Feasibility of agent based modelling of articular cartilage including a conceptual representation of its structure

The new model of articular cartilage has two characteristics, namely: i not use fibre-reinforced composite material idealization ii Provide a framework for that it does probing the micro

Feasibility of agent-based modelling of articular cartilage including a conceptual

representation of its structure

Quang Thien Duong

M.Sc (Applied Mechanics) (Ho Chi Minh University of Natural Sciences) - 2003

Thesis submitted in accordance with the regulations for the Degree of

Doctor of Philosophy in the Science and Engineering Faculty, Queensland University

of Technology according to QUT requirements

proteoglycans constrained within a 3D network of collagen fibrils Because of the complexity

of the cartilage structure, the relationship between its mechanical behaviours at the scale level and its components at the micro-scale level are not completely understood The research objective in this thesis is to create a new model of articular cartilage that can

macro-be used to simulate and obtain insight into the micro-macro-interaction and mechanisms underlying its mechanical responses during physiological function The new model of articular cartilage has two characteristics, namely: i) not use fibre-reinforced composite material idealization ii) Provide a framework for that it does probing the micro mechanism

of the fluid-solid interaction underlying the deformation of articular cartilage using simple rules of repartition instead of constitutive / physical laws and intuitive curve-fitting

Even though there are various microstructural and mechanical behaviours that can be studied, the scope of this thesis is limited to osmotic pressure formation and distribution and their influence on cartilage fluid diffusion and percolation, which in turn governs the deformation of the compression-loaded tissue

The study can be divided into two stages In the first stage, the distributions and concentrations of proteoglycans, collagen and water were investigated using histological protocols Based on this, the structure of cartilage was conceptualised as microscopic osmotic units that consist of these constituents that were distributed according to histological results These units were repeated three-dimensionally to form the structural model of articular cartilage In the second stage, cellular automata were incorporated into

of health and loading These behaviours are illuminated at the microscale level using the called neighbourhood rules developed in the thesis in accordance with the typical requirements of cellular automata modelling Using these rules and relevant Boundary Conditions to simulate pressure distribution and related fluid motion produced significant results that provided the following insight into the relationships between osmotic pressure gradient and associated fluid micromovement, and the deformation of the matrix For example, it could be concluded that:

so-1 It is possible to model articular cartilage with the agent-based model of cellular automata and the Margolus neighbourhood rule

2 The concept of 3D inter connected osmotic units is a viable structural model for the extracellular matrix of articular cartilage

3 Different rules of osmotic pressure advection lead to different patterns of deformation in the cartilage matrix, enabling an insight into how this micromechanism influences macromechanical deformation

4 When features such as transition coefficient were changed, permeability (representing change) is altered due to the change in concentrations of collagen, proteoglycans (i.e degenerative conditions), the deformation process is impacted

5 The boundary conditions also influence the relationship between osmotic pressure gradient and fluid movement at the micro-scale level

and material systems

Table of Contents v

List of Figures xiii

List of Equations xxi

List of Diagrams xxiii

List of Tables xxiv

List of Matlab Programs xxv

List of Abbreviations xxv

List of Papers and Posters xxvi

Declaration xxvii

Acknowledgements xxviii

Chapter 1 Introduction 1.1 The focus of the thesis 1

1.2 Research gap in articular cartilage computational modelling associated research 1 1.2.1 Current experimental studies of articular cartilage 3

1.2.1.1 General review 3

1.2.1.2 Some main factors governing the deformation of articular cartilage 6

1.2.2 Numerical modelling of cartilage and related simulations 12

1.2.3 The gap in research 17 1.3 Significance and benefits 19

Chapter 3 The conceptual in-silico structural model of articular cartilage

3.4 Virtual clay simulation 85

Chapter 4 Determination of cartilage matrix components

4.2.3.1 An example for calculating collagen concentration from absorbance values 124

4.5 Calculating osmotic pressure and collagen meshwork stress for articular cartilage 139

Chapter 5 Deformable cellular automata of articular cartilage

5.2.2 Visualization of the change of the thicknesses during the deformation process 158

6.2 The set of rules using in the model of articular cartilage 169

Appendixes (all appendixes are stored in the CD)

Appendix 1 New approach for articular cartilage and expected outcomes

Appendix 5.3 Water (g/cm3) Appendix 5.4 Thickness (cm)

Appendix 6.1 Main program for mode 1

Appendix 6.2 Main program for mode 2

Appendix 6.3 Sub program odd_step_active

Appendix 6.4 Sub program even_step_active

Appendix 6.5 Sub program CombineLayer

Appendix 6.6 Sub program odd_step_computation

Appendix 6.7 Sub program even_step_computation

Appendix 7 The graphs

Fig 2.1: The human knee showing the location of articular cartilage 29 Fig 2.2: The scheme of collagen meshwork entrap proteoglycans 30

Fig 2.3: The four zones of articular cartilage(James et al., [2001]) 32

Fig 2.4 a: TEM of normal general matrix structure as fixed on-bone in its pre-equilibration control state b: in its post-equilibration state Arrows indicate radial direction Bars, 1

Fig 2.5 a: TEM of normal general matrix structure as fixed on-bone in its pre-equilibration control state; b: in its post-equilibration state Arrows indicate radial direction Bars, 1

Fig 2.6: Structure of a collagen molecule (Shipman et al., [1985]) 38

Fig 2.8: Schematic illustration of the collagen meshwork within the matrix (Broom, [1982])

a) full thickness (from surface to calcified zone)

b) one fibril of the middle zone

Fig 2.9: SEM of the softened general matrix (Chen and Broom, [1999])

Arrow indicates radial direction Bar, 1micoron

a) Pre-swelling control state as fixed on-bone

relatively short distances in the transverse direction (Chen and Broom, [1999]) 43

Fig 2.12: A: The 3-D view of human articualar cartilage; B,C,D: The 2D sections from Fig A

Fig 2.16: Volume change in articular cartilage depends on water flow 57 Fig 2.17: An osmotic unit with collagen acts as membrane and proteoglycan

Chapter 3

Fig 3.1: The conceptual process to create the innovative model of articular cartilage in this

represents the collagen fibre network; the swollen balloons analogise the fluid swollen

Fig 3.4: Model of osmotic unit with collagen acts as membrane and proteoglycan

Fig 3.5: Comparing the structure of osmosis unit and the principle of the Margolus

Chapter 4

Fig 4.2: Cryostat section

a) before cutting with using cryostat, b) on a microscope slide 99

Fig 4.4: A digital image taken before staining with Safranin O 105

Fig 4.6: Sample image showing the optical density (proteoglycans) variation in a stained

Fig 4.7: Variation of absorbance (proteoglycans) with cartilage thickness 108

Fig 4.10: Variation of proteoglycans from medial to lateral (surface bone) direction in a

Fig 4.13: Variation of proteoglycans with section position (superior-inferior) from superficial

Fig 4.14: Variation of average proteoglycans with section position (superior -inferior) in a

Fig 4.15: Variation of average proteoglycans with section position (medial-lateral) 115 Fig 4.16: The response of absorbance value depending on concentration of Direct Red

121

Fig 4.19: Sample image showing the optical density (collagen) variation in a stained cartilage

Fig 4.20: Variation of absorbance (collagen) with cartilage thickness 126 Fig 4.21: Variation of collagen from medial to lateral (surface bone) direction in a patella

129 Fig 4.22: Variation of collagen from superior to inferior (surface bone) direction in a patella

129 Fig 4.23: Variation of collagen with section position (medial-lateral) from superficial to

Fig 4.24: Variation of collagen with section position (superior-inferior) from superficial to

135 Fig 4.28: Variation of water from superior to inferior (surface bone) direction in a patella

135 Fig 4.29: Variation of water with section position (medial-lateral) from superficial to

Fig 4.30: Variation of water with section position (superior-inferior) from superficial to

Fig 4.31: Variation of average water with section position (superior -inferior) in a cartilage

Fig 4.32: Variation of average water with section position (medial-lateral) 137 Fig 4.33 O PG, of solutions of proteoglycans extracted from human adult femoral head cartilage as a function of fixed charge density (FCD) 139 Fig 4.34: Relationship between tensile stress of collagen and its concentration in the normal articular cartilage matrix based on publish data (Basser et al., [1998]) 144

Fig 4.35: The spectral profile of dilute dissolved Safranin-O in range from 0 to 0.25mg/ml

148 Fig 4.36: The spectral profile of 7-micrometer cartilage section in 5 random positions from

Chapter 5

Fig 5.3: The patellae bone is used to generate the bottom of the cartilage model 160 Fig 5.4: The surface of the bone i.e the bottom of the cartilage model 161 Fig 5.5: The grid in the whole matrix with interpolation linear 162 Fig 5.6: Mapping content value via a look up table 163

Chapter 6

Fig 6.4: Moving plate is in the top Next to the moving plate is the active layer 174 Fig 6.5: Indenter move to layer 2 There is no more water in layer 2 175 Fig 6.6: The moving plate makes the alternative active and inactive layers in whole from

Fig 6.8a: The first four cells of layer 1 and the first four cells of layer 2 make 3D- block 1 187 Fig 6.8b: The next four cells of layer 1 and the next four cells of layer 2 make 3D-block 2 187

Fig 6.11 T x,  y and thickness 193

Chapter 7

Fig 7.2: Strain at A, B, C,and D with Pathway 1, Mode 1, Rule 1, Boundary condition 1D,

Fig 7.15: Dead-alive map in layer 05, step 16, under condition of rule 03 256

Fig 7.22: Strain versus time plots with the = 0.3, Pathway 1 265 Fig 7.23: Strain versus time plots with the = 0.3, Pathway 2 266 Fig 7.24: The cartilage matrix with the two regions when loading 268

Fig 7.26: The two - value curve comparing to one value and the experimental curves

270 Fig 7.27: The two values in boundary and internal area bring in-silico model closer to real

Fig 7.30: The deformations with the two values in boundary area and internal area 274

Page 65

Chapter 4

Page 91

Page 91

Page 98

Page 98

Diagram 4.1: Calculating water concentration at each voxel 134

Chapter 5

Diagram 5.1: The process of determining the deformation of the model 152

Diagram 5.2: The process of applying the experimental results into the in-silico model 154

Diagram 5.3: The progress of visualization of the thickness 159

Table 4.2: The rate of collagen plus proteoglycans from layer 1 to layer 10 134

Chapter 6

MainPrM1 (Main Program for Mode 1) 573

BEM: Boundary Element Method

CA: Cellular automata

CS: Chondroitin Sulphate

CT: Computed Tomography

FCD: Fixed Charge Density

FDM: Finite Difference Method

FE: Finite Element

FEA: Finite Element Analysis

FEM: Finite Element Method

IGFs: Insulin-like growth factors

KS: Keretan Sulphate

MFM: Meshfree Method

MRI: Magnetic Resonance Imaging

PEG: Polyethylene glycol

PG: Proteoglycan

PLM: Polarised Light Microscope

SEM: Scanning Electron Microscope

TEM: Transmission Electron Microscopy

List of Papers and Posters

Q.T Duong, K.Yusuf, A Oloyede

3D Mapping of Bovine Cartilage Proteoglycans: A Preliminary Analysis towards the Mapping of Human Articular Cartilage Proteoglycans

Proceeding of the IHBI Inspires 2008, Gold Coast- Australia, 4-5 December 2008

at any higher education institution To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where explicitly stated otherwise and due reference is made

Signed:

Date:

his time to teach me, both consciously and un-consciously, how good the thesis is done I

am grateful to him for his contributions of time, ideas to make me much stronger during the Ph.D journey, give me my Ph.D experience productive and stimulating His wisdom and maturity in research was the recourse that helps me during the tough time of the Ph.D pursuit I appreciate the values he teach me to finish the work It has been an honour to be his student

The members of the PC2 laboratory such as Ms Melissa Johnston, Senior Technician (Medical Engineering), Mr Jonathan James, supervising technician (civil, mechanical) have contributed to my histological experiments The biology lab in Q block, Gardens Point, where I spent a lot of time was the place I learn how to divide the samples into several-micrometer thin slices using cryostat, and how to store these slices I am especially thankful for the help of Mr Donald Geyer, Mr Joseph Kan, senior technician in Histology Department, who guided me in the lab I also want to take this chance to say thank you to the member in the lab in R block and IHBI lab such as Dr Joshua Bowden who taught me how to perform image processing with ImageJ (you are really a guru in this field), Dr Travis Klein in Cartilage Regeneration Laboratory, who help me to perform the calibration to work out absorbance coefficient of collagen and proteoglycans, Dr Loc Duong, Electron microscopist, Concrete petrologist and Dr Christina Theodoropoulos, Microscopy Applications Specialist (Biological) in Analytical Electron Microscopy Facility unit, Faculty of

codes Your enthusiasm, intensity, willingness and amazing ability made me much more skilful in programming with Matlab

In my attempted correcting my English, I thank the following people for helpful works: Diane Kolomeitz, Research Office Coordinator, Research Portfolio in BEE, Lynda Lawson, Language and Learning Advisor, International Student Services, Denise Scott in WriteTouch company, Timothy Bodisco, Ph D student, Benjamin Henley, my brother-in-law Thank you so much for your patience of listening to my explanation of the complex and difficulty concepts in my thesis to give me the best English that makes sense

For the thesis, I am so thankful to my reading committee members for their time, interesting and helpful comments I also want to take this chance to thank the other members of my final seminal for their time

I gratefully acknowledge the funding sources that made my Ph.D work possible I was funded by QUT for my years

my side and give me the motivation and inspiration to finish the thesis

Thank you

Quang Thien Duong

Chapter 1 Introduction

1.1 The focus of the thesis

This thesis concerns numerical modelling of cartilage with emphasis on capturing the micro-scale fluid movement and distribution processes governing its deformation Cellular automata and neighbourhood rules of repartition are combined to simulate pressure development and redistribution within the loaded cartilage matrix The expected outcome is a model that combines to the structural modelling of the tissue leading to an understanding of the complex transient osmotic / hydrostatic pressure processes underlying the macro-scale deformation of normal intact cartilage This innovative approach will also produce a means of numerically probing the internal mechanism of the tissue beyond experimental capabilities and a conceptual framework for studying the biological and biochemical interaction between cartilage and its external

environment in-vivo

1.2 Research gap in articular cartilage computational modelling associated research

Articular cartilage is a relatively thin translucent soft tissue that covers the end

of long mammalian bones such as the femur Basically, articular cartilage consists of three major or load bearing components, namely collagen fibre meshwork, proteoglycans and water The 3-D network of mostly type II collagen fibrils entraps an aggregation of fluid-swollen proteoglycans

So far, models of articular cartilage namely numerical, mathematical and experimental that have been developed based on the observed characteristics

of the tissue and intuition are unable to fully contribute and extend our current understanding of the complex mechanism underlying its biomechanical behaviours It is conceptualized that these mechanisms are microscopic biochemical and physical processes occurring at the local (i.e microscopic) level, leading to phenomena and processes at the global (i.e macroscopic) level; and are practically impossible to study with simple classical laboratory experiments, most of which are constrained by the need to maintain the delicate cartilage structure and ethical standards

Consequently, these experiments have only provided researchers with only macroscopic characteristics of the tissue leading to significant limitation in the understanding of the health to disease transitional processes Furthermore, the current mathematical and numerical models have not provided the knowledge that would unlock the micro-mechanism of cartilage deformation because they depend on intuitive analysis and interpretation of the results of the limited experimental observations They are also incapable of evolutionary processes that oscillate between order (healthy condition) and disorder (degeneration) in

a tissue like articular cartilage that is necessary to study and understand the initiation and progression of cartilage breakdown A new approach is needed

This thesis attempts to create and examine an innovative in-silico model of the

tissue that will potentially facilitate a detailed study of the micro-scale events underlying cartilage deformation, function and degeneration This will be limited, at this stage, to the modelling of the extracellular matrix and simulation

of the osmotic process and consequential pressure distribution governing fluid flow that in turn substantially determines the deformation and load bearing characteristics of articular cartilage

1.2.1 Current experimental studies of articular cartilage

1.2.1.1 General review

For years, many experimental studies have been conducted toward understanding the cartilage material Generally, we can classify the experiments into those designed to understand: 1) Fluid solid interaction (percolation) and permeability of loaded articular cartilage to the flow of fluid and how flow rates are generated during compression 2) Effect of load and rates of loading, friction properties of articular cartilage and the mechanism of lubrication in cartilage cartilage contact 3) Osmotic swelling processes 4) Effect of degeneration, for example removal of proteoglycans and the behaviour of degenerated articular cartilage

Wright and Dawson [1976] conducted experiments to analyse the mechanism

of lubrication of the synovial fluid on articular cartilage The results indicated that the load borne by the tissue could rise up to four times body weight in the hip and go up to twenty five times body weight in the knee during a walking

cycle This type of loading pattern further emphasizes the need for insight into the way and manner the cartilage matrix manages the applied load especially with regards to the fluid behaviour

Higginson and Snaith [1979] described an experiment in which the compressive stiffness of bovine and human articular cartilage under 1D constraints were measured leading to the conclusion that the creep behaviour of cartilage is well-defined and is in accordance with the earlier published studies of Freeman [1973], and Mow and Mansour [1977] Also, the dynamic response to oscillating load is almost linear elastic (Radin and Paul [1970], and Johnson et al., [1977]), demonstrating that the response can be described by an elastic modulus which

is much higher (~0.1 GPa) than those previously reported by e.g., Hori and Mockros [1976], who obtained an average of 0.084 minimum GPa and a maximum of 0.167 GPa for the compressive stiffness of cartilage

Another experimental study was conducted by Armstrong and Mow, [1982] Considering articular cartilage as a porous, permeable material, Armstrong and Mow, [1982] determined the intrinsic equilibrium modulus of the cartilage matrix and its permeability to fluid flow These authors noted that the visual or histological appearance of a cartilage specimen is not a good indicator of its

ability to function as the bearing material in the intact joint It is noted that this study did not lead to establishing the spatial and temporal distribution of

the permeability of the tissue which is important to its function and structural

differentiation, and which can provide an understanding of the dissimilarities in fluid flow and redistribution between healthy and degenerated cartilage Oloyede and Broom, [1991] and [1994b] tested cartilage with an innovative experimental apparatus, namely consolidometer, in order to obtain the transient pattern of the hydrostatic excess pressure and strain in the loaded matrix under confined and unconfined conditions These experiments demonstrated that the osmotically active fluid in cartilage behaves similarly to soils and clays and that load bearing is relative to the amount and rate of fluid exudation from a loaded sample While these experiments provide a window into the intramatrix fluid behaviour, they carry a significant limitation in that they are unable to provide information across the spatially complex environment of the cartilage matrix, as the data are limited to the response in the bottom layer only In another paper, Oloyede et al., [2004] used a consolidation test to determine the role of articular cartilage lipids in its load-bearing function, leading to the conclusion that the delipidized fluid-saturated articular cartilage is stiffer than its unaltered structurally intact counterpart with consequence for cartilage compliance during function Again, the results were confined to the bottom layer of the tissue with a concomitant loss in full spatial information of the intramatrix microscopic redistribution of the fluid, and osmotic/hydrostatic pressures, which is required for a thorough understanding

of the mechanisms underlying this change in stiffness

1.2.1.2 Some main factors governing the deformation of articular cartilage

Cartilage under mechanical tensile strained conditions

Williamson et al [2003] investigated tensile mechanical properties of articular cartilage in which the variations with growth and their relationships to collagen network components were enhanced Observing the growth from the foetus to the adult, the authors revealed that equilibrium and dynamic tensile moduli and strength of cartilage samples increased by an average of 391-1060%, whereas the strain at failure decreased by 43% Another result was that the collagen concentration (per wet weight) increased by 98%, and the pyridinoline cross-link concentration increased by 730%, while the glycosaminoglycan concentration remained unchanged or decreased slightly from the foetus to the adult Therefore, they concluded that the variation in the contents of collagen and pyridinoline cross-link, but not sulfated glycosaminoglycan influenced strongly the growth-associated variation in tensile moduli and strength of catilage

Huang et al [2001] in their study also stated that in general, collagen content and microstructural architecture dictated the intrinsic tensile propertiesof the cartilage matrix while proteoglycan content and molecular organizationas well

as water content are the main factors governing the intrinsic compressiveproperties

Li et al [2005] used stress relaxation measurements in the axial direction

(normal to the articular surface) to investigate the role of viscoelasticity

of collagen fibers in the cartilage matrix in compression and tension They concluded that for axial tension, collagen viscoelasticity accounted for most of the stress relaxation, while the effects of fluid pressurization on the tensile stress were negligible Li and Herzog [2004] and Henshaw et al [2006] provided the same results in their studies

Based on these researches, it is clear that the collagen structure plays an important role in the tensile deformation of articular cartilage

Li et al [2005] also found that for axial compression, the dominant mechanism for stress relaxation arose from fluid pressurization, not from collagen fibers Two characteristics of fluid namely hydraulic conductivity and osmotic pressure will be considered below

Cartilage under compression loads

Hydraulic conductivity

Schmidt et al [1990] conducted two biomechanical tests namely the viscoelastic creep test and a slow constant-rate uniaxial tension test In these tests, two enzymatic proteoglycan extraction procedures consisting of chondroitinase ABC treatment and a sequential enzymatic treatment with chondroitinase ABC, trypsin, and Streptomyceshyaluronidase were used The results of the viscoelastic creep test indicated that response of cartilage included two distinct phases: an initial phase (t < 15 s), characterized by a rapid

increase in strain following load application, and a late phase (15 s < t < 25,000 s), characterized by a more gradual increase in strain Considering the enzymatic proteoglycan extraction procedures during the tests, the authors found that the glycosaminoglycan content of the tissue was an important factor influencing the kinetics of the creep response

In another research, Oloyede et al., [1992] conducted compression tests on cartilage alone and cartilage-on-bone both at strain-rates from 10-5 sec-1 to 103sec-1 in order to investigate the compressive properties of articular cartilage

stiffness reached a limiting value Obviously, strain rate is a factor that influences the deformation of cartilage After analysing the characteristics of the strain field resulting from both low and high velocities, the authors concluded that cartilage matrix stiffness was controlled by two fundamentally different mechanisms of deformation

Based on the study of Oloyede et al., [1992], Rasanen, [1992] proposed the relationship between the apparent flow-independent stiffness of articular cartilage and the deformation rate The author also investigated whether the strain rate-stiffness curve will at some time scale to some "instant" rigidity However, as in the shear geometry, this study assumed that fluid flow was negligible which is not the case at rates below impact speeds Time-dependent

behaviour is expected to reflect the intrinsic properties of the cartilage matrix Mark et al [2001] simulated the effect of variable ramp strain-rates on the unconfined compression stress relaxation response of articular cartilage By investigating the abilities of the linear biphasic poroviscoelastic model and the linear biphasic poroelastic model, the authors concluded that there were two viscoelastic behaviours of articular cartilage, depending on strain-rates With a fast ramp strain-rate, a fluid flow-independent (intrinsic) viscoelastic mechanism is the main factor governing the behaviour, while under the condition of slow strain-rate, a fluid flow-dependent (biphasic) viscoelastic mechanism will influence the behaviour of the cartilage matrix

However, according to McCutchen, [1959], there is a type of lubrication in joint that is achieved at the cartilage-cartilage interface resulted from the action of the pressurized fluid This fluid is exuded from of any two contacting layers of the cartilage matrix under a compressive axial load From this hypothesis, McCutchen, [1982] suggested that the cartilage matrix is poroelastic, not viscoelastic In order to examine how the matrix fluid contributes to the provision of the mechanism in McCutchen, [1959], based on the suggestion of McCutchen, [1982], Oloyede and Broom, [1994b] conducted a research to obtain the response of articular cartilage to compression Using a new apparatus for testing the tissue in its unconfined state, the authors could measure simultaneously its strain and fluid excess pore pressure In order to