 weeks Torsion 8 weeks

Carter et al. No, Reasonable, No, No, Principal strain & hydrostatic stress

external bridging too early but inhibited final healing due to high pressure

strain limits too low failed to bridge

Claes and Heigele No, Reasonable, No, No, Strain & hydrostatic pressure

bridging too early, unstable tissue predictions

but inhibited final healing due to high pressure

strain limits too low failed to bridge

Prendergast et al. Yes, No, Yes, Yes, Deviatoric strain & fluid velocity

bony external callus prior to bridging

failed to bridge due to high fluid velocity

mature bone formation in gap prior to bridging

including bone formation in gap area

Deviatoric strain No, Reasonable, No, No, external bridging too early but too quick threshold values not

transferable to torsionextreme 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.

Chapter 4

52

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.


53

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.

55

5 Bone regeneration during distraction osteogenesis:

Mechano-regulation by shear strain and fluid velocity

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

5

5

Chapter 5

56

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

Bone regeneration during distraction osteogenesis: Mechano-regulation by shear strain and fluid velocity

57

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 mechano-regulation 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

Chapter 5

58

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).


59

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

extextF RTcccRT 24 22 −⎟⎠⎞⎜

⎝⎛ +=Δπ (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

Chapter 5

60

⎟⎟⎠

⎞⎜⎜⎝

⎛

+−=

Jnn

ccf

fFF 10,

0,0, (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

nii

n

niii

n

vg

vgEE

4

41 (Eq 5-3)

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),

n

nn vg

cellscells ][][ 1 =+ (Eq 5-4)

Grann

nn

n Ecells

cellscellsEcellscellsE ⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛= +∗

++

+max

1max1

max

11 ][

][][][][ (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.


61

nf,0cF,0

(meq/mm3)Volume growth

Bone growth rate

Fibrous tissue 0.8 7×10-2 ~ 15-20 %

Cartilage 0.8 3.5×10-2 ~ 5-7 %Appositionalbone growth

Endochondralbone growth

T = 298 K

Model parameters Resulting growth

0.8 3.5×10-2 ~ 5 µm/day a

R =8.3145 Nmm/mmolK c-ext = 1.5×10-4 mmol/mm3

0.8 5.25×10-2 ~ 25µm/day b

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.

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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.


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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.

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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).


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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.

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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.


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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

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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


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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,

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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.

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6 A mechano-regulatory bone-healing model based on cell phenotype specific activity

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

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6

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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

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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

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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.

Initial cell Migration Differentiation ApoptosisCell density Cell size rate Doubling Mitotic

Periosteum µm µm/min time hours fractionMSC 5 •105 (1) 106 (3) 10 (4) 60 (3,11) 12-24 (14,15-17) 0.50 (14,15,19)

FB 108 (4) 18 (7) 40 (11,12) 12-16 (18) 0.45 (18,20)

CC 2•105 (5,1-2) 25 (5,8-9) low 35 (1-2) 0.2 (16) 14-21 days (21-22) 4.5*norm (25)

OB 2.7•104 (6) 20 (10) 10 (13) 20 (14) 0.35 (14,20) 8-10 doubling (23,24) 35 (6,26)

Matrix Degradationmm3/cell pg/cell h µm/day mm3/cell

FT 5•10-6 (27,28) 0.3 (29) 5•10-6 (27,28)

C 5•10-6 (27,28) 3 (30) 5•10-6 (27,28)

B 3•10-6 (27,28) 15 (1,31,32) 3•10-6 (27,28)

Production

Maturation time cells/mm

Max cell density Proliferation rate

cells/mm3

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


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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.

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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 cell-culture 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

iiAPDspace

iiiPRiii

i cfcFc

ccfccDttc )(),(1)()(),(

41 Ψ−Ψ−⎟⎟⎠

⎞⎜⎜⎝

⎛−Ψ+∇∇=

∂∂

−

x (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 Ψ = i, ( )∑≠−=

−=ikk

kkDD cfF,41

(Eq 6-2)

If Ψ ≠ i, iiDD cfF =


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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

jijDMspace

jijPM

j mcfm

mcf

ttm

)(1)(),(

Ψ−⎟⎟⎠

⎞⎜⎜⎝

⎛−Ψ=

∂

∂ x (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 Transport Proliferation Differentiation Apoptosis

D f PR f D f AP

Periost Marrow External Callus mm2/it day -1 day -1 day -1

MSC 0.5 0,30 0,05 0,005 0.65 0,60 0,30 0,05FB 0,0 0,0 0,0 0,0 0.50 0,55 0,20 0,05CC 0,0 0,0 0,0 0,0 0.0 0,20 0,10 0,10OB 0,0 0,0 0,0 0,0 0.20 0,30 0,15 0,15

Matrix Initial Production Degradationconc f PM f DM

day -1 day -1

FT 0,0 0,20 0,05C 0,0 0,05 0,05B 0,0 0,10 0,05

Initial cell density

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.

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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 mechano-regulation 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.


79

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 .

0.375.3FVSSstim += (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,

∑∑==

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅+⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

BCFTjj

jGT

BCFTjj E

mm

EmmE,, max,,

max (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).

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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.

TissueFibrous Tissue Cartilage Immature

BoneMature Bone

Young’s modulus (MPa) 15750 a 2 1 2 f 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

Porosity 0.04 c 0.8 0.8 0.8 0.8 m 0.8 0.8

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); gArmstrong 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).


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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.

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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.


83

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.

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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 steady-state non-union (Figure 6-6c). The loading magnitude also had an effect on the predicted cell


85

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 fibro-cartilaginous.

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 time-point 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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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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6.4.1 Predictive capacity The predictive capacity and potential of the new model was assessed by showing that delayed- and 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

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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 re-vascularization 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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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 impaired-cartilage 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.

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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

7

7

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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

Determining the most important cellular characteristics for fracture healing, using design of experiments methods

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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-

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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:

∑∑==

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅+⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

BCFTjj

jGT

BCFTjj E

mm

EmmE,, max,,

max (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


95

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 Transport Proliferation Differentiation ApoptosisD f PR f D f AP

Periost Marrow External Callus mm2/it day -1 day -1 day -1

MSC 0.5 0.30 0.05 0.005 0.65 0.60 0.30 0.05FB 0.0 0.0 0.0 0.0 0.50 0.55 0.20 0.05CC 0.0 0.0 0.0 0.0 0.0 0.20 0.10 0.10OB 0.0 0.0 0.0 0.0 0.20 0.30 0.15 0.15

Matrix Initial Production Degradationconc f PM f DM

day -1 day -1

FT 0.0 0.20 0.05C 0.0 0.05 0.05B 0.0 0.10 0.05

Initial cell density

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.

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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 High Low1 Initial concentration Periost 0.8 0.22 Marrow 0.5 0.13 Outer 0.1 04 callus 0.05 05 Proliferation MSC 0.8 0.56 FB 0.7 0.37 CC 0.4 0.18 OB 0.5 0.159 Differentiation MSC 0.4 0.110 FB 0.3 0.0511 CC 0.2 0.02512 OB 0.3 0.0513 Migration MSC 0.8 0.414 FB 0.7 0.315 CC 0.1 016 OB 0.4 0.0517 Apoptosis MSC 0.2 0.0518 FB 0.2 0.0519 CC 0.2 0.0520 OB 0.2 0.0521 Matrix production FT 0.3 0.0522 CC 0.1 0.0123 B 0.2 0.0224 Matrix degradation FT 0.1 0.01525 CC 0.1 0.01526 B 0.1 0.015

Factors Levels

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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x Factors High Mid Low1 Initial MSC concentration P 0.8 0.5 0.22 M 0.5 0.3 0.13 Proliferation FB 0.7 0.5 0.34 CC 0.4 0.25 0.15 OB 0.5 0.325 0.156 Matrix production FT 0.5 0.3 0.17 CC 0.1 0.05 0.018 B 0.15 0.08 0.059 Matrix degradation FT 0.1 0.05 0.0110 C 0.1 0.05 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

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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

iiT yySS

1

2)( (Eq 7-2)

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

iFiFiF yyNSS

1

2)( (Eq 7-3)

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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conditions at each level of each factor, and Fiy is the mean outcome parameter at each level of each factor. The sum of the squares of the error was

∑=

−=Fi

iiTE SSSSSS

1 (Eq 7-4)

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 ( ) %100⋅TF SSSS . 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.

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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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x Source SSF % TSS SSF % TSS SSF % TSS SSF % TSS SSF % TSS1 Origin conc Periost 4.48E-01 1.0 1.89E+00 0.2 8.02E+00 0.6 8.57E+00 0.3 1.40E+02 1.12 Marrow 1.57E-01 0.3 2.29E+00 0.3 1.29E-01 0.0 3.91E+00 0.1 1.52E+01 0.13 Outer 1.57E-01 0.3 9.88E+00 1.2 2.55E+00 0.2 2.63E+01 1.0 6.55E+01 0.54 callus 4.48E-01 1.0 1.21E+01 1.5 1.28E-01 0.0 6.73E+01 2.5 4.51E+01 0.45 Proliferation MSC 2.14E+00 4.6 7.38E-01 0.1 2.94E+00 0.2 3.61E+00 0.1 7.90E+01 0.66 FB 4.67E+00 10.0 3.57E+01 4.4 1.53E+01 1.2 6.60E+00 0.2 4.20E+01 0.37 CC 1.12E-01 0.2 1.71E+01 2.1 2.54E+00 0.2 1.09E+02 4.0 1.07E+03 8.78 OB 1.12E-01 0.2 3.38E+01 4.1 1.19E+02 9.2 1.08E+02 4.0 1.01E+03 8.29 Differentiation MSC 5.34E-01 1.1 2.08E+01 2.5 2.38E+01 1.8 5.62E+01 2.1 1.58E+02 1.310 FB 1.96E+00 4.2 4.26E+00 0.5 2.28E+01 1.8 3.11E+00 0.1 1.92E+02 1.611 CC 5.34E-01 1.1 1.11E+01 1.4 1.07E+00 0.1 6.21E-01 0.0 4.78E+02 3.912 OB 3.74E-03 0.0 9.50E+00 1.2 6.39E-01 0.0 8.54E+00 0.3 5.13E+01 0.413 Migration MSC 1.12E-01 0.2 2.80E+00 0.3 4.70E+00 0.4 4.32E+00 0.2 1.62E+02 1.314 FB 1.01E+00 2.2 3.70E+00 0.5 8.21E-02 0.0 1.29E+01 0.5 1.97E+02 1.615 CC 1.12E-01 0.2 4.10E+00 0.5 1.34E+00 0.1 4.57E+00 0.2 2.21E+02 1.816 OB 1.96E+00 4.2 5.88E+00 0.7 4.53E+00 0.3 1.78E+01 0.7 2.27E+02 1.817 Apoptosis MSC 5.34E-01 1.1 1.11E+00 0.1 2.03E+00 0.2 5.73E+00 0.2 5.86E+01 0.518 FB 1.12E-01 0.2 8.57E+00 1.0 2.40E+00 0.2 5.97E+01 2.2 5.09E+02 4.119 CC 1.12E-01 0.2 1.14E+00 0.1 5.06E-01 0.0 7.06E+00 0.3 1.96E+02 1.620 OB 5.34E-01 1.1 3.00E+00 0.4 1.18E-02 0.0 3.75E+01 1.4 8.65E+01 0.721 Matrix prod. FT 3.74E-03 0.0 9.15E-02 0.0 3.89E+00 0.3 1.61E+01 0.6 7.76E+00 0.122 CC 8.08E-28 0.0 1.61E+02 19.7 2.14E+00 0.2 1.16E+03 42.9 1.96E+03 15.923 B 1.95E+01 41.8 1.41E+02 17.2 1.01E+03 77.6 2.86E+02 10.6 3.75E+02 3.024 Matrix deg. FT 1.57E-01 0.3 6.19E+00 0.8 1.78E+00 0.1 3.97E+01 1.5 4.03E-01 0.025 CC 6.21E+00 13.3 1.65E+02 20.1 1.61E+00 0.1 1.81E-01 0.0 2.40E+02 1.926 B 3.02E-01 0.6 3.28E-02 0.0 9.83E+00 0.8 7.67E+00 0.3 2.44E+01 0.2

ANOVA Sequential normal healing

Time to complete healing

_______________ Amount of bone formation_______________Early phase Mid phase Late phase

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

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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).

x Source SSF % TSS SSF % TSS SSF % TSS SSF % TSS SSF % TSS1 Origin conc Periost 1.99E-01 0.8 8.42E+01 1.0 3.92E-03 0.9 1.75E+01 2.3 1.79E-01 5.82 Marrow 1.99E-01 0.8 2.75E+02 3.4 1.27E-02 3.0 2.27E+01 3.0 5.65E-02 1.83 Proliferation FB 4.22E+00 16.9 5.49E+01 0.7 4.95E-03 1.2 1.23E+01 1.6 5.20E-02 1.74 CC 1.86E-01 0.7 7.51E+02 9.2 6.17E-04 0.1 5.32E+01 7.1 2.69E-01 8.65 OB 1.06E+00 4.2 1.37E+03 16.8 5.33E-02 12.7 2.38E+01 3.2 4.07E-01 13.16 Matrix prod. FT 4.39E-01 1.8 3.29E+01 0.4 3.79E-03 0.9 2.64E+00 0.4 6.09E-01 19.67 CC 2.35E+00 9.4 1.54E+03 18.9 1.17E-03 0.3 5.61E+02 75.3 1.06E+00 34.08 B 1.22E+01 48.9 2.65E+02 3.3 3.18E-01 75.8 3.99E+01 5.3 3.68E-02 1.29 Matrix deg. FT 2.08E+00 8.4 1.58E+02 1.9 5.98E-03 1.4 1.34E+00 0.2 5.55E-02 1.8

10 CC 1.85E+00 7.4 2.90E+03 35.7 2.41E-03 0.6 4.65E+00 0.6 1.36E-01 4.4

ANOVA Sequential normal healing


______________ Amount of bone formation_______________Early phase Mid phase Late phase

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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a)


35

45

55

65

75

High Mid Low

Levels

Out

com

e pa

ram

eter

(day

s)

Prol OBMatProd CMatDeg CAverage

b)

Amount of bone formation - Early phase

0.5

0.6

0.7

0.8

High Mid Low

Levels

Out

com

e pa

ram

eter

s (%

bon

e)

Prol OBMatProd BAverage

c)

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0.4

0.6

0.8

1

High Mid Low

Levels

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com

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eter

(% b

one)

Prol OBMatProd FTMatProd CAverage

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.

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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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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.

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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.

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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

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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

Remodeling of fracture callus in mice can be explained by mechanical loading

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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 mid-diaphyseal 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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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

k

n

kk UxxftxP μ),(),(

1∑=

= , (Eq 8-1)

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

),( txPCRRp

tr

maxmaxresorb −= , (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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clr Vdt

txdV=

),( , (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

)),((),(

trf ktxPdt

txdV−= τ , (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

γρ ),( txEE max= , (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

osteoiposteo Vn ρ= , (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.

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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 1.5 · 10-3 c

Maximum elastic modulus Bone

Emax GPa 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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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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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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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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21 days 28 days 42 daysMechnical testing

Stiffness % 50% (± 15 %) 61% (± 16%) 74% (± 20%)

μCT (mm3)woven bone 4.5 (± 0.7) 2.7 (± 0.8) 1.0 (± 0.3)

lamellar bone 2.6 (± 0.4) 3.3 (± 0.5) 3.1 (± 0.3)

μCT dual cortex (DC)DC / total mice 0 / 9 6 / 9 6 / 6

Histomorphometry (mm2)woven bone 1.8 (± 0.4) 0.5 (± 0.3) 0.4 (± 0.2)

lamellar bone 0.3 (± 0.1) 1.6 (± 0.5) 1.7 (± 0.4)cartilage 0.06 (± 0.09) 0.08 (± 0.2) 0.06 (± 0.1)

Experimental data

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


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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

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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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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

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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 rat-femur 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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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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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 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.

9

9

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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-of-experiments 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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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

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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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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 non-unions 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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(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 bio-physical models may appear to contradict the validation requirements described above. Still,


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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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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.


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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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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


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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.

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A

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.

iijiPMspace

iijiPM

iiAPiDspace

iiiPRiii

i

ff

fFfDt

tx

φφφφ

φ

φφφφ

φφφφ

≠≠

−−

Ψ−⎟⎟⎠

⎞⎜⎜⎝

⎛−Ψ

+Ψ−Ψ−⎟⎟⎠

⎞⎜⎜⎝

⎛−Ψ+∇∇=

∂∂

)(1)(

)(),(1)()(),(

41

(Eq A-1)

where iφ represents the variable of degree of freedom i. jφ 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

iiAPDspace

iiiPRiii

i cfcFc

ccfccD

ttc

)(),(1)()(),(

41 Ψ−Ψ−⎟⎟⎠

⎞⎜⎜⎝

⎛−Ψ+∇∇=

∂∂

−

x (Eq A-2)

{~~ }~~y

Transport

{~~~~~~~~ }~~~~~~~~y

Proliferation

{~~ }~~y

Differentiation

{ }y

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 Ψ = i, ( )∑≠−=

−=ijj

jjDD cfF,41

(Eq A-3)

If Ψ ≠ i, iiDD cfF =

Appendix A

136

Similarly, the matrix equation can be divided into parts specifying matrix production and degradation, as

jijDMspace

jijPM

j mcfm

mcf

ttm

)(1)(),(

Ψ−⎟⎟⎠

⎞⎜⎜⎝

⎛−Ψ=

∂

∂ x (Eq A-4)

{~~~~~~~~~~~~ }~~~~~~~~~~~~y

Production {~~~~~ }~~~~~y

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:

∂c∂tfffffff=5D c

` a

5 ci xb

, tb c

(Eq A-5)

where c is the concentration, t represents time, x is the two dimensional space xb= xb

x,y` a

, and D is the concentration dependent diffusivity. Deriving the finite element solution The strong form of the model equation is:

ZV

∂c∂tfffffffdV +Z

S

nbA Jc

dS = 0 (Eq A-6)

where V is any volume whose surface is S, Jc=@D c

` a

A 5 c xb

, tb 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:

ZV

δξ∂c∂tfffffff+ ∂

∂xffffffffA Jc

f g

dV = 0 (Eq A-7)

Appendix A

137

where δξ is an arbitrary, suitable continuous scalar field. After re-writing Eq A-7, and using the divergence theorem again:

ZV

δξ∂c∂tffffffff g

@∂δξ∂xfffffffffffffA Jc

h

j

i

kdV +ZS

δξ nbA Jc

dS = 0 (Eq A-8)

In the element formulation, the flux J was described as:

Jc=@ s D φ

b c ∂φ∂xfffffffff g

(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:

xb= N N T r

b` a

xb

N (Eq A-10) 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:

δφ = N NA δφ N (Eq A-11)

where NN(r) represents the interpolation functions. The discretized equation can then be written as:

ZV

N N s ∂φ∂tfffffffff g

+∂N N

∂xffffffffffffffffA s D ∂φ

∂xfffffffff g

H

J

I

KdV =ZS

N N q dS (Eq A-12)

where qa@n

bA Jc

. The backward Euler method (modified crank-Nicholson operator) was used for time integration, with time points t and t+ Δt:

( )ttttt uut

u −Δ

= Δ+Δ+1

& (Eq A-13)

Where u are the nodal variables, resulting in:

ZV

N N sφ@φt

Δtffffffffffffffffffff g

+∂N N

∂xffffffffffffffff

A s D ∂φ∂xfffffffff g

H

J

I

KdV =ZS

N N q dS (Eq A-14)

Appendix A

138

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:

ZV

N N sΔtfffffff∂φ

d e

+∂N N


A s ∂D∂φffffffffff∂φ

∂xfffffffff g

∂φ +∂N N


A s D δ∂φ∂xffffffffffffff g

H

J

I

KdV (Eq A-15)

Re-arranging and using the interpolation in Eq. A-10 gives:

ZV

sΔtfffffffd e

N N N M +∂N N

∂xffffffffffffffffA s D ∂N M

∂xfffffffffffffffff+ s ∂N N

∂xffffffffffffffff∂D

∂φffffffffff∂φ

∂xfffffffff g

N M

H

J

I

KdV (Eq A-16)

A3 Implementation To solve user defined elements in ABAQUS, the elemental contribution to the residual (F N ) at degree of freedom N, to the overall residual RN has to be defined. Additionally, the elemental contribution to the Jacobian KNM must be defined as:

K NM = @∂F N

∂u Mffffffffffffff (Eq A-17)

The elements were 4 noded with 4 integration points. The shape functions were defined as:

N N rbb c

=

1 4+ 1@ r1

b c

1@ r2

b c

1 4+ 1 + r1

b c

1@ r2

b c

1 4+ 1 + r1

b c

1 + r2

b c

1 4+ 1@ r1

b c

1 + r2

b c

h

l

l

l

l

l

l

l

l

l

l

l

j

i

m

m

m

m

m

m

m

m

m

m

m

k

(Eq A-18)

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:

det J` a

=dxdr1

fffffffffA

dydr2

ffffffffff@

dxdr2

ffffffffffA

dydr1

ffffffffff g

(Eq A-19)

The total derivatives with respect to x in the global coordinate system were calculated for each degree of freedom as:

dN N

dxfffffffffffffff= dN N

dr1

fffffffffffffffffA dydr2

ffffffffff@ dN N

dr2

fffffffffffffffffA dydr1

fffffffffff g

det J` a( and dN N

dyfffffffffffffff= dN N

dr2

fffffffffffffffffA dxdr1

ffffffffff@ dN N

dr1

fffffffffffffffffA dxdr2

fffffffffff g

det J` a( (Eq A-20)

The contribution to the residual (F) and the overall system matrix (K) were calculated for each degree of freedom as:

Appendix A

139

F i` a

= F i` a

old@X

iW int det J

` a

N iA s dφ

dtfffffffff g

+ D A dN i

dxfffffffffffffAdφdxffffffff+ dN i

dyfffffffffffffAdφdyffffffff

h

j

i

k

h

l

j

i

m

k (Eq A-21)

K i , jb c

=Xi

Xj

W int det J` a

N i N jA

sΔtfffffff+ D A dN i

dxfffffffffffffAdN j

dxffffffffffffff+ dN i

dyfffffffffffffAdN j

dyffffffffffffff

h

j

i

k+

dDdtffffffffffAN jAdN i

dxfffffffffffffAdφdxffffffff+ dN i

dyfffffffffffffAdφdyffffffff

h

l

l

l

l

l

l

l

l

j

i

m

m

m

m

m

m

m

m

k

(Eq A-22)

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).

F i` a

=F i` a

old@X

W int det J` a

A N iA s

dφdtfffffffff g

+D A dN i

dxffffffffffffAdφdxffffffff+ dN i

dyffffffffffffAdφdyffffffff

h

j

i

k

h

l

j

i

m

k

+W int ΝiAA A φ 1@ φ

φ lim

fffffffffffh

j

i

k@W int ΝiAB A φ @W int Ν

iAE A φ

+W int ΝiAPm 1@ φ

φ max

ffffffffffffh

j

i

kφ X @ W int ΝiADmA φ Aφ X

h

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

j

i

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

k

(Eq A-23)

K i , jb c

=XX

W int det J` a

N i N jA

sΔtfffffff+D A dN i

dxffffffffffffAdN j

dxfffffffffffff+ dN i

dyffffffffffffAdN j

dyfffffffffffff

h

j

i

k

h

l

j

i

m

k+

dDdtffffffffffA N jAdN i

dxffffffffffffAdφdxffffffff+ dN i

dyffffffffffffAdφdyffffffff+

@W int N i N j A A 1@ 2 Aφφ lim

fffffffffffh

j

i

k+W int N i N jAB +W int N i N j

AE

@W int N i N jAPmA

φ X

φ lim

fffffffffff+ W int N i N jADmAφ X

h

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

j

i

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

k

(Eq A-24)

A5 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

Appendix A

140

equations, φ is the concentration of the current variable, and φX 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:

1 D MSC AMSC B MSC E MSC 1 0 0 01 DFB AFB BFB EFB 1 0 0 01 DCC ACC BCC ECC 1 0 0 01 DOB AOB BOB EOB 1 0 0 01 0 0 0 0 0 PmFT DmFT 11 0 0 0 0 0 PmC DmC 11 0 0 0 0 0 PmB DmB 1

h

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

j

i

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

k

z~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x

Solub Diffus Prolif Differ Apop Cmax MatP MatD Mmax

MSCFBCCOBFTCB

(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.

141

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-lower-the-better signal to noise ratio, (S/N), as

)1log(10/ 2iy

NS −= (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

iiT NSNSSS

1

2)//( (Eq B-2)

where n was the number of experiments, S/Ni the signal-to-noise ratio for the ith treatment condition, NS / was the overall mean of S/N. For each factor, the sum of the squares of deviation about the mean was

∑=

−=n

iFiFiF NSNSNSS

1

2)//( (Eq B-3)

Where F is the factors, and n is the number of levels, NFi is the number of experiments at each level of each factor, FiNS / 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.

factorDOFSSMS FF = (Eq B-4)

∑=

−=F

iiTE SSSSSS

1 (Eq B-5)

factorDOFSSMS EE = (Eq B-6)

The fraction of the variance explained by each factor is calculated from the F value as EF MSMSF = . The percentage of the total sum of squares represents the approximate contribution of each factor to the variance and was calculated as ( ) %100% ⋅= TF SSSSTSS .

Appendix B

142

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

P M O C MSC FB CC OB MSC FB CC OB MSC FB CC OB MSC FB CC OB FT C B FT C Bx1 x2 x3 x4 x5 x6 x7 x8 x9 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 -12 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -13 -1 -1 -1 -1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 14 -1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 15 -1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 -16 -1 -1 -1 1 -1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 -17 -1 -1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 18 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 19 -1 -1 1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 1 -1 -1 1 1 -1 1 1 -1 -1 1

10 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 111 -1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -112 -1 -1 1 -1 1 1 -1 1 1 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 -113 -1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 1 114 -1 -1 1 1 -1 1 -1 1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 115 -1 -1 1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 1 1 1 1 -1 1 1 -1 -1 -116 -1 -1 1 1 1 1 1 -1 -1 1 1 -1 1 -1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -117 -1 1 -1 -1 -1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 118 -1 1 -1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 119 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -120 -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -121 -1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 1 122 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 123 -1 1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 1 1 -1 1 1 -1 1 -1 -124 -1 1 -1 1 1 1 1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 -125 -1 1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 1 1 -1 -126 -1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 -127 -1 1 1 -1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 1 1 -1 -1 1 128 -1 1 1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 129 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 -1 1 1 1 -1 -1 1 -130 -1 1 1 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 -131 -1 1 1 1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 1 -1 1 1 -1 1 1 -1 132 -1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 1 -1 -1 -1 1 1 -1 133 1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 1 -1 1 1 1 -1 -1 1 -134 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 -135 1 -1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 1 -1 136 1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 1 1 -1 1 -1 -1 -1 1 1 -1 137 1 -1 -1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 1 -1 1 1 -1 -138 1 -1 -1 1 -1 1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 -139 1 -1 -1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 1 140 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 141 1 -1 1 -1 -1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 142 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 143 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 1 1 -1 1 -1 -144 1 -1 1 -1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 -145 1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 -1 146 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 147 1 -1 1 1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -148 1 -1 1 1 1 1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -149 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 1 1 1 1 -1 -1 -1 1 1 150 1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 151 1 1 -1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 -1 1 1 1 1 -1 1 1 -1 -1 -152 1 1 -1 -1 1 1 -1 1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -153 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 1 1 -1 1 1 -1 -1 154 1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 155 1 1 -1 1 1 -1 -1 1 1 -1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -156 1 1 -1 1 1 1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 -157 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 -158 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 -159 1 1 1 -1 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 160 1 1 1 -1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 161 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -162 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 -163 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 164 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Apoptosis Matrix prod. Matrix deg.Initial MSC conc. Proliferation Differentiation Migration

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.

Appendix B

143

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

P M FB CC OB FT C B FT C Exp x1 x2 x3 x4 x5 x6 x7 x8 x9 x10

1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -12 -1 -1 -1 -1 0 0 0 0 0 03 -1 -1 -1 -1 1 1 1 1 1 14 -1 0 0 0 -1 -1 -1 0 0 05 -1 0 0 0 0 0 0 1 1 16 -1 0 0 0 1 1 1 -1 -1 -17 -1 1 1 1 -1 -1 -1 1 1 18 -1 1 1 1 0 0 0 -1 -1 -19 -1 1 1 1 1 1 1 0 0 0

10 0 -1 0 1 -1 0 1 -1 0 111 0 -1 0 1 0 1 -1 0 1 -112 0 -1 0 1 1 -1 0 1 1 013 0 0 1 -1 -1 0 1 0 1 -114 0 0 1 -1 0 1 -1 1 -1 015 0 0 1 -1 1 -1 0 -1 0 116 0 1 -1 0 -1 0 1 1 -1 017 0 1 -1 0 0 1 -1 -1 0 118 0 1 -1 0 1 -1 0 0 1 -119 1 -1 1 0 -1 1 0 -1 1 020 1 -1 1 0 0 -1 1 0 -1 121 1 -1 1 0 1 0 -1 1 0 -122 1 0 -1 1 -1 1 0 0 -1 123 1 0 -1 1 0 -1 1 1 0 -124 1 0 -1 1 1 0 1 -1 1 025 1 1 0 -1 -1 1 0 1 0 -126 1 1 0 -1 0 -1 1 -1 1 027 1 1 0 -1 1 0 -1 0 -1 1

Matrix degradation

Initial MSC concentration

Proliferation Matrix production

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.

Appendix B

144

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

Exp. No.


S/NGeneral normal healing

S/NEARLY STAGE

Bone formS/N

MID STAGE

Bone formS/N

LATE STAGE

Bone formS/N

AVERAGE Bone form

S/N

1 20 -26.021 6 15.563 0.819 -1.736 0.991 -0.081 1.000 -0.003 0.936 -0.5712 24 -27.604 7 16.902 0.810 -1.829 0.988 -0.105 1.000 0.000 0.933 -0.6053 82 -38.276 6 15.563 0.289 -10.769 0.101 -19.895 0.081 -21.834 0.157 -16.0714 41 -32.256 6 15.563 0.326 -9.747 0.599 -4.455 0.990 -0.088 0.638 -3.9035 92 -39.276 5 13.979 0.323 -9.818 0.138 -17.173 0.006 -44.620 0.156 -16.1516 73 -37.266 5 13.979 0.354 -9.009 0.250 -12.029 0.325 -9.760 0.310 -10.1747 24 -27.604 7 16.902 0.785 -2.098 0.972 -0.245 1.000 0.000 0.919 -0.7328 24 -27.604 7 16.902 0.823 -1.693 0.981 -0.166 1.000 -0.002 0.935 -0.5879 54 -34.648 6 15.563 0.213 -13.449 0.275 -11.211 0.895 -0.963 0.461 -6.72810 70 -36.902 6 15.563 0.352 -9.059 0.132 -17.613 0.036 -28.912 0.173 -15.22411 41 -32.256 7 16.902 0.826 -1.666 0.838 -1.537 0.980 -0.177 0.881 -1.10012 56 -34.964 7 16.902 0.782 -2.132 0.822 -1.697 0.924 -0.683 0.843 -1.48313 71 -37.025 7 16.902 0.832 -1.602 0.559 -5.057 0.856 -1.349 0.749 -2.51314 63 -35.987 7 16.902 0.669 -3.492 0.658 -3.640 0.900 -0.919 0.742 -2.59115 64 -36.124 6 15.563 0.298 -10.507 0.171 -15.358 0.683 -3.312 0.384 -8.31416 83 -38.382 7 16.902 0.310 -10.178 0.160 -15.893 0.663 -3.568 0.378 -8.45517 49 -33.804 7 16.902 0.778 -2.178 0.237 -12.513 0.945 -0.492 0.653 -3.69818 56 -34.964 7 16.902 0.778 -2.185 0.199 -14.009 0.355 -8.993 0.444 -7.05219 67 -36.521 6 15.563 0.350 -9.115 0.512 -5.808 0.865 -1.261 0.576 -4.79520 72 -37.147 7 16.902 0.277 -11.163 0.398 -8.000 0.810 -1.828 0.495 -6.10921 59 -35.417 6 15.563 0.348 -9.173 0.553 -5.147 0.894 -0.975 0.598 -4.46322 66 -36.391 6 15.563 0.331 -9.599 0.380 -8.414 0.782 -2.131 0.498 -6.06023 36 -31.126 7 16.902 0.661 -3.602 0.318 -9.949 0.993 -0.065 0.657 -3.64824 41 -32.256 7 16.902 0.788 -2.068 0.240 -12.404 0.979 -0.185 0.669 -3.49225 33 -30.370 6 15.563 0.435 -7.223 0.808 -1.850 0.998 -0.019 0.747 -2.53226 34 -30.630 6 15.563 0.216 -13.298 0.873 -1.178 0.992 -0.069 0.694 -3.17427 118 -41.438 7 16.902 0.624 -4.095 0.165 -15.628 0.003 -51.858 0.264 -11.56728 46 -33.255 7 16.902 0.851 -1.399 0.287 -10.845 0.942 -0.520 0.693 -3.18129 59 -35.417 7 16.902 0.648 -3.769 0.195 -14.191 0.661 -3.599 0.501 -5.99830 49 -33.804 7 16.902 0.835 -1.561 0.308 -10.221 0.926 -0.667 0.690 -3.22431 49 -33.804 6 15.563 0.388 -8.233 0.536 -5.418 0.948 -0.463 0.624 -4.09832 81 -38.170 7 16.902 0.196 -14.148 0.094 -20.505 0.338 -9.423 0.209 -13.57733 43 -32.669 7 16.902 0.729 -2.743 0.765 -2.330 0.983 -0.151 0.826 -1.66534 91 -39.181 7 16.902 0.615 -4.216 0.201 -13.947 0.004 -48.983 0.273 -11.26935 37 -31.364 6 15.563 0.188 -14.531 0.708 -2.995 0.995 -0.043 0.630 -4.00836 34 -30.630 6 15.563 0.487 -6.241 0.851 -1.405 0.995 -0.046 0.778 -2.18537 98 -39.825 6 15.563 0.162 -15.835 0.077 -22.294 0.007 -43.182 0.082 -21.75038 63 -35.987 7 16.902 0.737 -2.652 0.250 -12.032 0.877 -1.143 0.621 -4.13439 34 -30.630 7 16.902 0.806 -1.870 0.580 -4.738 0.988 -0.108 0.791 -2.03540 82 -38.276 7 16.902 0.479 -6.401 0.123 -18.227 0.040 -28.048 0.214 -13.40841 82 -38.276 6 15.563 0.222 -13.091 0.317 -9.968 0.720 -2.858 0.420 -7.54542 35 -30.881 5 13.979 0.353 -9.049 0.656 -3.656 0.999 -0.011 0.669 -3.48743 41 -32.256 7 16.902 0.815 -1.778 0.282 -10.993 0.985 -0.129 0.694 -3.17244 62 -35.848 7 16.902 0.700 -3.101 0.222 -13.073 0.493 -6.136 0.472 -6.52745 60 -35.563 6 15.563 0.815 -1.780 0.194 -14.259 0.322 -9.834 0.444 -7.06146 59 -35.417 7 16.902 0.598 -4.460 0.180 -14.880 0.626 -4.067 0.468 -6.59047 43 -32.669 6 15.563 0.160 -15.907 0.399 -7.982 0.993 -0.065 0.517 -5.72648 53 -34.486 6 15.563 0.351 -9.100 0.439 -7.155 0.931 -0.621 0.574 -4.82949 25 -27.959 5 13.979 0.784 -2.119 0.971 -0.251 1.000 -0.001 0.918 -0.74050 46 -33.255 7 16.902 0.758 -2.407 0.392 -8.144 0.941 -0.524 0.697 -3.13551 69 -36.777 6 15.563 0.346 -9.222 0.390 -8.173 0.850 -1.417 0.529 -5.53852 83 -38.382 6 15.563 0.181 -14.853 0.099 -20.060 0.006 -44.482 0.095 -20.41153 73 -37.266 6 15.563 0.352 -9.063 0.140 -17.076 0.163 -15.757 0.218 -13.21454 70 -36.902 6 15.563 0.266 -11.509 0.428 -7.369 0.861 -1.297 0.518 -5.70755 64 -36.124 7 16.902 0.539 -5.362 0.693 -3.183 0.902 -0.896 0.712 -2.95656 35 -30.881 7 16.902 0.815 -1.776 0.906 -0.858 0.987 -0.117 0.903 -0.89157 84 -38.486 6 15.563 0.297 -10.556 0.142 -16.951 0.332 -9.588 0.257 -11.81058 90 -39.085 6 15.563 0.260 -11.710 0.115 -18.807 0.009 -40.729 0.128 -17.86459 30 -29.542 7 16.902 0.725 -2.797 0.822 -1.705 1.000 0.000 0.849 -1.42460 43 -32.669 6 15.563 0.785 -2.097 0.768 -2.294 0.986 -0.127 0.846 -1.45061 31 -29.827 7 16.902 0.680 -3.352 0.647 -3.777 1.000 -0.004 0.776 -2.20862 30 -29.542 7 16.902 0.741 -2.603 0.849 -1.421 1.000 0.000 0.863 -1.27663 88 -38.890 6 15.563 0.306 -10.287 0.120 -18.450 0.078 -22.135 0.168 -15.49964 81 -38.170 6 15.563 0.255 -11.861 0.136 -17.356 0.286 -10.887 0.225 -12.939

mean -34.44 16.13 -6.66 -9.19 -7.69 -6.32SST 818.20 46.57 1300.03 2694.67 12328.48 1780.00SSE 817.21 45.67 1300.03 2694.67 12328.48 1780.00MSE 408.61 22.83 650.02 1347.33 6164.24 890.00

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.

Appendix B

145

Higher level experiment The results from each of the outcome analyses and the calculated signal-to-noise ratio for the higher level experiment was Exp. No.


S/NGeneral normal healing

S/NEARLY STAGE

Bone formS/N MID STAGE

Bone form S/NLATE

STAGE Bone form

S/N AVERAGE Bone form S/N

1 20 -26.021 6 15.563 8.42E-01 -1.494 0.999 -0.007 1.000 0.000 0.947 -0.4722 31 -29.827 6 15.563 5.02E-01 -5.988 0.796 -1.976 1.000 -0.002 0.766 -2.3153 30 -29.542 6 15.563 4.83E-01 -6.321 0.215 -13.357 0.021 -33.446 0.240 -12.4064 82 -38.276 7 16.902 7.46E-01 -2.550 0.978 -0.189 1.000 0.000 0.908 -0.8385 19 -25.575 5 13.979 5.95E-01 -4.505 0.670 -3.473 0.851 -1.398 0.706 -3.0276 32 -30.103 7 16.902 7.57E-01 -2.417 0.228 -12.843 0.064 -23.895 0.350 -9.1287 40 -32.041 5 13.979 5.91E-01 -4.561 0.694 -3.177 0.889 -1.019 0.725 -2.7968 70 -36.902 7 16.902 8.29E-01 -1.625 0.807 -1.865 1.000 0.000 0.879 -1.1239 32 -30.103 7 16.902 6.10E-01 -4.296 0.200 -13.998 0.014 -36.915 0.275 -11.228

10 61 -35.707 7 16.902 8.16E-01 -1.761 0.229 -12.791 0.712 -2.950 0.586 -4.64311 23 -27.235 6 15.563 6.90E-01 -3.224 0.247 -12.159 0.185 -14.659 0.374 -8.54612 47 -33.442 7 16.902 4.65E-01 -6.652 0.319 -9.918 0.973 -0.240 0.586 -4.64713 32 -30.103 5 13.979 7.30E-01 -2.737 0.264 -11.563 0.931 -0.620 0.642 -3.85414 55 -34.807 7 16.902 5.65E-01 -4.959 0.865 -1.264 1.000 -0.004 0.810 -1.83315 42 -32.465 7 16.902 7.59E-01 -2.392 0.741 -2.609 0.877 -1.142 0.792 -2.02316 35 -30.881 6 15.563 5.90E-01 -4.587 0.234 -12.610 0.950 -0.443 0.591 -4.56317 50 -33.979 7 16.902 8.22E-01 -1.699 0.784 -2.111 0.883 -1.083 0.830 -1.62118 84 -38.486 7 16.902 6.23E-01 -4.114 0.737 -2.647 0.955 -0.396 0.772 -2.24919 33 -30.370 7 16.902 8.03E-01 -1.910 0.962 -0.335 0.992 -0.066 0.919 -0.73320 25 -27.959 5 13.979 6.74E-01 -3.433 0.232 -12.700 0.736 -2.665 0.547 -5.24021 54 -34.648 6 15.563 3.87E-01 -8.256 0.500 -6.024 0.990 -0.085 0.626 -4.07522 55 -34.807 6 15.563 6.79E-01 -3.368 0.627 -4.060 0.861 -1.300 0.722 -2.82923 49 -33.804 7 16.902 5.10E-01 -5.856 0.191 -14.358 0.069 -23.204 0.257 -11.81024 70 -36.902 7 16.902 7.33E-01 -2.693 0.794 -1.998 0.904 -0.874 0.811 -1.82225 61 -35.707 7 16.902 5.99E-01 -4.453 0.871 -1.197 1.000 0.000 0.823 -1.68826 40 -32.041 6 15.563 8.09E-01 -1.842 0.268 -11.441 0.897 -0.941 0.658 -3.63527 60 -35.563 4 12.041 6.04E-01 -4.377 0.776 -2.202 0.866 -1.254 0.749 -2.515

mean -32.492 15.892 -3.78 -6.403 -5.504 -4.136SST 336.92 44.61 81.98 745.95 3014.79 310.84

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.

147

R

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163

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 niet-helende 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

Samenvatting

164

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.

165

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

167

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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