an innovative agent based cellular automata … · an innovative agent-based cellular automata...

AN INNOVATIVE AGENT-BASED

CELLULAR AUTOMATA FRAMEWORK FOR

SIMULATING ARTICULAR CARTILAGE

BIOMECHANICS

Jamal Kashani

MSc, BSc

Submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy

School of Chemistry, Physics and Mechanical Engineering

Faculty of Science and Engineering

Queensland University of Technology

2017

An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics i

Keywords

Articular cartilage

Agent-based method

Cellular automata

Porous media

Porosity

Single-phase multi-component material

Hybrid agent

Intra-agent rule

Extra-agent rule

Intra-agent evolution

Semi-permeable structure

Diffusion

Deformation

Consolidation

Wet salt

Dissolution

Virtual microscope

Microscopic mechanisms

ii An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics

An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics iii

Abstract

Articular cartilage is a porous, heterogeneous and semi-permeable biological

tissue that contains multiple components, which makes the tissue difficult to

characterise mechanically. Any attempt to insert a transducer inside the tissue via

piercing will damage the structure leading to unrepresentative data. Traditional

laboratory experiments and mathematical models so far have been limited in

describing fundamental insights into the complex behaviours of the articular

cartilage. This thesis aims to facilitate conducting cartilage characterisation

experiments on a computational platform to create a potential “virtual microscope”

that can monitor the constituent components inside the tissue. To this end, it

introduces a new computational cellular automata agent (hybrid agent) and a new

category of rules (intra-agent rules) that can be used to create emergent structures

that would more accurately represent single-phase multi-component materials. The

novel hybrid agent carries the characteristics of the system’s elements and it is

capable of changing within itself, while responding to its neighbours as they also

change. The performance of the hybrid agent under one-dimensional cellular

automata formalism is studied where growing patterns that demonstrate the striking

similarities with the porous saturated single-phase structures are generated.

The concepts of hybrid agent and intra-agent evolution are used to simulate

diffusion in articular cartilage. The spatial maps of diffusion at different times are

provided in colour-coded pictures and the simulated results are validated against

published magnetic resonance imaging (MRI) experimental data. The presented

novel agent-based approach is also used to simulate one-dimensional consolidation

of the articular cartilage, where the spatio-temporal patterns of fluid and solid are a

iv An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics

primary consideration. Qualitative and quantitative comparison of results with

experimental data shows that this novel approach can accurately and efficiently

simulate aspects of articular cartilage function. It demonstrates the potential of

hybrid agent and intra-agent evolution to enhance agent-based techniques for porous

materials and other areas of research. The feasibility of using this method for non-

biological systems is also demonstrated by simulating the wet salt dissolution

process.

The findings of this research demonstrate that (i) the enhanced agent offers an

improvement in simulating single-phase porous structures where the constituents are

practically inseparable up to the ultramicroscopic levels, (ii) the approach provides

insight into the microscopic mechanisms governing the functions of the tissue, and

(iii) this research proposes a viable opportunity for in silico experiments that can

facilitate provision of input data for numerical methods.

An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics v

List of Publications

Journal articles

Jamal Kashani, Graeme Pettet, Yuantong Gu, Lihai Zhang, Adekunle

Oloyede. An agent-based method for simulating porous fluid-saturated

structures with indistinguishable components. Journal of Physica A:

Statistical Mechanics and its Applications, 2017. 483: p. 36-43.

Jamal Kashani, Graeme Pettet, Yuantong Gu, Adekunle Oloyede. An agent-

based methodology to study load-carriage in healthy articular cartilage.

Under review in the Journal of the Mechanical Behavior of Biomedical

Materials.

Jamal Kashani, Graeme Pettet, Yuantong Gu, Adekunle Oloyede. An

innovative computational approach to study diffusion into articular cartilage.

Under review in the Journal of Annals of Biomedical Engineering.

Jamal Kashani, Graeme Pettet, Yuantong Gu, Adekunle Oloyede. Cellular

automata simulation of diffusion into degenerated articular cartilage (In

preparation).

Jamal Kashani, Graeme Pettet, Yuantong Gu, Adekunle Oloyede. The effect

of degeneration on the internal fluid flow of articular cartilage under static

loading: A cellular automata study (In preparation).

Conference and poster presentations

Jamal Kashani, Lihai Zhang, Yuantong Gu, Adekunle Oloyede. Feasibility of

agent-based method in collecting functional qualitative data of articular

cartilage. The 7th International Conference on Computational Methods

(ICCM2016), 1-4 August 2016, Berkeley, USA

Jamal Kashani, Yuantong Gu, Zohreh Arabshahi and Adekunle Oloyede.

An innovative agent-based approach to simulation of the functional

characteristics of the articular cartilage. Poster presentation at the IHBI

Inspires Postgraduate Student Conference 2016.

vi An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics

An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics vii

Table of Contents

Contents

Keywords .................................................................................................................................. i

Abstract ................................................................................................................................... iii

List of Publications ...................................................................................................................v

Table of Contents ................................................................................................................... vii

List of Figures ......................................................................................................................... xi

List of Tables ........................................................................................................................ xvi

List of Abbreviations ........................................................................................................... xvii

Acknowledgements ............................................................................................................... xix

Chapter 1: Introduction ...................................................................................... 1

1.1 Background .....................................................................................................................1

1.2 Research Problem ...........................................................................................................2

1.3 Aim and objectives .........................................................................................................4 1.3.1 Research questions ...............................................................................................4 1.3.2 Outcomes ..............................................................................................................5

1.4 Significance ....................................................................................................................5

1.5 Thesis Outline .................................................................................................................7

Chapter 2: Literature Review ........................................................................... 11

2.1 Introduction ..................................................................................................................11

2.2 Articular cartilage structure ..........................................................................................12

2.3 Articular cartilage load bearing ....................................................................................15

2.4 Articular cartilage degeneration ...................................................................................16

2.5 Mechanical properties of the articular cartilage ...........................................................17

2.6 Experimental methods for data collection ....................................................................19 2.6.1 Classical laboratory experiments ........................................................................19 2.6.2 Non-invasive methods ........................................................................................21

2.7 computational methods .................................................................................................22 2.7.1 Mixture models ...................................................................................................24 2.7.2 Continuum approach ..........................................................................................27 2.7.3 Finite Element method .......................................................................................28

2.8 Agent-based methods....................................................................................................30 2.8.1 Cellular automata ...............................................................................................33 2.8.2 Lattice gas automaton .........................................................................................36

2.9 Techniques to develop porous structures ......................................................................40

2.10 Inadequacy of current agents and rules for articular cartilage ......................................42

Chapter 3: New Agent and Rule ....................................................................... 45

3.1 Introduction ..................................................................................................................45

viii An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics

3.2 Hybrid agent ................................................................................................................. 45

3.3 Intra-agent rule ............................................................................................................. 46

3.4 Adaptation of the hybrid agent for porous materials .................................................... 49

Chapter 4: Using hybrid agents to create porous structures ......................... 53

4.1 Introduction .................................................................................................................. 53

4.2 Methodology ................................................................................................................ 53 4.2.1 Hybrid agent ...................................................................................................... 53 4.2.2 Extra-agent rule.................................................................................................. 55 4.2.3 Intra-agent rules ................................................................................................. 55 4.2.4 Two-dimensional domain .................................................................................. 58

4.3 Results and discussion ................................................................................................. 61 4.3.1 Effect of different rules ...................................................................................... 61 4.3.2 Effect of initial seed ........................................................................................... 69

Chapter 5: Diffusion throughout the articular cartilage ................................ 73

5.1 Introduction .................................................................................................................. 73

5.2 Material and methods ................................................................................................... 74 5.2.1 Adaptation of the hybrid agent for diffusion of the articular cartilage .............. 74 5.2.2 The matrix model ............................................................................................... 75 5.2.3 Rules .................................................................................................................. 77 5.2.4 Corresponding time step to experimental time .................................................. 83 5.2.5 Simulation of the degenerated articular cartilage .............................................. 86

5.3 Results .......................................................................................................................... 88 5.3.1 Corresponding simulation time step to real time ............................................... 88 5.3.2 Effect of kc on results ......................................................................................... 89 5.3.3 Diffusion spatial maps for healthy articular cartilage ........................................ 90 5.3.4 Diffusion patterns of the degenerated articular cartilage ................................... 96

5.4 Discussion .................................................................................................................... 98

Chapter 6: Deformation of the articular cartilage ........................................ 103

6.1 Introduction ................................................................................................................ 103

6.2 Material and methods ................................................................................................. 104 6.2.1 Overview of the axial loading of confined articular cartilage ......................... 104 6.2.2 Adaptation of the hybrid agent for deformation of the articular cartilage ....... 105 6.2.3 Articular cartilage model ................................................................................. 106 6.2.4 Loading scenario .............................................................................................. 107 6.2.5 Rules ................................................................................................................ 109 6.2.6 Degenerated tissue model ................................................................................ 120 6.2.7 Simulations ...................................................................................................... 120 6.2.8 Corresponding time step to experimental time ................................................ 121

6.3 Results ........................................................................................................................ 122 6.3.1 Corresponding simulation time step to real time ............................................. 122 6.3.2 Effect of kc on results ....................................................................................... 123 6.3.3 Validation of the healthy articular cartilage simulation ................................... 124 6.3.4 Healthy cartilage results ................................................................................... 125 6.3.5 Degraded matrix results ................................................................................... 132

6.4 discussion ................................................................................................................... 136

Chapter 7: Dissolution of wet salt ................................................................... 139

An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics ix

7.1 Introduction ................................................................................................................139

7.2 Material and methods .................................................................................................140 7.2.1 Overview of the dissolution process .................................................................140 7.2.2 Rock salt model using hybrid agent .................................................................140 7.2.3 Rules .................................................................................................................141 7.2.4 Simulation ........................................................................................................145 7.2.5 Global properties ..............................................................................................146 7.2.6 Traditional salt and water agents’ simulation ...................................................146 7.2.7 Number of simulation runs ...............................................................................147

7.3 Results and discussion ................................................................................................149 7.3.1 Number of required simulation runs .................................................................149 7.3.2 Hybrid agent dissolution results .......................................................................153 7.3.3 Effect of parameter change ...............................................................................157 7.3.4 Effect of initial conditions ................................................................................159 7.3.5 Dissolution similar to the real condition ..........................................................162

Chapter 8: Discussion ...................................................................................... 165

Chapter 9: Conclusion, limitations and future work .................................... 169

Bibliography ........................................................................................................... 173

Appendices .............................................................................................................. 192

Appendix A ………………………………………..………………….……….…. 192

Appendix B ……………………………………..………………………….…….. 194

Appendix C …………………………………………..………………….……….. 207

Appendix D ……………………………………………..…………………….….. 212

Appendix E …………………………………………..………………….……….. 215

Appendix F …………………………………………..………………….……….. 216

x An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics

An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics xi

List of Figures

Figure 1.1: Overview of the thesis chapters. ................................................................ 9

Figure 2.1 Zones of articular cartilage showing distribution of components

[36]. .............................................................................................................. 14

Figure 2.2 The main difference between effective stress (a) and mixture (b)

approaches [135]. ......................................................................................... 23

Figure 2.3 FE method flow chart. .............................................................................. 28

Figure 2.4 Examples of a regular two-dimensional lattice. A: Moore

neighbourhood with Manhattan distance 1 (r=1). B: von Neumann

neighbourhood (r=1). The grey cells are the neighbourhood for the

black cell (central cell). ................................................................................ 34

Figure 2.5 Margolus neighbourhood. .......................... Error! Bookmark not defined.

Figure 2.6 The lattice used in the HPP model. The four arrows a, b, c and d

indicate the possible movement directions of a particle. ............................. 37

Figure 2.7 Collision rules in HPP[201]. Two particles experiencing a head-on

collision are deflected in the perpendicular direction. ................................. 37

Figure 2.8 The hexagonal lattice used in the FHP model. Each particle can

move along six directions [195]. .................................................................. 38

Figure 2.9 All possible collisions of the FHP variants: empty cells are

represented by thin lines, occupied cells by arrows [195]. .......................... 39

Figure 2.10 A: a lattice consists of solid and fluid agents in red and white

respectively. B: same lattice when agents were grouped. Thick lines

shows groups borders. C: same lattice when groups were considered

as elements of the system. ............................................................................ 44

Figure 3.1 Illustration of the application of the intra-agent and extra-agent

rules. Extra-agent rules define interaction between agent and

environment beyond the agent itself such as neighbours, while the

intra-agent rule is applied to each agent individually to determine

intra-agent evolution. ................................................................................... 48

Figure 3.2 Intra-agent change of hybrid agent H when it contains two

constituents. ................................................................................................. 48

Figure 3.3 A conception of a hybrid agent to represent a porous non-saturated

material. The hybrid agent is the combination of space, fluid and solid

sub-agents. It illustrates a key concept of the philosophical notion of

the new agent-based approach. .................................................................... 51

Figure 3.4 Intra-agent change of a hybrid agent. Blue and grey show fluid and

solid respectively. A and B: Hybrid agent is transformed into a full

fluid and solid agent respectively. C: fs of the agent decreases. D: fs of

the agent increases. ...................................................................................... 51

xii An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics

Figure 4.1 Immediate neigbours of a hybrid agent and their possible

permutations. A: a hybrid agent and its immediate neighbours. Agents

contain characteristic of properties A and B. B: Possible arrangements

of properties A and B for hybrid agents shown in image A. ....................... 54

Figure 4.2 1D automata Rules 22 and 73. White and black cells represent

properties A and B respectively. .................................................................. 55

Figure 4.3 Initial state of the 2D domain. Cells located in the first row contain

agents while other cells in the lattice are empty. ......................................... 58

Figure 4.4 Traditional 1DCA growing patterns. A and B: Patterns generated by

traditional agents and Rules 22 and 73 respectively. Numbers on the

left side of the patterns show the row or generation number. ...................... 62

Figure 4.5 A, B and C: patterns generated using extra-agent rules 22 and intra-

agent rule sets 1, 2 and 3 (patterns (iii), (iv) and (v) respectively) after

50 generations. ............................................................................................. 63

Figure 4.6 A, C and E: First 15 generation of hybrid agent generated patterns

using extra-agent Rule 73 and intra-agent rule sets 1, 2 and 3

respectively, starting with a hybrid agent with equal characteristics of

properties A and B (AB=1). B, D and F are corresponding patterns to

A, C and E after 50 iterations. ...................................................................... 68

Figure 4.7 Patterns resulted from applying extra-agent Rule 22 and intra-agent

rule set 1 after 50 iterations. A, B, C and D: patterns resulting from

initial seeds with AB ratio equal to 0.01, 1, 100 and 0 respectively. ........... 70

Figure 4.8 Patterns resulting from applying extra-agent Rule 22 and intra-

agent rule set 2 after 50 iterations. A, B, C and D: patterns resulting

from initial seeds with AB ratio equal to 0.01, 1, 100 and 0

respectively. ................................................................................................. 71

Figure 4.9 Patterns resulting from applying extra-agent Rule 22 and intra-

agent rule set 3 after 50 iterations. A, B, C and D: patterns results

from initial seed with AB ratio equal 0.01, 1, 100 and 0 respectively. ........ 72

Figure 5.1 Layered weight fraction distribution of fluid in the normal human

knee articular cartilage based on relative distance from the surface

[251]. ............................................................................................................ 76

Figure 5.2 Schematic illustration of the lattice. Blue, red and yellow show

cells containing agents filled with marked fluid, impervious agents

which are blocked to the fluid, and the articular cartilage agents

respectively. ................................................................................................. 77

Figure 5.3 2D von Neumann neighbourhood (r=1). Central agent located at

cell C interacts with agents located at cells East (E), West (W), North

(N) and South (S) at each time step. ............................................................ 78

Figure 5.4 Interactions of the EX cell with its neighbours in one time step. EX

cell interacts with the blue cells at each time step when all lattice cells

were selected to be central cells one by one. ............................................... 79

Figure 5.5 A: Vertical profile points. The mean marked fluid concentrations of

the layers in the middle-third, shown by the dash line, were used to

calculate profile points. B: Horizontal profile points. .................................. 84

An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics xiii

Figure 5.6 Partial degeneration of the articular cartilage. Red and blue show

degenerated and healthy regions. ................................................................. 87

Figure 5.7 Discrepancy between simulated and experimental results based on

CV(RMSE) and time steps for the horizontal (A) and vertical (B)

profiles. ........................................................................................................ 88

Figure 5.8 Horizontal, vertical and total error based on time step

corresponding to 2 hours. ............................................................................. 89

Figure 5.9 Diffusion into human articular cartilage at different times. A:

Percentage of marked fluid in the lattice at time steps 810, 1620 and

2430 B: Contrast agent diffusion after 2, 4 and 6 hours immersion

(taken with permission from [13]). .............................................................. 91

Figure 5.10 Percentage of depth concentration of marked agent at T=810, 1620

and 2430 and contrast agent in the human knee articular cartilage

after 2, 4 and 6 hours of immersion [13]. .................................................... 93

Figure 5.11 Depth- and time-dependent profiles of marked fluid concentration

for the surface, bottom, 1/3, ½ and ²/3 thickness depth. .............................. 95

Figure 5.12 Simulated concentration of the marked fluid at time steps 1632

and 3264, corresponding to 1 and 2 hours respectively (kc=0.025) for

the healthy articular cartilage model (A), the partially degenerated

model (B) and 70% solid resorption (C) based on percentage. ................... 97

Figure 6.1 Loading and boundary conditions for predicting consolidation

response of confined articular cartilage. A: Loading via a porous

indenter (loading scenario 1). B: Loading via an impervious indenter

(loading scenario 2).................................................................................... 104

Figure 6.2 Hybrid agent containing indistinguishable fluid and solid and

separable dead quantity (empty space). ..................................................... 105

Figure 6.3 Initial arrangements of the lattice cells. Impervious agents were

located in brown cells. Blue were empty cells. Articular cartilage

agents were located at yellow cells. ........................................................... 106

Figure 6.4 Layered distribution of the mass fraction of the fluid (A) [290] and

volume fraction of the fluid (B) in the normal bovine articular

cartilage based on relative distance from the surface of the tissue. ........... 107

Figure 6.5 Initial arrangements of the lattice cells in loading scenario 1 (A)

and loading scenario 2 (B). Impervious agents were located in the red

cells. Blue were empty cells. Articular cartilage agents were located at

yellow cells. ............................................................................................... 108

Figure 6.6 2D Margolus neighbourhood. The cells partitions alternate between

blocks indicated by solid lines, and dashed lines at odd and even steps

respectively. ............................................................................................... 110

Figure 6.7 A block consist of four cells and one agent at each cell. Agents

contained S1, S2, S3 and S4 solid, Fim1, Fim2, Fim3 and Fim4

immobile fluid, and Fm1, Fm2, Fm3 and Fm4 moveable fluid. ................ 117

Figure 6.8 Processes to keep size of the agents constant. A: Agents transfer

their extra volumes to their immediate top neighbours. B: Volumes are

xiv An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics

transferred from top agents to fill dead spaces of their immediate

bottom neighbours. Green shows filled space of the standard cell size

by indistinguishable fluid and solid. Yellow and white show extra

volume and dead space respectively. ......................................................... 119

Figure 6.9 CV(RMSE) of simulated strain when kc was equal to 0.05, based

on time step corresponding to one (1) minute. ........................................... 123

Figure 6.10 Effect of kc on CV(RMSE) and time step corresponding to one (1)

minute experimental time. .......................................................................... 124

Figure 6.11 Comparison between experimental [287] and predicted strain

values when kc=0.05. ................................................................................. 125

Figure 6.12 Spatial fs distributions in the lattice at time steps 0, 90, 450, 900,

5400 and 13500, corresponding to 0, 1, 5, 10, 60 and 150 minutes

(kc=0.05). ................................................................................................... 126

Figure 6.13 Spatial fluid volume fraction distributions of the healthy model at

different times, subjected to the loading scenario 1 (kc=0.05). ................. 127

Figure 6.14 Fluid mass over solid mass in the entire lattice (blue) and in the

bottom layer of the lattice (red) over time. ................................................ 128

Figure 6.15 Profiles of the strain and exuded fluid volume percentage for the

second loading scenario based on time steps. ............................................ 129

Figure 6.16 Distribution of the fluid volume fraction in the healthy matrix

based on the percentage during deformation process at time steps 0,

30, 90, 180, 1350 and 8100, corresponding to 0, 20 seconds, 1 minute,

2 minutes, 15 minutes and 1.5 hours respectively (kc=0.05). .................... 130

Figure 6.17 Distribution of the fs in the healthy matrix based on the percentage

during deformation process at time steps 0, 30, 90, 180, 1350 and

8100, corresponding to 0, 20 seconds, 1 minute, 2 minutes, 15 minutes

and 1.5 hours respectively (kc=0.05). ........................................................ 131

Figure 6.18 Strain versus time steps for the degenerated and healthy model of

the articular cartilage under loading scenario 1 (A) and loading

scenario 2 (B). ............................................................................................ 132

Figure 6.19 Distribution of the fluid volume fraction in the degenerated matrix

when kc=0.05 based on the percentage during deformation process at

time steps 0, 30, 90, 180, 450 and 2700. .................................................... 134

Figure 6.20 Distribution of the fs in the degenerated matrix during

deformation process when kc=0.05 at time steps 0, 30, 90, 180, 450

and 2700, corresponding to 0, 20 seconds, 1, 2, 5 and 30 minutes

respectively. ............................................................................................... 135

Figure 7.1 2D Moore neighbourhood. Central cell (cell C) is surrounded with

North East (NE), North (N), North West (NW), East (E), West (W),

South East (SE), South (S) and South West (SW) cells. ............................ 142

Figure 7.2 Interaction of two agents when two equal portions (λ1 and λ2) are

exchanged. .................................................................................................. 143

Figure 7.3 Integration of the agent 1 and portion λ. ................................................ 144

An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics xv

Figure 7.4 Rock salt block and its surrounding water. Locations of selected

cells (Cells 1-4) and areas (area 1 and 2) are shown in yellow and with

a white dashed line respectively. ............................................................... 149

Figure 7.5 Errors of salt concentration at area1 and 2 versus number of

simulation runs for the traditional technique. ............................................ 150

Figure 7.6 Errors of salt concentration at areas 1 and 2 versus number of

simulation runs for the simulation number 1 using hybrid agent. ............. 151

Figure 7.7 Simulation runs’ error at hybrid agents 1-4 located at cells 1-4 over

25000 time steps based on their salt concentration in the first 20

consecutive simulation runs. ...................................................................... 153

Figure 7.8 Distribution of salt concentration in the lattice at different time

steps based on percentage when TR, λs and λf equal to 0.7, 0.5 and 0.2

respectively. ............................................................................................... 155

Figure 7.9 Global salt concentration at vertical layers. Position 0 represents a

vertical layer from top to bottom of the lattice, which passes through

the centre of the salt block. Positions -1 and 1 present vertical layers

located at margins of the lattice. ................................................................ 156

Figure 7.10 The number of porous medium agents over 6000 iterations for

various values of TR and λs. ...................................................................... 158

Figure 7.11 Salt concentration in the surrounding water for various values of

TR and λs. .................................................................................................. 159

Figure 7.12 Distribution of salt concentration in the lattice for simulations 1

and 5 at time steps 1000 and 2000. ............................................................ 160

Figure 7.13 A: Number of porous medium agents versus time step in

simulations 1 and 5. B: Salt concentration in the surrounding water in

simulations 1 and 5 at various time steps................................................... 161

Figure 7.14 The number of porous medium agents over 2000 iterations for the

different salinity of the surrounding fluid. ................................................. 162

Figure 7.15 Concentration of salt based on percentage at time steps 10000 and

50000.......................................................................................................... 164

xvi An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics

List of Tables

Table 4.1 Rules and initial seed of the generated patterns. ........................................ 60

Table 5.1 Total, vertical and horizontal errors and corresponding time steps to

2, 4 and 6 hours diffusion time for different values of kc. ........................... 90

Table 5.2 CV(RMSE) of horizontal and vertical profiles corresponding to 2, 4

, 6 and 8 hours of immersion. ....................................................................... 94

Table 7.1 Value of parameters for different simulations. ....................................... 145

Table 7.2 Location of the selected cells and centre of the areas from the lattice

margins. ...................................................................................................... 149

An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics xvii

List of Abbreviations

MRI magnetic resonance imaging

CT computed tomography

ABM agent-based methods

NIR Near infrared

SEM scanning electron microscope

PLM polarized light microscopy

AFM atomic force microscopy

1D One dimensional

3D three dimensional

2D two dimensional

dGEMRIC delayed gadolinium enhanced MRI of cartilage

FCD fixed charge density

pQCT peripheral quantitative computed tomography

PDE partial differential equation

FE finite element

FEA finite element analysis

CA cellular automaton

LGA Lattice gas automaton

LB Lattice Boltzmann

wt% Weight percentage

xviii An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics

Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet

requirements for an award at this or any other 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 due reference is made.

Signature:

Date: 12/05/2017

QUT Verified Signature

An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics xix

Acknowledgements

The past four years at the Queensland University of Technology have, for the

most part, been productive, especially in terms of personal growth. But this time has

not been without its challenges, the most notable of which involved the departure of

my former principal supervisor. This unexpected development threw me into a

temporary state of confusion and uncertainty.

I would like to thank my principal advisor, Professor Adekunle Oloyede, for

his mentorship during the first three years of my study, in which I was introduced to

the real concept of “Doctor of Philosophy” and meaning of the word “Research”. I

wish to express my appreciation to my continuing principal supervisor, Professor

YuanTong Gu, for taking over the supervisor role, and for his patience. I also would

like to express my deep gratitude to my associate supervisor, Professor Graeme

Pettet, who although joining the supervisory team in the final stages, did a great job

in organizing my dissertation and providing valuable critiques.

I am extremely appreciative of the helpful comments made by the panel

members of my final and confirmation of candidature seminars: Professor Troy

Farrell, Dr Paul Wu and Dr Lihai Zhang. I would also like to thank Dr Hayley

Moody who, although not part of my supervisory team, was always there to answer

my questions.

I wish also to extend my thanks to Mr Samuel Zimmer (International students

counsellor), Professor Helen Klaebe (Dean of Research & Research Training), Ms

Susan Gasson (Research Student Centre) and Dr Deborah Peach (Student

Ombudsman), Ms Karyn Gonano, Dr Christian Long, Ms Sophie Abel (QUT

Language and Learning), Mrs Lissy Alvaran Jaramillo, Miss Elaine Reyes (HDR

xx An innovative agent-based cellular automata framework for simulating articular cartilage biomechanics

office), and Mrs Diane Kolomeitz, who provided editing services in accordance with

the Guidelines for Editing Research Theses in Australia.

I would like to thank all of my friends, officemates and colleagues, including

Sherrie Bernoth, Grant Gardiner, Maryam Shirmohammadi, Scott Bernoth, Ali

Azimi, Azadeh Azarmehr, Paul Bradley, Ashkan Heidarkhan Tehrani, Edith Wilson,

Ron Hodge, Keivan Bamdad, Mehdi Amirkhani, Chaturanga Bandara and Isaac

Afara for their support during the course of my PhD.

Finally, I would like to say thank you to my parents, my sisters and my lovely

wife, Zohreh, as without her patience and support I could not have accomplished the

long and sometimes overwhelming journey towards the completion of my thesis.

Chapter 1: Introduction 1

Chapter 1: Introduction

1.1 BACKGROUND

Articular cartilage is a connective thin tissue that covers the bony surface of

joints to protect them against contact load and friction [1]. It is a porous, fluid-

saturated, osmotically active, soft gel-like tissue that performs the physiological

function of load bearing, load spreading and lubrication. The behaviour of articular

cartilage as a mechanical system depends on its extracellular matrix, which consists

of collagen fibres, proteoglycans and fluid [2]. The chemically active fluid is the

major component of articular cartilage (60-85% wt%), while proteoglycans and

collagen make up its solid skeleton and occupy 5-10% (wt%) and 15-22% (wt%)

respectively [3, 4]. These components of articular cartilage are distributed non-

uniformly throughout its extracellular matrix and mix at a molecular level [5],

causing the cartilage to become highly heterogeneous, anisotropic and single-phase

where solid and fluid are indistinguishable. This level of complexity in the structure

of articular cartilage makes it difficult to understand how it responds to internal and

external stimuli such as mechanical loads and chemical modifications. The complex

responses of articular cartilage result from interactions between components of the

articular cartilage at the molecular level [6-8]. These interactions form micro-

mechanisms that govern the complex behaviour of the tissue.

Classical laboratory experiments are unsuitable for generating observations of

changes inside articular cartilage during its function since any attempt to place a

transducer inside the articular cartilage via piercing, damages this delicate structure

and results in unrepresentative tissue. Therefore, only data from the tissue margins is

2 Chapter 1: Introduction

practically and accurately available through traditional experiments. Non-invasive

experimental methods such as magnetic resonance imaging (MRI) and computed

tomography (CT), can provide structural data from inside the articular cartilage such

as spatial distribution of the proteoglycans and collagens, and orientation of the

collagen fibres, without disturbing its structure [9-12]. However, as non-destructive

methods indirectly collect internal information of the articular cartilage via a

radioactive tracer [13-15], only limited functional data for articular cartilage can be

obtained [16, 17]. Experimental data may be used to develop mathematical methods

which are usually based on differential equations, which describe characteristics of

the system as a set of functions and parameters [18, 19]. Despite the fact that current

computational methods are highly efficient at the prediction of articular cartilage

performance and description of the current observations, they rely on experimental

data and methods, such as curve-fitting, to estimate their required parameters. They

are also often insufficient to explain modality of formation and the reasons behind

observed behaviours.

1.2 RESEARCH PROBLEM

Biological systems can be conceptualised as emergent structures since their

characteristics are functions of adaptations to their environment [20]. These emergent

states are arguably manifestations of interactions that are currently not measurable by

traditional experiments or accessible through conventional numerical methods such

as finite element analysis [21, 22]. The interactions within and between various

components determine the observable responses in space and time, such that the

emergent structures are practically beyond classical computational modelling

frameworks that are based on strict geometrical and mathematical formalisms [23,


24]. Mechanisms behind responses of a system may be captured using agent-based

methods (ABM) where a collection of interactions between autonomous entities,

called agents, form the system [25]. A set of rules determines both individual agent

behaviour and the interactions of the agents with each other [26]. ABM have been

used to simulate functions of porous materials with distinguished components

including rock and clay [27, 28] where agents are identified by their states, e.g. 0 and

1 or black and white, and the system operates like mixtures of individual agents. This

is unlike the situation in many biological systems, including articular cartilage, where

components of the system are mixed at an ultra-microscopic level and are practically

inseparable. Despite the capability of ABM to capture complex macro-scale

responses by considering micro-mechanisms of a system, there has been very limited

research to use ABM for the single-phase porous materials, due to the inadequacy of

current agents to represent such complex structures.

The knowledge gap is that the micro-scale activities relating to the articular

cartilage functions are not well understood. Currently available agent-based

modelling techniques even cannot provide a realistic structural model that

demonstrates the single-phase, multi-component nature of the tissue. This situation

calls for an enhanced agent-based approach to create the microscale architectures of

these complex systems, upon which further computational analysis can be

performed.


1.3 AIM AND OBJECTIVES

The aim of this study is to develop a virtual microscope to probe underlying

mechanisms of the single-phase porous materials, such as articular cartilage, in-

silico. The following objectives are met:

Identify potential techniques used to investigate porous materials through a

literature review.

Develop a new agent to represent single-phase multi-component structures

more practically.

Develop a new category of rules to control evolution of the new agent.

Develop an agent-based model to simulate diffusion throughout the articular

cartilage and investigate fluid dynamics.

Develop an agent-based model to simulate one-dimensional consolidation of

articular cartilage and study internal change and fluid flow qualitatively.

Develop a model to simulate dissolution of wet salt.

1.3.1 Research questions

A series of research questions are addressed in this thesis:

What level of complexity is necessary for an agent to be able to

represent a single-phase multi-component material?

How can single-phase multi-component structures be created

realistically by agent-based methods?

How do components of articular cartilage interact with each other

during tissue functions?


Can agent-based methods provide a framework to conduct “virtual

experiments” on articular cartilage?

1.3.2 Outcomes

The outcomes of this research are:

Presentation of a more complex agent that is capable of intra-agent

evolution.

An approach for creating emergent structures to represent single-

phase multi-component materials.

A framework for simulating articular cartilage functions and

collecting internal functional data on a microscopic-scale. It

introduces a new paradigm and advanced interpretation of soft tissue

modelling in general.

A modelling approach that facilitates further insight into degenerated

cartilage.

Extension of the technique to other porous single-phase materials.

1.4 SIGNIFICANCE

This study is the first to present an agent that is able to realistically

demonstrate single-phase multi-component structures. The agent presented in this

thesis adds a new level of complexity to typically developed cellular automata agents,

due to the ability of intra-agent evolution of the hybrid agent by means of an intra-

agent rule, which is a new category of rules. This contributes knowledge that


potentially leads to new approaches in the modelling of the articular cartilage and

similar fluid-saturated tissues and structures.

The implementation of this agent-based method extends our capacity for

simulation and leads to a “soft probing tool” for the internal working of the articular

cartilage matrix. This new approach results in a better understanding of the

interrelationships between cartilage components and their adaptation to external

stimuli. This new computer-based probing utility will be adaptable to the study of

how cartilage adapts itself for physiological efficacy during ageing, and when it

carries localised areas of degradation such as focal defects that characterise the

conditions of osteoarthritis, avascular necrosis and traumatic injuries.

The ability to probe the real-time response of articular cartilage during

function can potentially provide a “virtual microscope” into the internal workings of

the system, to provide critical knowledge in the area of cartilage biomechanics. This

methodology would provide spatial and temporal functional data that could then

facilitate other models, such as finite element, mesh free, course-grained particle and

smooth particle hydrodynamics. It contributes knowledge that will potentially lead

to a new approach in the simulation of articular cartilage without using mechanical

formulas, differential equations and experimental curve fitting for determining

required parameters. It creates a virtual tissue that carries characteristics of real

cartilage, allowing for observation of internal micro-scale processes. This “virtual

microscope” provides direct probing of the underlying tissue and micro-mechanisms.

It extends capability and advances knowledge in computational biomechanics beyond

that available from current methods such as numerical models and traditional

experiments.


Articular cartilage undergoes progressive changes due to ageing, overloading

and trauma. The mechanisms involved in articular cartilage disease and degeneration

are largely unknown [29]. One of the ethical limitations of investigating articular

cartilage degeneration is the impossibility of studying cartilage through changes from

its normal healthy condition to the disease state. It is apparent that in order to

increase our current knowledge on joint diseases and degeneration, new methods

that extend capability beyond current experimental and in silico models are required.

This research is significant in its potential to meet this critical need. It will present

the framework for simulating a near-realistic degradation process with the critical

capacity for practitioners to simulate this health-to-disease transformation in silico.

Furthermore, it will be possible to simulate events such as spatio-temporal

collagen fibril disruption and proteoglycan loss at any desired rate and study the

effect on the load processing property of the tissue.

1.5 THESIS OUTLINE

This thesis specifically addresses the question of whether it is possible to

conduct computational experiments for complex single-phase multi-component

materials such as articular cartilage. It is hypothesised that an agent-based technique

can be adapted for the complex porous structures where behaviour of the system at

macroscopic level stems from microscopic behaviours and interactions of its

components.

Figure 1.1 shows an overview of the thesis chapters. Chapter 1 of this thesis

provides an introduction to the research including a brief background, research

problem, research questions and outcomes, followed by aim, objectives and the


significance of the study. The current experimental and computational methods of

investigating articular cartilage are critically reviewed in Chapter 2. This includes a

discussion on current experimental and theoretical methods and introducing agent-

based modelling techniques which lead to establishing a need for a more complex

agent. Chapter 3 includes development of a new agent that is suitable for single-

phase multi-component materials, and intra-agent rules to control the evolution of

hybrid agents. The agent and category of rules, which are developed in Chapter 3, are

implemented in the next four chapters. Emergent structures have been developed in

Chapter 4 to represent porous semipermeable single-phase materials. Diffusion into

the articular cartilage and related fluid dynamics are simulated in Chapter 5. One-

dimensional consolidation of articular cartilage under axial static load is simulated in

Chapter 6. Simulation of dissolution of rock salt as a non-biological porous single-

phase material is presented in Chapter 7. The approach and findings of the thesis are

discussed in Chapter 8 and finally, Chapter 9 presents the main conclusion,

acknowledging limitations and the implications of where this study may be extended

in the future.


Introduction (Chapter 1) Current methods (Chapter 2)

Develop enhanced agent

Develop new rule

(Chapter 3)

Create emergent

structure of porous

non-phasic material

(Chapter 4)

Application 2

Simulate deformation

of cartilage

(Chapter 6)

Application 1 Simulate diffusion

into cartilage

(Chapter 5)

Application 3 Non-biological porous

single-phase material

(Chapter 7)

Figure 1.1: Overview of the thesis chapters.

Chapter 2: Literature Review 11

Chapter 2: Literature Review

2.1 INTRODUCTION

Porous materials contain a solid skeleton and void spaces (pores) which are

usually invisible to the naked eye. Knowledge of the pore distribution is required to

predict porous media performance in any given application. Porosity, which is the

fraction of the volume that is occupied by pores over total volume, is one of the

quantities that characterises porous media [30, 31]. Permeability is another critical

property of porous materials that indicates the ability of fluid flow through media

[32] where materials with high permeability allow fluids to transmit through it

quickly.

Pores are filled with fluid in fluid-saturated porous materials such as soils, clay

and biological tissues where pores may be distributed in a disorderly manner,

rendering their structures highly heterogeneous [30]. The pores are also usually

interlinked, forming continuous three-dimensional (3D) channels for diffusion,

percolation and exudation of fluid during deformation and in the case of biological

materials, during physiological functions such as nutrient exchange, waste excretion,

and osmosis [31].

Articular cartilage is a fluid-saturated avascular soft tissue that facilitates load

bearing and lubrication in mammalian joints. To date, much experimental and

theoretical work has been done to investigate the complex structure and functions of

articular cartilage. However, understanding articular cartilage functions at the micro-

scale level is beyond the scope of current methods. The aim of this thesis is to adapt

agent-based methods (ABM) in order to demonstrate micro-mechanisms behind

articular cartilage functions. Adapted ABM provides a capable tool to probe the

12 Chapter 2: Literature Review

tissue internally. In order to establish a need for this tool and understand its critical

role, we must first examine the literature to determine:

What are the structural characteristics of articular cartilage?

What methods are available to probe and investigate articular cartilage?

To what extent can current methods capture and manifest internal functions

and structures of the articular cartilage?

What specific methods can potentially be used to capture and understand the

underlying process of the single-phase porous materials?

What are the limitations of existing methods to be used for, or adapted to,

articular cartilage?

In order to address these questions, a critical review of the literature in the areas of

articular cartilage biology, related laboratory experimental methods, theoretical

modelling of cartilage and computational modelling of porous media needs to be

conducted.

2.2 ARTICULAR CARTILAGE STRUCTURE

Articular cartilage is the hyaline tissue that covers bony joint surfaces to

facilitate load bearing and lubrication within the synovial environment. As it contains

no blood vessels, diffusion through the tissue from surrounding synovial fluid

provides nutrition [33, 34]. The thickness of articular cartilage is different from one

joint to another, ranging from 2 to 4 mm [35]. Fluid, including water and mobile

ions, is the major component of cartilage’s porous structure. It constitutes 65% to

80% of the total wet weight for healthy cartilage. Collagen and proteoglycan create a

solid skeleton of the tissue where they constitute approximately 75% and 25% of the


dry tissue by weight respectively [3, 4]. The tissue also contains approximately 3%

living cells (chondrocytes) [3]. Constituents of the cartilage are distributed non-

uniformly and vary through the depth from the surface. The concentration of the

proteoglycans through the depth of the tissue follows a bell-shaped profile in which it

has low values near the articular surface and increases to a maximum at 50% to 80%

depth and then decreases until the area adjacent to the subchondral bone [36]. The

relative concentration of fluid decreases from approximately 80% in the area close to

the articular surface to 65% near subchondral bone [37, 38]. The greatest density of

collagen fibres is close to the surface of the cartilage while it decreases closer to the

bone. The orientation of collagen fibrils are perpendicular at the area of attachment

to the bone and gradually become parallel with the cartilage surface [7].

Four hypothetical layers in cartilage structure are identified based on the

distribution of components including superficial, intermediate (middle or

transitional), deep (radial) and calcified zones [39, 40] (Figure 2.1). Although the

relative height of each zone varies according to the joint and depends on age and

species, superfacial, middle and deep zones occupy 10-20%, 40-60% and 30-40% of

the total depth of the cartilage respectively [37]. The superficial zone is in contact

with the synovial fluid and contains densely packed collagen and relatively low

quantities of fluid and proteoglycan. The middle zone has the highest proportion of

proteoglycan and fluid among the four zones. The amount of proteoglycan decreases

while the amount of collagen increases from the middle zone to deep zone. The

calcified zone is devoid of proteoglycan and acts as a boundary between cartilage

and the underlying subchondral bone. It plays a significant role in securing the tissue

to subchondral bone by fixing the deep zone collagen fibrils to the bone [7, 37, 40,


41]. The zonal architecture is vital for optimal load processing and protection of

living cells (chondrocytes) from mechanical damage [42, 43].

Figure 2.1 Zones of articular cartilage showing distribution of components [44]

(reprinted from Clinics in Sports Medicine, vol. 28, H. G. Potter, B. R. Black, L. R.

Chong, New techniques in articular cartilage imaging, pp. 77-94, Copyright (2009),

with permission from Elsevier (Appendix E)).

The complex structure and behaviour of articular cartilage are a result of how

the components, namely water, collagen, proteoglycans and mobile ions, interact

with each other at an ultra-microscopic level. The water is joined to the

proteoglycans and collagens by different degrees of molecular attraction. A small

proportion of the water is contained in the intracellular space. Approximately 20-

30% of the tissue water is contained in between collagen fibrils (interstitial water)

and appears as a gel, while the remainder is stored in the pore space of the tissue [45-

47]. Proteoglycans have a negative charge and the collagen fibres form a fibrous 3D

meshwork that entraps the fluid-swollen proteoglycans. The fixed charge density

(FCD) on proteoglycans attracts mobile counterions such as Na+ [45]. The existence

of proteoglycans in extrafibrillar space results in increasing concentration of mobile


cations to keep electro-neutrality [48]. While the collagen meshwork acts as semi-

permeable membrane [49], this excess of mobile ions (mobile positive charges) due

to the presence of FCD yields a pressure difference between intrafibrillar and

extrafibrillar spaces [50]. This pressure difference generates a higher fluid pressure

in proteoglycan-containing extrafibrillar space, which is known as the Donnan

osmotic pressure [46]. This osmotic pressure influences mechanical properties of

cartilage such as compressive stiffness.

As articular cartilage is an avascular tissue, any molecules including nutrients

and wastes cannot move in/out of the tissue via blood vessels. It is believed that

diffusion through the articular cartilage conducts significant and vital role of

transport of nutrients and waste products [35, 51-53]. Diffusion of water and ions is

important to the physiological function of articular cartilage. Diffusion also plays a

key role in load-bearing of the articular cartilage [54, 55] where fluid molecules

diffuse through the matrix due to hydrostatic and osmosis pressure [34, 56]. Loading

and unloading increase fluid percolation and diffusion through the tissue [33]

resulting in volume change. The characteristic change in volume leads to time-

dependent pore size changes with concomitant decrease change in the average

permeability and fluid flow related [57].

2.3 ARTICULAR CARTILAGE LOAD BEARING

Articular cartilage is subjected to the various types of loads such as

compressive load in everyday activities. Articular cartilage is a saturated tissue in

which fluid flows relative to a deformation. The loading of the articular cartilage

causes an immediate increase in interstitial water pressure, which develops a

hydrostatic pore pressure over the osmotic pressure [37]. This causes the fluid to


flow out of the extracellular matrix to increase osmotic pressure resulting in the pores

within the matrix to be narrow, which increases resistance to flow and decreases the

cartilage’s permeability [58]. Fluid is osmotically driven back into the extracellular

matrix when the load is removed.

Articular cartilage responds differently to slow and impact loads [59]. When a

load is slowly applied to the articular cartilage, it might exhibit consolidation-type

deformation [58, 60] where the fluid initially carries the entire applied pressure and

transiently shares it with the solid skeleton [61, 62]. It has been demonstrated that

articular cartilage’s consolidation behaviour differs from that of soils and clays in the

initial stages of loading [58, 62-64]. The specific response of the tissue is viewed as a

correspondence of the swelling resistance, which is a pressure caused by the osmotic

process involving the swollen proteoglycans that are entrapped within the 3D

collagen meshwork.

When articular cartilage is subjected to impact loading, the load is immediately

carried by tissue stiffness, which is caused by swelling of the proteoglycans. The

fluid cannot exude out of the matrix due to low permeability of the tissue and the

limited deformation time [59].

2.4 ARTICULAR CARTILAGE DEGENERATION

Articular cartilage is a resilient tissue which can tolerate great loads and

stresses throughout a lifetime of activity. Certain daily life activities, impact and

torsional loadings and ageing can result in joint degeneration [65]. Due to the lack of

blood vessels in the tissue, it has a limited ability to repair degraded tissue.

Osteoarthritis (OA) is a degenerative joint disease that affects the functional

quality of tissue. It is a major cause of chronic pain and disability [66], which


affected almost 15% of Australians (about 3.3 million people) in 2011 [67]. It has

also been estimated that the number of people in Australia with arthritis will be

almost double by 2050. It costs the economy approximately $24 billion annually

[68]. OA is characterised by the slow progressive degradation of articular cartilage

[69]. The OA process is directly related to the degradation of collagen networks and

loss of proteoglycan in terms of content and chemical composition [70, 71].

Macroscopically, this is manifested as an increase in water content and permeability,

and reduction in the compressive stiffness of the tissue [71]. In spite of extensive

research, the OA process is not yet well understood [29, 65].

2.5 MECHANICAL PROPERTIES OF THE ARTICULAR CARTILAGE

The mechanical properties of articular cartilage in bulk such as stiffness,

Young’s modulus and shear modulus have been calculated in early experimental

studies on cartilage [72-74]. The mechanical properties of cartilage under

compressive loading are provided by its solid skeleton, with significant contribution

from fluid flow throughout the matrix, which results from an osmotic process in the

matrix [56, 62, 75-78]. Consequently, permeability, which plays a significant role in

fluid flow, becomes one of the major physiological characteristics of the cartilage.

McCutchen [79] sliced bovine cartilage from the shoulder into two pieces and

measured the permeability of each piece. He found that the permeability of the piece

that was near the surface was 7.65 x 10-16

m4/N.S versus 4.3 x 10

-16 m

4/N.S for the

next layer below it. Based on this, he argued that permeability decreased with the

depth of articular cartilage. Mow and Mansour [80, 81] investigated the uniaxial

creep-like behaviour of articular cartilage. They stated that compressing the tissue

decreases its permeability due to increasing frictional resistance of the solid skeleton


to the flow, as a consequence of compaction of the collagen meshwork. Therefore,

the permeability of articular cartilage has an inverse relation with compressive strain.

More research also shows inhomogeneity of the permeability [82, 83], anisotropy of

the permeability during compression [84], and correlation of the permeability with

the state of degeneration [85]. Correlation between fluid content of the articular

cartilage and permeability was also confirmed so that as water content increases, the

tissue becomes more permeable [86]. In a series of experiments [43, 51, 87],

Maroudas showed that permeability varies in depth and inversely with the FCD and

collagen content of the cartilage matrix.

The 3D meshwork of collagen fibrils and entrapped proteoglycan makes

articular cartilage structure highly heterogeneous and anisotropic [88]. Consequently,

bulk properties are not realistic and provide very rough estimations since spatial

distributions cannot be addressed.

Due to the heterogeneous structure of cartilage, several researchers calculated

physical properties of cartilage such as permeability and compression modulus, with

assumptions of depth and/or strain dependency, based on curve fitting [82, 84, 89-

96]. They created equations as a function of depth and/or strain to be fitted on each

measured mechanical property with some assumptions such as homogeneity. Such

curves generally were generated according to experiments on partial-thickness

sections of articular cartilage [82] or examined intact tissue to be analysed as

multiple layers [89-91]. For instance, Schinagl et al. 1997 [90] took displacement and

strain data of full-thickness cartilage specimens in different depths and compression

levels. Stress-strain data for each layer were fit to a finite deformation stress-strain

relation to determine the equilibrium confined compression modulus in each tissue

layer. Chen et al. [82] tested both full-thickness and partial-thickness cartilage, and


compression modulus (HA0) was fitted to the experimental results by the expression:

HA0(z) [MPa]=1.44 exp(0.0012·z) where z is the depth measure in µm. However,

since the properties of distinct layers differ from and vary significantly from the full

thickness cartilage [82], the second issue is another assumption about the impact of

each layer on the calculation of physical properties of full-thickness cartilage. The

discrepancies in the depth/strain-dependent properties of full-thickness articular

cartilage and specific regions of tissue illustrate complexity in cartilage behaviour

[82]. In addition, anisotropy and heterogeneity of properties, which stem from the

heterogeneous architecture of the tissue, cannot be captured by depth/strain

dependency. Correlation between physical properties and biochemical compositions,

which vary in depth and position, has been proved [7, 97], confirming that other

undefined factors influence these properties [96]. Since the physiological

characteristics of the tissue are governed by the interactions between proteoglycans,

collagen fibril meshwork and fluid content, experimental curve fitting can only

provide an estimated range of the physical properties of tissue [98].

2.6 EXPERIMENTAL METHODS FOR DATA COLLECTION

2.6.1 Classical laboratory experiments

Extensive laboratory experiments have been conducted for many years to

investigate articular cartilage structure and function. Mechanical properties of

articular cartilage were measured in the laboratory (Section 2.5). The thickness of

articular cartilage can be measured destructively via needle probing [99] and

stereophotographic techniques [100], and non-destructively via ultrasonic [101] and

near-infrared (NIR) techniques [102]. Hydrostatic pore pressure of the loaded tissue

under consolidation condition at the margin adjacent to the subchondral bone was


measured using a miniature consolidometer [64]. Morphology and microscopic

structure of articular cartilage, including distribution of collagen and proteoglycan

[103-105], can be destructively evaluated via histology, which involves sectioning,

staining, electron microscopy and image processing [106]. Imaging methods such as

scanning electron microscopy (SEM), polarised light microscopy (PLM) and atomic

force microscopy (AFM) have long been used for investigating the morphology

[107-109] and mechanical properties, e.g. elastic modulus, [110, 111] of the articular

cartilage surface at the microscopic levels. These imaging instruments furnish a

detailed and realistic structure of the surface topography of articular cartilage with a

sub-nanometre resolution but are only able to provide information up to a very

limited depth of tissue [112].

The amount of FCD in the articular cartilage was measured using titration

where the thin slices of the tissue were immersed in saline and allowed to equilibrate

after several quantitated additions of acid or alkali to the solution [113, 114]. The pH

of the solution after equilibrium and an independent measure of isoelectric point of

the tissue were used to calculate the FCD of the cartilage. The tissue FCD was also

calculated from independent measurements of streaming potential, hydraulic

permeability and specific conductivity [51, 115].

Obtaining data from articular cartilage using classical laboratory methods is

limited only to the margins and data in bulk, which make them inadequate for

providing data from inside the tissue without destroying it. Probing underlying tissue

by mechanical tools causes damage to the delicate interweaving structure that

determines the functionality of the articular cartilage [116].


2.6.2 Non-invasive methods

Recently, non-destructive techniques, such as Magnetic Resonance Imaging

(MRI) and Computed Tomography (CT) scan, were used to determine physical

characteristics of cartilage, such as thickness [12, 102, 117]. Traditional MRI

techniques were used for diagnosis by detecting morphological changes of cartilage

such as tears and tissue narrowing [118]. MRI and CT scan techniques generate

different contrasts between components based on absorbing and emitting radio

frequency energy [119]. The contrast between components is generated by the

difference between the absorbance of X-ray radiation [120]. As constituents of the

articular cartilage are intermixed at a molecular level [5], MRI and CT scan

techniques cannot directly distinguish components of the tissue. Advanced MRI

techniques can assess composition of cartilage indirectly [121, 122] using contrast

agents such as Hexabrix [12], sodium iodide [123], tantalum oxide nanoparticles

[124] and gadopentetate [125]. For example, delayed gadolinium enhanced MRI of

cartilage (dGEMRIC) uses fixed charge density (FCD) within the tissue to measure

proteoglycan content quantitatively. In this technique, the negatively charged

contrast agent Gd(DTPA)2-

is diffused throughout the cartilage prior to scanning. The

contrast agent accumulates in an inverse relationship with proteoglycan content due

to repulsive forces between negatively charged molecules (contrast agent and

proteoglycans) [126]. Sodium MRI uses sodium cations (Na+) as a contrast agent that

can be attracted by negatively charged proteoglycans. As a result, distribution of

sodium cations illustrates proteoglycan content [127]. The T2 mapping technique,

which is based on excitation and relaxation times of water molecules [128], is used

for measuring water content in cartilage [129] and indirect evaluation of collagen

content and orientation of fibres [9, 121].


Non-invasive methods and a radioactive tracer can be used to investigate

functions of the articular cartilage. For example, peripheral quantitative computed

tomography (pQCT) is used to study diffusion into cartilage where quantitative data,

such as contrast agent concentration curves at certain times, and qualitative

information, such as spatial and temporal distribution of contrast agent, were

provided [13, 14, 130].

Although non-invasive techniques are able to provide useful structural and

functional information about the tissue, they are limited to indirect interpretations

that may affect the accuracy of the obtained images [131]. For instance, dGEMRIC

might overestimate proteoglycan content in the deep zone [1] and Sodium MRI

suffers from the difficulty of generating MR signal due to the low concentration of

sodium cations in comparison with H ions within the tissue [121]. In addition, they

are confined to collect data of certain functions. To illustrate this, these techniques

are not able to observe the time-dependent internal behaviour of the tissue under

external loads.

2.7 COMPUTATIONAL METHODS

Computational methods were developed to observe the behaviour of the tissue

based on theoretical models and numerical methods. Mechanical theories and

physical laws were used to determine governing equations and formalisms of porous

media, which are usually based on differential equations that determine

characteristics of the medium through a choice of functions and associated

parameters. Then, numerical methods such as finite elements (FE) are used to solve

the equations. While experimental methods can only obtain data with limited insight

from the tissue margins, theoretical models provide an opportunity to study the


underlying mechanisms and related functional properties of the full-thickness

cartilage such as fluid flow, osmotic pressure, fluid pressure and stress within the

solid skeleton [132-134]. The simplest articular cartilage model is a single phase and

one-component material [72] in which articular cartilage is assumed to be an

isotropic and linearly elastic solid. Some significant properties of articular cartilage

including fluid flow and consolidation-type behaviour, cannot be taken into account

in a single phase model [135]. Consequently, this model is only applicable to simple

tissues.

In general, time-dependent behaviour of the articular cartilage can be

explained based on two approaches: mixture and effective stress [5]. In the mixture

formulation, the total stress is broken up into stresses created in each constituent

(Figure 2.2). The effective stress, which is based on Biot consolidation theory [61],

uses a control volume that is large enough compared to the size of the pores and so

can be treated as homogeneous, while at the same time it is small enough compared

to the scale of medium, for it may be considered as infinitesimal [61] (Figure 2.2).

Since cartilage can be considered as a fluid saturated porous and deformable material

[79, 136], poroelasticity theory has been used to build up cartilage models.

Figure 2.2 The main difference between effective stress (a) and mixture (b)

approaches [137].


2.7.1 Mixture models

Mixture models have been developed based on the theory of composites and

superimposed behaviour of components. Different models have been created based

on mixture theory. Biphasic [91] models separate cartilage structure into two

different phases: fluid and solid. The solid phase consists of collagen fibrils and

proteoglycans, in which collagen fibrils reinforce proteoglycan aggregates. In the

original biphasic theory, which was presented by Mow et al. [91], both solid and

fluid phases are assumed to be incompressible and non-dissipative, while the solid

phase is assumed to be linearly elastic. Since fluid can flow into and out of the

model, stresses in the solid matrix, fluid pressures and local consolidation strains can

be determined as a function of time. In the biphasic model of cartilage, the behaviour

of the tissue is a function of three fundamental parameters: permeability of the

cartilage, elastic modulus and Poisson ratio of the solid phase. The cartilage tissue is

assumed to be fully saturated and the porosity ϕf and solidity ϕs are defined as ϕf =Vf

/(Vf+Vs) and ϕs=Vs /(Vf+Vs) such that ϕf + ϕs =1 where Vf and Vs are fluid and solid

volume fractions respectively. The governing differential equations for the linear

biphasic theory are as follows [138]:

The continuity equation of the mixture: . (ϕf ѵf + ϕs ѵs ) = 0

The momentum equation of the solid phase: .σs +Пs= 0

The momentum equation of the fluid phase: .σf +Пf= 0

The constitutive relations for the solid phase: σs = - ϕs p I + λs es I + 2 µs εs

The constitutive relations for the fluid phase: σf = - ϕf p I

The diffusive momentum exchange: Пs = - Пf = K(ѵf - ѵs)

The diffusion drag coefficient: K=( ϕf)²

𝑘


In the above equations, superscripts s and f refer to the solid and fluid phases

respectively. is the gradient operator, Ѵ is velocity vector, σ is the stress tensor, П

is the diffusive momentum exchange between the phases, I is the identity tensor, p is

the apparent pressure, λs and µs are the elastic Lame constants of the solid phase, ε is

the strain tensor, e is the dilatation of the solid phase and 𝑘 is hydraulic permeability

of the tissue. However, biphasic models cannot address the time-dependent

behaviour of articular cartilage accurately [89].

Later, an extension to biphasic theory was proposed by Mak [139] in which

solid phase was assumed to be viscoelastic and the fluid as inviscid. Simon [140]

assumed that the solid phase had hyperelastic properties. In another study, Lanir

[141, 142] hypothesised that deformation of the solid structure and fluid flow

determine swelling behaviour of the cartilage tissue. Therefore, he added solute

concentration effects to the biphasic model by adding a deformation-dependent

pressure term to the standard biphasic equations. In his ‘bi-component’ model, the

solid component includes a collagen meshwork and the fluid phase includes water.

The fluid phase contains proteoglycan molecules that are trapped in the collagen

meshwork and generate a Donnan osmotic pressure, which is coupled to mechanical

deformation via volumetric strain. This model can address swelling proteoglycans

and distention of the collagen meshwork in loaded and unloaded conditions.

However, Lanir used a composite-structure concept and ignored diffusion of ions by

assuming that the ionic distribution is always in equilibrium, which cannot be

justified in unsteady state conditions.

Diffusion of the ionic component was considered in the triphasic model

presented by Lai et al. [143]. The model includes the two fluid and solid phases

(biphasic), and an ion phase. The ion phase is salt, including mobile anions and


cations to represent physicochemical activities in the cartilage matrix, while chemical

expansion stress (-Tc) is calculated from ion concentration:

Chemical expansion stress: 𝑇𝑐 = 𝑎0 𝑐𝐹𝑒−𝑘(𝜆±/𝜆∗±) √𝑐(𝑐 + 𝑐𝐹) − 𝑃∞

Total mixture stress: 𝜎 = −𝑝𝐼 − 𝑇𝑐𝐼 + λs 𝑒s I + 2 µs εs

In the above equations 𝑎0 is the charge-to-charge activity parameter, 𝑐𝐹 is fixed

charge density, λs and µs are the elastic Lame constants of the solid phase, I is the

identity tensor, 𝑝 is fluid pressure 𝜆 ± is mean activity coefficient, 𝜆 ∗ ± is mean

activity coefficient for external solution (𝜆 ±/𝜆 ∗ ±= 1 for ideal solution) and 𝑃∞ is

osmotic pressure due to the proteoglycans in tissue. Although ion diffusion is

considered in this triphasic model, a comparison between the ‘bi-component’ model

of Lanir and the triphasic model, demonstrated that they behave almost the same in

respect of generated stress and strains in the tissue [144].

In the quadriphasic model [145], the ion phase in the triphasic model was

divided into two different phases: anion and cation phases. Like the triphasic model,

electro-neutrality was applied to the whole model. The three phase multi-species

model, presented by Gu et al. [146], is another extension for the triphasic model.

This model includes a solid phase, a fluid phase and ionic phases that consist of

several ionic species. Each ionic phase can contain one or more species. These

species always remain within their initial phase. However, a drawback of the

triphasic model and its extensions is, since they need electro-neutrality, the physics

inside the extracellular matrix are not acceptable for standard analysis [5].

Mixture theory has been built based on the principle of superposition of

components, therefore, the validity of the theory depends on the separation of

components [147]. The collagen fibrils include water bound to the fibrils at a

molecular scale, while proteoglycans and collagen entangle molecularly [35]. Mobile


ions are also attracted to both collagen and proteoglycans. As a result, all

components of the extracellular matrix are intermixed on a molecular scale.

Therefore, phase boundaries do not exist and some parameters used in mixture

theory, such as porosity, cannot be defined accurately [5, 148].

2.7.2 Continuum approach

This approach uses an effective stress formulation for poroelastic materials. When

a pressure is applied to porous fluid-saturated media, the load is transferred to the

fluid immediately and this causes an increase in hydrostatic pressure [64]. As a

result, the pressure of the fluid is increased to a certain maximum magnitude, called

the maximum hydrostatic pore pressure. Then, this excess pore pressure gradually

declines to zero over a period of time, which is determined by porous media

properties such as permeability and compressive stiffness of the solid skeleton [63].

Simon et al. [149] used an effective stress formulation to model the deformation

of the intervertebral disc. Simon later, in another study, demonstrated equivalence

between both poroelastic formulations (mixture and effective stress). Oloyede and

Brown [76, 150] generalised the effective stress approach (consolidation approach)

by assuming that the flow of fluid through tissue obeys Darcy’s law of percolation.

In traditional mathematical models, constitutive laws (equations), e.g. Fick's law of

diffusion and Darcy's law for porous flow, are used to define the macroscopic

behaviour of the tissue. These equations also use parameters taken to be properties of

material [151] and determination of the material properties of cartilage still relies on

macroscopic experimental results, e.g. experimental curve fitting.


2.7.3 Finite Element method

The theoretical models of articular cartilage include partial differential equations

(PDEs), which are discretised and numerically solved by different techniques such as

the finite element method (FEM) [152]. In FEM, a complex mathematical problem is

discretised in time and space into a system of algebraic equations, which can easily

be solved. The material parameters can be defined as complex functions of both time

and space at each discrete point. Figure 2.3 shows the steps of using the FE method.

Figure 2.3 FE method flow chart.

Create the domain

structure of the tissue

Choosing a mathematical model e.g. continuum model, biphasic

model

Creating the mathematical equations

based on theoretical model

Determining material

parameters

Setting up the system of partial differential equations for

system

Dividing the domain into finite elements (meshing)

Solving the system of algebraic equations and obtaining the

results for each element


A linear biphasic model using the FE method [138] was used to simulate

mechanical behaviour of articular cartilage during fluid exudation. However, it leads

to a need to solve a linear system of equations. If a linear solver is used to solve a

model under large deformation, results may be considerably incorrect [153]. To

address this problem, the fibril-reinforced model has been presented [154]. Although

the fibril reinforced model is able to capture the material nonlinearity of the cartilage,

it cannot capture the geometric nonlinearity [153]. Li et al. [155] added nonlinearity

to the cartilage fibrils and presented a nonlinear fibril reinforced poroelastic model.

In another study fibril orientations were considered and deformation, fluid pressure

and fluid flow of the tissue were studied using a commercial finite element analysis

(FEA) software (ABAQUS, Simulia Corp. USA) [156]. Julkunen et al. [132]

combined quantitative microscopy with the fibril-reinforced poroviscoelastic model

to study the relationships between proteoglycan and collagen content of the cartilage

and estimate mechanical properties of the cartilage using curve fitting. In another

study, Julkunen et al. [157] used a combination of MRI data and fibril-reinforced

poroviscoelastic FEA to estimate mechanical properties of the cartilage e.g. non-

fibrillar (proteoglycan) and fibrillar (collagen) modulus and permeability. Simon et

al. developed FE models considering the solid skeleton to be hyperelastic [140, 158,

159]. Frijns [160] used a quadriphasic model and FEA to simulate swelling and

compression behaviour of the intervertebral disc. Numerical overlay effective stress

(NOLES) modelling methodology was another FEA study, in which a collagen

meshwork entrapped swollen proteoglycans [161]. In the NOLES method, the

physical model of Broom and Marra [162], which includes a network of strings filled

with inflated balloons, was adapted in an FEA model of cartilage to simulate the

load-bearing structure of cartilage. Sun et al. [163] developed an FEM model based


on triphasic theory to investigate a one-dimensional (1D) compression and free

swelling of cartilage.

Current computational methods are based on strict geometrical and

mathematical operators [23]. Despite that they are highly efficient at the observation

of porous media performance and description of the current states, they have very

limited capability to explain modality of formation and the reasons behind observed

situations and behaviours. There have been several studies that are geared towards

the determination of the physics and internal micro-mechanisms of their deformation

characteristics including microscale diffusion, percolation, swelling, solid structural

reorganisation and cellular responses to external stimulation and internal dynamics.

However, they still rely on experimental curve fitting that cannot capture the

heterogeneity of the cartilage, to assume the required parameters. Therefore, new

methods of determination of tissue properties beyond the conventional experimental

and computational, with the ability to consider the internal interaction of

components, are required.

2.8 AGENT-BASED METHODS

Computational models traditionally rely on equation-based methods that are

difficult to apply to complex systems such as biological systems. Heterogeneity,

variations and interdependencies of such systems are difficult to formulate

mathematically [164]. To this end, ABM has been developed in order to study

complex and unpredictable phenomena. ABM includes discrete autonomous and

self-directed individuals, named agents, with a set of characteristics and behaviours

[165]. Agents are situated in an environment which they interact with other agents.

ABM is a discrete ‘bottom-up’ method, in which individual behaviours and


interactions between agents in discrete times (time steps) are considered. Rules of

behaviour are defined for each individual agent and the whole system behaviour

appears via the local interactions among the agents, based on defined behaviour rules

at the agent level [166, 167].

The advantages of using ABM over traditional computational techniques for

biological systems can be summarised in two statements: Ability to capture complex

systems, and flexibility.

- Capturing complex systems: Complex systems result from the

interactions of individual entities that are beyond superimposing individuals to

create the whole where the topology of interactions is complex and

heterogeneous. These characteristics of complex systems make them

unpredictable and difficult to understand. Such heterogeneous and non-linear

systems are too difficult to be defined and captured by aggregate differential

equations. Aggregate equations often assume global homogenous mixing and

tend to smooth out fluctuations [24, 168], consequently an actual system might

have considerable deviations from differential equation prediction. On the

contrary, complicated, non-linear and discrete behaviours of a system’s elements

can be considered by ABM, which makes it potentially capable of capturing

complexities.

- Flexibility: ABM is highly flexible spatially and temporally. Agents

are able to be physically mobile or immobile. A diversity of options can also be

defined with the neighbourhood. Complexity of the system can be adjusted by

adding or removing agents to/from the system while behaviours and relations of

agents can be changed. Levels of description and aggregation of the system can

also be tuned [24, 169].


ABM is suitable for capturing emergent systems, in which the system arises

through interactions among smaller and simpler elements of the system that do not

exhibit property and behaviour of the whole system [24, 170]. Biological systems are

considered emergent systems since the behaviour of the systems as a whole, results

from discrete phenomena happening at the cellular level [171]. In particular,

interactions between constituents of articular cartilage –proteoglycan, collagen and

fluid- at a molecular level, govern the complex response of the tissue to internal and

external stimulus. The complexity of biological materials underlines the use of

ABMs in a computational situation to create the structure and elucidate the

microscale underlying such a complex system [18, 172, 173].

The individual agents create an agent-based system. Agent-based methods are

divided into two categories: lattice-free and lattice-based techniques. Agent positions

are restricted to a regular two or three-dimensional lattice in lattice-based

approaches, while in the case of lattice-free models, agent positions and orientations

are not limited in space [174]. Lattice-free techniques are more flexible and allow

more complex and accurate coupling between agents and their environment.

However, computational costs and technical difficulties, such as developing

interaction between agents, limit the lattice-free approaches. Lattice-based

techniques are more practical and computationally efficient and due to the existence

of the lattice, each cell interacts with a limited number of neighbours; therefore, the

need for elaborate interaction testing between agents are eliminated [175, 176]. In the

following sections, two of the most common lattice-based techniques, lattice gas

(LG) and cellular automata (CA), are discussed.


2.8.1 Cellular automata

A CA is a particular class of ABM, which is based on local interactions

between agents located at regular locations [177]. CA has three main principals:

Structure, local interaction and agent states.

Structure: The environment of the CA is formally defined by a cellular space,

called lattice, consisting of regularly arranged locations, named cells. Each cell is a

position where an agent represents an individual. Consequently, every cell in the

lattice can address a location of an agent. A typical example of the CA structure is a

checkerboard or grid, in which cells are arranged regularly.

Local interaction: Agents as individuals can only interact with others in the

neighbourhood around them based on local rules, which can be modified to simulate

different conditions and to investigate mechanisms [177, 178]. The interacting

individuals are named neighbours. Local interaction highly depends on how

neighbours are defined.

Agent states: Agents are represented by their states [179]. Agents have limited

options for their states where numbers such as 0, 1 and 2, lights on or off, and

colours such as black and white may demonstrate states of the agents [26, 178].

Different states, for example, may represent various materials [180-182] or urban

structures [183]. The state of the agent can change depending on the state of the

agent and its neighbours.

A neighbourhood is applied to shape the lattice. Figure 2.4 demonstrates two

classical widely used neighbourhoods: Moore and von Neumann. The two-

dimensional (2D) Moore neighbourhood with Manhattan distance 1 includes the

eight cells (26 in 3D) surrounding a central cell (Figure 2.4A), while the central cell

in the 2D von Neumann neighbourhood with Manhattan distance 1 is surrounded by


four cells (14 in 3D) orthogonally. The von Neumann and Moore neighbourhoods

have been used for prediction of highly complex physical and biological processes,

e.g. tumour development, fluid flow and diffusion of solute into a solvent [181, 184-

188].

A B

Figure 2.4 Examples of a regular two-dimensional lattice. A: Moore neighbourhood

with Manhattan distance 1 (r=1). B: von Neumann neighbourhood (r=1). The grey

cells are the neighbourhood for the black cell (central cell).

Partitioning is another technique to generate a neighbourhood. In partitioning

CA or block CA, the lattice of cells is divided into non-overlapping partitions

(blocks) and all cells within a block interact with each other at each time step

according to the transition rule. The Margolus neighbourhood is the well-known

partitioning technique in which the nearest eight (in 3D) or four (in 2D) cells make

one block (Figure 2.5). Each block moves one cell to the right and down at even time

steps and then moves back at odd steps where in odd and even steps, each cell

belongs to different blocks and objective cells are common between blocks.

Information propagates due to objective cells.


Figure 2.5 Margolus neighbourhood.

As an effective approach, CA addresses many scientific problems by providing

an efficient method to simulate specific phenomena, in which conventional

computational techniques are hardly applicable. For example, permeability of a

membrane [189], bond interactions among molecules [190], drug release [180] and

dissolution of a solute in a solvent in different conditions, e.g. temperature and solute

concentration, [181, 188, 191] were simulated using von Neumann neighbourhood.

Moore neighbourhood was used to understand Chagus disease evolution [184],

tumour development [185] and the cracking process of rock [192]. Margolus

neighbourhood has already been used to simulate deformation in clays [23, 193] and

diffusion and fluid flow in porous media [194]. A Margolus neighbourhood can also

be used to simulate a Toffoli-Margolus (TM) gas model, in which the system obeys a

simple rule: rotate every block clockwise on even steps of the simulation, and

counter-clockwise on odd ones, except in the case that a block contains two

diagonally opposite agents [178]. The TM gas model has successfully been used to

simulate gas propagation inside a container and other fluid dynamic studies [178,

195].


2.8.2 Lattice gas automaton

A Lattice Gas Automata (LGA) is a system of identical particles where the

particles move on a discrete spatial lattice, which is an array of points (sites),

arranged in a regular crystallographic fashion [196]. The basic idea of LGA is that

different microscopic interactions between constitutive components can lead to the

same form of macroscopic behaviours [197]. This method includes a lattice, where

the sites on the lattice can take a certain number of states, which are different

particles with certain velocities. The state at each given site is Boolean, in which a

site is either empty or occupied by a particle. No more than one particle is allowed to

be at each node (exclusion principle). Evolution of the lattice is done in discrete time

steps. The state of the site itself and neighbouring sites before the time step can

determine the state at a given site after each time step. Propagation and collision are

two processes that are conducted at each time step. In the propagation step, each

particle will move to a neighbouring site determined by the velocity of the particle.

In the collision step, if multiple particles reach the same site, collision rules

determine location and velocity of the particles. These collision rules are required to

maintain mass conservation, and conserve the total momentum [26, 197]. LGA and

its derivation, Lattice Boltzmann (LB), which is based on solving Boltzmann’s

equation to simulate fluid flow [198], were successfully used to simulate fluid flow

[197-201].

The first and the simplest LGA model was introduced by Hardy, Pomeau and

de Pazzis (HPP) [202]. HPP is a 2D LGA model over a square lattice where a

particle is allowed to move along four directions, e.g. north, south, east and west

(Figure 2.6). The basic idea of HPP was to create an automaton that obeys

conservation laws at the microscopic level. A single particle has a ballistic motion.


The collisions between particles are strictly local, in which only particles of a single

site are involved. There is only one collision configuration for HPP. When a pair of

particles enter a node from opposite directions and the other two directions are

empty, a head-on collision takes place, which rotates both particles by 90º in the

same direction (Figure 2.7) [197, 203]. All other configurations stay unchanged

during the collision step. The way that directions of particles change in HPP, makes

it very similar to a TM gas [195].

Figure 2.6 The lattice used in the HPP model. The four arrows a, b, c and d indicate

the possible movement directions of a particle.

Before collision After collision

Figure 2.7 Collision rules in HPP[203]. Two particles experiencing a head-on

collision are deflected in the perpendicular direction.


HPP is easy to implement on the computer, as only the information from the

four neighbours are required at each collision and propagation step. Despite the

simple required calculations and simulation abilities of HPP, it does not obey the

Navier-Stokes equations in the macroscopic scale due to the inadequate degree of

rotational symmetry of the lattice [197]. This weakness prevents the HPP from being

applied to a great number of fluid problems. Other drawbacks of HPP are the existing

additional conserved quantities except mass and momentum. For instance, the

difference in the number of parallel and anti-parallel particles to a lattice axis does

not change by collisions or propagation. These conserved quantities limit the

dynamics of the model and have no match in the real world.

Frisch, Hasslacher and Pomeau (FHP) [204] introduced a lattice gas model

based on a hexagonal lattice, where a particle is allowed to move along six directions

(Figure 2.8). Higher lattice symmetry in FHP compensates the HPP drawbacks and

leads to the Navier-Stokes equation in the macroscopic level. The additional

properties of the FHP model are as follow [197]:

1. Sites are linked to six nearest neighbours (nodes) located all at the same distance

with respect to the central node (Figure 2.8).

2. As shown in Figure 2.9, there are several collision configurations.

Figure 2.8 The hexagonal lattice used in the FHP model. Each particle can move

along six directions [197].


Figure 2.9 All possible collisions of the FHP variants: empty cells are represented by

thin lines, occupied cells by arrows [197].

FHP has been used to analyse the physical phenomena in micro-scale such as

osmosis [200], in which the model considered two species and a semi-permeable

membrane was located between diffusing fluid entities, and fluid flow in

heterogeneous porous media [205], in which permeability fields were created by

distributing obstacles within the media. Similar to HPP, FHP is computationally-

friendly. However, since the exclusion principle leads to a Fermi-Dirac local

equilibrium distribution [206], there is a high level of statistical noise for many

applications of FHP [207, 208]. Ensemble and space average should be used to


reduce statistical noise, which results in a much larger required lattice size than the

original problem [209].

In order to reduce noise and the massive computational work of LGA,

McNamara and Zanetti proposed the Lattice Boltzmann (LB) model, in which

Boltzmann’s equation [210] controls the time evaluation of sites and Boolean

variables are replaced with their ensemble average [211]. Consequently, the Boolean

site populations become real numbers between 0 and 1. The LB has successfully

been used to simulate a variety of complex phenomena in porous media such as fluid

flow [212], soot combustion [213] and non-Darcy flow in disordered porous media

[214].

2.9 TECHNIQUES TO DEVELOP POROUS STRUCTURES

Computational methods, e.g. LGA, LB and CA, require detailed structural

models to simulate functions of the media. Imaging methods such as CT scanning,

provide realistic structural models based on X-ray absorption. CT images with a

resolution about 1 mm have been used to measure bulk properties of phasic porous

media such as density and porosity [215, 216]. In more recent research, high-

resolution 2D images of porous materials such as soil and rock were obtained, using

X-ray micro-tomography and Micro-CT scans in order to reconstruct a 3D

representation of the porous medium where pore space topology was characterised

[212, 217-222].

On the other hand, it is important to construct models that closely mimic the

heterogeneity of real porous structures while at the same time they are

computationally efficient. Realistic structures that have been reconstructed by


imaging techniques are extremely complex, thus huge computational effort is

required [223]. In order to simplify the model and make it more practical, many

researchers use conceptual models to develop porous structures. For instance, in the

schematic model that is one of the simplest models, pores are distributed regularly or

irregularly in a lattice or space and connected by throats in which radii of throats are

set by drawing randomly [224-227]. Overlapping spheres and packed sphere models

both used randomly distribution of spherical balls or disks to create porous

structures, while space between solid balls or disks creates pores [228, 229]. Many

studies used simple structural models of fluid saturated porous materials [32, 230],

while more complex patterns generated by CA rules – e.g. voting rules [178, 231],

phase transition rules [26] and random drive [232] – were used to construct porous

medium structures [27, 182, 233]. Several researchers used rules for evolving two-

phase flow and phase separation to create porous structures [28, 213].

However, the above-mentioned structural models (both realistic and

conceptual) are limited to phasic materials where a clear boundary between

components of the medium is required to be able to create a porous structure.

Moreover, the porosity of the porous media in conceptual models cannot be less or

more than certain values in order to keep long-range connectivity between pores

[230, 234, 235]. This limits conceptual models to porous materials with a limited

range of porosity.

Fuzzy random models of pore structure [236], which are based on fuzzy set

theory [237], allocates a porosity between 0 to 1 to each agent in the lattice. User-

specified statistical distribution of porous medium was used for allocating porosity to

each pixel. In this model, each agent is identified by a number, which describes the

volume fraction of the pore relative to the total unit volume. Low porosity structures


can be modelled while pore connectivity can be assured. However, this model is

limited to the systems that sum of the proportions of components, for example, pore

and solid skeleton, in the agent equals one. For instance, if the sum of fractions of the

components within an agent is variable over time (sum of fractions is not always

equal to one), this agent is inadequate. In addition, it is too difficult to use the fuzzy

model for the complex structures with variable porosity in time, where the porosity

of the structure is anisotropic and changes spatially irregularly.

2.10 INADEQUACY OF CURRENT AGENTS AND RULES FOR

ARTICULAR CARTILAGE

A critical review of agent-based techniques, which have been used for fluid

flow through a porous medium, identifies the significant gaps in current rules and

agents that form the focus of this work. Rules are a set of logical decisions that

govern the activity of constituent elements of a system. State, direction of movement,

velocity or location of an agent may change according to rules of the system [173,

238]. A rule also determines how an agent interacts with its global or local

environments, e.g. neighbours [169, 239]. Base-level rules such as deterministic,

probabilistic and stochastic, determine behaviours of agents and provide agent

responses to the environment, while higher-level rules, e.g. learning automata, define

rules to change rules for adaptation purposes and handling unanticipated situations

[165, 240]. However, current rules are only able to determine relations between

agents at the extra-agent environment level. Although the state of the agents can be

changed, and an agent can be converted to another agent (for example, 0 is changed

to 1, which is representative of another element of the system), current rules are not


capable of having the same level of control on change within the agents as extra-

agent change. They only affect systems on an inter-agent scale by changing agents’

arrangements in the system, while they have very limited capacity for intra-agent

evolution.

An agent as a basic element of the system is characterised by states. The

simplest agent-based model of a porous medium requires at least two agent states to

represent constituents of the medium, e.g. 1 to represent solid and 0 to represent

fluid. Solid agents are fully impervious and act as obstacles to fluid agents that form

void space within the media. The agents may have various states in more complex

simulations with more components, but the porous system is still generated by

mixing fluid and impervious agents [179]. Although ABM has been successfully

used to create porous structural patterns as well as simulations of complex porous

systems [27], the simulation of porous systems can be argued to operate like a

mixture of obstacles and open pathways where the structure contains agents, which

are either solid or space. This does not change the fact that no matter the rules of

simulation adopted for solid and space agents, the result will always be a mixture of

distinguishable solid and space agents. However, this is unlike the situation in many

biological systems such as articular cartilage, where the system contains multiple

components, which are intermingled at the molecular level [5], and consequently

elements of the system at any level (size) consist of practically inseparable solid and

fluid.

In addition, properties such as semi-permeability and porosity, and functions

like percolation and dilution, become meaningful when both porous (fluid) and

impervious (solid) components exist. Since each agent represents only one

component, those properties and functions are meaningless for a single agent. They


are defined at a bulk level where a group of agents with a variety of states including

both solid and fluid are included. Figure 2.10A shows a lattice with solid (in red) and

fluid (in white) agents with uneven distribution. Solid and fluid agents are distinct

from each other in the lattice where fluid agents are fully permeable and solids are

fully impervious. Figure 2.10B shows a lattice after grouping its agents. Thick lines

show the boundaries of the groups. Since each group consists of several solid and

fluid agents with different arrangements, bulk properties and behaviours can be

defined for each group individually. Figure 2.10C shows the same lattice when

groups are considered as basic elements that create the system where solid and fluid

agents are sub-elements. The properties can be defined for each element and

distribution of properties can be determined. However, since the elements of the

system were created by superimposing fluid and solid sub-elements, the same rules

and methods of rearrangement as Figure 2.10A are used to update the porous and

impervious agents in Figure 2.10C. Therefore, the behaviour of such a system is

obtained by summing the contribution from porous and impervious agents and

therefore, properties of the groups are still at a bulk level and the system suffers from

the same limitations as the system in Figure 2.10A.

A B C

Figure 2.10 A: a lattice consists of solid and fluid agents in red and white

respectively. B: same lattice when agents were grouped. Thick lines shows groups

borders. C: same lattice when groups were considered as elements of the system.

Chapter 3: New Agent and Rule 45

Chapter 3: New Agent and Rule

3.1 INTRODUCTION

This chapter presents an enhanced agent (hybrid agent) that can be adapted to

represent single-phase multi-component materials. It presents the methodology that

describes how a hybrid agent evolves when within agent (intra-agent) changes occur.

It also includes a new category of rules (intra-agent rules) that enable intra-agent

evolution of the hybrid agent.

3.2 HYBRID AGENT

Each hybrid agent contains within it the system constituents. This agent carries

characteristics of all constituents where the characteristic of the agent is a

combination of all carried characteristics. A hybrid agent is identified by its

constituents and their quantities, which define the state of the hybrid agent. The

constituents can be physical components such as materials, or abstract properties

such as attributes or a combination of both. It is not necessary for the quantities of

the constituents of a hybrid agent to sum to one. For example, if a system consists of

customers and sellers, each hybrid agent includes both customer and seller. When a

seller purchases an item for himself, he is a customer because he purchased the item

and paid for it while simultaneously he sold the item as a seller. Therefore in this

situation, an agent must carry attributes of a customer and a seller at the same time.

46 Chapter 3: New Agent and Rule

A hybrid agent is capable of changing within itself. The intra-agent evolution

of the hybrid agent may be indicated by changing quantities of the constituents

within the agent. For example, a hybrid agent can consist of prey and predator

attributes in a fish life study [241], in which a fish can gradually transform from egg

to fish larvae and then adult. Eggs can be prey for both larvae and adult fish, larvae

can be hunted by adult fish, and bigger adult fish may hunt smaller adult fish as well.

If each fish, including egg, fish larvae and various sizes of adult fish, is considered

as an agent, intra-agent evolution of the agents equals the growth of fish that change

their behaviour and role in the system (be a prey, hunter or both). In such a system, a

fish as an agent always remains a fish, while its attributes as a prey or a predator

change when it grows and transforms gradually from egg to a full-size adult. In this

example, intra-agent attributes of the hybrid agents are evolved in time, even without

extra-agent interaction. In order to determine gradual intra-agent changes of the

hybrid agent, a new category of rules is required.

3.3 INTRA-AGENT RULE

This is a new intra-agent control mechanism, which enables intra-agent

evolution of the hybrid agent. Although a hybrid agent can interact with its

neighbours based on extra-agent rules such as traditional neighbourhood rules, an

intra-agent rule is required to determine change within the agent. Extra-agent rules

determine relations between agents, often involving spatial rearrangements of the

agents in the system, while intra-agent rules define how a hybrid agent evolves. A

hybrid agent is capable of evolving itself in time, where such transformation is based

on intra-agent rules. As intra-agent rules are applied to the agents in the system


individually, the intra-agent evolution of each hybrid agent is distinct from or

independent of other agents in the system. An intra-agent rule is applied to the agent

itself. Figure 3.1 shows a hybrid agent (agent A) with two neighbours (agents B and

C) and where intra- and extra-agent rules are applied. Extra-agent rules operate at the

environment level outside the agent (extra-agent environment), while intra-agent

rules are applied to the environment inside the agent. The hybrid agent may change

based on intra-agent rules as a consequence of interactions with its neighbours.

However, a hybrid agent is capable of evolving in time without interaction with any

external element. Figure 3.2 presents some probable states of hybrid agent H

consisting of two constituents, where the quantities of the agent’s constituents

change without any interaction with neighbours, thus illustrating the operation and

expected consequences of the intra-agent rule. The change within the agent H is

dictated by the intra-agent rule in the way that the quantity of one of the constituents

can be replaced by any other one (Figure 3.2A); consequently, the hybrid agent

transforms to a single component agent with the same agent size, which is equal to

the sum of its constituents before transformation. The quantity of both constituents

can be decreased or increased (Figures 3.2B and 3.2C respectively), which changes

the agent size with or without changing the ratio of constituents’ quantities. The

quantity of one constituent can be increased and other one decreased, while the sum

of quantities of the constituents in the agent may change (E) or may not change (D).

When the size of the agent that might be determined by the sum of the quantities of

its constituents is changed, the fractions of the quantities may be unchanged (same as

Figure 3.2B). Therefore, the intra-agent environment evolves and changes without

necessarily changing quantity fractions (such as volume or mass fraction) of the

components in the agent.


Figure 3.1 Illustration of the application of the intra-agent and extra-agent rules.

Extra-agent rules define interaction between agent and environment beyond the agent

itself such as neighbours, while the intra-agent rule is applied to each agent

individually to determine intra-agent evolution.

Figure 3.2 Intra-agent change of hybrid agent H when it contains two constituents.

A: one constituent is vanished and replaced by another without agent size change. B

and C: quantity of the constituents increased and decreased respectively with agent

size change. D: Change of quantity of the constituents without agent size change. E:

Change of quantity of the constituents with agent size change.


3.4 ADAPTATION OF THE HYBRID AGENT FOR POROUS MATERIALS

The hybrid agent can be adapted to represent a porous medium with multiple

components, where the quantities of the components such as volume or weight

determine the state of the agent. The simplest representative form of non-saturated

porous material consists of three constituents: “solid”, “fluid” and “space”. Fluid fills

all empty voids in the saturated porous medium; therefore, the hybrid agent, which

represents a non-saturated porous structure, can be used to represent a saturated

porous material if all “space” in the agent is replaced by fluid (quantity of the space

equals zero). Figure 3.3 shows the conception of the hybrid agent, where a hybrid

agent results from hybridization of solid, fluid and space agents. This hybrid agent

carries properties of solid, space and fluid within it and can simultaneously exhibit

the characteristics of solid, fluid and space in time, while it is neither fully solid nor

fluid or space. The components within the hybrid agent are not necessarily separable.

The quantity of the space is zero in the hybrid agent, which represents saturated

porous materials. Each hybrid agent is identified by quantities of its constituents. If

the sum of the quantities of a property of the agent is always constant, the ratio of

quantities might be used to identify an agent. For example, a hybrid agent that

represents a saturated porous material can be identified by weight ratio of fluid to

solid, if the sum of weights is constant where the volume of the agent may change

due to exchanging weight ratio.

Permeability is one of the critical properties of porous materials that indicates

the ability of fluid flow through media [32] where materials with high permeability

allow fluids to transmit through it quickly. Permeability of a porous structure

depends on its resistance to fluid flow through it where the amount of solid and pores

(fluid) present in the structure is a determining factor for this resistance [242]. The


resistance to fluid flow varies between zero, i.e. no solids are present or huge value

of porosity, and infinity where in the absence of the pores (fluid) in the porous

medium, the resistance to fluid flow is infinite resulting in zero permeability [243].

In order to identify hybrid agent for the porous materials, and distinguish different

agents from one another in a saturated porous system with constant size of the hybrid

agents, a variable “fs” is defined as the ratio of quantity of the fluid to solid. The

variable fs is equal to zero if the hybrid agent transforms to an agent with the

characteristics of a solid which demonstrates infinite resistance to fluid flow. It is

equal to infinity when the hybrid agent transforms to an agent with the properties of a

fluid agent which represents an open pathway with zero resistance to fluid flow.

Increasing proportion of the fluid to solid causes an increase in the fs value and a

decrease in the resistance to fluid flow. Consequently, the ratio of fs has a negative

correlation with resistance to the fluid flow in the hybrid agent and, therefore,

reflects permeability of the agent. As a hybrid agent which is representative of

porous materials locally changes and transforms to another hybrid agent if its

permeability changes, evolution of the hybrid agent occurs by changing and updating

its fs over time as the systemic evolution intra- and extra-agent rules determine.

Figure 3.4 shows possible fs changes for an agent where the sum of quantities of the

components (agent size) is always constant. The agent can transform to a full fluid or

full solid characteristic agent (A and B respectively), or decrease its solid

characteristic leading to an increase in its fs (C), or a decrease in its fluid

characteristic leading to a decrease in its fs (D). Unlike current existing agents that

have finite states (0 and 1 to represent solid and fluid [27, 189]), a hybrid agent has

infinite states due to an infinite number of values for fs.


Figure 3.3 A conception of a hybrid agent to represent a porous non-saturated

material. The hybrid agent is the combination of space, fluid and solid sub-agents. It

illustrates a key concept of the philosophical notion of the new agent-based

approach.

Figure 3.4 Intra-agent change of a hybrid agent. Blue and grey show fluid and solid

respectively. A and B: Hybrid agent is transformed into a full fluid and solid agent

respectively. C: fs of the agent decreases. D: fs of the agent increases.

Chapter 4: Using hybrid agents to create porous structures 53

Chapter 4: Using hybrid agents to create

porous structures

4.1 INTRODUCTION

The philosophical premise of this chapter is the notion that in order to create

appropriate models to study the microscale responses of a complex system such as

articular cartilage, a representative structural model is required. The first challenge in

the study of biological porous materials is to determine the structure of the medium

at the microscopic level. In this chapter, one-dimensional cellular automata (1D CA)

[26, 244] are adapted to generate growing two-dimensional (2D) patterns. Arbitrary

1D CA rules are used to generate rows in a 2D domain, which have striking

morphological and characteristic similarities with the porous semi-permeable fluid-

saturated single-phase structures. In the simplest class of traditional 1D CA, each

agent has two possible states (black or white) and the evolution depends only on the

states of the agent, and its left and right immediate neighbours (the nearest

neighbours). The traditional 1D CA generates individual strips (rows) of the agents

over the time steps [245].

4.2 METHODOLOGY

4.2.1 Hybrid agent

The hybrid agent consists of two constituents, property A and property B, where

they are indistinguishable in the agent. Each cell indicates position of only one

hybrid agent, which remains in the same cell over the entire time of the simulation. A

hybrid agent and its immediate neighbours are presented in Figure 4.1A. The agent

54 Chapter 4: Using hybrid agents to create porous structures

and its left and right neighbours contain 𝛼C, 𝛼L and 𝛼R quantities of property A, and

𝛽C, 𝛽L and 𝛽R quantities of property B respectively. This arrangement can be broken

down into eight permutations (Figure 4.1B) to demonstrate all possible

neighbourhood arrangements if each cell contains only one component (A or B)

instead of an agent carrying both components A and B. Left, central and right agents

in Figure 4.1A are parents of the permutation cells located in the left, central and

right respectively. Each permutation includes three cells, one cell in the middle

(central cell), surrounded by two cells, one on the left and one on the right. The state

of each permutation cell can be A or B. Permutation cells contain property A or B

equal in quantity to their corresponding parent agents. Therefore, cells at each

permutation are identified by their state (A or B) and the quantity of the contained

component, referred to as cell value. For example, cells at permutation 3 in Figure

4.1B, which from left to right consist of 𝛼L property A (equal to the quantity of

property A in its parent agent), 𝛽C property B (equal to the quantity of property B in

its parent agent) and 𝛼R property A (equal to the quantity of property A in its parent

agent), have states of A, B and A, and cell values equal 𝛼L, 𝛽C and 𝛼R respectively.

A

𝛼L , A 𝛽L, B

𝛼C , A 𝛽C , B

𝛼R , A 𝛽R , B

B 𝛼L

A

𝛼C

A

𝛼R

A

𝛼L

A

𝛼C

A

𝛽R

B

𝛼L

A

𝛽C

B

𝛼R

A

𝛽L

B

𝛼C

A

𝛼R

A

𝛼L

A

𝛽C

B

𝛽R

B

𝛽L

B

𝛼C

A

𝛽R

B

𝛽L

B

𝛽C

B

𝛼R

A

𝛽L

B

𝛽C

B

𝛽R

B

1 2 3 4 5 6 7 8

Figure 4.1 Immediate neigbours of a hybrid agent and their possible permutations. A:

a hybrid agent and its immediate neighbours. Agents contain characteristic of

properties A and B. B: Possible arrangements of properties A and B for hybrid agents

shown in image A.


4.2.2 Extra-agent rule

Two well known, existing elementary cellular automata rules (Rules 22 and 73

[26] that are shown in Figure 4.2) have been used as extra-agent rules for the hybrid

agent in which white and black represent properties A and B respectively. As the

states of the permutations of a hybrid agent and its nearest neighbours are determined

as property A or B, the extra-agent rule can be used to determine the state of the next

generation of the central cells of the permutations. For instance, the type and value of

cells in permutation no.5 (Figure 4.1B) from left to right cells are property A (𝛼L),

property B (𝛽C) and property B (𝛽R) respectively. Since white and black in Figure 4.2

represent property A and property B respectively, cell arrangements of permutation

no 5 are equal to a white cell with black and white cells on its left and right

respectively, where the state of next generation of such a neighbourhood

arrangement will be black (property B) if Rule 22 is used as the extra-agent rule and

white (property A) for Rule 73 (Figure 4.2).

4.2.3 Intra-agent rules

The first intra-agent rule is size of agents are constant. This size constraint results

in a constant quantity of the properties A and B combined across all cells over the

Rule 22

Rule 73

Figure 4.2 1D automata Rules 22 and 73. White and black cells represent properties

A and B respectively.


entire simulation time. Therefore, agents containing properties A and B can be

identified by the ratio of the quantities of the property A to property B, named AB.

The intra-agent rules determine the value of property A or property B of the next

generation of the central cells for each permutation, as well as the quantity of the

property A and property B in the next generation of the central parent agent. In order

to investigate the influence of the intra-agent rules on the generated pattern, three

individual arbitrary intra-agent rule sets are used. Each rule set includes two rules: (i)

one to determine the value of the next generation cells of the permutations, and (ii)

an arbitrary rule to determine quantities of the properties A and B, and the intra-agent

AB ratio of the next generation of the central parent agent based on the state and

value of the next generation cells of the eight permutations of the parent agents. Rule

sets are defined as follow:

Intra-agent rule set 1: (i) The value of the cell in the next generation is equal to

the minimum values of the cell and its immediate neighbours. For example, the value

of the next generation of permutation no.5 is equal to the minimum of 𝛼L, 𝛽C, and

𝛽R.

(ii) The sum of property A values of the next generation of the permutations

divided by the sum of property B values of the next generation of the permutations

determines the intra-agent AB ratio of the given hybrid agent in the next generation.

To illustrate the intra-agent rule set 1, when extra-agent Rule 22 and intra-agent rule

set 1 are used, the states of the central cell in the next generations of permutation 1

to 8 in Figure 4.1 are A, B, A, B, A, A, A and B respectively. Based on the intra-

agent rule set 1 (i), values of the next generation cells of the permutation 1 to 8 are

equal to λ1, λ2, λ3, λ4, λ5, λ6, λ7 and λ8 where λ1 = min(𝛼L , 𝛼C and 𝛼R), λ2 = min(𝛼L ,

𝛼C and 𝛽R), λ3 = min(𝛼L , 𝛽C and 𝛼R), λ4 = min(𝛽L , 𝛼C and 𝛼R), λ5 = min(𝛼L , 𝛽C and


𝛽R), λ6 = min(𝛽L , 𝛼C and 𝛽R), λ7 = min(𝛼L , 𝛽C and 𝛽R) and λ8 = min(𝛽L , 𝛽C and 𝛽R).

Therefore, based on rule set 1 (ii), the AB ratio of the next generation of the central

agent of the parent agents equals to λ1 + λ3 + λ5 + λ6 + λ7

λ2 + λ4 + λ8.

Intra-agent rule set2: (i) The value of the next generation of a permutation

equals the value of the central cell without considering values of its immediate

neighbours. For example, the value of next generation central cell of permutation no.

5 equals to 𝛽C.

(ii) Similar to the rule set 1, the ratio of the sum of property A to property B

values of the next generations of the permutations determines AB ratio of the given

hybrid agent in the next generation.

Intra-agent rule set 3: (i) The property A or property B value of the next

generation central cell equals the minimum of the values of the cell and its

immediate neighbours (same as intra-agent rule set 1).

(ii) AB ratio of the given hybrid agent in the next generation is determined as

an accumulation of the property A quantity of the central parent agent and property

A values of the next generations of its permutations, divided by the similar

accumulation of the property B. To illustrate this, the next generation of the central

hybrid agent in Figure 4.1 using intra-agent rule set 3 and Rule 22 as extra-agent rule

is calculated as below:

Based on Rule 22, the value and state of the next generations of the

permutations 1 to 8 are λ1 and A, λ2 and B, λ3 and A, λ4 and B, λ5 and A, λ6 and A, λ7

and A, and λ8 and B respectively where λ1 to λ8 are equal to min(𝛼L , 𝛼C and 𝛼R),

min(𝛼L , 𝛼C and 𝛽R), min(𝛼L , 𝛽C and 𝛼R), min(𝛽L , 𝛼C and 𝛼R), min(𝛼L , 𝛽C and 𝛽R),

min(𝛽L , 𝛼C and 𝛽R), min(𝛼L , 𝛽C and 𝛽R) and min(𝛽L , 𝛽C and 𝛽R) respectively. Based

on intra-agent rule set 3, ratio AB of the next generation is the summation of initial


property A quantity in the central parent agent and sum of the property A values of

the permutations, divided to summation of initial property B quantity in the central

parent agent and the sum of the property B values of the permutations, therefore:

𝐴𝐵 =𝛽𝑐 + λ1 + λ3 + λ5 + λ6 + λ7

𝛼𝑐 + λ2 + λ4 + λ8 .

4.2.4 Two-dimensional domain

The 2D lattice, which creates the 2D domain, innitially consists of empty

cells except cells in the first row, which contain agents (Figure 4.3). At each time

step, only the row including empty cells, which is beneath a row containing agents, is

changed. The evolution of the 2D lattice can be demonstrated as an orthogonal

growth structure by starting with the generation zero in the first row (initial state) and

successive growing generations on the next rows where the rows are perpendicular to

the direction of growth [246, 247].

Figure 4.3 Initial state of the 2D domain. Cells located in the first row contain agents

while other cells in the lattice are empty.


The adapted 1D CA approach for hybrid agents involves the concept of

combining simultaneous intra-agent and extra-agent evolutions where an emerging

structure is determined by the evolution of hybrid agents located in one row (one

generation) of the 2D lattice to create the next row (next generation).

Pattern generation starts from the first generation of the agents (first row of

the cells), in which all hybrid agents contained only property A (AB=infinity) except

the agent located in the cell in the centre of the row (initial seed). In this chapter, the

patterns generated by black and white agents using local Rules 22 and 73 [26]

(traditional patterns), and patterns using hybrid agents, and various intra- and extra-

agent rules and initial seed, were generated. Applied rules, agent type and initial seed

were explained in Table 4.1, in which patterns (i) and (ii) are generated by traditional

black and white agents, initiated by a black seed and using local Rules 22 and 73 [26]

respectively. Hybrid agents are employed in patterns (iii) to (xvii) where an agent

with equal quantity of properties A and B (AB=1) were selected as the initial seed for

patterns (iii) to (viii). In order to study the effect of the initial seed on a generated

pattern, AB of the initial seed in patterns (iii), (iv) and (v) was changed to 0.01, 100

and 0 in patterns (ix) to (xi), (xii) to (xiv) and (xv) to (xvii) respectively. In order to

investigate the effect of the rules on a generated pattern, patterns (iii) to (vii) were

generated, in which Rule 22 was used as extra-agent rule in patterns (iii), (iv) and (v)

and where it was replaced by Rule 73 in patterns (vi), (vii) and (viii). Intra-agent rule

set 1 was employed in patterns (iii), (iv), (ix), (xii) and (xv). Intra-agent rule set 2

was employed in patterns (iv), (vii), (x), (xiii) and (xvi), and rule set 3 was employed

in patterns (v), (viii), (xi) and (xiv) and (xvii). In order to investigate effects of an

initial seed characteristic on the generated pattern, patterns (ix), (x), (xi), (xii), (xiii),

(xiv), (xv), (xvi) and (xvii) were created, in which initial seeds with AB equals to


0.001, 100 and 0 were studied. Programs in Matlab (Mathworks Inc, MA, USA) have

been developed to create the patterns (Appendix A).

Table 4.1 Rules and initial seed of the generated patterns.

Pattern Agent type Intra-agent rule Extra-agent rule Initial seed

(i) Traditional - Rule 22 Black (B)

(ii) Traditional - Rule 73 Black (B)

(iii) Hybrid Rule set 1 Rule 22 AB=1

(iv) Hybrid Rule set 2 Rule 22 AB=1

(v) Hybrid Rule set 3 Rule 22 AB=1

(vi) Hybrid Rule set 1 Rule 73 AB=1

(vii) Hybrid Rule set 2 Rule 73 AB=1

(viii) Hybrid Rule set 3 Rule 73 AB=1

(ix) Hybrid Rule set 1 Rule 22 AB=0.01

(x) Hybrid Rule set 2 Rule 22 AB=0.01

(xi) Hybrid Rule set 3 Rule 22 AB=0.01

(xii) Hybrid Rule set 1 Rule 22 AB=100

(xiii) Hybrid Rule set 2 Rule 22 AB=100

(xiv) Hybrid Rule set 3 Rule 22 AB=100

(xv) Hybrid Rule set 1 Rule 22 AB=0

(xvi) Hybrid Rule set 2 Rule 22 AB=0

(xvii) Hybrid Rule set 3 Rule 22 AB=0


4.3 RESULTS AND DISCUSSION

4.3.1 Effect of different rules

The adapted 1D CA are used to create a 2D growing structure, in which the 1D

strip in the growing edge grows and adds a new row to the 2D lattice. The parent

row, which consists of parent agents, forms the growing edge at each time step. The

2D pattern emerges in a strict 1D pattern formation process since the 2D pattern is

evolved only along the direction of growth, which is in the direction perpendicular to

the strips. In such patterns, the second dimension results from growth, which is an

accumulation of the strips along the growing edge.

The growing patterns generated by traditional (black and white) agents and Rules

22 and 73 (patterns (i) and (ii) in Table 4.1) are presented in Figures 4.4A and 4.4B

respectively. They show the first 50 iterations (generations) of the patterns where the

black and white cells can be representative of solid and fluid components, thus

representing a saturated porous material. As a consequence of only two possible

values for a cell (black and white), generated patterns are checkerboard-like. Rules

only change arrangements of the black and white cells to generate various patterns.

Although these patterns demonstrate pores (fluid) and skeleton (solid), they cannot

fully represent biological tissues in which solid and fluid are indistinguishable and

intermixed up to the ultra-microscopic scale. In addition, since black and white cells

represent impervious (obstacle) and fully porous (open pathway) cells in the lattice,

individual cells are not capable of carrying the semi-permeability characteristics

from which the extent of permeability of a cell can be determined.


Figure 4.5 shows the first 50 generations of the growing patterns (iii), (iv) and (v)

in Table 4.1 which were generated by extra-agent Rule 22 and intra-agent rule sets

1, 2 and 3 respectively. Patterns show the distribution of the AB ratio in the 2D

lattice. The AB ratio of the agents is shown by colour-coded images according to the

legend attached to the pictures. Red shows agents with AB ratio greater than 3,

which include full fluid characteristic agents as well, and dark blue demonstrates

agents with AB ratio equal to one. The patterns were initiated with an equal

properties A and B quantity seed (AB=1) in the centre of the row while the rest of the

agents of the first generation contained only property A (AB=∞). The first generation

of all generated patterns consists of initial seed in the middle (shown in blue) and

hybrid agents contain only property A (shown in red) on the left and right. Second

generations were created based on the first generation of the agents and applied rules.

In the same way, the next generations were created based on previous generation

agents’ conditions (properties A and B content) and arrangement, and applied rules.

While the initial seed was the only agent in the first generation that contained

A B

Figure 4.4 Traditional 1DCA growing patterns. A and B: Patterns generated by

traditional agents and Rules 22 and 73 respectively. Numbers on the left side of the

patterns show the row or generation number.


property A, property A containing agents were expanded to the left and right

symmetrically for the next generations.

Single-property region Single-property region

Uniform region A Transition region


Uniform region B Transition region


Uniform region C Transition region

Figure 4.5 A, B and C: patterns generated using extra-agent rules 22 and intra-agent

rule sets 1, 2 and 3 (patterns (iii), (iv) and (v) respectively) after 50 generations.


The property B content agents in the first generation (shown in red in the figure

4.5) evolved to agents containing both properties A and B, and the quantities of their

properties A and B which indicated with AB ratio, were changed thereafter for the

several generations until stability in which the AB ratio of the agent was unchanged

thereafter. The evolution of a property A containing hybrid agent to unchanged

properties A-B containing agent, creates single-property, transition and uniform

regions. Agents carry one of the properties A or B in the single-property region.

Agents contain both properties A and B in the transition region where the AB

ratio of an agent was changed in consecutive generations. The AB ratios of agents

located in the transition area were different from their previous generation. The

transition region was oblique to the direction of growth and located between the

single-property region (in red) and the region where all agents carried both properties

A and B with equal AB ratio, namely, the uniform region. The number of generations

that a full property A hybrid agent required to reach the transition region depended

on the agent’s distance from the initial seed, where closer agents evolved in fewer

generations. The number of transient generations (transient steps) in which AB ratio

of the agent was changing in consecutive generations, was equal for all first

generation agents in Figures 4.5A and B. However, transition steps for the Figure

4.5C were variable, where agents close to the initial seed required more steps. The

uniform regions in Figures 4.5A, B and C were golden, green and golden triangular

shapes respectively representing AB ratio equal to 2, 1.5 and 2.

If properties A and B respectively represent fluid and solid in the hybrid agents,

agents that create the patterns contain inseparable fluid and solid. As a result, fluid

and solid in the generated patterns cannot be divided from one another. This is in line


with the structure of single-phase multi-component materials in which the constituent

components of the medium being inseparable at any size.

Permeability of a porous structure depends on its resistance to the fluid flow

through it where the amount of solid and pores (fluid) present in the structure is a

determining factor for this resistance [242]. The resistance to fluid flow varies

between zero and infinity, where in the absence of the pores (fluid) in the porous

medium, the resistance to fluid flow is infinite resulting in zero permeability [243].

The ratio of AB illustrates ratio of fluid quantity to solid quantity (fs) in the agent in

which a hybrid agent with fs ratio equal to zero and infinity contains zero fluid and

solid quantity respectively, resulting in infinity and zero fluid flow resistance

respectively. Consequently, the ratio fs has a negative correlation with resistance to

the fluid flow in the hybrid agent. The fs of a hybrid agent, therefore, reflects local

permeability of the pattern since ratio of fs controls resistance to fluid flow through

the hybrid agents. The porosity of an agent which is ratio of the fluid content in the

agent to the quantity of the fluid and solid combined can also be extracted from the

ratio of fs where porosity is equal to the fs divided by one plus fs (Agent porosity =

fs

1+fs). As hybrid agents located in different cells are able to have various fs values,

resulting in different permeability and porosity, the generated patterns are also

capable of heterogeneous property distribution.

A hybrid agent with fs ratio greater than zero but not equal to infinity carries

characteristics of both solid and fluid so that the agent is neither fully fluid nor solid.

This agent is partially open and demonstrates semi-permeable characteristics, in

which resistance to the fluid flow is greater than zero but not infinite. The fs of the

agents located at transition and uniform regions are not equal to zero or infinity,

indicating that agents contained both fluid and solid. Therefore the semi-permeable


area in the patterns consists of both the transition and uniform regions. The semi-

permeable area for the Figures 4.5A and B are inside an isosceles triangle whose the

vertex is located at the initial seed in the centre of the first row, and the last

generation agents form its base. The length of the base of the semi-permeable

isosceles triangle for the Figure 4.5C pattern is approximately two-thirds of its

counterparts in the Figures 4.5A and 4.5B. Comparison between patterns that were

generated by different intra-agent rule sets (Figure 4.5) indicated that changing the

intra-agent rule might result in different patterns for the semi-permeable region

(Figure 4.5 A and B versus C) or in different degrees of permeability (Figure 4.5 A

versus B).

Figure 4.6 presents generated growing patterns using hybrid agents, extra-agent

Rule 73 and intra-agent rule sets 1, 2 and 3 after 15 steps (A, C and E respectively)

and 50 steps (B, D and E respectively), based on coloured distributions of the AB

ratio according to the attached legend. First generation agents in all patterns evolved

and formed single-property, transition and uniform regions. In order to make the

generated patterns suitable for porous structures, properties A and B were considered

as fluid and solid respectively. Consequently, the distribution of AB ratio shows the

distribution of fs. The fs of the agents located in the uniform region were equal to

1.5, 2 and 1.5 for intra-agent rule sets 1, 2 and 3 respectively. Intra-agent rule sets 1

and 2 created a pattern with a semi-permeable triangular region in the centre, similar

to Figures 4.5A and B. The pattern in their single-property region includes

successive full fluid and full solid hybrid agents, similar to the pattern generated by

traditional Rule 73, and black and white agents (Figure 4.4B). Intra-agent rule set 3

generated a pattern where, except for the first generation agents, all agents in the

pattern contained both solid and fluid. Due to differences between the fs of the


agents located at the uniform region in the patterns generated by intra-agent rule sets

1 and 2, these patterns represent different semi-permeable structures. However,

patterns in their single-property region were the same and similar to the pattern

generated by Rule 73 and traditional black and white agents (Figure 4.4B). Unlike

rule sets 1 and 2, the pattern generated by rule set 3 has no similarity to the

traditional Rule 73 pattern. The first generation of the agents in Figures 4.6 E and F

evolved to semi-permeable agents and formed a uniform pattern with fs equal to 1.5

after three generations, except for the immediate neighbours of the initial seed, which

took five generations.

The effect of an extra-agent rule is investigated by comparison of patterns in

Figure 4.5 with Figure 4.6. Patterns generated in Figure 4.5 and Figure 4.6 used

extra-agent Rule 22 and 73 respectively, while Figures 4.5A and 4.6B both used

intra-agent rule set 1, Figures 4.5B and 4.6D used intra-agent rule set 2, and Figures

4.5C and 4.6F used intra-agent rule set 3. Since all patterns started with the same

seed (AB=1), the difference between patterns generated by the same intra-agent rule

was due to the applied extra-agent rule. Comparison of the patterns shows

discrepancies in the single-property region, transition and uniform regions between

generated patterns by the same intra-agent rules, but different extra-agent rules.

Extra-agent Rule 22 generated full property A or fluid agents in the single-property

region, similar to traditional black and white agents (Figure 4.4) while extra-agent

Rule 73 resulted in the generation of consecutive full property A (fluid) and full

property B (solid). It is concluded that the intra-agent rule had no effect on the

single-property region except for rule set 3, which could change the number of the

agents in the single-property region. The intra-rule set 3 increased the number of full

property A agents when it is combined with extra-agent Rule 22, while it decreased


the single-property region to only first generation agents when it is combined with

extra-agent Rule 73.

Figure 4.6 A, C and E: First 15 generation of hybrid agent generated patterns using

extra-agent Rule 73 and intra-agent rule sets 1, 2 and 3 respectively, starting with a

hybrid agent with equal characteristics of properties A and B (AB=1). B, D and F are

corresponding patterns to A, C and E after 50 iterations.

Extra-agent rules also affected agents in the transition region, in which the

AB ratio (fs) of the agents gradually decreased using extra-agent Rule 22 while

extra-agent Rule 73 resulted in fluctuation in the change of AB ratio of the agents.

The transient region of Rule 73 was also thinner than Rule 22. The AB ratio of the

A B

C D

E F


agents located in the uniform region decreased for intra-agent rule sets 1 and 3 when

extra-agent Rule 22 was swapped with Rule 73, while it resulted in an increase of

AB ratio when intra-agent rule set 2 was combined with extra-agent rule 73. The

results confirm that swapping extra-agent rules results in changing the patterns,

which shows the significant role of extra-agent rules in creating various patterns.

4.3.2 Effect of initial seed

Figures 4.7, 4.8 and 4.9 present coloured distributions of AB ratio of the growing

patterns, which were generated using extra-agent Rule 22 and intra-agent rule sets 1,

2 and 3 respectively when the AB ratio of the initial seed was equal to 0.01, 1, 100

and 0. Figure 4.7 shows generated patterns by rule set 1 using various characteristics

for the initial seed. When the AB ratio of the initial seed was greater than zero,

generated patterns were different at the transient region and vertex of the uniform

region for the first few generations of the agents close to the initial seeds (Figures

4.7A, B and C). The rest of the patterns, including the uniform and single-property

regions, were the same. Figure 4.7D presents the pattern generated by an initial full

property B seed (AB=0), in which red shows agents with full property A

characteristic and blue shows full property B agents. This pattern is similar to the

generated pattern by traditional black and white agents and Rule 22 (Figure 4.4A).

Based on the Figure 4.7 patterns, it is concluded that rule set 1 has minimal effect on

the generated pattern when initial seed carries full property B characteristics.


Figure 4.7 Patterns resulted from applying extra-agent Rule 22 and intra-agent rule

set 1 after 50 iterations. A, B, C and D: patterns resulting from initial seeds with AB

ratio equal to 0.01, 1, 100 and 0 respectively.

The patterns generated by rule set 2 and an initial seed containing both

properties A and B (Figures 4.8A, B and C) were similar except for initial seed agent

and second generation of the seed. Intra-agent rule set 2 decreased effects of the

initial seed on the generated pattern significantly when initial seed carried both

properties A and B characteristics (0 < AB < ∞). Using a full property B seed

generated a pattern (Figure 4.8D) consisting of full property A and full property B

hybrid agents with the arrangement similar to the traditional Rule 22 and traditional

black and white agents (Figure 4.4A).

A B

C D


Figure 4.8 Patterns resulting from applying extra-agent Rule 22 and intra-agent rule

set 2 after 50 iterations. A, B, C and D: patterns resulting from initial seeds with AB

ratio equal to 0.01, 1, 100 and 0 respectively.

Figure 4.9 presents the patterns generated using intra-agent rule set 3, starting

with various initial seeds. The AB ratio of the agents located in the uniform region

and the transition region of these patterns are similar. Comparison between the

patterns shows discrepancies in the first few generations in the area close to the

initial seed in the transition and uniform regions. Increasing the AB ratio of the initial

seed, increases the number of discrepant generations (Figure 4.9C versus Figures

4.9A, B and D). Unlike intra-agent rule sets 1 and 2, even a seed with full property B

characteristic (Figure 4.9D) could generate similar patterns as seeds with both

property A and B characteristics. Therefore, changing the initial seed affects only

early generations of agents, which are close to the initial seed. Despite some

differences between patterns in the initial steps, a similarity of pattern demonstrates a

A B

C D


minimal effect of differing initial seed characteristics. It shows that the role of a

chosen rule to create a pattern is more significant than the choice of initial seed.

A combination of extra and intra-agent rules increases the varieties of

patterns generated. Unlike traditional black and white agents where each local rule

can generate only one pattern, implementing intra-agent rules and hybrid agents

provides an opportunity for the creation of various patterns for each extra-agent rule.

In addition, characteristics of the initial seed can be varied in terms of the proportion

of property A and B (solid and fluid) characteristics using hybrid agents. Various

initial seed characteristics provide more patterns without a need to change rules.

However, generated patterns using different seeds may be slightly different from one

another, depending on the choice of intra-agent rule. Therefore, although the initial

seed characteristic is important, rules play a more significant role in generating

patterns.

A B

C D Figure 4.9 Patterns resulting from applying extra-agent Rule 22 and intra-agent rule

set 3 after 50 iterations. A, B, C and D: patterns results from initial seed with AB ratio

equal 0.01, 1, 100 and 0 respectively.

Chapter 5: Diffusion throughout the articular cartilage 73

Chapter 5: Diffusion throughout the

articular cartilage

5.1 INTRODUCTION

The aim of this chapter is to simulate free diffusion throughout the articular

cartilage, where the tissue was immersed in a hypotonic solution. Due to a lack of

blood vessels in articular cartilage, the flow of fluid through the articular cartilage

transports and distributes nutrients to the tissue cells (chondrocytes) [55]. This

diffusive process is also vital for removal of waste products from the tissue [37, 248].

Poor nutritional supply and insufficient waste product removal are the main cause of

tissue degeneration and other related diseases [249-251]. Therefore, knowledge of

diffusion through the articular cartilage is crucial in the understanding of tissue

diseases and degeneration as well as cellular nutrition in the tissue.

During the free diffusion process, the surrounding fluid percolates into the

articular cartilage matrix and diffuses through it, resulting in a gradual replacement

of the initially resident fluid in the articular cartilage matrix. Simultaneously, fluid

inside the cartilage matrix moves out and diffuses into the surrounding fluid. The

fluid molecules move into the articular cartilage under hypotonic conditions due to

changing osmotic pressure in the tissue.

In this chapter, the hybrid agent introduced in Chapter 3 is adapted to represent

the structure of articular cartilage, and free diffusion throughout the tissue, is

simulated by means of cellular automata (CA) and the intra- and extra-agent rules.

The amount of fluid inside the articular cartilage is assumed to be significantly

smaller than the surrounding fluid. Therefore, the process of free diffusion has no

impact on the surrounding fluid.

74 Chapter 5: Diffusion throughout the articular cartilage

In order to validate and calibrate the CA model, the simulated results were

compared with experimental diffusion results of human knee articular cartilage,

taken from the literature [13], in which enhanced computed tomography (CECT) and

peripheral quantitative computed tomography (pQCT) had been used. The CA lattice

in the simulation was developed based on the width and thickness of the

experimental samples [13] (4.0mm and 1.99 ± 0.38 mm respectively).

Validated rules were then used to simulate full and cleft-like partial

degenerated matrix, in which a percentage of solid content was replaced by fluid in

the degenerated regions. Colour-coded maps of diffusion were obtained and

compared with the healthy model.

5.2 MATERIAL AND METHODS

5.2.1 Adaptation of the hybrid agent for diffusion of the articular cartilage

Fluid and solid skeleton are considered to be the two major constituents of

articular cartilage. Hybrid agents are adapted to represent articular cartilage where it

consists of indistinguishable solid and fluid within it. In order to address diffused

fluid, the fluid within the hybrid agent included two types of fluids, named unmarked

and marked fluids. Unmarked fluid represents the initial fluid resident in the articular

cartilage, before starting the diffusion process. Marked fluid represents the

surrounding fluid that would diffuse into the articular cartilage. Although the adopted

hybrid agent contained three constituents - solid, marked and unmarked fluids -

marked and unmarked fluids exhibited the same behaviour in the intra- and extra-

agent environments. At the beginning of the simulation (T=0) agents that represent


articular cartilage contained only unmarked fluid. The diffusion process relates to the

movement of the fluid molecules throughout the tissue which can be determined by

the weight of the exchanged fluid. Therefore, weights of resident solid and fluids in

the hybrid agents were considered and agents were identified by their mass quantity

of the fluids and solid.

5.2.2 The matrix model

A two-dimensional (2D) cellular automata lattice, consisting of 30 x 60 cells, was

employed to represent the extracellular matrix of the cartilage. Each cell in the lattice

contained one hybrid agent, therefore the quantity of solid and fluid in the cell was

equal to that of its contained agent. The lattice dimensional ratio of 30/60 (0.5)

corresponded to the thickness-to-width ratio of the experimental samples (1.99

4=

0.498) as used in the literature [13]. The initial quantity of solid and fluid in the

hybrid agents located at different rows of the cells in the lattice (layers) were

determined based on experimental data of the layered weight fraction of fluid in

normal human knee articular cartilage from the literature [252] (Figure 5.1). For

example, agents located in the first row cells, which represented the articular

cartilage surface, contained 78% fluid and 22% solid, while the last row of the lattice

cells, which were attached to the subchondral bone contained 56% fluid and 44%

solid.


Figure 5.1 Layered weight fraction distribution of fluid in the normal human knee

articular cartilage based on relative distance from the surface [252].

In order to define boundary conditions, one layer of cells was added to the top,

bottom, left and right side of the articular cartilage lattice. In this simulation,

diffusion was allowed from the top, left and right margins of the articular cartilage

lattice, while the bottom margin was blocked because of the assumed effect of the

subchondral bone that results in this region being impervious [43]. Therefore, cells

added to the bottom contained impervious agents, which could not exchange fluid

with other agents, and cells added to the sides and top contained agents filled with

marked fluid (Figure 5.2), which could exchange fluid with articular cartilage agents.

Due to an inconsiderable amount of fluid in the cartilage matrix in comparison with

the surrounding environment, boundary agents at the top and sides contained 100%

marked agent over the diffusion process. The progression of the time-dependent flow

within the matrix (diffusion) was followed by tracking marked fluid. The simulation

ended when all the initial fluid (unmarked) in the hybrid agents was replaced by

marked fluid. A program in Matlab (Mathworks Inc, MA, USA) was developed to

simulate the diffusion process over the time steps (Appendix B).


5.2.3 Rules

This simulation incorporates a novel concept of a simultaneous combination of

intra-agent and extra-agent responses, where intra-agent and extra-agent rules apply.

The intra-agent rules determine the change within the hybrid agent (intra-agent

evolution) in which the quantity of the contained components of the agent may

change. The extra-agent rules determine extra-agent interactions, e.g. interaction of

an agent with its neighbours in the lattice and rules that apply to the entire lattice.

The following extra- and intra-agent rules were used for free diffusion throughout of

the articular cartilage matrix:

Extra-agent rules:

ER5.1: The 2D von Neumann neighbourhood was implemented for interaction

between neighbours, in which each cell interacts with its orthogonally-adjacent

neighbours [26]. Figure 5.3 shows the von Neumann neighbourhood with Manhattan

distance r=1 in which central agent (agent C) located at cell C can interact with the

agents N, W, S and E, located in the cells at its north, south, east and west. The von

Neumann neighbourhood defines a regular lattice that enables very efficient

visualisations of diffusion processes [253].

Marked fluid cell Impervious cell Articular cartilage cell

Figure 5.2 Schematic illustration of the lattice. Blue, red and yellow show cells

containing agents filled with marked fluid, impervious agents which are blocked to

the fluid, and the articular cartilage agents respectively.


A synchronous or parallel updating method was used, in which all cells

interacted at the same time [254]. To do this, a pseudo-lattice was created in parallel

with the real lattice, where lattice size was equal to the real lattice but cells contained

empty agents (zero quantity of solid, marked and unmarked fluids). At each time

step, all cells in the real lattice were selected as the central cell one-by-one and in

order from left towards right and from top row to bottom of the lattice to guarantee

interaction of all agents in the lattice with their neighbours. Interactions might

change the state of the agents as a consequence of changing quantities of contained

marked and unmarked fluids within the agent. In order to provide equal conditions

for all interactions, any change in the agents’ components due to interaction with the

neighbours including the quantity of gained or lost marked and unmarked fluids, was

recorded in the pseudo-lattice, while agents in the real lattice remained unchanged.

After selecting all agents as central cells, agents in the real lattice were updated by

considering lost or gained marked and unmarked fluids, which were stored at

pseudo-lattice cells. Then, the stored quantities of the components in the pseudo-

lattice cells were changed to zero to prepare the pseudo-lattice for the next time step.

To illustrate this, quantities of the marked and unmarked fluid of the hybrid agent

located at cell EX (yellow cell in Figure 5.4) were determined based on the sum of

following interactions: interactions between En+1, Dn+2, En+2, Fn+2 and EX (En+2 was

N

E C W

S

Figure 5.3 2D von Neumann neighbourhood (r=1). Central agent located at cell C

interacts with agents located at cells East (E), West (W), North (N) and South (S) at

each time step.


the central cell), interactions between Dn+2, Cn+3, Dn+3, EX and Dn+4 (Dn+3 was central

cell), interactions between En+2, Dn+3, EX, Fn+3 and En+4 (EX was central cell),

interactions between Fn+2, EX, Fn+3, Gn+3,Fn+4 (Fn+3 was central cell) and interactions

between EX, Dn+4, En+4, Fn+4 and En+5 (En+4 was central cell). As a result of the

interaction of different central cells with their neighbours, cell EX interacted with the

blue cells in Figure 5.4. It shows the van Neumann neighbourhood with Manhattan

distance 2 (r=2) [255]. The quantity of the gained or lost marked and unmarked fluid

due to the above-mentioned interactions were accumulated in the EX cell in the

pseudo-lattice, where lost was shown by a negative quantity and gain was shown by

a positive quantity. After the last neighbourhood interaction, when the last right cell

in the last row was selected as a central cell, the quantity of the components in the

EX agent at real lattice were changed based on the accumulated quantities in the EX

cell in the pseudo-lattice, by summing quantities. As lost and gained were shown by

negative and positive quantities, the quantity of a component in the cell was

decreased if the cell lost the component more than it gained during interactions with

its neighbours.

Figure 5.4 Interactions of the EX cell with its neighbours in one time step. EX cell

interacts with the blue cells at each time step when all lattice cells were selected to be

central cells one by one.

A B C D E F G H

n

n+1 Cn+1 Dn+1 En+1 Fn+1 Gn+1

n+2 Cn+2 Dn+2 En+2 Fn+2 Gn+2

n+3 Cn+3 Dn+3 EX Fn+3 Gn+3

n+4 Cn+4 Dn+4 En+4 Fn+4 Gn+4

n+5 Cn+5 Dn+5 En+5 Fn+5 Gn+5

n+6

n+7


ER 5.2: The proportion of the marked fluid in the cells located in the top and

sides of the lattice that initially were filled with marked fluid remained unchanged.

Due to the small value of fluid inside the articular cartilage matrix in comparison

with surrounding fluid, it was assumed that diffusion of the fluid out of the articular

cartilage matrix did not change the surrounding fluid concentration if the surrounding

fluid and initial fluid inside articular cartilage were considered as two different

fluids. Therefore, cells located in the top and sides of the lattice contained 100%

marked fluid during the entire simulation.

ER 5.3: The number of articular cartilage cells in the lattice (Figure 5.3) and total

mass of the agents located in the articular cartilage cells did not change during the

diffusion process simulation. As immersing articular cartilage in the fluid causes no

change in the volume and weight of the tissue, articular cartilage agents remained at

steady state.

Intra-agent rules:

IR 5.1: Only marked and unmarked fluids could move in and out of the agent.

Solid content of the hybrid agent was constant over the simulation time. Collagen

fibres and proteoglycan forms the solid skeleton of the articular cartilage where

collagen forms an interwoven meshwork and proteoglycans are trapped inside the

meshwork. As a result, the arrangement of the collagen fibres and trapped

proteoglycans changes only when the cartilage matrix deforms [256]. Immersing the

articular cartilage in a hypotonic environment results in a time-dependent flow within

the matrix. It is assumed that mass change of the model due to the diffusion is very

small and ignorable in comparison with the initial size of the model. Considering

these facts about articular cartilage, solid could not be exchanged between agents and

agents could only exchange fluid.


IR5.2: Size of the agents was constant. Quantities of the solid and fluid (marked

and unmarked combined) in a hybrid agent were constant at all time steps. As it is

assumed that there is no tissue mass change due to the diffusion, replacing resident

fluid in a hybrid agent with the surrounding fluid did not change the mass of the

hybrid agent.

IR5.3: The amount of fluid that moves into a cell equals the amount of fluid that

moves out of the cell. According to IR5.1 and IR5.2, agent mass does not change and

agents could only gain or lose fluid. Therefore, the amount of lost and gained fluid

must be balanced. However, gained and lost fluids may contain unequal proportions

of marked and unmarked fluids, resulting in changing marked and unmarked fluid

proportions after the exchange.

IR5.4: Only certain proportions of contained fluid in an agent could move out as

a consequence of fluid exchange with neighbours at each time step. This rule resulted

from the semi-permeable structure of the articular cartilage [50], which limits fluid

flow in time. The hydraulic permeability of the articular cartilage depends on fluid

and solid content [42, 51, 257], in which the permeability increases with decreasing

solid content and increasing the water content of the tissue. According to Darcy’s

Law, low permeability results from high resistance to fluid flow where a value of

zero permeability would give rise to infinite resistance [243]. The resistance to fluid

flow through a porous medium is highly related to the amount of solid and fluid

present [242]. At one extreme, when there is no solid present, the medium is fully

porous and the resistance to fluid flow is zero. At the other, when the medium

contains fully solid and there is no fluid or pore, the medium is impervious and the

resistance is infinite. Consequently, the resistance to fluid flow through a porous

medium has a direct relation with the ratio of solid value to fluid value.


The proportion of the fluid in the agents that could move out at each time

step, named exchangeable fluid, depended on the resistance to fluid flow through the

agent where the amount of exchangeable fluid has an inverse relation with the

resistance. The resistance is infinite and the amount of movable fluid equals zero

when an agent contains only solid. The resistance is zero and all of the fluid in the

agent can move out if the agent contains 100% fluid and 0% solid. Therefore,

parameter fs, which is defined as the ratio of fluid quantity to solid in the agent, has a

negative correlation with the fluid flow resistance and positive correlation with the

amount of exchangeable fluid. It was used to determine exchangeable fluid of an

agent as follows:

𝐸𝐹 = 𝑘𝑐 𝑓𝑠 𝑓𝑐 (Eq. 5.1)

Where EF is the exchangeable fluid of the agent, kc was a constant value, fs was

the quantity of the marked and unmarked fluid combined, divided by solid quantity

in the agent, and fc was quantity of the fluid resident in the agent, including both

marked and unmarked.

According to the Eq. 5.1, the amount of exchangeable fluid was 𝑘𝑐 𝑓𝑠 of the

resident fluid in the agent. As the quantity of the exchangeable fluid could not be

more than the whole resident fluid in the cell (fc), 𝑘𝑐 𝑓𝑠 was always smaller than or

equal to one (𝑘𝑐 𝑓𝑠 ≤ 1). Therefore, kc must be smaller or equal to the inverse of the

total resident fluid in the agents (𝑘𝑐 ≤1

fs ).

Since the fluid, marked and unmarked combined, and the solid content of each

agent were constant during the diffusion process, kc was a determining factor to

make permeability variable. If hydraulic permeability of the tissue was variable in

time or direction, the constant kc would be variable in time or might depend on an


agent’s interactive neighbourhood. Free diffusion neither changes solid and fluid

contents in the articular cartilage nor deforms the tissue, resulting in a constancy of

hydraulic permeability during the diffusion process [57]. In addition, permeability of

an undeformed tissue is isotropic [83, 93]. Consequently, kc of the agents was time

and direction independent for free diffusion. In this chapter, three different values,

0.1, 0.05 and 0.025, were used for kc to investigate the effect of kc on diffusion.

5.2.4 Corresponding time step to experimental time

Finding a corresponding time step to real time was a challenging task and a

determining factor for the validation of the simulation. Quantitative simulated results

at different time steps were compared with their experimental counterparts in order

to find the corresponding time step that produces the least discrepancy. To this end, a

vertical profile, which presented concentration of the marked agent along articular

cartilage depth, and a horizontal profile, which demonstrated concentration of the

marked fluid from side to side, were calculated at all time steps individually and

were compared with the experimental horizontal and vertical profiles to find the best

match. The vertical profile was calculated as the mean of the marked fluid

concentration of the middle-third cells of the lattice rows from top to bottom of the

lattice (Figure 5.5A). In order to match the number of measured points of the profile

with the experimental, which was 15 points, the mean of every two layers was

considered as one point. The horizontal profile was calculated as the mean of the

marked fluid concentration of the agents in cells located in the columns from left to

right (Figure 5.5B). The first column from the left and the first two columns from the

right (marginal columns) were not included in the profile calculation as marginal

pixels also were discarded in the experimental results [13]. The profile points were


calculated as an average for three columns to produce 19 points, corresponding with

the experimental horizontal profile.

A B

Figure 5.5 A: Vertical profile points. The mean marked fluid concentrations of the

layers in the middle-third, shown by the dash line, were used to calculate profile

points. B: Horizontal profile points.

The concentration of the contrast agent in the literature [13] was based on mM,

and diffusion took over 12 hours where there was no change in concentration after 24

hours. In order to make it comparable with the simulation results, concentrations

relative to the maximum concentration were calculated. To do this, the

concentrations of the contrast agent profiles at each point were divided by

corresponding maximum concentration (24 hours) to calculate relative contrast agent

concentration profiles. For example, relative concentration equals one indicated

replacement of total fluid by the radioactive fluid.

The deviation between simulated and actual (reference) data was calculated by

root mean square error (RMSE), which was calculated as the square root of the mean

of the squares of the deviations [258-260] (Eq. 5.2). In order to facilitate comparison


between data sets with different scales, RMSE was normalized. RMSE normalized to

the mean of the measured data, is called the coefficient of variation of the root mean

square error (CV(RMSE)) [261-263] (Eq. 5.3). In this chapter, experimental data

from the literature [13] was used as a reference and since vertical and horizontal

profiles consisted of 15 and 19 measurement points respectively, CV(RMSE) was

used.

𝑅𝑀𝑆𝐸 = √∑ (𝑆𝐶𝑘−𝐸𝐶𝑘)2𝑛

𝑘=1

n (Eq. 5.2)

CV(RMSE) =

√∑ (𝑆𝐶𝑘−𝐸𝐶𝑘)2𝑛

𝑘=1n

MSC (Eq. 5.3)

Here, SCk is the simulated relative concentration of the marked fluid at point k,

ECk is the relative concentration of the contrast agent at point k extracted from

experimental results in the literature [13], n is the total number of points of the

profile and MSC was the mean of the simulated relative concentration of the marked

fluid at the profile points.

CV(RMSE) was calculated for the horizontal and vertical profiles at all time

steps using 2, 4 and 6 hours experimental results. The total simulation error of the

profile was defined as the sum of profile errors at time steps corresponded to 2, 4 and

6 hours, calculated as follows:

𝑇𝑃𝐸𝑖 = 𝐶𝑉(𝑅𝑀𝑆𝐸)𝑖,2ℎ + 𝐶𝑉(𝑅𝑀𝑆𝐸)2𝑖,4ℎ + 𝐶𝑉(𝑅𝑀𝑆𝐸)3𝑖,6ℎ (Eq. 5.4)

Here, i is the simulation time step corresponding to two hours, TPEi is the total error

of the profile when i time steps is equal to 2 hours, CV(RMSE)i,2h , CV(RMSE)2i,4h

and CV(RMSE)3i,6h are CV(RMSE) at time steps i (corresponding to 2 hours), 2i


(two times greater than time steps i), and 3i (three times greater than time steps i)

respectively.

The sum of total errors of the vertical and horizontal profiles created a total

error of the simulation for a selected corresponding time step. The total error was

calculated as follows:

𝑇𝑆𝐸𝑖 = 𝑇𝐻𝐸𝑖 + 𝑇𝑉𝐸𝑖 (Eq. 5.5)

Where TSEi was the total error when time step i was equal to 2 hours, THEi and

TVEi are horizontal and vertical profile errors when time step i was equal to 2 hours

respectively.

The total error was calculated for different time steps as corresponding to 2 hours

real time. The least total error determined the best time step corresponding to 2

hours.

5.2.5 Simulation of the degenerated articular cartilage

Solid content decreases in the final stage of the degeneration of articular

cartilage where reduced quantity is replaced by fluid, resulting in an increasing fluid

content in the matrix [264]. In this chapter, it is assumed that the increase of fluid

content results from loss of the matrix. Therefore, the solid and fluid contents of the

healthy cartilage lattice (Section 5.2.2) were modified at degenerated regions to

simulate full and partial degenerated tissues. In order to develop the partially

degenerated model, a 70% of the solid content of the agents located at three cleft-like

regions, according to Figure 5.6, was replaced by fluid. The width of each


degenerated region was two cells, located at 1

12 ,

1

2 and

3

4 total width (60 cells) from

the left with half, full and two-third depths of the thickness respectively (Figure 5.6).

In the full degenerated model, the solid contents of all agents in the lattice were

decreased by 70%. The same extra- and intra-agent rules as healthy articular cartilage

(Section 5.2.3) were used and corresponding time steps for real time were based on

corresponding time steps for the healthy tissue. As the fluid content of the

degenerated agents was increased, their fs were also increased. The region near the

articular cartilage surface carried a maximum value of the fs where the healthy tissue

contained approximately 80% fluid and 20% solid (fs=4). Fluid and solid contents

were changed to 94% and 6% respectively, by removing 70% of the solid and

replacing it with fluid. Therefore, fs dropped from approximately 4 to 15.67. The

value of kc equalled 0.025 for the degenerated model simulations to ensure 𝑘𝑐 𝑓𝑠 was

always less than one.

Figure 5.6 Partial degeneration of the articular cartilage. Red and blue show

degenerated and healthy regions.


5.3 RESULTS

5.3.1 Corresponding simulation time step to real time

Figures 5.7 shows the CV(RMSE) for horizontal and vertical profiles of

simulation based on time steps when kc=0.1. Simulated horizontal profile error

reached minimum discrepancy with 2, 4 and 6 hours experimental results [13] at 794,

1336 and 1842 time steps respectively (figure 5.7A). Consequently, the

corresponding time step for 2 hours experimental time based on minimum

CV(RMSE) of 2, 4 and 6 hours, equals 794, 668 and 614 respectively. The minimum

CV(RMSE) of the simulated vertical profile based on 2, 4 and 6 hours experimental

diffusion time occurred at 988, 1622 and 2203 time steps (figure 5.7B), which meant

2 hours was corresponding to 988, 811 and 734 time steps respectively. The range of

time steps equivalent to 2 h experimental time was from 614, based on horizontal

profile at 6 h, to 988, based on vertical profile at 2 h.

A B

Figure 5.7 Discrepancy between simulated and experimental results based on

CV(RMSE) and time steps for the horizontal (A) and vertical (B) profiles.

Figure 5.8 shows the simulated horizontal, vertical and total error based on a

time step corresponding to 2 hours. The time steps that generated the minimum error

for the horizontal and vertical profiles were equal to 710 and 856. The errors sharply


decreased until the minimum, then slightly increased. Total error decreased

considerably until time step 810, where it reached to the minimum, then moderately

increased. Since the minimum total error (810 time steps) was corresponding to 2

hours, each time step was equal to 8.889 seconds when kc was equal to 0.1.

Figure 5.8 Horizontal, vertical and total error based on time step corresponding to 2

hours.

5.3.2 Effect of kc on results

Three values, 0.1, 0.05 and 0.025, were chosen for kc to investigate the effect

of kc on the error and the simulation time step corresponding to the real time. Table

5.1 shows total error, time steps that provided the least total error, minimum

CV(RMSE) for the simulation results compared to 2, 4 and 6 hours horizontal and

vertical profiles, and their corresponding time steps for kc chosen values. The value

of kc practically had no effect on the CV(RMSE)s and total error. All time steps

related to kc equals 0.05 were two times greater than kc equals 0.1. In the same way,

time steps were double when kc changed from 0.05 to 0.025. Therefore, the value of

kc did not change the value of the generated error, while time steps were changed

corresponding to the inverse value of the kc.


Table 5.1 Total, vertical and horizontal errors and corresponding time steps to 2, 4

and 6 hours diffusion time for different values of kc.

Kc v

alue

To

tal error

CV(RMSE)

Best tim

e step (eq

ual 2

h)

Time step

Min

imu

m h

orizo

ntal (2

h) %

Min

imu

m h

orizo

ntal (4

h) %

Min

imu

m h

orizo

ntal (6

h) %

Min

imu

m v

ertical (2h

) %

Min

imu

m v

ertical (4h

) %

Min

imu

m v

ertical (6h

) %

ho

rizon

tal (2h

)

ho

rizon

tal (4h

)

ho

rizon

tal (6h

)

vertical (2

h)

vertical (4

h)

vertical (6

h)

0.1 0.604 11.2 7.5 5.8 8.7 5.3 3.8 810 816 1354 1829 999 1628 2207

0.05 0.604 11.2 7.5 5.8 8.7 5.3 3.8 1821 1632 2709 3720 1998 3256 4414

0.025 0.604 11.2 7.5 5.8 8.7 5.3 3.8 3242 3264 5419 7440 3996 6512 8829

5.3.3 Diffusion spatial maps for healthy articular cartilage

The diffusion patterns of marked agents into the lattice for kc equals 0.1 at time

steps equal 810, 1620 and 2430, corresponding to 2, 4 and 6 hours experimental

diffusion time, were presented in Figure 5.9A. The colour-coded maps show the

spatial distribution of the marked fluid concentration in the lattice based on the mass

percentage (mass of the marked fluid over total fluid mass) at a given time step. Each

colour represented a certain percentage of the marked fluid concentration as

indicated in the legend attached to the pictures. Red depicts regions containing 100%

marked fluid, while blue depicts areas with very little marked fluid. When all initial

fluid in an agent was replaced with marked fluid, the concentration of marked fluid

in the agent was equal to 100% and is shown in red. Similarly, 50% concentration

meant unmarked fluid (initial fluid) and marked fluid had equal proportions in the

agent (shown in yellow).


Initially (at T=0), concentration of the marked fluid in the lattice was zero (not

shown in the figure). Then the marked fluid percolated into the lattice, resulting in an

increased proportion of marked fluid over time (T=810, 1620 and 2430). Images at

T=1620 and 2430 indicated a more significant increase in the accumulation of the

marked fluid in the regions near the surface rather than near the sides. It also showed

that diffusion from the surface along the tissue depth was considerably higher than

from the sides towards the inside of the tissue.

Figure 5.9 Diffusion into human articular cartilage at different times. A: Percentage

of marked fluid in the lattice at time steps 810, 1620 and 2430 B: Contrast agent

diffusion after 2, 4 and 6 hours immersion [13] (reprinted from Osteoarthritis and

Cartilage, vol. 17, T.S Silvast, J.S. Jurvelin, M.J. Lammi, J. Töyräs, pQCT study on

diffusion and equilibrium distribution of iodinated anionic contrast agent in human

articular cartilage – associations to matrix composition and integrity, pp. 26-32,

Copyright (2009), with permission from Elsevier (Appendix F)).

A

T=821 T=1620 T= 2430

B

Percentage of contrast agent concentration Contrast agent concentration (mM)


Figure 5.9B shows experimental results of contrast agent diffusion into human

articular cartilage [13] at time points 2, 4 and 6 hours (left to right), corresponding to

time steps in Figure 5.9A. The legend on the bottom right shows contrast agent

concentration based on mM, in which red illustrated maximum concentration (15

mM) and light blue demonstrated zero concentration. In order to compare the

experimental with the simulated data, the percentage of the contrast agent

concentration (left legend) was calculated based on the ratio of contrast agent

concentration to maximum concentration in which 15 mM was equal to 100%, and

10 and 5 mM were equal to 67% and 33% respectively.

Comparison between CA and experimental diffusion colour-coded maps

demonstrated symmetrical patterns of diffusion into the cartilage. At simulated 810

time step and its experimental corresponding time (2 hours), the concentration in the

area near the tissue surface was high and fluid could not penetrate very deeply during

this time. Red colour depth was approximately one-third of the thickness from the

surface in both pictures. At T=1620 (4 hours) the concentration of marked fluid in

the simulated picture and contrast agent in the experimental image were increased

significantly up to the centre of the lattice (articular cartilage) along its depth, while

at T=2430, which is equal to 6 hours, only the region close to the bone did not

undergo a significant concentration change.

Marked fluid in the simulation and contrast agent in the experimental test

were diffused into the articular matrix and penetrated along the depth of the tissue.

The simulated depth-dependent diffused fluid concentration profiles (in percentage)

when kc=0.1 at 810, 1620 and 2430 time steps, and their experimental counterparts at

two, four and six hours [13] are shown in Figure 5.10, in which the average

concentration of the diffused fluid of the lattice rows or layers determined profile


points. The CV(RMSE) were equal to 17.3%, 5.2% and 5.9% for 810, 1620 and 2430

time steps respectively (Table 5.2). According to the literature that the experimental

data was taken from [13], the immersed articular cartilage reached equilibrium after

24 hours, when change in contrast agent concentration was insignificant afterwards.

The experimental profiles indicated the average concentrations of the diffused fluid

along tissue depth at various times, based on the maximum concentration that were

reached at equilibrium time (here 24 hours).

Figure 5.10 Percentage of depth concentration of marked agent at T=810, 1620 and

2430 and contrast agent in the human knee articular cartilage after 2, 4 and 6 hours

of immersion [13].

Simulated profiles at all selected times followed the same trend of the

experimental results where the concentrations of the marked fluid were

approximately 85%, 92% and 95% at time steps 810, 1620 and 2430 respectively at

the surface. The concentrations of the marked fluid then dropped gradually with

depth up to approximately 12 %, 43% and 67% at time steps 810, 1620 and 2430

respectively. The largest discrepancy between simulated and experimental results


occurred at the region close to the bone at 810 time steps, corresponding to 2 hours,

where the simulated concentration of the diffused fluid (mark fluid) was about half of

the experimental. However, there was not a considerable discrepancy between

experimental and simulated results from the surface till almost half thickness depth at

810 time steps.

Table 5.2 showed that CV(RMSE) of the both vertical and horizontal profiles

decreased in time. Vertical profile at 810 time steps had also the highest CV(RMSE)

among the vertical and horizontal profiles at various time steps where it was

approximately 70% greater than its counterpart in the horizontal profile (18.1%

versus 11.2%). The high error of the vertical profile at time step 810 dropped to 5.3%

after 1620 time steps, corresponding to 4 hours. While CV(RMSE) of the horizontal

profiles slightly decreased from 11.2% at time step 810, corresponding to 2 hours to

8.9% at time step 3240 (corresponding to 8 hours), the error of the vertical profile

remarkably dropped from 18.1% at time step 810 to 4.3% at time step 3240. Despite

the relatively high error at the vertical profile at time step 810, the simulated results

compared reasonably well with experimental data, which substantiated the validity of

the results of the CA simulation.

Table 5.2 CV(RMSE) of horizontal and vertical profiles corresponding to 2, 4 , 6

and 8 hours of immersion.

Vertical p

rofile

2 h

ours

Vertical p

rofile

4 h

ours

Vertical p

rofile

6 h

ours

Vertical p

rofile

8 h

ours

Horizo

ntal p

rofile

2 h

ours

Horizo

ntal p

rofile

4 h

ours

Horizo

ntal p

rofile

6 h

ours

Horizo

ntal p

rofile

8 h

ours

Corresponding

time step

810 1620 2430 3240 810 1620 2430 3240

CV(RMSE) % 18.1 5.3 5.8 4.3 11.2 10.0 9.9 8.9


Figure 5.11 shows depth-dependent bulk concentration percentage of marked

fluid collected at various areas in depth (layers) including surface, 1/3 thickness depth,

middle (½ thickness depth), 2/3 thickness depth and bottom. Overall, the

concentrations were lower towards the bottom regions (close to the bone) in time.

The curve representing concentration at the surface illustrated that unmarked fluid

was replaced by marked fluid rapidly and after about 500 time steps, over 90% of the

unmarked fluid moved out of this region. The profile of concentration at the bottom

layer followed different trends over time and took a significantly longer time to

replace the majority of initial fluid with marked fluid. All curves demonstrated

growth of marked fluid over time while the rate of increase over time dropped

considerably with depth.

Figure 5.11 Depth- and time-dependent profiles of marked fluid concentration for the

surface, bottom, 1/3, ½ and ²/3 thickness depth.


5.3.4 Diffusion patterns of the degenerated articular cartilage

As the same rules and lattice size as healthy articular cartilage were used for

the degenerated model, the corresponding time step to the real time for the

degenerated articular cartilage was chosen equal to the healthy matrix. Therefore,

according to Table 5.1 and selected value of kc for the degenerated models’

simulations (kc = 0.025), 3264 time step corresponded to 2 hours or one time step

was equivalent to 2.2 seconds. Figure 5.12 shows the concentration of the marked

fluid of the healthy (5.12A), cleft-like partial degeneration (5.12B) and complete

degeneration (5.12C) at 1632 and 3264 time steps (corresponding to one and two

hours immersion respectively). The legend attached to the pictures shows

concentration of the marked fluid based on the percentage of the total fluid. The

image at 1632 time step illustrates that the concentration of the marked fluid was

higher near the surface and decreased gradually in depth towards the bottom. In

addition, fluid concentrations at bottom left and right sides were greater than 80%

and this decreased to less than 10% at the bottom middle of the lattice at T=1632.

Marked fluid diffused progressively towards the bottom and the centre of the lattice

in time where concentration of the marked fluid reached approximately 20% at the

bottom centre at T=3264. Partial degeneration (Figure 5.12B) caused marked fluid

concentration to increase in degenerated regions. According to images at T=1632 and

T=3264 in Figure 5.12B, concentration of the marked fluid at the middle, cleft-like

degenerated regions was approximately two times greater than concentrations at

nearby healthy areas. Faster penetration of the marked fluid via degenerated regions

resulted in higher marked fluid concentration in the entire lattice, particularly at the

bottom, in comparison with the healthy model, where areas around degenerated

regions contained more marked fluid than their counterparts in the healthy model. It


also shows that the deeper degenerated cleft-like region had more influence on the

diffusion spatial map.

A

T= 1632 T= 3264

B

T= 1632 T= 3264

C

T= 1632 T= 3264

Figure 5.12 Simulated concentration of the marked fluid at time steps 1632 and

3264, corresponding to 1 and 2 hours respectively (kc=0.025) for the healthy

articular cartilage model (A), the partially degenerated model (B) and 70% solid

resorption (C) based on percentage.


Figure 5.12C showed that diffusion was much faster in the full degenerated

model than the healthy and partially degenerated models. At T=1632, the

concentration of the marked fluid was about 50% in the small area located at the

centre bottom of the lattice, while the concentration of the rest of the matrix model

was above 70%. The diffusion process was almost completed after 3264 time steps.

Comparing Figures 5.12A, B and C indicates that diffusion throughout the fully

degenerated matrix was the fastest, followed by the partial degenerated and healthy

models. Therefore, degeneration increased diffusion rate and reduced the time of the

diffusion process (time to equilibrium).

5.4 DISCUSSION

The aim of this chapter was to provide a temporal and spatial map of diffusion

throughout the healthy and degenerated articular cartilage models. The 2D CA model

of the human knee articular cartilage was developed based on information of tissue

composition, including fluid distribution, extracted from literature [252]. A hybrid

agent, as presented in Chapter 3, was adopted for articular cartilage as a single-phase,

porous, saturated material. The free diffusion of surrounding fluid throughout the

cartilage matrix was simulated for the healthy, partially solid skeleton degraded and

fully degraded articular cartilage, where spatial maps of the diffusion at different

times were prepared. The simulation results of the healthy articular cartilage were

verified against experimental data from the literature [13] where tracking and

imaging of a radioactive tracer could show a diffusion map of the articular cartilage,

as transport of the radioactive tracer into the articular cartilage happens largely by

diffusion [14].


There were a number of simplifications made in developing the CA model.

One overall simplification was that articular cartilage had two constituents: solid and

fluid where proteoglycans and collagens were not considered as two individual

components. They both formed the solid skeleton and solid content of the agent.

Another simplification related to the distribution of articular cartilage components.

The lattice that represented cartilage matrix was developed based on a layered

distribution of the solid and fluid. Therefore, lateral-medial compositions were

assumed to be constant and only superior-inferior composition change was taken into

account.

The CA simulation time was based on discrete time steps. Mathematical analysis

was used to find the time step corresponding to real diffusion time that created the

minimum discrepancy between experimental and simulated results. The horizontal

and vertical profiles of the diffusion of a contrast agent into the articular cartilage at

2, 4 and 6 hours [13] were selected as reference times to calculate error of any

corresponded time step to the real time. The number of measurement points of the

horizontal profile was greater than vertical profile (19 versus 15). The total error of

the simulation consisted of deviation of the simulated results from both horizontal

and vertical profiles. If the sum of least squares was used to calculate total error

[265], it gave the horizontal profile more weight than vertical profiles, due to their

unequal measurement points. This limitation could be removed by RMSE, as least

squares were divided by the number of measurement points. However, RSME gave

less weight to errors with smaller absolute values than errors with larger absolute

values [266]. Since the simulated profiles may have various measurement ranges,

RMSE could create unequal allocated weight to the profiles. To overcome this

limitation, RMSE normalised to the mean (CV(RMSE)) was used to calculate errors


of simulated profiles. The total error, which included CV(RMSE) of the horizontal

and vertical profiles at different times, was considered to find the corresponding time

step to the real experimental time. The CV(RMSE) of the simulated results showed

satisfactory results with CV(RMSE) ranging from 18.1% to 4.3% for the vertical and

horizontal profiles at different times (Table 5.2).

Change of the diffused fluid (marked fluid) in the lattice in time indicated the rate

of diffusion through the model of the articular cartilage, where faster change

demonstrated a higher rate of the diffusion. According to the simulation results in

Figure 5.10, diffusion of the fluid through the area close to the surface of healthy

articular cartilage was faster than through the area near the bottom (deep cartilage).

As with previous studies, which found that the diffusion coefficient that controls

diffusion rate based on Fick’s law [267], decreased with depth, and it correlated

positively with the fluid content and negatively with solid content, in particular,

proteoglycan content [14, 15, 268-271].

Spontaneous degeneration of the articular cartilage causes destruction of the

cartilage matrix solid skeleton including proteoglycan and collagen [272-274]. It

results in solid skeleton resorption and increasing fluid content in the cartilage matrix

[274-276]. In order to simulate degenerated articular cartilage, the healthy CA model

was modified where a percentage of solid content of the agents was replaced by

fluid. Therefore, agents that represented degenerated tissue contained less solid and

more fluid that their counterparts in the healthy lattice. The validated rules for the

healthy tissue were used to simulate diffusion throughout the degenerated tissue.

The degenerative structural changes affect articular cartilage functions. It has been

showed experimentally that the diffusion of solutes increased with articular

degeneration [15, 270, 277]. The diffusive transport also became faster after


enzymatically degraded articular cartilage, which only digests proteoglycan [270,

278, 279]. Along with the previous studies, the results of simulation demonstrated

much higher diffused fluid concentration in the lattice for the fully degenerated

lattice at all given time steps, in comparison with the healthy model (Figures 5.12A

and B). Simulated results in Figure 5.12 also show that diffusion of fluid throughout

the degenerated regions was faster than in their healthy counterparts. As the

degenerated hybrid agent contained more fluid and less solid than its non-

degenerated counterpart, this result was in agreement with the previous studies,

which confirmed the positive correlation of diffusion rate with the fluid content [14].

Computational methods, such as finite element analysis (FEA), are based on

physical laws, such as Fick’s laws of diffusion [280], and algebraic equations where

experimental data was used to calculate required parameters such as diffusion

coefficient [14, 281-283]. Current differential equation-based computational methods

are therefore dependent on the experimental techniques to determine their required

parameters. The diffusion coefficient could be determined experimentally on macro-

scale (on bulk) using radioactive or fluorescent tracer tracking, sectioning and

imaging the tissue [270, 271, 284, 285], in which the diffusion coefficient was

extracted from a spatial map of the tracer movement into the tissue. The required

diffusion spatial map of the tracer also could be obtained by non-destructive

methods, such as MRI and CT scan [14, 278]. The presented approach in this chapter

could provide a spatial map of diffusion (Figures 5.9 and 5.12) and diffusion

statistical data (Figures 5.10 and 5.11) at any given time. The agreement of the CA

simulation results with the experimental data confirmed validity and applicability of

the approach, including adopted hybrid agent, intra- and extra-agent rules, to


simulate the articular cartilage function. Consequently, this CA simulation could

provide results that only experimental techniques were able to obtain.

On the other hand, movement of the tracer into the tissue greatly depends on the

charge and size of the tracer molecules [14, 55, 284], agitation of the fluid [43], and

temperature [286], resulting in different spatial and temporal diffusion maps for

different tracers and environmental conditions. In this simulation, constant kc could

also control the number of time steps corresponding to the real (experimental)

diffusion time (Section 5.3.2). Different values of kc resulted in different real time

equivalent for one time step, where greater values for the kc made equilibrium time

shorter. The dependence of the corresponding time step to kc made the model

suitable for simulation of various tracers and conditions, where different values could

be allocated to kc.

Chapter 6: Deformation of the articular cartilage 103

Chapter 6: Deformation of the articular

cartilage

6.1 INTRODUCTION

Articular cartilage deforms under compressive loading where the fluid inside

the cartilage’s matrix is exuded over period of loading and deformation [56]. The

rate of outflow is determined by the permeability, which is one of the major

physiological characteristics of the cartilage [42]. The fluid flow and diffusion of

solutes are the dominant mechanisms of load support in the articular cartilage [42,

85]. The characteristic change in volume also leads to time-dependent pore size

changes with concomitant decrease in related fluid flow and the average permeability

[42]. Earlier investigation of joint lubrication indicated that articular cartilage might

exhibit consolidation-type deformation [60], where fluid is gradually moved out from

saturated structure of the tissue without replacement of exuded fluid by air.

Despite many studies about unloaded articular cartilage architecture, at present,

little quantitative data is available about the internal morphologic reactions of the

tissue under loading [287]. Study of consolidation of the articular cartilage is

beneficial in order to determine the fundamental mechanical properties of the tissue

[133]. In this chapter, therefore, the responses of articular cartilage in a confined

compression test were studied, using principles of cellular automaton (CA), hybrid

agent, intra- and extra-agent rules. The model and boundary conditions were defined

in such a way as to provide strain comparison with the experimental data from the

literature [288]. The spatio-temporal patterns of fluid and solid were a primary

consideration in this chapter.

104 Chapter 6: Deformation of the articular cartilage


6.2.1 Overview of the axial loading of confined articular cartilage

Loading scenario 1 (Figure 6.1A) shows one-dimensional (1D) consolidation,

where a piece of articular cartilage is tightly bounded by solid impermeable walls

laterally and at the base [64, 289]. The top surface of the articular cartilage is

subjected to a compressive axial static load via a uniform porous indenter. Fluid

exudes from the top surface of the tissue in the loading scenario 1, while the solid

walls restrict fluid exudation from the bottom and sides. As the articular cartilage is

confined laterally, strain develops only along the loading axis. Therefore, the

geometry ensures axial (one-dimensional) deformation of the tissue [290]. In the

loading scenario 2 (Figure 6.1B) which is an indentation test, the axial static load

was applied to the surface of the tissue via an impervious indenter with the width of

one-third of the articular cartilage width. As the solid walls in the sides and bottom,

and the impervious indenter restrict fluid exudation, fluid can flow out from the

unloaded surface of the tissue.

Porous Load

indenter Load Impervious

indenter

Articular cartilage Articular cartilage

Solid wall Solid wall

A B

Figure 6.1 Loading and boundary conditions for predicting consolidation response

of confined articular cartilage. A: Loading via a porous indenter (loading scenario 1).

B: Loading via an impervious indenter (loading scenario 2).


6.2.2 Adaptation of the hybrid agent for deformation of the articular cartilage

Adapted hybrid agent contained indistinguishable fluid and solid as inseparable

fluid and solid were assumed to be major constituents of the articular cartilage. The

quantities of the fluid and solid in the hybrid agent are indicated by the fluid and

solid volumes assigned to the agent. During the process of deformation, the indenter

progressively moves into space originally occupied by the tissue. To facilitate

progressive volume change of the lattice, the agents located in the surface (first layer

of the lattice) may contain fluid, solid and a ‘dead space’. This dead space can be

filled by the impervious indenter or empty space. For example, when the tissue is

compressing via an impervious indenter, the dead space may be created in the agents,

which are located right below the indenter where the indenter fills the dead space of

the agents. In the case of indentation via a porous indenter, dead space is filled with

empty space in which, as the indentation continues, the volume of the dead space

(here empty space) is increased.

Figure 6.2 shows an adapted hybrid agent for deformation where dead space

can be identified from inseparable fluid and solid. At the beginning of the process, all

representative agents for the articular cartilage contain zero dead space while agents

located in the surface of the articular cartilage matrix may contain dead space during

the deformation process. As this dead space was just used to compensate reduced

volumes of the fluid and solid in the under-sized agents, it has no impact on the agent

performance.

Dead space

Indistinguishable fluid and solid

Figure 6.2 Hybrid agent containing indistinguishable fluid and solid and separable

dead quantity (empty space).


6.2.3 Articular cartilage model

A two-dimensional (2D) cellular automata (CA) lattice of cells, consisting of 32 x

36 cells, was employed to represent articular cartilage, solid walls and the indenter

(Figure 6.3). Agents representing extracellular matrix of the articular cartilage were

depicted in yellow cells (30 x 30 cells). The articular cartilage agents were

surrounded by impervious agents at the bottom, and left and right sides, depicted in

the brown cells. The lattice consisted of five rows of empty cells at the top (blue

cells). All cells and agents have equal volume (size).

Cell1,1 Cell1,2 Celli,j Cell2,1 5 cells Cell2,2 30 cells 1 cell

1 cell 30 cells 1 cell

Figure 6.3 Initial arrangements of the lattice cells. Impervious agents were located in

brown cells. Blue were empty cells. Articular cartilage agents were located at yellow

cells.

The initial distribution of fluid and solid in the articular cartilage agents were

determined, based on the known layered weight distribution of fluid and solid in the

bovine knee articular cartilage [291]. Figure 6.4A shows the layered distribution of


the fluid, where the layers near the surface of the tissue contain maximum fluid

(approximately 85% of the total mass) and layers close to subchondral bone

contained the minimum fluid (approximately 70% of the total mass).

The volume of an object is equal to its mass divided by density. According to the

literature [292], densities of the solid skeleton of the bovine articular cartilage and

fluid are equal to 1.323 g/mm2 and 1.0 g/mm

2 respectively. The volume fractions of

the fluid in the layers were calculated by means of the layered distribution of the

mass fraction of the fluid [291], and fluid and solid densities. Figure 6.4B shows

volume fraction of fluid from the surface to the bone in the healthy bovine knee

cartilage.

A B

6.2.4 Loading scenario

The tissue model was subjected to two loading scenarios to reach 37% strain.

The lattice consisted of empty cells at the top (blue cells) in loading scenario 1

(Figure 6.5A) to represent a porous indenter where empty cells allowed fluid to flow

through them. The deformation of the matrix occurred by moving empty cells into

Figure 6.4 Layered distribution of the mass fraction of the fluid (A) [290] and volume

fraction of the fluid (B) in the normal bovine articular cartilage based on relative

distance from the surface of the tissue.


the lattice. Empty cells were expanded during the deformation process via filling the

dead space of the agents located in the surface. If the empty space occupies the total

volume of the agent, the agent is removed from the lattice and the cell is changed to

an empty cell.

The lattice in loading scenario 2 included 10 cells-width impervious agents

on the top centre of the articular cartilage matrix cells and empty cells on their left

and right (red and blue cells in Figure 6.5B). The impervious agents progressively

moved into the articular cartilage matrix (yellow cells) where the dead space of the

agents below these impervious agents was gradually filled by the agents’ impervious

property over the deformation process. The empty cells on the top also allowed fluid

to move out of the matrix.

The entire matrix is directly under the load in loading scenario 1, therefore all

articular cartilage agents were loaded agents. In loading scenario 2, only agents

under the indenter were loaded agents as they are directly under the load. All agents

located in yellow cells in Figure 6.5A are loaded agents, while only agents inside the

thick line rectangle are loaded ones in Figure 6.5B.

A B

Figure 6.5 Initial arrangements of the lattice cells in loading scenario 1 (A) and

loading scenario 2 (B). Impervious agents were located in the red cells. Blue were

empty cells. Articular cartilage agents were located at yellow cells.


6.2.5 Rules

Deformation of the articular cartilage results in movement of both fluid and solid

skeleton in the loaded tissue. The fluid was assumed to be incompressible and elastic

deformation of the solid skeleton was also assumed to be ignorable in comparison

with exuded fluid volume. The hybrid agent, intra- and extra-agent rules were used to

facilitate the consolidation of the lattice representing articular cartilage, in which the

sum of the volumes of fluid, solid and dead space (if there was any in the agent)

determined the size of the agent. The following extra- and intra-agent rules were

used:

Extra-agent rules: Extra-agent rules were categorised into two groups: (i)

Independent extra-agent rules that were applied to the lattice without considering

intra-agent rules, (ii) dependent extra-agent rules which were in conjunction with

intra-agent rules. The following independent extra-agent rules were used in this

study:

ER 6.1: The volume of the lattice deformation (bulk deformation) was equal to the

volume of the exuded fluid. This rule was extracted from incompressibility of the

fluid inside the articular cartilage, and ignoring the elastic deformation of the solid

skeleton during consolidation. The strain of the lattice (bulk strain) was calculated as

follows:

Strain =VD

V0 (Eq. 6.1)

Where V0 is the sum of the loaded agents’ volumes before deformation of the matrix,

and VD is exuded fluid volume at any given time.

ER 6.2: 2D Margolus neighbourhood [26, 178] was implemented for

interaction between neighbours. The lattice was divided into non-overlapping

partitions (block) where the nearest four cells made one block. The boundaries of


blocks change at odd and even time steps as demonstrated at Figure 6.6. Each block

moves one cell to the right and down at even time steps and then moves back at odd

steps. All cells within a block interact with each other at each time step. As blocks do

not have any overlap in a Margolus neighbourhood, the priority of selecting blocks

computationally for interaction does not influence the results. Therefore, once the

lattice is divided into non-overlapping blocks, according to the Margolus

neighbourhood, regardless of which interaction of whichever block was taken into

account first, different interaction permutations have the same results as long as

interaction in all blocks were considered. Therefore, the Margolus neighbourhood

creates a synchronous method of updating.

Cell1,1, cell1,2, cell2,1 and cell2,2 in Figure 6.3 are in one block at the even time

steps. Both indicates i and j of the celli,j in the upper left cell of the blocks are even at

even time steps while they are odd at odd time steps.

Figure 6.6 2D Margolus neighbourhood. The cells partitions alternate between

blocks indicated by solid lines, and dashed lines at odd and even steps respectively.

ER 6.3: Only fluid could be exchanged between unloaded agents. As elastic

deformation of the unloaded area is ignorable during consolidation-type indentation

of the articular cartilage [59], the exchange between unloaded agents was limited to

the fluid.


ER 6.4: Impervious agents could not exchange solid or fluid with their

neighbours.

ER 6.5: Empty cells could accept both fluid and solid.

ER 6.6: Exuded fluid could move to an empty cell. If an empty cell contained

only fluid without any solid at a time step, it lost the fluid and became empty cell

again for the next time step.

Intra-agent rules: Intra-agent rules (intra rules) controlled changes within the agent.

They determine how an intra-agent environment evolved during the deformation

process. The location of the interactive neighbours (agents in the same block) may be

taken into account for an intra rule. Following are the intra rules:

IR 6.1: Sizes of all agents in the lattice were equal at each time step. Sizes of the

agents were also equal to the volumes of agents at the previous time step. This rule

was similar to the rule IR 5.2 (Section 5.2.3).

IR 6.2: A certain proportion of the initial fluid of the loaded agents moved out

during the consolidation process. The quantity of the fluid that moved out from the

agent depended on the initial quantity of fluid and solid in the agent. This rule was

derived from the dependency of the exuded fluid from a loaded tissue to its solid

content concentration.

The articular cartilage under static loading loses fluid to reach osmotic

equilibrium [48, 49, 63, 64, 161], which the value of the osmotic pressure directly

correlates with FCD in the tissue [75, 293]. Previous studies proved that the value of

FCD agrees closely with the concentration of the proteoglycans [96, 115]. As solid

included proteoglycans, solid concentration has a positive correlation with FCD. The

loaded tissue loses fluid until the FCD increases sufficiently for osmotic equilibrium

[46]. A loaded articular cartilage with a low value of FCD, such as degenerated


tissue, needs to lose more fluid to reach equilibrium than a tissue with a higher value

of FCD, such as a healthy tissue under the same load [63]. Therefore, fluid lost due

to loading has a positive correlation with the proportion of the fluid in the tissue.

A proportion of the initial resident fluid in the loaded agent would exude from the

agent at the end of the deformation process in order for the agent to reach osmotic

equilibrium again. Accordingly, fluid at each loaded agent was divided into two

parts: remaining, which could not leave the lattice, and movable, which could move

out of the lattice. The maximum quantity of the movable fluids in the agents was at

the beginning of the deformation process while it was equal to zero at the end of the

process. The agents that contained greater proportions of the fluid lost more fluid.

Therefore, the quantity of the movable fluid in the hybrid agent had a negative

correlation with the solid concentration in the agent and was calculated as:

𝐸𝐹𝑖 =𝑇𝐸𝐹∗ 𝑠𝑐𝑖

∑ (𝑠𝑐𝑖)𝑛𝑘=0

(Eq. 6.2)

Here, sci is solid concentration in the agent i (solid quantity divided by fluid and

solid quantities combined), and EFi is total quantity of the exuded fluid from agent i,

TEF is total quantity of the exuded fluid from the matrix during entire deformation

and n is total number of loaded agents in the lattice.

IR 6.3: Only a certain proportion of the existing movable fluid in the agent was

able to move out of the agent at each time step. This rule was similar to the rule IR

5.4 (Section 5.2.3). It was derived from the semi-permeable property of the articular

cartilage structure [49], where the semi-permeable membrane is formed by the

proteoglycan-collagen structure (solid skeleton) [294]. The higher density of the

solid skeleton results in a more compact structure, more resistance to the fluid flow

and lower permeability [3]. As a result, the proportion of the movable fluid that was


able to move out (exchangeable fluid) in this rule was defined to have a direct

relation with the ratio of fluid quantity to solid quantity (fs) in the agent as follows:

𝐸𝐹𝑡 = 𝑘𝑐 𝑓𝑠(𝑡) 𝑓𝑚(𝑡) (Eq. 6.3)

Here, EFt is exchangeable fluid of the agent at time step t, kc is a constant value,

fs(t) is the quantity of the fluid (moveable and remaining fluids combined) over solid

quantity in the agent at time step t, and fm(t) is quantity of the movable fluid in the

agent at time step t.

The quantity of the agent’s exchangeable fluid cannot be greater than the quantity

of the agent’s moveable fluid. Consequently, 𝑘𝑐 𝑓𝑠(𝑡) in Eq.6.3 must always be

equal or less than one (𝑘𝑐 𝑓𝑠 ≤ 1). Therefore, kc must be smaller or equal to the

inverse of the quantity of the fluid (moveable and remaining fluids combined) over

solid quantity in the agents (𝑘𝑐 ≤1

fs ).

IR 6.4: If an agent contained 100% dead space and it was located right below an

indenter, the cell where the agent was located was changed according to the property

of the indenter. For example, if tissue was indented via an impervious indenter, an

agent (cell) which contained 100% dead space was changed to an impervious agent

(cell) and added to the impervious indenter. For compression via a porous indenter,

the cell which contained 100% dead space agent was changed to an empty cell and

was added to the empty cells, which represented a porous indenter.

Dependent extra-agent rules:

ER 6.7: The amount of moveable fluid that moved out from an agent (rule IR

6.3) was shared between its neighbours where agents that had less ratio of moveable

fluid quantity to solid quantity received more fluid share. Since solid, including

proteoglycan and collagen, and fluid create an osmotic unit, this rule was derived


from osmosis phenomenon [295], which is the movement of fluid from a low

concentration of solute to a higher concentration of solute.

Proteoglycan molecules are highly negatively charged and are trapped inside

the collagen meshwork, which acts as a semi-permeable membrane [49]. The

concentration of the proteoglycans contributes mainly to the characteristic osmotic

pressure of the articular cartilage [296]. In this simulation, proteoglycan was

considered as a proportion of the solid content. The concentration of the solid in the

agent, therefore, reflects the concentration of the proteoglycan. When there is

movable fluid in an agent due to loading or gaining from the neighbours, solid

concentration decreases and extra fluid (movable fluid) in the agent tries to move out

in order for the agent to reach the equilibrium again. Lower solid concentration

results in higher gradient towards outside the agent. Therefore, agents that contain

more movable fluid in comparison with their neighbours potentially can receive less

fluid.

In this rule, gained fluid proportions of the neighbours from the exuded

movable fluid of an agent were related to the inverse ratio of the movable fluid

quantity to the solid quantity of the agent.

ER 6.8: The proportion of each neighbour of an agent from the agent’s

exuded movable fluid depended on the neighbour location relative to the agent.

Based on previous findings in the literature [57, 83], the hydraulic

permeability of the articular cartilage along radial and axial directions are equal

when the tissue is uncompressed. Although hydraulic permeability of the articular

cartilage decreases with compression in general, hydraulic permeability along the

radial direction progressively decreases more significantly than the axially directed

permeability [57, 83, 84, 150].


In this rule, the ability of fluid movement towards horizontal, vertical and

angled directions were equal at the beginning of the simulation, when the lattice had

not deformed yet and agents had not lost any fluid. Consequently, any exuded fluid

was shared equally between neighbours at the beginning of the deformation process.

When an agent lost fluid, its vertical fluid share ability decreased in comparison with

horizontal direction, to reflect the difference between the axial and radial

permeability of the real compressed tissue. As the ratio of fluid quantity to solid

quantity in the agent (fs) reflects fluid content in the agent, comparative changes of

fluid movement from an agent towards its neighbours located at its horizontal,

vertical or oblique directions (directional constants) were determined according to

the agent’s initial and current fs as follows:

𝐾𝑉

𝐾𝐻=

𝑓𝑠𝑖

𝑓𝑠𝑖𝑛𝑖 (Eq. 6.4a)

𝐾𝑂 =𝐾𝑉+ 𝐾𝐻

2 (Eq. 6.4b)

Here KV, KH and KO are directional constants along vertical, horizontal and oblique

directions, fsi is the fluid quantity to solid quantity of the agent at time step i and fsini

is the ratio of the fluid quantity to the solid quantity of the agent at the beginning of

the deformation (Time step = 0).

To illustrate moveable fluid exchange between agents, fluid distributions in

one block including one agent at each cell (Figure 6.7) are determined. Agents 1 to 4

contain S1, S2, S3 and S4 solid, Fim1,Fim2, Fim3 and Fim4 immobile fluid, and Fm1, Fm2,

Fm3 and Fm4 of moveable fluid respectively. At time T, the amounts of fluid that

move out from agents 1, 2, 3 and 4 are determined according to rule IR 6.3 and

equation Eq. 6.3 as follows:

𝐹𝐿1 = 𝑘𝑐 𝑓𝑠1 𝑓𝑚1





Where FL1, FL2, FL3 and FL4, are amounts of fluid that moves out from agents 1, 2, 3

and 4 respectively, kc is a constant value, fs1, fs2, fs3 and fs4 are ratio of the fluid

quantity (including movable and immobile fluids combined) to solid quantity in the

agents 1, 2, 3 and 4 respectively, and fm1, fm2, fm3 and fm4 are quantities of the

movable fluid in the agents 1, 2, 3 and 4 respectively.

Exuded fluid from one agent was shared between other agents of its block

according to rules ER 6.7 and ER 6.8. The amount of fluid that each agent gains

depends on the inverse ratio of its movable fluid quantity to solid quantity (rule ER

6.7), and location of the agent related to the agent that distributed its fluid (rule ER

6.8). Fluid from agent 1 moved along horizontal, vertical and oblique directions to

reach agents 2, 3 and 4 respectively; consequently, exuded fluid of agent 1 is

distributed between agents 2, 3 and 4 with the proportions of 𝐾𝐻

𝑓𝑠𝑚2 ,

𝐾𝑉

𝑓𝑠𝑚3 and

𝐾𝑂

𝑓𝑠𝑚4

respectively. The fluid that is gained by agents 2, 3 and 4 is as follows:

𝐹𝐺2 = 𝐹𝐿1 (

𝐾𝐻𝑓𝑠𝑚2


+ 𝐾𝑉

𝑓𝑠𝑚3+

𝐾𝑂𝑓𝑠𝑚4

)

𝐹𝐺3 = 𝐹𝐿1 (

𝐾𝑉𝑓𝑠𝑚3


+ 𝐾𝑉

𝑓𝑠𝑚3+


)

𝐹𝐺4 = 𝐹𝐿1 (



+ 𝐾𝑉

𝑓𝑠𝑚3+


)

Where FG2, FG3 and FG4 are amount of gained fluid by agents 2, 3 and 4 respectively,

FL1 is the quantity of the fluid that is exuded from agent 1, KO, KV and KH are

directional constants towards oblique, vertical and horizontal directions respectively,


and fsm2, fsm3 and fsm4 are the ratios of movable fluid quantity to solid quantity in

the agents 2, 3 and 4 respectively.

Other agents in the block also lose and gain fluid in the same way as agent 1.

Their lost fluids are shared between neighbours, similar to agent 1. The initial fluid

in the agent combined with the fluids gain from and lose to the neighbours

determines the fluid quantity of the agents as follow:

𝐹𝑛1 = 𝐹𝑖1 − 𝐹𝐿1 + 𝐹𝐿2 (



+ 𝐾𝑂

𝑓𝑠𝑚3+


) + 𝐹𝐿3 (



+ 𝐾𝑂

𝑓𝑠𝑚2+


) + 𝐹𝐿4 (



+ 𝐾𝑉

𝑓𝑠𝑚2+


)

𝐹𝑛2 = 𝐹𝑖2 − 𝐹𝐿2 + 𝐹𝐿1 (



+ 𝐾𝑉

𝑓𝑠𝑚3+


) + 𝐹𝐿3 (



+ 𝐾𝑂

𝑓𝑠𝑚2+


) + 𝐹𝐿4 (



+ 𝐾𝑉

𝑓𝑠𝑚2+


)

𝐹𝑛3 = 𝐹𝑖3 − 𝐹𝐿3 + 𝐹𝐿1 (



+ 𝐾𝑉

𝑓𝑠𝑚3+


) + 𝐹𝐿2 (



+ 𝐾𝑂

𝑓𝑠𝑚3+


) + 𝐹𝐿4 (



+ 𝐾𝑉

𝑓𝑠𝑚2+


)

𝐹𝑛4 = 𝐹𝑖4 − 𝐹𝐿4 + 𝐹𝐿1 (



+ 𝐾𝑉

𝑓𝑠𝑚3+


) + 𝐹𝐿2 (



+ 𝐾𝑂

𝑓𝑠𝑚3+


) + 𝐹𝐿3 (



+ 𝐾𝑂

𝑓𝑠𝑚2+


)

Where Fn1, Fn2, Fn3 and Fn4 are the fluid quantity in the agents 1, 2, 3 and 4

respectively after the fluid exchange, Fi1, Fi2, Fi3 and Fi4 are initial fluid quantity

(before exchange) in the agents 1, 2, 3 and 4 respectively.

Agent 1

S1 , Fim1 ,

Fm1

Agent 2

S2 , Fim2 ,

Fm2

Agent 3

S3 , Fim3 ,

Fm3

Agent 4

S4 , Fim4 ,

Fm4

Figure 6.7 A block consist of four cells and one agent at each cell. Agents contained

S1, S2, S3 and S4 solid, Fim1, Fim2, Fim3 and Fim4 immobile fluid, and Fm1, Fm2,

Fm3 and Fm4 moveable fluid.


ER 6.9: In this rule, indistinguishable fluid and solid from one agent could

move to another agent to ensure the volume of the agents in the lattice would not be

increased or decreased (to maintain rule IR 6.1). The standard size of the agent was

defined as the sum of the volumes of fluid and solid before deformation of the

matrix. Fluid exchange between neighbours might result in size changes of the agent

and makes it unequal to the standard size. If fluid loss of an agent was greater than its

fluid gained, the size of the agent was below standard size and the agent contained

dead space to compensate the difference between standard and current size of the

agent. Whereas, if the agent gained fluid more than lost, then its size exceeded the

standard agent size. As the lattice was confined at the sides and bottom, dead spaces

of the agents were filled by solid and fluid from the immediate top neighbours. Extra

volumes including solid and fluid combined were also transferred from the agents to

their immediate top neighbours. Figure 6.8 shows agents located at a column from

the top layer (agent 1) to the bottom (agent n) where thicker lines indicate the

standard size of the agents. Agents might have dead spaces (E1, E2,… En) or extra

volumes (M1, M2,…, Mn). The proportions of fluid and solid quantities in the extra

volumes were the same as the corresponding agents. For example, the ratio of the

fluid quantity to the solid quantity (fs) at Mn, Mn-1,…, M2 was equal to fs at Bn, Bn-

1,…, B1 respectively. Figure 6.8.A shows redistribution of the extra volumes in one

column of the lattice. Agent Bn was located right above the impervious agent at the

bottom of the lattice, while agent B1 was located right under an empty cell. As the

lattice was confined at all margins except the top, extra volume Mn in agent Bn only

could move towards the top, so Mn moved to agent Bn-1. If Bn-1 agent contained dead

space, Mn could fill it first; if Bn-1 became oversized after receiving Mn, extra volume

of the Bn-1 (Mn-1) would be transferred to the agent Bn-2. This transfer of the extra


volumes was continued up to the top agent (B1). If a top agent became oversized, its

extra volume would be transferred to the empty cell on top of it and the former

empty cell became a cell that contained an agent. Figure 6.8.B demonstrates how

dead spaces of the agents were filled by solid and fluid from their immediate top

neighbours. Indistinguishable fluid and solid from agent Bn-1 filled the dead space of

agent Bn (En). Transfer of the fluid and solid from Bn-1 to Bn might create En-1 dead

space in the Bn-1. Dead space of agent Bn-1 (if there was any) was filled by solid and

fluid from its immediate top neighbour (Bn-2). This process was continued until the

dead space of agent B2 (E2) was filled by agent B1. If the size of agent B1 located at

the top was less than or equal to E2, agent B1 was changed to an agent containing

100% dead space. As a result of this rule, the size of all articular cartilage agents

would be equal where agents located at the top might contain dead space.

A B

Figure 6.8 Processes to keep size of the agents constant. A: Agents transfer their

extra volumes to their immediate top neighbours. B: Volumes are transferred from

top agents to fill dead spaces of their immediate bottom neighbours. Green shows

filled space of the standard cell size by indistinguishable fluid and solid. Yellow and

white show extra volume and dead space respectively.


6.2.6 Degenerated tissue model

The composition of the articular cartilage is changed due to degeneration

where solid skeleton decays and fluid content increases [264, 272, 297, 298]. In order

to develop a degenerated articular cartilage model, solid content of all lattice agents

was decreased by 40 % mass, and decreased solid in the agents was replaced by

fluid. According to the intra-agent rule IR 6.3 and Eq. 6.3, the value of the kc must be

always smaller than or equal to the inverse of the quantity of fluid over solid quantity

in the agents (𝑘𝑐 ≤1

fs). As 40% mass of the solid content of the healthy model was

removed to generate a degenerated model, the maximum fluid proportion was

increased from 85% total mass (88.2% total volume) for the healthy model (Figure

6.4) to 91% total mass (93% total volume) for the degenerated model. As quantities

of the fluid and solid were based on volume, maximum fs of the agents in the lattice

was increased from 7.47 to 13.3. According to the initial conditions, 𝑘𝑐 ≤ 0.13 for

the healthy model and 𝑘𝑐 ≤ 0.075 for the degenerated model. However, fluid

content proportion and fs ratio of the agents located in the unloaded area are

increased as a result of gaining fluid during the deformation process. Consequently, a

lower value for the kc was required to meet the condition of the intra-agent rule IR

6.3. The value of the kc was equalled to 0.05 for the degenerated model simulation

to ensure kc was always less than 1

fs.

6.2.7 Simulations

Both healthy and degenerated models were subjected to two loading

scenarios till 37% strain. The deformation process causes fluid flow through the

lattice, where movable fluid flows out from the lattice. When there is no movable

fluid in the lattice, the system reaches equilibrium and deformation process is


completed. The strain-time result of the healthy model subjected to loading scenario

1 was validated against experimental data [288]. The validated extra- and intra-agent

rules for the healthy model under loading scenario 1 were used for the simulations of

loading scenario 2 and deformation of the degenerated articular cartilage under

loading scenario 2. Programs in Matlab (Mathworks Inc, MA, USA) were developed

to simulate the consolidation-like behaviours of the healthy and degenerated articular

cartilage models (Appendix C).

As the size of an agent was measured according to its volume, the quantity of

the fluid and solid in an agent indicated their volumes. Densities of the fluid and

solid (1.0 and 1.323 g/mm2 respectively [292]) were used to convert quantities of

fluid and solid in an agent from volume to mass.

6.2.8 Corresponding time step to experimental time

The simulated 37% strain results of the healthy model under loading scenario 1

(compression via a porous indenter) were validated against the experimental strain-

time profile from the literature [288]. Experimental strains at several time points

were compared with their simulated counterparts to find the best corresponding time

steps. The previous studies of the consolidation of the healthy articular cartilage

showed that the strain reaches approximately 65% of the total after 15 minutes and

95% after 90 minutes [64, 150, 288, 299]. As deformation of the articular cartilage

was faster during the first 15 minutes, experimental measured points were selected

with one (1) minute interval for the first 15 minutes (15 points) and 5 minutes

interval from 15 minutes to 90 minutes (15 points). Coefficient of variation of the

root mean square error (CV(RMSE)) [261-263] (Eq. 6.5) was used to illustrate the

deviation between simulated and experimental data [288].


CV(RMSE) =

√∑ (𝑆𝑆𝑘−𝐸𝑆𝑘)2𝑛

𝑘=1n

MSS (Eq. 6.5)

Here, SSk is the simulated strain at measured point k, ESk is experimental strain

at point k, n is the total number of points of the profile and MSS is the mean of the

simulated strain at the profile points.

CV(RMSE) was calculated for various time steps that corresponded to the one

(1) minute experimental time. If, for example, time step T1 was selected to be

corresponding to 1 minute, then twofold, threefold and tenfold T1 were

corresponding to 2, 3 and 10 minutes. The time step that generated the least value of

CV(RMSE) was chosen as the corresponding time step to the experimental time.

6.3 RESULTS

6.3.1 Corresponding simulation time step to real time

The CV(RMSE) describes the discrepancy between simulated and

experimental results or error of the simulation where lower values for the

CV(RMSE) indicate less simulated result diversion from the experimental results.

Figure 6.9 shows the CV(RMSE) for the simulated strain, when kc=0.05 based on

time steps corresponds to one minute. The CV(RMSE)-time step profile has negative

slope until 90 time step, where the CV(RMSE) reached to its minimum value (2.6%),

and then the slope become positive. It illustrates that the best match between

experimental and simulation results occurred when 90 time steps were equal to one

minute. Consequently, one time step was equal to approximately 0.67 seconds.


Figure 6.9 CV(RMSE) of simulated strain when kc was equal to 0.05, based on time

step corresponding to one (1) minute.

6.3.2 Effect of kc on results

Seven values, 0.08, 0.07, 0.06, 0.05, 0.04, 0.03 and 0.02 were chosen for kc to

investigate the effect of kc on the simulation error and the time step corresponding to

the real time. Figure 6.10 shows simulation error (CV(RMSE)) and time step

corresponding to one minute experimental time for various values of kc. The

CV(RMSE) was increased from 2.3% to 2.8% (an approximately 20% increase)

when kc decreased from 0.08 to 0.02 (a 300% decrease), indicating the value of kc

had minimal effect on the CV(RMSE). The time steps corresponding to one minute

were 48, 58, 71, 90, 117, 164 and 257 for kc equal to 0.08, 0.07, 0.06, 0.05, 0.04,

0.03 and 0.02 respectively. The corresponding time steps were increased

approximately 50% (from 48 to 71) and 40% (from 117 to 164) when kc decreased

25% in both of them (from 0.08 to 0.06, and from 0.04 to 0.03 respectively). It

illustrates that time steps were decreased non-linearly and significantly, by increasing

the value of kc.


Figure 6.10 Effect of kc on CV(RMSE) and time step corresponding to one (1)

minute experimental time.

6.3.3 Validation of the healthy articular cartilage simulation

The full consolidation of the confined articular cartilage under an axial load via

porous indenter takes several hours [150]. In the CA simulation, when kc was equal

to 0.05, the lattice reached 99.9% of the total strain after approximately 13500 time

steps (2.5 hours). The strain curve of the CA simulation (kc=0.05) and corresponding

experimental result for loading scenario 1 and healthy cartilage are plotted in Figure

6.11. Both simulated and experimental profiles show that fluid immediately starts

flowing out of the matrix, as indicated in strain-time step curve resulting in rapid

growth of strain at the beginning of the deformation process. The strain growth rate

was decreased considerably after about 2000 time steps (approximately 22 minutes)

and the strain curve became almost level after 6000 time steps (approximately 65

minutes). According to Figure 6.10, CV(RMSE) was also equal to 2.6% for kc equals

0.05 which demonstrates a great agreement between experimental data from the

literature [288] and simulated results in Figure 6.11.


Figure 6.11 Comparison between experimental [287] and predicted strain values

when kc=0.05.

6.3.4 Healthy cartilage results

6.3.4.1 Loading scenario 1

Figure 6.12 shows the spatial distribution of the fluid volume to solid volume (fs)

in the lattice subjected to the loading scenario 1 at time steps 0, 90, 450, 900, 5400

and 13500, corresponding to 0, 1, 5, 10, 60 and 150 minutes. The legend attached to

the pictures shows the ratio of the fs. The kc was equal to 0.05, therefore, according

to Figure 6.10, 90 time steps correspond to one minute. Overall, the comparison

between the image at the beginning of the deformation process (T=0) and 2.5 hours

after applying load (T=13500) shows that the fs ratio in all regions of the lattice was

decreased significantly, due to consolidation. The images at 0, 90 time steps (1

minute) and 450 time steps (5 minutes) show that shortly after loading, the fs was

decreased significantly in the layers close to the porous indenter (surface layers) and

the layers between surface and bone (middle layers) of the lattice, while the fs was

progressively increased in the layers near the bone (bottom layers). The decrease of

the fs in the surface and middle layers was continued until end of the consolidation


process. After an initial increase of the fs in the bottom layers, the value of the fs

dropped until the end of the process (images at 900, 5400 and 13500 time steps).

According to Figure 6.11, the lattice reached to over 34% strain (92% of the total

37% strain) at 5400 time steps (one hour) and 36.96% strain (99.9% of the total

strain) after 13500 time steps. Comparison between pictures at T=5400 and T=13500

shows a very slight change in distribution of the fs in the lattice.

T=0 T=90 (1 min) T=450 (5 min)

T=900 (10 min) T=5400 (60 min) T=13500 (150 min)

Figure 6.12 Spatial fs distributions in the lattice at time steps 0, 90, 450, 900, 5400

and 13500, corresponding to 0, 1, 5, 10, 60 and 150 minutes (kc=0.05).

Figure 6.13 shows simulated spatial distribution of the fluid volume fraction in the

healthy matrix (kc=0.05) subjected to loading scenario 1 on the basis of the

percentage at time steps 0, 90, 450, 900, 5400 and 13500. The legend attached to the

pictures shows fluid content based on the percentage of the fluid fraction. Initially at

T=0, surface and middle layers contained more fluid in comparison with the bottom

Porous indenter

Porous indenter

Porous indenter

Porous indenter

Porous indenter

Porous indenter


layers. While the volume of fluid in the surface layers was decreased gradually from

the beginning of loading until the equilibrium time, the fluid increased progressively

in the bottom layers at the early stages of the deformation (images at T=0, 90 and

450) and reached a peak after approximately 450 time steps. The fluid volume in the

bottom layers dropped after the peak time until the end of the process. At the time

close to the equilibrium time (T=13500), all agents in the lattice lost significant

amounts of fluid.

T=0 T=90 (1 min) T=450 (5 min)

T=900 (10 min) T=5400 (60 min) T=13500 (150 min)

Figure 6.13 Spatial fluid volume fraction distributions of the healthy model at

different times, subjected to the loading scenario 1 (kc=0.05).

Figure 6.14 shows the ratio of the fluid mass over solid mass in the entire matrix

(FS) and in the layer of the lattice, which is adjacent to the bone (FSB) over time. The

profiles of the FS and FSB depict the change of ratio of fluid mass to solid mass in a

bulk level. The FS reduced sharply in the early steps of the process and continued

Porous indenter

Porous indenter

Porous indenter

Porous indenter

Porous indenter

Porous indenter


reducing until the end of the process, whereas FSB underwent a sharp increase and

reached its peak after about 450 time steps. FSB dropped significantly after its peak.

Both FS and FSB were slightly decreased when the matrix reached about 80% of its

total strain (at about 3000 time steps). At the time close to the end of deformation, FS

and FSB were almost level.

Figure 6.14 Fluid mass over solid mass in the entire lattice (blue) and in the bottom

layer of the lattice (red) over time.

6.3.4.2 Loading scenario 2

An impervious indenter with 10 cells width was used for the indentation of the

healthy model up to 37% strain. As the same lattice, kc and intra- and extra-agent

rules as loading scenario 1 (Section 6.3.4.1) were used, a time step corresponding to

the real time equal to that used in loading scenario 1 was employed (90 time steps

was equal to one minute). The lattice under loading scenario 2 reached 99.9% of its

total strain after 8100 time steps (90 minutes).

Figure 6.15 shows profiles of the strain and percentage of the exuded fluid

volume over 5000 time steps for the second loading scenario when kc equals 0.05.

The strain escalated significantly from the beginning of the deformation process until

approximately 27% strain (300 time steps). The slope of the exuded fluid profile

during this period was considerably less than the strain curve. As the indenter was


impervious, movable fluid from the loaded area exuded out of the lattice via

unloaded agents. The difference between the strain and exuded fluid indicates that

the rate of fluid movement from the loaded area to the unloaded area was higher than

the fluid exudation rate from the lattice at the early stages of the indentation.

Consequently, a considerable amount of the moved fluid from the loaded area still

remained in the unloaded area. The slope of the exuded fluid was greater than the

strain profile from 300 to 4000 indicating that more fluid volume moved out of the

lattice than transferred from the loaded agents to the unloaded agents. From 4000

time steps until the end of the indentation process, both strain and exuded fluid

profiles had almost the same rate of growth, meaning the exudation rate of fluid from

the matrix was equal to the fluid movement rate from the loaded area to the unloaded

one. The exuded fluid reached 100% at almost the same time that strain reached to

37%.

Figure 6.15 Profiles of the strain and exuded fluid volume percentage for the second

loading scenario based on time steps.

Figure 6.16 shows the spatial distribution of the fluid volume fraction in the

lattice under loading scenario 2 at time steps 0, 30, 90, 180, 1350 and 8100, based on

the percentage. When the matrix was subjected to the load, fluid immediately started


moving to the unloaded agents from the loaded agents, causing an increase in fluid

percentage in the unloaded area close to the loaded area (T=30). It also resulted in

temporary swelling of the unloaded area (T=30 and 90). Migration of the fluid from

loaded area to unloaded area continued while simultaneously, fluid moved out from

the top surface of the unloaded area. At the equilibrium time, the loaded area lost

fluid significantly, resulting in an increasing solid percentage and decreasing fluid

content, while the percentages of fluid and solid content in the unloaded area were

equal to the initial condition (T=0).

T=0 T=30 (20 second) T=90 (1 min)

T=180 (2 min) T=1350 (15 min) T=8100 (90 min)

Figure 6.16 Distribution of the fluid volume fraction in the healthy matrix based on

the percentage during deformation process at time steps 0, 30, 90, 180, 1350 and

8100, corresponding to 0, 20 seconds, 1 minute, 2 minutes, 15 minutes and 1.5 hours

respectively (kc=0.05).

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Figure 6.17 shows the spatial distribution of the fluid volume to solid volume

(fs) in the lattice under loading scenario 2 at various time steps. Initially, surface and

middle layers had greater fs values than bottom layers. Indentation resulted in a

significant decrease of the fs in the loaded agents, while unloaded agents had the

same fs as the initial time. However, the fs of the unloaded regions close to the

loaded area initially underwent a temporary increase due to fluid flow from loaded

area to unloaded area (T=30).

T=0 T=30 (20 second) T=90 (1 min)

T=300 (200 seconds) T=1350 (15 min) T=8100 (90 min)

Figure 6.17 Distribution of the fs in the healthy matrix based on the percentage

during deformation process at time steps 0, 30, 90, 180, 1350 and 8100,

corresponding to 0, 20 seconds, 1 minute, 2 minutes, 15 minutes and 1.5 hours

respectively (kc=0.05).

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6.3.5 Degraded matrix results

The strain-time step profile of the degenerated and healthy articular cartilage

models under loading scenarios 1 and 2 are shown in Figures 6.18A and 6.18B

respectively. The degenerated model under both loading scenarios had a greater

strain growth rate in the early stages of the consolidation and reached the total strain

(37%) much faster than the healthy model. The healthy and degenerated models

reached 99.9% of the 37% strain after 13500 and 5570 time steps respectively when

they were subjected to loading scenario 1. The healthy and degenerated models under

loading scenario 2 reached 99.9% of total strain after 8100 and 2700 time steps

respectively. The lattice under a wider indenter took more time steps to reach 99.9%

of total strain in both healthy and degenerated models. This indicates that the healthy

matrix was stiffer than the degenerated matrix. These results are consistent with

previous research that proved lower stiffness of the degenerated and degraded

articular cartilage than in healthy tissue [76, 288, 300].

A B

Figure 6.18 Strain versus time steps for the degenerated and healthy model of the

articular cartilage under loading scenario 1 (A) and loading scenario 2 (B).


Figure 6.19 shows spatial distributions of the fluid volume fraction in the

degenerated lattice under loading scenario 2 at time steps 0, 30, 90, 180, 450 and

2700, corresponding to 0, 20 seconds, 1, 2, 5 and 30 minutes respectively. The

legend attached to the pictures shows fluid content based on the percentage of the

fluid fraction. The initial fluid content in the degenerated model (T=0) was higher

than in the healthy model (Figure 6.16) due to replacing removed solid with fluid.

The same as the healthy model, fluid from the loaded area flowed towards the

unloaded area immediately after starting indentation. As a result of this internal fluid

flow, the loaded area was compressed and lost a significant amount of fluid while the

unloaded area swelled, where closer positions to the indenter noticeably swelled

greater than farther positions (images at T=30 and 90). Near system equilibrium

(30000 time steps), the distribution of the fluid in the unloaded area was similar to

the initial distribution (T=0) while fluid content in the loaded area was considerably

lower than that in the initial condition.

Despite similarities of the fluid flow trend between healthy and degenerated

models, comparison of the degenerated images at T=0, 30 and 90 with their

counterparts in the healthy model show that a smaller percentage of the initial fluid

remained in the loaded area in the degenerated model. The equilibrium time for the

degenerated model was also significantly shorter than the healthy model (2700 time

step versus 8100 time step), which indicates a higher rate of fluid outflow for the

degenerated model. Therefore, the fluid internally flowed faster in the degenerated

model than the healthy model. As internal fluid flow in the articular cartilage matrix

has a positive correlation with the permeability of the matrix [42], it can be

concluded from the images that the degenerated model was more permeable than the

healthy one, which is consistent with previous published research [63, 86].


T=0 T=30 (20 seconds) T=90 (1 min)

T=180 (2 min) T=450 (5 min) T=2700 (30 minutes)

Figure 6.19 Distribution of the fluid volume fraction in the degenerated matrix when

kc=0.05 based on the percentage during deformation process at time steps 0, 30, 90,

180, 450 and 2700.

Figure 6.20 shows spatial distributions of the fluid volume divided by solid

volume (fs) in the degenerated lattice under loading scenario 2 at time steps 0, 30,

90, 180, 450 and 2700. The legend attached to the pictures shows the ratio of the fs.

The fs of the loaded area decreased over the entire loading process. The fs of the

unloaded area increased temporarily, shortly after loading (T=30 and 90), due to the

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fluid movement from the loaded area to the unloaded one, then decreased gradually

to reach the initial condition at the equilibrium time. The indentation resulted in a

significant decrease of the fs in the loaded area. Similar to the fluid fraction images

(Figure 6.19), the change of the fs in the degenerated model was quicker than the

healthy model.

T=0 T=30 (20 seconds) T=90 (1 min)

T=180 (2 min) T=450 (5 min) T=2700 (30 minutes)

Figure 6.20 Distribution of the fs in the degenerated matrix during deformation

process when kc=0.05 at time steps 0, 30, 90, 180, 450 and 2700, corresponding to 0,

20 seconds, 1, 2, 5 and 30 minutes respectively.

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6.4 DISCUSSION

Hybrid agents as presented in Chapter 3, were adapted for articular cartilage to

simulate consolidation-like behaviour of the tissue. The 2D CA model of bovine

knee articular cartilage was developed where the constituents of articular cartilage

were simplified to fluid and solid. The layered mass distribution of the fluid and solid

of healthy articular cartilage [291] and density of the fluid and solid skeleton of

bovine articular cartilage [292] were used to create the matrix model based on

volume distributions of the fluid and solid in layers. The lattice, which represented

the articular cartilage matrix, was surrended on the bottom, and left and right sides,

with impervious agents simulating confining boundary conditions. Two boundary

conditions were used at the top: (i) Empty cells (Figure 6.5A) to mimic a porous

indenter, (ii) 10 cells in the centre containing impervious agents and empty cells on

their left and right (Figure 6.5B) to mimic an impervious indenter. Fluid could flow

out only via empty cells in the top. As a result of intra- and extra-agent rules, both

solid and fluid could move in the lattice, resulting in compression in the loaded area

and temporary swelling in the unloaded area. The healthy model was used for

development of the degenerated model.

The comparison between the experimental strain-time curve of the healthy

articular cartilage under axial static load via a porous indenter [301] and the

corresponding CA strain-time step curve (Figure 6.11) shows a great agreement

between simulated and experimental results (CV(RMSE) = 2.6 %). It proved

validation of the CA model and applied intra- and extra-agent rules.

The matrix model was deformed via a 30 cell-width porous indenter, equal to the

width of the articular cartilage matrix in the the first loading scenario and a 10 cell

wide impervious indenter in the loading scenario 2. The healthy model reached 15%


and 25% strain after 600 and 1950 time steps respectively when it was compressed

via the porous indenter (Figure 6.11), while it took 40 and 190 time steps via the

impervious indenter to reach the same strain values (Figure 6.15). A correlation

between size of the indenter (diameter or width) and strain-time profile was observed

where a narrower indenter resulted in faster deformation. This is in agreement with

previous studies [302, 303], which stated dependency of the magnitude of the

indentation on the size of the indenter where indenters with the smaller diameters had

higher strain growth rate.

Since pores are occupied by fluid in porous saturated materials, volume fraction of

the fluid depicts porosity, which is defined as the fraction of the volume that is

occupied by pores over total volume [30, 31]. Therefore, Figures 6.13, 6.16 and 6.19

depict the spatial distribution of the porosity in the articular cartilage matrix during

deformation at different times. The porosity change of the matrix at any location

during the consolidation process can be measured by comparing spatial distributions

of the porosity at given times.

According to Darcy’s law, permeability has a negative correlation with fluid flow

resistance where a value of zero permeability results from infinite resistance [243].

On one hand, the resistance to fluid flow through a porous medium is highly related

to the amount of solid and fluid present [242]. On the other hand, the ratio of fs

shows the amount of fluid and solid in the hybrid agent which is in the range of zero

to infinity. When there is no solid present, the agent is fully porous and the fs equals

infinity where resistance to fluid flow is zero. When the agent contains only solid and

there is no fluid, the agent is impervious and the fs equals zero where the resistance is

infinite. Therefore, the fs of the hybrid agent has a negative correlation with the

resistance of the agent to the fluid flow and reflects permeability of the agent. The


spatial distribution of the fs at different times during deformation (Figures 6.12, 6.17

and 6.20) shows how permeability of the matrix changes during deformation.

Computational techniques that are used for simulation of the articular cartilage

functions, including consolidation are based on theoretical models such as biphasic

[91] and numerical methods such as finite element analysis (FEA) [133] (Section

2.7). The required parameters for the formulation of the theoretical models, such as

permeability and porosity, are extracted from experimental results using estimation

methods such as curve fitting (Section 2.4). The presented CA approach in this

chapter can provide data such as a spatial map of the porosity and resistance to fluid

flow at any given time during consolidation. As the validity of the CA against

experimental was confirmed, required parameters for theoretical models can be

potentially extracted from the presented CA approach.

Chapter 7: Dissolution of wet salt 139

Chapter 7: Dissolution of wet salt

7.1 INTRODUCTION

Rock salt (halite) is a sedimentary mineral form of sodium chloride that usually

forms in hot and dry climates where ocean water evaporates. Previous measurements

showed that rock salt has porosity range from 0.1% to 11% [304-308] and

permeability in the range of 10−16

to 10−22

m2 [306, 307, 309, 310]. It not only

dissolves into the water but also absorbs humidity and water due to its hygroscopic

properties and porous structures [304, 308], thus even dry rock salt contains a small

percentage of water content [311]. Salt is an ionic compound and consists of

negatively charged chloride and positively charged sodium ions. A water molecule is

also considered as a polar molecule, where it has a negative charge near the oxygen

atom and positive charge near the hydrogen. Polar water and sodium chloride

molecules create a chemical bond that makes wet rock salt a single-phase multi-

component mineral material. Existing water in the halite structure also plays a

significant role in controlling the behaviour and properties of the rock salt, such as

ductility, creep, deformation, diffusion and solution [311, 312]. Rock salt dissolves

into water where the dissolution process changes rigid (solid) salt molecules into

dissolved ones.

During the water dissolution process of rock salt, water molecules penetrate

into the structure of the halite and salt molecules simultaneously diffuse into the

solution [313]. Therefore, the concentration of salt in the fluid is increased as a result

of the dissolution of salt into the water, while penetrated water dilutes the wet salt

structure. Dilution of the salt continues until rock salt disintegrates, whereas

140 Chapter 7: Dissolution of wet salt

diffusion of the salt molecules continues until the solute distributes uniformly and the

solution becomes homogenous.

Two simultaneous phenomena (water percolation through rock salt and

diffusing salt into water) make the process of dissolution complex. At present, little

is known about mechanisms involved in fluid percolation and diffusion through rock

salt [306, 314]. In this chapter, naive diffusion, which is one of the most primitive

stochastic models of diffusion, hybrid agents, intra- and extra-agent rules are used to

simulate the dissolution process of rock salt into water. Traditional methods of naive

cellular automata (CA) diffusion, including individual salt and water agents (black

and white agents) and neighbourhood rules, are used to compare results of the

presented approach and traditional methods.


7.2.1 Overview of the dissolution process

The process of dissolution of a block of wet rock salt when fully immersed in

the water is simulated. The rock salt block contains 10% water and 90% salt and is

surrounded by water in all directions. The dissolution process ends when the block of

salt is solved and concentration of the salt in the solution becomes uniform.

7.2.2 Rock salt model using hybrid agent

A two-dimensional (2D) lattice with 80 x 80 cells was created to represent the

entire environment, where each cell contained one hybrid agent. The hybrid agents in

this case study contain salt and water, without distinguishing them where the agents


were identified by their proportions of salt and water mass. Therefore, the size of an

agent was determined by the sum of water and salt mass in the agent. In the absence

of salt or water in the agent, a zero proportion was allocated to the absent component

in the agent. For example, if an agent contained only water, it was identified by

100% water and 0% salt. A 30 x 30 block of agents were placed in the centre of the

lattice containing 10% water and 90% salt to represent the wet salt block, surrounded

by agents containing only water. Proportions of the components of an agent (salt and

water) could be changed as a result of gaining and losing salt or water. The

simulation was ended when all agents in the lattice carried the same proportions of

water and salt which presented the homogeneous solution. A program in Matlab

(MathWorks, MA, USA) was developed to simulate the dissolution process

(Appendix D).

7.2.3 Rules

Rules consisted of extra- and intra-agent rules. The interactions between agents

are determined by extra-agent rules while intra-agent rules control intra-agent

evolution, in which proportions of the fluid and solid within the agent may change.

The following extra- and intra-agent rules were used to simulate dissolution of the

rock salt into water:

Extra-agent rules:

ER7.1: Naive CA diffusion and Moore neighbourhood were implemented for

interaction between neighbours. Each agent in Naive CA diffusion interacts with one

of its neighbours randomly at each time step [315-317]. Since diffusion is performed

as a stochastic process that imitates Brownian motions, the Moore neighbourhood

was used to facilitate simulation of diffusion due to a greater number of neighbours


of a cell [318] in comparison with other neighbourhoods, such as von Neumann and

Margolus. This neighbourhood allows an agent to interact with neighbours located in

the cells that have common borders with the agent’s cell, as well as those located in

the corners’ cells. Figure 7.1 shows Moore neighbourhood, where an agent located in

the central cell (cell C) can interact with agents located in its North East (NE), North

(N), North West (NW), East (E), West (W), South East (SE), South (S) and South

West (SW). As a result of using Naive CA diffusion, the agent in the cell C interacts

with only one of the eight neighbouring agents (N, NE, NW, S, SE, SW, W and E)

randomly at each time step. Synchronous updating methods [254] were used to

ensure simultaneous interactions of the agents.

ER 7.2: Two equal portions of the interactive agents are exchanged when

they interact. Figure 7.2 shows agents 1 and 2 before and after interaction, in which

portions of λ1 and λ2 are swapped between the agents. The size (mass) of λ1 and λ2

are equal while their water and salt proportions are equal to agent 1 and agent 2

respectively.

NE N NW

E C W

SE S SW

Figure 7.1 2D Moore neighbourhood. Central cell (cell C) is surrounded with North

East (NE), North (N), North West (NW), East (E), West (W), South East (SE), South

(S) and South West (SW) cells.


Figure 7.2 Interaction of two agents when two equal portions (λ1 and λ2) are

exchanged.

ER 7.3: The value of the exchanged portion (λ) when two agents interact

depends on the attribute of the agent. As two fluid agents have a higher ability of

intermixing than two porous medium agents, the value of λ for fluid agents (λf) might

be greater than the porous medium agents (λs).

Intra-agent rules:

IR 7.1: Size of an agent is constant. As a result of fixed size agents over time

steps, quantity of the salt and water in the agent is equivalent to their corresponding

proportions.

IR 7.2: An attribute of an agent depends on water and salt proportions. If

proportion of salt in an agent is less than a certain proportion, named the threshold of

rigidity (TR), the agent shows behaviour and characteristics of a solution where salt

dissolves in water and creates a fluid. The agent stays rigid and shows behaviour of a

saturated porous medium on the condition that the salt proportion is greater than the

threshold of rigidity (TR).

IR 7.3: This rule is in conjunction with rule ER 7.2. When an agent receives a

portion from another agent as a result of swapping two equal portions with another


agent, the water and salt proportions of the agent may be changed because of the

received portion. Figure 7.3 shows that agent 1, containing Wi and Si proportions of

water and solid respectively, receives portion λ containing Wj and Sj proportions of

water and solid respectively. As water and salt are indistinguishable in the agent,

water and salt of λ and agent 1 will be integrated resulting in Wk and Sk proportions

of water and solid for agent 1. According to the law of conservation of mass,

quantities (mass) of the water and salt after integration are equal to the sum of water

and salt mass in agent 1 (before integration) and portion λ. Wk and Sk are determined

based on the size of λ and initial fluid and salt proportions of agent 1 and λ as

follows:

{𝑊𝑘 = (1 − 𝜆) ∗ 𝑊𝑖 + 𝜆 ∗ 𝑊𝑗

𝑆𝑘 = (1 − 𝜆) ∗ 𝑆𝑖 + 𝜆 ∗ 𝑆𝑗 (Eq. 7.1)

Here, λ is the size of the received portion based on total agent size (0 ≤ λ ≤ 1); 𝑊𝑖 ,

𝑊𝑗 , 𝑆𝑖 and 𝑆𝑗 are initial (before integration) water and salt proportions of agent 1

and λ; and 𝑊𝑘 and 𝑆𝑘 are proportions of water and salt of agent 1 after integration.

Figure 7.3 Integration of the agent 1 and portion λ.


7.2.4 Simulation

Various simulations were carried out for different values of λf, λs and TR

according to Table 7.1, where if at least one of the interactive agents was a porous

medium agent, the exchange portion was assumed equal to λs. The λf, TR, salt block

and surrounding fluid were equal for simulations 1 and 2 while two different λs were

applied to investigate the effect of λs. The value of TR differs simulations 3 and 4

from 1 and 2 respectively. Simulation 5 studies the effect of initial water content of

the salt block where salt block agents contain 10 times less water than simulation 1.

In order to make the amount of salt in the block equal to other simulations, the

dimension of the rock salt block was changed to 27 x 30 cells from 30 x 30 cells to

compensate for increasing salt proportion in the block. Rock salt was immersed into

distilled water in simulations 1-5 while surrounding water contained 10% salinity in

simulation 6. Simulation number 7 replicated the dissolution process when low value

and relatively high value were allocated to λs and TR respectively.

Table 7.1 Value of parameters for different simulations.

Simulation

number

λf λs TR Initial salt

proportion in

the rock salt

Salt proportion in

the surrounding

fluid

Rock salt

block

dimension

1 0.5 0.2 0.7 90% 0 30 x 30

2 0.5 0.1 0.7 90% 0 30 x 30

3 0.5 0.2 0.3 90% 0 30 x 30

4 0.5 0.1 0.5 90% 0 30 x 30

5 0.5 0.2 0.7 99.9% 0 27 x 30

6 0.5 0.2 0.7 90% 20 % 30 x 30

7 0.5 0.0001 0.9 99% 0 30 x 30


7.2.5 Global properties

The hybrid agent is able to carry intra-agent properties where extra-agent (global)

properties extract from intra-agent ones. Global salt and fluid proportions (S and W

respectively) of a group of agents are equal to the sum of the fluid or salt mass of the

agents divided by the sum of total mass (fluid and solid combined) of the agents.

They are calculated as follows:

𝑆 =∑ ( 𝑠𝑖 )

𝑛

𝑖=1

∑ ( 𝑠𝑖 + 𝑤𝑖 ) 𝑛

𝑖=1

, 𝑊 =∑ ( 𝑤𝑖 )

𝑛

𝑖=1

∑ ( 𝑠𝑖 + 𝑤𝑖 ) 𝑛

𝑖=1

(Eq. 7.2)

Here, S and W are global proportions of the salt and water respectively, n is the

number of agents, and si and wi are salt and water proportions of the agent i.

Global proportions of the salt and fluid are functions of salt and fluid proportions

within the agents. As the global salt proportion describes the salt mass fraction in an

area, it illustrates salt concentration.

7.2.6 Traditional salt and water agents’ simulation

The dissolution process of the wet rock salt was carried out using individual

salt and water agents for comparison purposes with simulation number 1 in Table

7.1. The simulation was performed for a 80 x 80 lattice where each cell contains one

salt or water agent. The lattice had a square 30 x 30 cells in the centre to represent

wet salt. As the proportion of salt to water in the wet salt block was 9 to 1 (90% salt

and 20% water), central square cells consist of 810 salt and 90 water agents to

represent 20% water in the salt structure. Agents in the central square are arranged

randomly, wherein red and blue states represent solid and fluid particles. All cells

outside the central square contain a fluid agent (blue state) to represent surrounding

water. The Moore neighbourhood (Figure 7.1) and a Naive diffusion model are used,


in which each agent exchanges its state with one of its neighbours (N, NE, NW, S,

SE, SW, W and E) randomly at each time step. In order to compensate partial

exchange between interactive hybrid agents, an interaction probability was added to

the neighbourhood rule. For example, in order to compensate λf = 0.2, in which 20%

of the agent was exchanged, a probability of 20% was considered for the traditional

agents’ interaction where the central agent swapped its state with one of the eight

neighbours with a probability of 20%. The simulation was ended when salt agents

distributed almost uniformly in the lattice.

7.2.7 Number of simulation runs

The accuracy of the simulation over multiple simulation runs is one of the

important issues in computational simulations. If the model is deterministic, then

even a single run is enough and repetition of simulation results in the same output as

the first run. On the other hand, the predictions of various simulation runs are not

exactly the same if the model is stochastic. Taking the average of multiple simulation

runs is the accepted method to determine prediction of the stochastic models [319],

while the number of required runs to produce results with enough accuracy is another

issue. The traditional and one of the most acceptable techniques is to choose an

arbitrary large number, e.g. 10 [320], 20 [184] and 1000 [321], and assume it is an

adequate “number” to address the issue. Although the average value of multiple runs

is more accurate than a single run, it is not possible to check whether the number of

runs is large enough or not. Consequently, choosing an arbitrary number is not well-

motivated mathematically. Another technique is to repeat the simulation, until the

mean values converge and additional runs do not change the mean values

significantly [319].


In this chapter, the least squares method was used to compare results of

consecutive runs. The sum of squared differences between the mean of previous run

values, and the recent run at all time steps, describes the difference between recent

run and previous runs or error of the recent run, based on the average of the previous

runs:

𝐸 = ∑ (𝑥𝑖−𝑦𝑖)2𝑛

𝑖=1 (Eq. 7.3)

where E is the error of the recent run, n is total time steps, xi is the value of the recent

run at time step i and yi is mean of values of the previous runs at time step i.

Simulation number 1 (Table 7.1) using hybrid agents, and a traditional agent

simulation (Section 7.2.6) with corresponding probability (20%), are selected in

order to determine the required number of simulation runs where error for

consecutive runs are measured. To this end, salt proportions within four hybrid

agents located at different cells (cells 1-4 in Figure 7.4) and salt concentration at two

areas (areas 1 and 2 in Figure 7.4) of the lattice were measured during 25000 time

steps. As traditional salt and water agents are able to carry only one state (red (salt)

or blue (water)), only global salt concentration could be calculated for the traditional

simulation. Locations of the selected cells based on distance from the top and left

margins of the lattice are demonstrated in Table 7.2. Cell 1 is close to the top left

corner of the lattice, far from the rock salt block, while cell 2 is located in the centre

of the lattice and salt rock block. Cell 3 is located at 11 cells distance from the block,

and cell 4 is located on the boundary between water and the salt block. Both area 1

and 2 included 10x10 cells. Area 1 is located at the top left corner of the lattice and

area 2 is at bottom right corner of the rock salt block (Figure 7.4), initially including

salt and water agents.


Figure 7.4 Rock salt block and its surrounding water. Locations of selected cells

(Cells 1-4) and areas (area 1 and 2) are shown in yellow and with a white dashed line

respectively.

Table 7.2 Location of the selected cells and centre of the areas from the lattice

margins.

Cell 1 Cell 2 Cell 3 Cell 4 Area 1 Area 2

Distance from top

(number of cells)

10 40 66 55 5 55

Distance from left

side(number of

cells)

10 40 48 25 5 55

7.3 RESULTS AND DISCUSSION

7.3.1 Number of required simulation runs

Naïve diffusion involves equiprobable selection of one of eight neighbouring

agents in the Moore neighbourhood [316]. This makes simulations probabilistic,


where each simulation run can be considered as an independent experiment.

Consequently, separate runs possibly generate different results [322]. In order to

overcome this stochasity, repetition of runs establishes validity of the results and

indicates a relative error involved in the simulations where an average result and

standard deviation can be obtained. The number of times to run a simulation plays a

significant role in the reliability of prediction and ensures that the simulation has

already converged on stable results [323].

Figure 7.5 shows traditional simulation errors of consecutive runs (iterations)

based on salt concentration in the areas 1 and 2 over 25000 time steps. Despite a high

value of error in a small number of runs, results converge quickly where the errors of

area 1 and 2 reached from 0.24 and 0.6 in the first run to approximately 0.0001 and

0.0008 after 10 runs. Area 1, which was located at a greater distance from rock salt

than area 2, demonstrated a smaller error than area 2 over various numbers of

simulation runs. It shows that farther areas from the rock salt block reach

convergence faster and have more accurate results at various runs.

Figure 7.5 Errors of salt concentration at area1 and 2 versus number of simulation

runs for the traditional technique.


Figure 7.6 shows simulation number 1 errors of consecutive simulation runs

based on the concentration of salt in the areas 1 and 2 over 25000 time steps. It

shows relatively low errors for area 1 during various iterations (4.8E-5, 4.28E-6 and

6.0E-7 for one, five and ten iterations), while errors of area 2 were generally higher

than area 1. The error of area 2 was dropped considerably over the simulation runs.

The error of area 2 was decreased from 8.5E-4 at one simulation run to 1.1E-4 and

9.2E-6 (one-eighth and one ninetieth of the single run respectively) after 5 and 10

simulation runs.

Figure 7.6 Errors of salt concentration at areas 1 and 2 versus number of simulation

runs for the simulation number 1 using hybrid agent.

Comparison between simulations using traditional agents (Figure 7.5) and

hybrid agents (Figure 7.6), both of which were based on global concentrations,

showed that using a hybrid agent could dramatically decrease the error of the


simulation (0.24 and 0.6 versus 4.8E-5 and 8.5E-4 for a single run). It took 10

iterations for the traditional agents to the errors of areas reached less than 0.001,

while the hybrid agent could achieve this with a single run. Therefore, simulations

using hybrid agents could converge considerably quicker than traditional agents,

which means hybrid agents’ simulations could generate more stable results than

traditional agents in equal simulation runs.

As hybrid agents are able to represent intra-agent properties, stability of the

simulation runs at various agents was tested by considering the intra-agent salt

proportion of the agents located at the cells. As the salt proportion of a hybrid agent

describes the salt fraction within the agent, it shows salt concentration within the

agent. Figure 7.7 shows consecutive simulation run errors at agents 1-4 located at

cells 1-4 over 25000 time steps based on their salt concentration. Agents located at

the centre and boundary of the rock salt block (agents 2 and 4 respectively)

demonstrated greater errors than agents located in the surrounding water (agents 1

and 3). The errors of the agents 2 and 4 were decreased from approximately 0.19 and

0.25 in the first run to 0.03 and 0.009 respectively after five runs. The lowest error

occurred at agent 1 (errors were equal to 0.0016 and 8.6E-5 after one and five runs

respectively), which was located at the greatest distance from the rock salt agents

among selected agents. The relatively high errors in the small number of simulation

runs were rapidly decreased by iteration. The agents’ errors were significantly

decreased until 10 simulation runs, when decrease in errors became slight.


Figure 7.7 Simulation runs’ error at hybrid agents 1-4 located at cells 1-4 over

25000 time steps based on their salt concentration in the first 20 consecutive

simulation runs.

According to Figures 7.6 and 7.7, salt concentration errors at areas (global errors)

were considerably lower than the agents (local errors). Consequently, the system

converged faster globally than locally. In addition, results confirmed that the number

of required runs for convergence was dependent on the location of selected area or

agent. Those that were located at a greater distance from the salt block generated

lower errors in general and required a lower number of simulation runs to converge.

7.3.2 Hybrid agent dissolution results

Dissolution of a rigid block of wet salt emerged from interactions between hybrid

agents and changes within the hybrid agents, where the evolution of the system was

represented by updating salt and water proportions of the agents using intra- and

extra-agent rules. This simulation was an example of using hybrid agents to provide


qualitative and quantitative data from a dissolvable, semi-permeable structure during

the dissolution process. Local salt concentration was defined as the proportion of the

salt (salt concentration) within the hybrid agent. The global salt concentration,

including salt concentration in a certain region emerged from local properties (Eq.

7.2).

In order to minimise simulation run error, the average value of ten simulation runs

has been taken. The colour-coded images of Figure 7.8 show distribution of salt

concentration in the lattice at time steps 0, 100, 1535 and 20000 for simulation

number 1 where λs, λf and TR equal 0.2, 0.5 and 0.7 respectively. Each colour depicts

salt concentration based on percentage according to the legends attached to the

images, where red shows the highest concentration in the pattern and dark blue

shows the lowest. The initial condition of the system has been shown at T=0, in

which the salt rock block (showed in red) has a square shape with 90% salt

concentration, while salt concentration of the other agents in the lattice (shown in

blue) equals zero (0), which demonstrates agents with 100% proportion of water.

Sharp corners of the wet salt block became round shortly after initiation of

dissolution process (picture at T=100), as in the real physical process of dissolution.

As a result of penetration of surrounding water and internal diffusion through the

structure of the rock salt, agents that formed wet salt started diluting. This changes

the initial homogeneous structure of the wet salt to a layered one, where salt

concentration of the agents was increased towards the centre of the block. When the

process starts, simultaneously with the percolation of water into the rock salt block,

salt dissolves and diffuses into the surrounding water. Images at T=100 and T=1535

show an increase of the salt concentration in the surrounding agents that initially

contained 100% water. The process of penetration of water into the rock salt block


continued until T=1535, when the salt concentration of all agents in the lattice was

less than threshold TR (TR equals 0.7 in simulation1). Consequently, all existing

agents behaved like fluid (λ=λf) after 1535 iterations. However, agents located at the

centre of the lattice still had a significant salt concentration. Evolution of the system

continued until T=20000, by which time all agents in the system reached almost

equal salt concentration (approximately 12.5%), which means a uniform distribution

of solid and fluid in the lattice. This is consistent with the real process of dissolution

of salt in water, which eventually results in a homogenous solution.

Global properties of the system can be driven from local properties (Section 7.2.5

and Eq. 7.2). In order to study global salt concentration over time, the lattice was

divided into 80 vertical layers (from top to bottom) and salt concentration was

T=0 T=100

T=1535 T=20000

Figure 7.8 Distribution of salt concentration in the lattice at different time steps

based on percentage when TR, λs and λf equal to 0.7, 0.5 and 0.2 respectively.


calculated for each vertical layer. Figure 7.9 shows salt concentration at vertical

layers for simulation number 1 at time steps 0 (initial condition), 100, 1535, 5000

and 20000 based on layer distance from the central layer. The initial condition

profile (T=0) shows the zero salt concentration of the layers far from the centre

suddenly increased to the maximum value in the layers around the centre (position

0), corresponding to the initial location of rock salt (Figure 7.8, T=0). The salt

concentrations of the layers were changed after starting the process where salt

concentration of the layers increased towards the central layer. All profiles reached

their peak in the central layer (position 0), while layers that are far from the middle

layer (position 1 and -1) have the minimum salt concentration. The maximum salt

concentration of profiles decreased over time, which resulted from penetration of

water into the salt block and dilution of the salt block agents. On the other hand, the

minimum value of the layers’ salt concentration (at position 1 and -1) increased over

time, stemming from the diffusing of salt in the surrounding water. The profile was

levelled off at equilibrium time (T=20000), which showed equal salt concentrations

in the layers.

Figure 7.9 Global salt concentration at vertical layers. Position 0 represents a

vertical layer from top to bottom of the lattice, which passes through the centre of the

salt block. Positions -1 and 1 present vertical layers located at margins of the lattice.


7.3.3 Effect of parameter change

During the process of dissolution, the size of the wet salt block changes due to

two contradictory phenomena that occur simultaneously. On one hand, since the salt

block acts like a semi-permeable porous medium, water percolation into the block

causes swelling, and increases the size of the block. On the other hand, salt dissolves

and salt molecules diffuse into the surrounding water, resulting in decaying salt

block. Parameter TR determines a threshold, above which a hybrid agent behaves

like a porous medium and below it, the agent becomes like a fluid. A low value for

TR describes a high capability of the porous medium-like agent for absorbing water

without attribute change, while a high value of TR shows a high tendency of the

hybrid agent to convert to a fluid-like agent after dilution. Figure 7.10 shows a

number of porous medium agents (salt block agents) for various TR and λs

(simulations 1-4) at time step 1 till 6000. The λf of all simulations were equal to 0.5

and the average value of ten simulation runs has been taken. The comparison

between simulations 2 and 4 shows that when λs kept constant and TR decreased

from 0.5 to 0.7, the number of required time steps for the disintegration of the salt

block became double (increased from 2000 to 4000). This means that lower TR

values resulted in the salt block persisting in the system for a longer time. Very low

values for TR, such as in simulation 3, may result in increasing the number of the

porous medium agents leading to swelling of the salt block, during early stages of the

dissolution process.

The number of porous medium agents, representing the size of the salt block,

diminished faster when a higher value of λs was allocated to the agents (simulation 1

versus simulation 2). Increasing λs accelerates dilution of the agent stemming from a

higher rate of percolation of water into the agent. Consequently, λs may be


considered as a parameter to reflect semi-permeability of the agent, in which

increasing λs makes the agent more permeable. Longer disintegration time for the salt

block means lower dissolution rate. Therefore, λs can control the dissolution rate of

the salt block.

It can be concluded that a rock salt block can swell if salt agents are highly

permeable and are able to stand rigid longer. The size increase will be intensified

when the low value of TR is accompanied with high magnitude of λs. This

overgrowth of the salt rock was along with previous experiments [324].

Figure 7.10 The number of porous medium agents over 6000 iterations for various

values of TR and λs.

Change of salt concentration in the surrounding water for simulation number

1-4 (Table 7.1) over 15000 time steps is showed in Figure 7.11. The fastest salt

concentration growth occurred in simulations 1 and 2, where both had the greatest

value of TR among simulations. Increasing λs from 0.1 (simulation 2) to 0.2

(simulation 1) slightly increased salt concentration growth rate. Simulation 1 versus

3, and simulation 2 versus 4 illustrate that the rate of salt concentration growth in the

surrounding water has a direct correlation with the value of the TR.


The salt concentration growth rate in the surrounding water before complete

disintegration of the salt block depends on the solubility of salt, where greater growth

rate shows higher solubility. Therefore, both TR and λs have a direct correlation with

the solubility of the salt.

Figure 7.11 Salt concentration in the surrounding water for various values of TR and

λs.

7.3.4 Effect of initial conditions

In this section, the effect of initial water proportion in the rock salt block, and

salinity of the surrounding water on the dissolution process were studied. In order to

study the effect of initial water proportion within the wet salt on the dissolution

process, simulation number 1 in Table 7.1 was compared with simulation number 5

where the only difference between them was initial salt proportion in the rock salt

(90% versus 99.9%). The salt block in simulation 5 contained 100 times less water

than simulation 1. The effect of salinity of surrounding water on the dissolution

process was investigated by comparing simulations 1 and 6, where parameters (TR,

λs and λf) of the simulations and salt block composition were the same. The salt


block was immersed in distilled water (zero salinity water) in simulation 1, while the

surrounding fluid contained 20% salt in simulation 6. Mean data of ten simulation

runs have been taken for all simulations.

Distribution of the salt concentration of the simulation numbers 1 and 5 at time

steps 1000 and 2000 are compared in Figure 7.12. The legends attached to the

pictures depict salt concentration based on percentage. The pictures show that the

concentration of the salt in the centre was lower in simulation 1 than simulation 5 at

both time steps. The distributions of salt concentration in the surrounding water were

almost the same for both simulations.

Simulation 1, T=1000 Simulation 5, T=1000

Simulation 1, T=2000 Simulation 5, T=2000

Figure 7.12 Distribution of salt concentration in the lattice for simulations 1 and 5 at

time steps 1000 and 2000.


Figure 7.13A shows change of the salt block size during simulations 1 and 5.

Although the size of the block in simulation 5 was initially smaller than simulation 1

(900 agents versus 810 agents) due to a greater proportion of the salt in the block,

salt blocked in simulation 5 was decayed slower than simulation 1. The size of the

salt block was equal in both simulations at 215 time steps, where salt blocks in

simulations 1 and 2 were approximately 60% and 70% of their initial sizes

respectively.

Figure 7.13B shows the global concentration of the salt in the surrounding

water during simulations 1 and 5. The value of the salt concentration was almost the

same for the simulations at various time steps. The system in both simulations also

reached equilibrium (salt concentration = 0.125) at the same time.

Different proportions of the water in the block resulted in different

disintegration times and decay rates. On the condition that there was the same

amount of the salt in the block, the initial proportion of the water had no effect on the

surrounding water salinity.

A B

Figure 7.13 A: Number of porous medium agents versus time step in simulations 1

and 5. B: Salt concentration in the surrounding water in simulations 1 and 5 at

various time steps.


The effect of salinity of the surrounding water on the rate of dissolution of the

rock salt block is seen in Figure 7.14, where the salt block was immersed in zero

salinity water in simulation 1 and 20% salt concentration in simulation 6. The salt

block consisted of 900 agents in both simulations at the beginning of the dissolution

(T=0). The salt rock was disintegrated after 1535 and 1850 time steps in simulations

1 and 6 respectively. The slope of the rock salt decay rate in simulation 1 was greater

than that in simulation 2. It means that rate of salt dissolution of surrounding water

with less salinity was higher. This finding is consistent with previous experimental

results in the literature [307, 325].

Figure 7.14 The number of porous medium agents over 2000 iterations for the

different salinity of the surrounding fluid.

7.3.5 Dissolution similar to the real condition

According to Figure 7.10, λs controls the dissolution rate of the salt block and

reflects permeability of agents (local permeability). As rock salt has a very low bulk

permeability [305, 306, 309, 310, 314] and relatively low dissolution rate [326, 327],


simulation number 7 in Table 7.1 was carried out, in which λs was equal to 0.0001.

The agents were assumed to stay rigid (behave like porous medium) up to 10% water

proportion (TR=0.90).

The colour-coded maps in Figure 7.15 show concentration of salt based on

percentage at time steps 10000 and 50000. Red depicts agents with 100% salt

proportion and dark blue depicts agents with 90% and less salt proportion in Figure

7.15 A. Therefore, due to the value of the TR (0.9), all dark blue in the images A and

B carry fluid-like attribute. The image at T=10000 illustrates that water could not

penetrate deep inside the block and a thin diluted boundary was created in the block

margins. Despite the decreasing size of the block due to dissolution at T=50000, the

thin diluted boundary still remained. This result is along with experimental

observations in the literature [324].

Figure 7.15 B shows salt concentration in the surrounding water at time steps

10000 and 50000. The legends attached to the pictures demonstrate salt

concentration based on percentage, in which red depicts 2% and more at T=10000

and 12% and more at T=50000. Overall, the salt concentration in T=50000 was

significantly greater than for T=10000. The concentration of salt near salt block was

noticeably higher than in farther regions, the same as a real situation.

Figure 7.15 provided spatiotemporal data of the dissolution process, in which

salt concentration of any point or area at any given time step could be determined.


A

T=10000 T=50000

B

T=10000 T=50000

Figure 7.15 Concentration of salt based on percentage at time steps 10000 and

50000.

Chapter 8: Discussion 165

Chapter 8: Discussion

Traditional agents are identified with their state, such as 0, 1 or black and

white. This simplicity limits them to only address phasic materials accurately where

different phases are addressed by agents with different states [26, 27, 178, 181]. The

agents used in today’s agent-based models operate like a mixture and are able to

carry one state or characteristic at a time. Therefore, when constituents of the system

are indistinguishable, the structure of the system and the way that components

interact become too complex to be simulated by traditional agents.

Over recent decades, a variety of agent-based techniques such as cellular

automata (CA) [227], lattice gas cellular automata (LGCA) [27, 200], lattice

Boltzmann (LB) [199, 328], lattice-Boltzmann discrete element method (LBDEM)

[329] and smoothed particle hydrodynamics (SPH) [330, 331], used rules or

equations and traditional simple agents for simulating porous media responses to

both external and internal stimuli where the complexity of the system required

complicated methods. The agents remained simple, despite the complexity of

equations and adapted rules for simulation of complex phenomena such as osmosis

and diffusion through a semi-permeable membrane [189, 200]. The same simple

agents were used for more complex phenomena that require integration of systems,

leading to complicated rules and formulation that increase the number of variables

and parameters and result in a more complicated input and output [332, 333]. In

spite of a rapid increase in computational power, adjusting parameters of complex

agent-based models based on known input and output (calibration), and fitting and

optimisation of results with experiments and observations of the environment

166 Chapter 8: Discussion

(validation), are still challenging problems [168, 333], thus limiting the agent-based

modelling approach relative to the level of complexity of rules and formulation.

The hybrid agent, which was developed in Chapter 3, contains constituents of

the system without any obligation to distinguish constituents. It provides us with an

opportunity to create more complex systems and structures such as single-phase

multi-component materials. The hybrid agent was adapted for porous structures,

where it carries the intrinsic properties of the materials such as porosity and

permeability. This creates the capability to define local properties for a porous

medium, which facilitates creating heterogeneous and anisotropic structures such as

articular cartilage.

The properties of the hybrid agent highly depend on values of the constituents

within the agent. The hybrid agent evolves when quantities of its constituents are

changed. In order to determine a control mechanism for intra-agent change of the

hybrid agent, intra-agent rules were introduced. The intra-agent rules enable the

hybrid agent to change gradually within itself. The presence of the components of the

system in the hybrid agent and intra-agent evolution adds a new level of complexity

to the agent. Collections of the hybrid agents’ evolutions (local evolutions) form

global change which shows the response of the system at the macroscopic level.

The hybrid agent is able to use some of the established simple neighbourhood

rules as extra-agent rules accompanied with intra-agent rules to create patterns of

emergent structures. Two well-known one-dimensional cellular automata (1D CA)

rules 22 and 73 were used as extra-agent rules in Chapter 4 to generate growing

patterns, which represent semi-permeable structures (Figures 4.5, 4.6, 4.7, 4.8 and

4.9). The patterns were beyond patterns that are generated by just mixing black and

white agents as representative of impervious (solid skeleton) and pores.

Chapter 8: Discussion 167

The methodology involving the combination of hybrid agents, intra- and

extra-agent rules, provided a unique opportunity for investigating the transient intra-

matrix diffusion of the articular cartilage in Chapter 5. For the first time, diffusion

and percolation of fluid into the cartilage as a single-phase material was successfully

investigated qualitatively (Figures 5.9) and quantitatively (Figures 5.10, 5.11) using

an agent-based method. The reasonably close agreement between simulated and

published experimental results of the healthy articular cartilage was shown in Table

5.1 and Figures 5.9 and 5.10. The success of this approach in the simulation of

diffusion throughout healthy articular cartilage suggests that it can be used for further

investigation of the functional characteristics of loaded and deforming articular

cartilage, and also tissues that are affected by degeneration and disease where current

methods are technically or ethically inadequate. The validated rules for the healthy

tissue model were applied to the partially and fully degenerated two-dimensional

(2D) model of articular cartilage, and spatial maps from inside the tissue at various

diffusion times were obtained (Figure 5.12).

The hybrid agent has been adapted to create the extracellular matrix of the

articular cartilage to simulate consolidation-type deformation in Chapter 6. The

simulation results were validated against experimental results in the literature. The

spatial and temporal variations of the porosity and fluid flow resistance, which

govern the deformation process at the microscale level, were obtained during the

articular cartilage’s function (Figures 6.12, 6.13, 6.16, 6.17 and 6.19). The

simulations demonstrate the capability of the approach for real-time qualitative and

quantitative observations of the intra-matrix activities of the healthy and degenerated

tissue, which was inaccessible in the past. As agents are micro-scale elements of an

agent-based structure, hybrid agents are capable of intra-agent evolution that

168 Chapter 8: Discussion

provides the feasibility of studying system change in time at a micro-scale level.

Therefore, micro-scale spatial and temporal data can be obtained in a manner

describable as using a “virtual microscope”.

The presented agent-based method was also extended to simulate dissolution of

wet rock salt as an example of a non-biological porous material in Chapter 7. The

effects of composition and permeability of the rock salt and salinity of the solvent

were investigated, where the flexibility of the approach has been demonstrated. The

system’s qualitative data, in the form of image and statistics (quantitative

information), at desired times has been obtained. The intra-agent condition of the

hybrid agent determines behaviours of the agent in the system (fluid- or porous

medium-like) and evolution of the system is a function of the evolution of the agents.

Therefore, intra-agent evolutions that are in a microscale level form the system

behaviour in a macroscale level.

Chapter 9: Conclusion, limitations and future work 169

Chapter 9: Conclusion, limitations and

future work

This thesis presented a computational framework based on an agent-based

method including an enhanced agent and new category of rules. The presented agent-

based method was validated against the articular cartilage functions and dissolution

of the rock salt in water. The presented enhanced agent and new category of rules

contributed, in creating a potential “virtual microscope”, which can potentially obtain

data from inside the complex experimentally-inaccessible structures.

The hybrid agent, which enables local micro-constituents to change in time and

space during agent-based simulations, has been developed and used in simulation to

demonstrate that it could lead to better effects in the creation of semi-permeable

porous materials from primitive agent characteristics. This agent offers improvement

in simulating biological single-phase porous structures, such as articular cartilage,

where the constituents of the tissue (fluid, proteoglycan and collagen) are practically

inseparable up to the ultramicroscopic levels. The hybrid agent and introduction of

the intra-agent rule would enable researchers to create emergent patterns that can

feed semi-permeable structures to models that are intended for analysis, such as mesh

free, finite element analysis, smooth particle hydrodynamics (SPH) and other micro-

agent based simulations.

The agent-based simulations of the articular cartilage consolidation and

diffusion conducted in this thesis were the first agent-based models developed to

investigate fluid flow and solid skeleton movement of the articular cartilage

extracellular matrix during its function. Agent-based methods provided the ability to

170 Chapter 9: Conclusion, limitations and future work

conduct experiments cost-effectively on the computer, where the cartilage was

probed virtually.

The in silico method in this thesis can represent single-phase multi-component

materials such as articular cartilage and wet salt on the computer. It provides the

ability to conduct experiments on the computer, where the issues of ethics can be

eliminated. It proposes a viable opportunity for in silico experiments that can

facilitate the provision of input data for numerical methods such as finite element

analysis, meshless and smoothed particle hydrodynamics. The combination of the

approach presented here with numerical methods can prepare a framework for

modelling and analysis of complex porous materials where the constituents of the

system may be indistinguishable in the manner of known mixtures.

In addition, this in silico method and outcomes will advance modelling of

articular cartilage’s extracellular matrix and extend methodologies for modelling and

investigating fluid-filled poroelastic functional tissues and materials. Also,

potentially, a cell (chondrocyte) model can be incorporated in order to study cell-

matrix interactions in health and disease. It leads to the understanding of the disease

and degeneration process and casts light into the development of disease such as

osteoarthritis.

The presented computational framework can potentially be used for other

biological tissues and processes such as kinetics of drug release from tablets and

implants, where drug mixes with solvent at the molecular level and affects body cells

via a diffusion process, oncology, when a cell gradually evolves to a cancer cell, and

tissue and tumour growth.

The scope of this research was limited to two-dimensional (2D) simulations. The

constituents of the articular cartilage were also simplified to solid and fluid, where

Chapter 9: Conclusion, limitations and future work 171

proteoglycans and collagens combined were considered solid. The role of

chondrocytes in the tissue was not taken into account and chondrocytes were

considered as a fluid. The layered distribution of the fluid and solid was used to

create the structure of the articular cartilage in 2D. The lateral-medial distribution of

the fluid and solid in the matrix was assumed to be uniform. The elastic deformation

of the indentation model in unloaded areas assumed to be ignorable. In the future,

more constituents of the articular cartilage, such as proteoglycans and collagens, can

be considered for the simulation. Moreover, the structure of the articular cartilage

can be created in three-dimensional (3D). Furthermore, more realistic degeneration,

such as increasing of fluid content due to both swelling and loss of matrix, can be

considered for the degenerated model.

In this thesis, one-dimensional consolidation and indentation of the bovine knee

cartilage and free diffusion of the human knee cartilage were simulated. In the future,

more functions, such as unconfined consolidation, dynamic loading and swelling of

the articular cartilage, can be studied. Articular cartilage from other joints such as

shoulder and spine can also be used.

The solid skeleton was removed uniformly to create degenerated models.

However, in the future, more realistic patterns of the degeneration can be used where

solid resorption occurs in a non-uniformly manner in the articular cartilage matrix.

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192 Appendices

Appendices

Appendix A

Matlab program for developing semi-permeable patterns, using hybrid agent,

local and global rules.

clear;clc;

profile on

profile clear

LN = 0 ; % Layer number

TL = 50;%50 ; % Total layer

NCL = 2*TL-1;%3 ; % number of cell in current layer

CeC = 1 ; % Capacity of the cell

PX = 0.5 ; % Percentage of decrease or increase

K0=0.15;

EPS=0.05 ;

INIfs= 1;%0.1 ; % initial condition

INIsf= 1/INIfs ; % initial condition

cellA(1,1:TL-1) = 0 ;

cellA(1,TL+1:NCL) = 0 ;

cellA(1,TL) = 1 ;

cell(1,1:TL-1) = 0 ;

cell(1,TL+1:NCL) = 0 ;

cell(1,TL) = INIsf ;

for Li = 1 : TL-1 ;

LN = LN + 1 ;

for j = 2 : NCL-1 ;

cellS = (cell ./ (1 + cell)) .* CeC ;

cellF = (1 ./ (1 + cell)) .* CeC ;

cellF(cellF(:,:) < 1e-25) = 0;

cellS(cellS(:,:) < 1e-25) = 0;

cellF(cellF(:,:) > 1-1e-25) = 1;

cellS(cellS(:,:) > 1-1e-25) = 1;

cellS1 = zeros(TL,NCL);

cellF1 = zeros(TL,NCL);















if cellS(LN,j-1) ~= 0 & cellS(LN,j) ~= 0 & cellS(LN,j+1) ~= 0

cellS1(LN+1,j) = 0 ;

A = [cellS(LN,j-1) cellS(LN,j) cellS(LN,j+1)] ;

cellF1(LN+1,j) = min(cellS(LN,j-1),min(cellS(LN,j),cellS(LN,j+1))) ; %

mean (A) ;

end ;

if cellS(LN,j-1) ~= 0 & cellS(LN,j) ~= 0 & cellF(LN,j+1) ~= 0


A = [cellS(LN,j-1) cellS(LN,j) cellF(LN,j+1)] ;

cellF2(LN+1,j) = min(cellS(LN,j-1),min(cellS(LN,j),cellF(LN,j+1))) ;

end ;

if cellS(LN,j-1) ~= 0 & cellF(LN,j) ~= 0 & cellS(LN,j+1) ~= 0


A = [cellS(LN,j-1) cellF(LN,j) cellS(LN,j+1)];

cellF3(LN+1,j) = min(cellS(LN,j-1),min(cellF(LN,j),cellS(LN,j+1))) ;

end ;

if cellS(LN,j-1) ~= 0 & cellF(LN,j) ~= 0 & cellF(LN,j+1) ~= 0

A = [cellS(LN,j-1) cellF(LN,j) cellF(LN,j+1)] ;

Appendices 193

cellS4(LN+1,j) = min(cellS(LN,j-1),min(cellF(LN,j),cellF(LN,j+1))) ;

cellF4(LN+1,j) = 0 ;

end ;

if cellF(LN,j-1) ~= 0 & cellS(LN,j) ~= 0 & cellS(LN,j+1) ~= 0


A = [cellF(LN,j-1) cellS(LN,j) cellS(LN,j+1)] ;

cellF5(LN+1,j) = min(cellF(LN,j-1),min(cellS(LN,j),cellS(LN,j+1))) ;

end ;

if cellF(LN,j-1) ~= 0 & cellS(LN,j) ~= 0 & cellF(LN,j+1) ~= 0

A = [cellF(LN,j-1) cellS(LN,j) cellF(LN,j+1)] ;

cellS6(LN+1,j) = min(cellF(LN,j-1),min(cellS(LN,j),cellF(LN,j+1))) ;


end ;

if cellF(LN,j-1) ~= 0 & cellF(LN,j) ~= 0 & cellS(LN,j+1) ~= 0

A = [cellF(LN,j-1) cellF(LN,j) cellS(LN,j+1)] ;

cellS7(LN+1,j) = min(cellF(LN,j-1),min(cellF(LN,j),cellS(LN,j+1))) ;


end ;

if cellF(LN,j-1) ~= 0 & cellF(LN,j) ~= 0 & cellF(LN,j+1) ~= 0


A = [cellF(LN,j-1) cellF(LN,j) cellF(LN,j+1)] ;

cellF8(LN+1,j) = min(cellF(LN,j-1),min(cellF(LN,j),cellF(LN,j+1))) ;

end ;

cellSf(LN+1,j) = cellS1(LN+1,j) + cellS2(LN+1,j) + cellS3(LN+1,j) +

cellS4(LN+1,j) + cellS5(LN+1,j) + cellS6(LN+1,j) + cellS7(LN+1,j) + cellS8(LN+1,j) ;

cellFf(LN+1,j) = cellF1(LN+1,j) + cellF2(LN+1,j) + cellF3(LN+1,j) +

cellF4(LN+1,j) + cellF5(LN+1,j) + cellF6(LN+1,j) + cellF7(LN+1,j) + cellF8(LN+1,j) ;

end ;

cellSf(LN+1,NCL) = 0 ;

cellFf(LN+1,NCL) = 0 ;

cellS(LN+1,:) = cellSf(LN+1,:) ; % + cellS(LN,:) ;

cellF(LN+1,:) = cellFf(LN+1,:) ; % + cellF(LN,:) ;

cellF(LN+1,1) = cellF(LN+1,NCL-1) ;

cellF(LN+1,NCL) = cellF(LN+1,2) ;

cellS(LN+1,1) = cellS(LN+1,NCL-1) ;

cellS(LN+1,NCL) = cellS(LN+1,2) ;

cellS(isnan(cellS))=0;

cellS(isinf(cellS))=1e30;

cellF(isnan(cellF))=0;

cellF(isinf(cellF))=1e30;

cell (LN+1,:) = cellS(LN+1,:) ./ cellF(LN+1,:) ;

cell(isnan(cell))=0;

cell(isinf(cell))=1e30;

end ;

CC=1 ./ cell;

imagesc(CC)

Fluid = 1 ./ (1+cell) ;

Solid = cell ./ (1+cell) ;

for i = 1 : TL

SF (i,1) = sum(Solid(i,:)) ./ sum(Fluid(i,:)) ;

end ;

194 Appendices

Appendix B

Matlab program for simulating diffusion throughout the articular cartilage.

clear;clc;

profile on

profile clear

KK1=0;

KK2=0;

KK3=0;

KK4=0;

KK5=0;

KK6=0;

TimeStep1 = 1 ;

TVC = 1 ;

KH = 40;

KV = 40;

KTop = 1;

KSide = 1;

TotalTimeSteps = 20000;

FSL = xlsread('TestDiffusion.xlsx');

Beta = 0.7;

FSL = (FSL + Beta) ./ (1- Beta) ;

XD1 = 14 ;

YD1 = 1 ;

Beta = 0;%0.2 ;

XCent = round (size(FSL,1)/2) ;

YCent = round (size(FSL,2)/2) ;

FSL(XCent-XD1:XCent+XD1, YCent-YD1:YCent+YD1) = (FSL(XCent-XD1:XCent+XD1, YCent-

YD1:YCent+YD1) + Beta) ./ (1- Beta) ;

XD1 = 1 ;

XD2 = 15 ;

YD1 = 5 ;

YD2 = 7 ;

Beta = 0;%0.2 ;



FSL(XD1:XD2, YD1:YD2) = (FSL(XD1:XD2, YD1:YD2) + Beta) ./ (1- Beta) ;

XD1 = 1 ;

XD2 = 20 ;

YD1 = 45 ;

YD2 = 47 ;

Beta = 0;%0.2 ;



FSL(XD1:XD2, YD1:YD2) = (FSL(XD1:XD2, YD1:YD2) + Beta) ./ (1- Beta) ;

XD1 = 2 ;

YD1 = 2 ;

Beta = 0;





XD1 = 2 ;

YD1 = 2 ;

Beta = 0;

XCent = round (size(FSL,1)/1.2) ;




XD1 = 2 ;

YD1 = 2 ;

Beta = 0;

XCent = round (size(FSL,1)/3.5) ;

YCent = round (size(FSL,2)/1.25) ;



x1=size(FSL,1);

y1=size(FSL,2);

xg=x1+10;

yg=y1+10;

FSLattice(1:xg,1:yg)=zeros ;

FSLattice(6:xg-5,6:yg-5)=FSL ;

clear('x1','y1')

Alpha = 1.*(FSLattice ./ KH).^1.0 ;

Appendices 195

AlphaH = 1.*(FSLattice ./ KH).^1.0 ;

AlphaV = 1.*(FSLattice ./ KV).^1.0 ;

APopulation(1:xg,1:yg) = zeros ;

BPopulation(1:xg,1:yg) = zeros ;

APopulation(6:xg-5,6:yg-5) = ((TVC .* FSLattice(6:xg-5,6:yg-5)) ./(1+FSLattice(6:xg-

5,6:yg-5))) ;

BPopulation(6:xg-5,6:yg-5) = (TVC ./(1+FSLattice(6:xg-5,6:yg-5))) ;

CPopulation (1:xg,1:5) = TVC ;

CPopulation (1:xg,yg-4:yg) = TVC ;

CPopulation (1:5,1:yg) = TVC ;

CPI = CPopulation ;

sum(sum(APopulation(6:xg-5,6:yg-5)))

sum(sum(CPopulation(6:xg-5,6:yg-5)))

for TimeStep = 1 : TotalTimeSteps

tic

APopulationNew = zeros(xg,yg) ;

CPopulationNew = zeros(xg,yg) ;

i = 6 ;

for j = 7 : yg-6 ;

TotalFAout = AlphaH (i,j) * APopulation(i,j)/4;

TotalFCout = AlphaH (i,j) * CPopulation(i,j)/4;

TotalMFOUT = TotalFAout + TotalFCout ;

APopulationNew(i,j) = APopulationNew(i,j) - 4 * TotalFAout ;

CPopulationNew(i,j) = CPopulationNew(i,j) - 4 * TotalFCout ;

APopulationNew(i,j-1) = APopulationNew(i,j-1) + TotalFAout ;

APopulationNew(i,j-1) = APopulationNew(i,j-1) - (TotalMFOUT/(APopulation(i,j-

1)+CPopulation(i,j-1)))*APopulation(i,j-1) ;

APopulationNew(i,j) = APopulationNew(i,j) + (TotalMFOUT/(APopulation(i,j-


CPopulationNew(i,j-1) = CPopulationNew(i,j-1) + TotalFCout ;

CPopulationNew(i,j-1) = CPopulationNew(i,j-1) - (TotalMFOUT/(APopulation(i,j-

1)+CPopulation(i,j-1)))*CPopulation(i,j-1) ;

CPopulationNew(i,j) = CPopulationNew(i,j) + (TotalMFOUT/(APopulation(i,j-


APopulationNew(i,j+1) = APopulationNew(i,j+1) + TotalFAout ;

APopulationNew(i,j+1) = APopulationNew(i,j+1) -

(TotalMFOUT/(APopulation(i,j+1)+CPopulation(i,j+1)))*APopulation(i,j+1) ;

APopulationNew(i,j) = APopulationNew(i,j) +


CPopulationNew(i,j+1) = CPopulationNew(i,j+1) + TotalFCout ;

CPopulationNew(i,j+1) = CPopulationNew(i,j+1) -

(TotalMFOUT/(APopulation(i,j+1)+CPopulation(i,j+1)))*CPopulation(i,j+1) ;

CPopulationNew(i,j) = CPopulationNew(i,j) +


APopulationNew(i,j) = APopulationNew(i,j) + (TotalMFOUT/(APopulation(i-

1,j)+CPopulation(i-1,j)))*APopulation(i-1,j) ;

CPopulationNew(i,j) = CPopulationNew(i,j) + (TotalMFOUT/(APopulation(i-

1,j)+CPopulation(i-1,j)))*CPopulation(i-1,j) ;

APopulationNew(i+1,j) = APopulationNew(i+1,j) + TotalFAout ;

APopulationNew(i+1,j) = APopulationNew(i+1,j) -

(TotalMFOUT/(APopulation(i+1,j)+CPopulation(i+1,j)))*APopulation(i+1,j) ;



CPopulationNew(i+1,j) = CPopulationNew(i+1,j) + TotalFCout ;

CPopulationNew(i+1,j) = CPopulationNew(i+1,j) -

(TotalMFOUT/(APopulation(i+1,j)+CPopulation(i+1,j)))*CPopulation(i+1,j) ;



end ;

i = xg-5 ;

for j = 7 : yg-6 ;



TotalMFOUT = (TotalFAout + TotalFCout) ;











196 Appendices













APopulationNew(i-1,j) = APopulationNew(i-1,j) + TotalFAout ;

APopulationNew(i-1,j) = APopulationNew(i-1,j) - (TotalMFOUT/(APopulation(i-




CPopulationNew(i-1,j) = CPopulationNew(i-1,j) + TotalFCout ;

CPopulationNew(i-1,j) = CPopulationNew(i-1,j) - (TotalMFOUT/(APopulation(i-




end ;

j = 6 ;

for i = 7 : xg-6 ;








































end ;

j = yg-5 ;

for i = 7 : xg-6 ;







Appendices 197


































end ;

i = 6 ;

j = 6 ;


































i = 6 ;

j = yg-5 ;







198 Appendices




























i = xg-5 ;

j = 6 ;






























i = xg-5 ;

j = yg-5 ;
















Appendices 199















for i = 7 : xg-6 ;

for j = 7 : yg-6 ;

TotalFAout = AlphaH (i,j) * APopulation(i,j)/4; % Total

TotalFCout = AlphaH (i,j) * CPopulation(i,j)/4; % Total TotalMFOUT =

(TotalFAout + TotalFCout) ; % share of each APopulationNew(i,j) =

APopulationNew(i,j) - 4 * TotalFAout ;



APopulationNew(i,j-1) = APopulationNew(i,j-1) -

(TotalMFOUT/(APopulation(i,j-1)+CPopulation(i,j-1)))*APopulation(i,j-1) ;




CPopulationNew(i,j-1) = CPopulationNew(i,j-1) -

(TotalMFOUT/(APopulation(i,j-1)+CPopulation(i,j-1)))*CPopulation(i,j-1) ;














APopulationNew(i-1,j) = APopulationNew(i-1,j) -

(TotalMFOUT/(APopulation(i-1,j)+CPopulation(i-1,j)))*APopulation(i-1,j) ;




CPopulationNew(i-1,j) = CPopulationNew(i-1,j) -

(TotalMFOUT/(APopulation(i-1,j)+CPopulation(i-1,j)))*CPopulation(i-1,j) ;













end ;

end ;

APopulation = APopulation + APopulationNew ;

CPopulation = CPopulation + CPopulationNew ;

for i = 6 : xg-5 ;

for j = 6 : yg-5 ;

if APopulation(i,j) < 0

Neighbour1 = 1 ;

Neighbour2 = 1 ;

Neighbour3 = 1 ;

200 Appendices

Neighbour4 = 1 ;

Neighbour5 = 1 ;

ExtraA = abs(APopulation(i,j)) ;

CPopulation(i,j) = CPopulation(i,j) + APopulation(i,j) ;

APopulation(i,j) = 0 ;

NAPos1 = 0 ;

PosCell1 = zeros(1,8) ;

clear('NCell1') ;

for i1 = -1 : 1 ;

for j1 = -1 : 1 ;

if i1 ~=0 | j1 ~= 0

NAPos1 = NAPos1 + 1 ;

NCell1(NAPos1) = APopulation(i+i1,j+j1) ;

if APopulation(i+i1,j+j1) > 0

PosCell1(NAPos1) = 1 ;

end ;

end ;

end ;

end ;

TotalNeighbourA1 = sum(NCell1 .* PosCell1 ) ;

if sum(TotalNeighbourA1) >= ExtraA

KK1=KK1+1 ;

TS1(KK1) = TimeStep ;

NAPos1 = 0 ;

for i1 = -1 : 1 ;

for j1 = -1 : 1 ;

if i1 ~=0 | j1 ~= 0


if PosCell1(NAPos1) == 1

APopulation(i+i1,j+j1) = APopulation(i+i1,j+j1) -

(APopulation(i+i1,j+j1) / TotalNeighbourA1) * ExtraA ;

CPopulation(i+i1,j+j1) = CPopulation(i+i1,j+j1) +

(APopulation(i+i1,j+j1) / TotalNeighbourA1) * ExtraA ;

end ;

end ;

end ;

end ;

else

Neighbour1 = 0 ;

end ;

if Neighbour1 == 0

NAPos2 = 0 ;


clear('NCell2') ;

for i1 = -2 : 2 ;

for j1 = -2 : 2 ;

if i1 ~=0 | j1 ~= 0





end ;

end ;

end ;

end ;



KK2=KK2+1 ;


NAPos2 = 0 ;

for i1 = -2 : 2 ;

for j1 = -2 : 2 ;

if i1 ~=0 | j1 ~= 0



APopulation(i+i1,j+j1) =

APopulation(i+i1,j+j1) - (APopulation(i+i1,j+j1) / TotalNeighbourA2) * ExtraA ;

CPopulation(i+i1,j+j1) =

CPopulation(i+i1,j+j1) + (APopulation(i+i1,j+j1) / TotalNeighbourA2) * ExtraA ;

end ;

end ;

end ;

end ;

else

Neighbour2 = 0 ;

end ;

end ;

Appendices 201

if Neighbour1 == 0 & Neighbour2 == 0

NAPos3 = 0 ;


clear('NCell3') ;

for i1 = -3 : 3 ;

for j1 = -3 : 3 ;

if i1 ~=0 | j1 ~= 0





end ;

end ;

end ;

end ;



KK3=KK3+1 ;


NAPos3 = 0 ;

for i1 = -3 : 3 ;

for j1 = -3 : 3 ;

if i1 ~=0 | j1 ~= 0







end ;

end ;

end ;

end ;

else

Neighbour3 = 0 ;

end ;

end ;

if Neighbour1 == 0 & Neighbour2 == 0 & Neighbour3 == 0

NAPos4 = 0 ;


clear('NCell4') ;

for i1 = -4 : 4 ;

for j1 = -4 : 4 ;

if i1 ~=0 | j1 ~= 0





end ;

end ;

end ;

end ;



KK4=KK4+1 ;


NAPos4 = 0 ;

for i1 = -4 : 4 ;

for j1 = -4 : 4 ;

if i1 ~=0 | j1 ~= 0







end ;

end ;

end ;

end ;

else

Neighbour4 = 0 ;

end ;

end ;

if Neighbour1 == 0 & Neighbour2 == 0 & Neighbour3 == 0 & Neighbour4

== 0

202 Appendices

NAPos5 = 0 ;


clear('NCell5') ;

for i1 = -5 : 5 ;

for j1 = -5 : 5 ;

if i1 ~=0 | j1 ~= 0





end ;

end ;

end ;

end ;



KK5=KK5+1 ;


NAPos5 = 0 ;

for i1 = -5 : 5 ;

for j1 = -5 : 5 ;

if i1 ~=0 | j1 ~= 0







end ;

end ;

end ;

end ;

else

Neighbour5 = 0 ;

end ;

end ;


== 0 & Neighbour5 == 0

NAPos6 = 0 ;

KK6=KK6+1 ;


PosCell6 = zeros(1,(xg-10)*(yg-10)) ;

clear('NCell6') ;

for i1 = 6 : xg-5 ;

for j1 = 6 : yg-5 ;


NCell6(NAPos6) = APopulation(i1,j1) ;

if APopulation(i1,j1) > 0


end ;

end ;

end ;



NAPos6 = 0 ;

for i1 = 6 : xg-5 ;

for j1 = 6 : yg-5 ;



APopulation(i1,j1) = APopulation(i1,j1) -

(APopulation(i1,j1) / TotalNeighbourA6) * ExtraA ;

CPopulation(i1,j1) = CPopulation(i1,j1) +

(APopulation(i1,j1) / TotalNeighbourA6) * ExtraA ;

end ;

end ;

end ;

else

Neighbour6 = 0 ;

end ;

end ;


== 0

end;

end ;

APopulation(1:5,:) = 0 ;

APopulation(xg-4:xg,:) = 0 ;

Appendices 203

APopulation(:,1:5) = 0 ;

APopulation(:,yg-4:yg) = 0 ;

if CPopulation(i,j) < 0

APopulation(i,j) = CPopulation(i,j) + APopulation(i,j) ;

CPopulation(i,j) = 0 ;

end ;

end ;

end ;

IntDatCollect = 1 ;

TimeStep1 = TimeStep1 + 1 ;

AN(:,:,TimeStep) = APopulation ; % Unmarked fluid

CN(:,:,TimeStep) = CPopulation ; % Marked fluid

Time(TimeStep,1) = toc ;

end;

clear 'CNLR'

clear 'Error'

Time=100; % required Timestep

X1 = round(yg/2) - 28 ; %17

X2 = round(yg/2) + 28 ; %19

Y1 = 6 ;

Y2 = size(AN,1)-5 ;

KT=0;

CONS = CN ./ (CN+AN); % Concentration of the marked fluid (Ratio of marked fluid to

total fluid)

CONS(isnan(CONS))=0 ;

Interval = 3;

for Time = 1:1:fix((TimeStep-1)/IntDatCollect)

k=0;

KT=KT+1;

ErrorLR (KT,1)= Time ;

for i=X1:Interval:X2

k=k+1;

CNLR (k,KT) = sum(sum (CONS (Y1:Y2,i:i+Interval-1,Time))) /

(Interval*(size(AN,1)-11)) ;

end;

end ;

clear 'CNLL'

WD=round ((size(AN,2)-10)/5);

X1 = round (yg/2) - round (WD/1) ;

X2 = round (yg/2) + round(WD/1) ;

KT=0;

for Time = 1:1:fix((TimeStep-1)/IntDatCollect)

k=0;

KT=KT+1;

ErrorD (KT,1)= Time ;

for i=7:2:size(AN,1)-5

k=k+1;

CNLL (k,KT) = sum(sum (CN (i:i+2,X1:X2,Time))) / (sum(sum (AN

(i:i+2,X1:X2,Time))) + sum(sum (CN (i:i+2,X1:X2,Time)))) ;

end;

end;

clear 'ErrorLR'

Exp24h =[11.36 11.59 11.64 11.5 11.5 11.5 11.5 11.57 11.45 11.64 11.87 11.8 11.55

11.68 11.86 11.95 11.9 11.82 11.68] ;

Exp24h = Exp24h' ;

Exp2h =[7.77 7.68 7.18 6.68 6.64 6.18 5.91 5.77 5.68 5.72 6.09 6.09 5.77 6.22 6.77

7.14 7.32 7.73 8.05] ;

Exp2h = Exp2h' ;

Exp2h = Exp2h ./ Exp24h ; %11.5 ;

Exp4h =[9.05 8.95 8.95 8.72 8.32 7.95 7.86 7.59 7.65 7.68 7.82 7.75 7.86 8.23 8.55

8.68 8.77 9.05 9.23] ;

Exp4h = Exp4h' ;

Exp4h = Exp4h ./ Exp24h ; %11.5 ;

Exp6h =[9.27 9.73 9.73 9.45 9.36 9.14 8.91 8.91 8.68 9 9.27 9 9.05 9.5 9.82 9.82 9.95

10.13 10.2] ;

Exp6h = Exp6h' ;

Exp6h = Exp6h ./ Exp24h ; %11.5 ;

Exp8h =[10.09 10.09 10.32 10.09 9.95 9.81 9.77 9.82 9.73 9.9 10.05 9.9 9.82 10.18

10.62 10.41 10.37 10.46 10.45] ;

Exp8h = Exp8h' ;

Exp8h = Exp8h + 0.04 ;

Exp8h = Exp8h ./ Exp24h ; %11.5 ;

ExpTimeLR = [1 ; 2 ; 3;4;5;6;7;8;9;10;11;12;13;14;15;16;17;18;19] ;

ExpTimeD = [1 ; 2 ; 3;4;5;6;7;8;9;10;11;12;13;14;15] ;

for i=1: size (CNLR,2);

ErrorLR (i,1) = i ;

204 Appendices

ErrorLR (i,2) = sum ((Exp2h - CNLR(:,i)) .^2) / size(Exp2h,1) ;




CVLR (i,1) = i ;

CVLR (i,2) = (sum ((Exp2h - CNLR(:,i)) .^2) / size(Exp2h,1))^0.5 /

mean(CNLR(:,i)) ;


mean(CNLR(:,i)) ;


mean(CNLR(:,i)) ;


mean(CNLR(:,i)) ;

end ;

for i=1: size (CNLR,2);

MAELR (i,1) = i ;

MAELR (i,2) = ( sum (abs(Exp2h - CNLR(:,i))) ./ size(Exp2h,1) ) / mean(CNLR(:,i))

;


;


;


;

end ;

clear 'ErrorD'

ExpD2h = [10.81 10 9.2 8.3 7.3 6.4 5.6 5.1 4.6 4.1 3.2 2.8 2.6 2.3 2.1] ;

ExpD2h = ExpD2h' ;

ExpD4h = [11.64 11.18 10.45 9.82 9.09 8.45 7.82 7.36 6.95 6.5 6.09 5.55 4.82 4.27

3.73] ;

ExpD4h = ExpD4h' ;

ExpD6h = [12.64 11.9 11.2 10.55 9.9 9.36 9 8.64 8.41 8 7.64 7.18 6.63 6 5.27] ;

ExpD6h = ExpD6h' ;

ExpD8h = [13.42 12.91 12.28 11.65 11.14 10.63 10.25 10.06 9.62 9.24 8.8 8.29 7.85

7.23 6.52] ;

ExpD8h = ExpD8h' ;

ExpD24h = [13.73 13.41 12.95 12.45 12 11.64 11.27 11.09 11 10.82 10.68 10.45 10.18

9.73 9.36] ;

ExpD24h = ExpD24h' ;

ExpD12h = [13.4 12.82 12.18 11.73 11.18 10.73 10.36 10.15 10 9.82 9.45 9.09 8.64 8

7.45] ;

ExpD12h = ExpD12h' ;

ExpD2h = ExpD2h ./ ExpD24h ; % 13.4;%Comp1 ; %ExpD24h ; %

ExpD4h = ExpD4h ./ ExpD24h ; %13.4;%Comp1 ; %ExpD24h ; %

ExpD6h = ExpD6h ./ ExpD24h ; %13.4;%Comp1 ; %ExpD24h ;%

ExpD8h = ExpD8h ./ ExpD24h ;

for i=1: size (CNLL,2);

ErrorD (i,1) = i ;

ErrorD (i,2) = sum ((ExpD2h - CNLL(:,i)) .^2) / size(ExpD2h,1) ;



CVD (i,1) = i ;

CVD (i,2) = (sum ((ExpD2h - CNLL(:,i)) .^2) / size(ExpD2h,1))^0.5 /

mean(CNLL(:,i)) ;


mean(CNLL(:,i)) ;


mean(CNLL(:,i)) ;

end ;

for i=1: size (CNLL,2);

MAED (i,1) = i ;

MAED (i,2) = ( sum (abs(ExpD2h - CNLL(:,i))) ./ size(ExpD2h,1) ) /

mean(CNLL(:,i)) ;


mean(CNLL(:,i)) ;


mean(CNLL(:,i)) ;


mean(CNLL(:,i)) ;

end ;

[a,b] = min(CVD(:,2:4)) ;

TimeStepCLR(:,3)=b ;

b(:,2) = round (b(:,2) ./ 2) ;

b(:,3) = round ( b(:,3) ./ 3) ;

minLD = min(b) ;

maxLD = max(b) ;

ErrorCLR(:,1)=a;

Appendices 205


[a,b] = min(CVLR(:,2:4)) ;


b(:,2) = round (b(:,2) ./ 2) ;

b(:,3) = round ( b(:,3) ./ 3) ;

minLR = min(b) ;

maxLR = max(b) ;

ErrorCLR(:,2)=a;


k=1;

for i=min(minLR, minLD)-100: max(maxLR,maxLD)+100;

TotalCVLR (k,1) = i ;

TotalCVLR (k,2) = CVLR (i,2) + CVLR (2*i,3) + CVLR (3*i,4) ;

TotalCVD (k,1) = i ;

TotalCVD (k,2) = CVD (i,2) + CVD (2*i,3) + CVD (3*i,4) ;

TotalCV (k,1) = i ;

TotalCV (k,2) = CVLR (i,2) + CVLR (2*i,3) + CVLR (3*i,4) + CVD (i,2) + CVD

(2*i,3) + CVD (3*i,4) ;

k=k+1 ;

end ;

[a,b] = min(TotalCV(:,2)) ;

Int2h = TotalCV(b,1) ;

LRfit(:,1) = Exp2h ;

LRfit(:,2) = CNLR(:,Int2h) ;


LRfit(:,4) = CNLR(:,2*Int2h) ;


LRfit(:,6) = CNLR(:,3*Int2h) ;

Dfit(:,1) = ExpD2h ;

Dfit(:,2) = CNLL(:,Int2h) ;


Dfit(:,4) = CNLL(:,2*Int2h) ;


Dfit(:,6) = CNLL(:,3*Int2h) ;

[a,b] = min(MAED(:,2:4)) ;


b(:,2) = round (b(:,2) ./ 2) ;

b(:,3) = round ( b(:,3) ./ 3) ;

minLD = min(b) ;

maxLD = max(b) ;

ErrorCLR(:,1)=a;


[a,b] = min(MAELR(:,2:4)) ;


b(:,2) = round (b(:,2) ./ 2) ;

b(:,3) = round ( b(:,3) ./ 3) ;

minLR = min(b) ;

maxLR = max(b) ;

ErrorCLR(:,2)=a;


k=1;

for i=min(minLR, minLD)-100: max(maxLR,maxLD)+100;

TotalMAELR (k,1) = i ;

TotalMAELR (k,2) = MAELR (i,2) + MAELR (2*i,3) + MAELR (3*i,4) ;

TotalMAED (k,1) = i ;

TotalMAED (k,2) = MAED (i,2) + MAED (2*i,3) + MAED (3*i,4) ;

TotalMAE (k,1) = i ;

TotalMAE (k,2) = MAELR (i,2) + MAELR (2*i,3) + MAELR (3*i,4) + MAED (i,2) + MAED

(2*i,3) + MAED (3*i,4) ;

k=k+1 ;

end ;

[a,b] = min(TotalMAE(:,2)) ;

MInt2h = TotalMAE(b,1) ;

CVLRfit (1,1) = (sum ((Exp2h - CNLR(:,Int2h)) .^2) / size(Exp2h,1))^0.5 /

mean(CNLR(:,Int2h)) ;

CVLRfit (1,2) = (sum ((Exp4h - CNLR(:,2*Int2h)) .^2) / size(Exp4h,1))^0.5 /

mean(CNLR(:,2*Int2h)) ;





CVDfit (1,1) = (sum ((ExpD2h - CNLL(:,Int2h)) .^2) / size(ExpD2h,1))^0.5 /

mean(CNLL(:,Int2h)) ;

206 Appendices

CVDfit (1,2) = (sum ((ExpD4h - CNLL(:,2*Int2h)) .^2) / size(ExpD4h,1))^0.5 /

mean(CNLL(:,2*Int2h)) ;





MAELRfit (1,1) = ( sum (abs(Exp2h - CNLR(:,MInt2h))) ./ size(Exp2h,1) ) /

mean(CNLR(:,MInt2h)) ;

MAELRfit (1,2) = ( sum (abs(Exp4h - CNLR(:,2*MInt2h))) ./ size(Exp4h,1) ) /

mean(CNLR(:,2*MInt2h)) ;





MAEDfit (1,1) = ( sum (abs(ExpD2h - CNLL(:,MInt2h))) ./ size(ExpD2h,1) ) /

mean(CNLL(:,MInt2h)) ;

MAEDfit (1,2) = ( sum (abs(ExpD4h - CNLL(:,2*MInt2h))) ./ size(ExpD4h,1) ) /

mean(CNLL(:,2*MInt2h)) ;





[a,b] = min(MAPED(:,2:4)) ;


b(:,2) = round (b(:,2) ./ 2) ;

b(:,3) = round ( b(:,3) ./ 3) ;

minLD = min(b) ;

maxLD = max(b) ;

ErrorCLR(:,1)=a;


[a,b] = min(MAPELR(:,2:4)) ;


b(:,2) = round (b(:,2) ./ 2) ;

b(:,3) = round ( b(:,3) ./ 3) ;

minLR = min(b) ;

maxLR = max(b) ;

ErrorCLR(:,2)=a;


Appendices 207

Appendix C

Matlab program for simulating deformation of the articular cartilage.

clear;clc;

profile on

profile clear

k=0;

TVC = 1 ;

TotalTimeSteps = 14000;

FSL = xlsread('Test.xlsx');

x1=size(FSL,1);

y1=size(FSL,2);

xg=x1+7;

yg=y1+2;

FSLattice(1:xg,1:yg)=zeros ;

XXP = 0 ;

FSLattice(7:xg-1,2:yg-1)= (FSL + XXP) ./ (1-XXP) ;

FSLattice = (FSLattice./(1+FSLattice)) ./ (1./ (1.323 .* (1+FSLattice))) ;

clear('FSL','x1','y1')

Alphaini = FSLattice ;

KH = (Alphaini ./ Alphaini) .^ 1 ;

KV=(Alphaini ./ Alphaini) .^ 1 ;

KO=(Alphaini ./ Alphaini) .^ 1 ;

KH(isnan(KH)) = 0 ;

KV(isnan(KV)) = 0 ;

KO(isnan(KO)) = 0 ;

APopulation(1:xg,1:yg) = zeros ;

BPopulation(1:xg,1:yg) = zeros ;

APopulationini(1:xg,1:yg) = zeros ;

BPopulationini(1:xg,1:yg) = zeros ;

APopulation(2:xg-1,2:yg-1) = ((TVC .* FSLattice(2:xg-1,2:yg-1)) ./(1+FSLattice(2:xg-

1,2:yg-1))) ;

APopulation(isnan(APopulation)) = 0 ;

BPopulation(7:xg-1,2:yg-1) = (TVC ./(1+FSLattice(7:xg-1,2:yg-1))) ;

BPopulation(isnan(BPopulation)) = 0 ;

APopulationini(2:xg-1,2:yg-1) = ((TVC .* FSLattice(2:xg-1,2:yg-1))

./(1+FSLattice(2:xg-1,2:yg-1))) ;

APopulationini(isnan(APopulationini)) = 0 ;

BPopulationini(7:xg-1,2:yg-1) = (TVC ./(1+FSLattice(7:xg-1,2:yg-1))) ;

BPopulationini(isnan(BPopulationini)) = 0 ;


Volumeini = sum(sum(APopulation))+sum(sum(BPopulation)) ;

DensityS=1;

DensityF=1.0 ;

TimeStep1 = 1 ;

DeltaV = 0 ;

Strain=0.37 ;

CP = 1.0 ;

LW = 30;

Alpha = Alphaini ;

KC = 20.0

ylow = yg/2-fix(LW/2) ;

yhigh = yg/2+round(LW/2) ;

DV = (sum(sum(APopulation(:,ylow:yhigh)))+sum(sum(BPopulation(:,ylow:yhigh) ./

DensityS))) * Strain ;

V0 = sum(sum(APopulation(:,ylow:yhigh)))+sum(sum(BPopulation(:,ylow:yhigh) ./

DensityS)) ;

APopulation = 0 .* APopulation ;

AlphaAi = APopulationini ./ (APopulationini + BPopulationini) ;

AlphaAi(isnan(AlphaAi))=0 ;

APopulation(:,ylow:yhigh) = DV .* AlphaAi(:,ylow:yhigh).^1.0 ./

sum(sum(AlphaAi(:,ylow:yhigh).^1.0)) ;

APopulationRemain = APopulationini - APopulation ;

Alpha = CP .* ( (APopulationRemain + APopulation) ./ (KC .* BPopulation) ) .^ 1.0 ;

Alpha(isnan(Alpha)) = 0 ;

Alpha(Alpha>1)=1;

AlphaS = ((BPopulation./BPopulation) ./ APopulation).^1.0

AlphaS(isnan(AlphaS)) = 0 ;

AlphaS(isinf(AlphaS)) = 1000 ;

TimeStep = 1 ;

AN(:,:,TimeStep) = APopulation ;

BN(:,:,TimeStep) = BPopulation ;

CN(:,:,TimeStep) = APopulationRemain ;

208 Appendices

FS(TimeStep,1) = ( sum(sum(APopulation)) + sum(sum(APopulationRemain)) ) /

sum(sum(BPopulation)) ;

FSBottom(TimeStep,1) = ( sum(APopulation(size(APopulation,1)-1,:)) +

sum(APopulationRemain(size(APopulation,1)-1,:)) ) /

sum(BPopulation(size(APopulation,1)-1,:)) ;

VCT(TimeStep,1) = DeltaV ;

StrainT(TimeStep,1) = 100 * DeltaV / V0 ;

for i = 2:yg-1

XL(i,1) = 6 ;

XL(i,2) = i ;

end ;

ExudeWaterpercent= 100 * ((sum(sum(APopulation)) + sum(sum(APopulationRemain))

) / sum(sum(APopulationini))) ;

for TimeStep = 2 : TotalTimeSteps+1

tic

TimeStep1 = TimeStep1 + 1 ;

AN(:,:,TimeStep1) = APopulation ;

CN(:,:,TimeStep1) = APopulationRemain ;

BN(:,:,TimeStep1) = BPopulation ;

FS(TimeStep1,1) = ( sum(sum(APopulation)) + sum(sum(APopulationRemain)) ) /

sum(sum(BPopulation)) ;

FSBottom(TimeStep1,1) = ( sum(APopulation(size(APopulation,1)-1,:)) +

sum(APopulationRemain(size(APopulation,1)-1,:)) ) /

sum(BPopulation(size(APopulation,1)-1,:)) ;

VCT(TimeStep,1) = DeltaV ;

BottomFluid(TimeStep,1) = sum(APopulation(xg-1,:)) ;

A0(TimeStep-1,1) = TimeStep -1 ;

Vnew = sum(sum(APopulation(2:xg-1,ylow:yhigh) + APopulationRemain(2:xg-

1,ylow:yhigh) + BPopulation(2:xg-1,ylow:yhigh)./ DensityS)) ;

StrainT(TimeStep,1) = 100 * (V0 - Vnew) / V0 ;

ExudeWaterpercent= 100 * ((sum(sum(APopulation)) + sum(sum(APopulationRemain))

) / sum(sum(APopulationini))) ;

ExudeWP(TimeStep,1) = 100 - ExudeWaterpercent ;

AL1 = 0 ;

AL2 = 0 ;

CNAL=0;

for i=2:size(XL,1)

AL1 = AL1 + Alpha(XL(i,1)+1,XL(i,2)) ;

AL2 = AL2 + sum(Alpha(XL(i,1)+1:xg-1,XL(i,2))) ;

CNAL = CNAL + xg-1 -XL(i,1) ;

end ;

ALSurface(TimeStep-1,1) = AL1 ;

ALTotal(TimeStep-1,1) = AL2/CNAL ;

CoreX = mod(TimeStep+1,2) ;

E = APopulation .* Alpha ;

EXU=1

for i = 2:xg-1

for j=2:yg-1

if i==XL(j,1) & j==XL(j,2) & DeltaV < DV

if j==yg-1

Alph = Alpha(i+1,j) ;

DeltaV = DeltaV + EXU * APopulation(i+1,j) * Alph ;

APopulation(i+1,j) = APopulation(i+1,j) * (1 - EXU * Alph) ;

E(i+1,j) = APopulation(i+1,j) .* Alpha(i+1,j) ;

else

Alph1 = Alpha(i+1,j) ;

Alph2 = Alpha(i+1,j+1) ;

DeltaV = DeltaV + EXU * APopulation(i+1,j) * Alph1 + EXU *

APopulation(i+1,j+1) * Alph2 ;

APopulation(i+1,j) = APopulation(i+1,j) * (1 - EXU * Alph1) ;

E(i+1,j) = APopulation(i+1,j) .* Alpha(i+1,j) ;

APopulation(i+1,j+1) = APopulation(i+1,j+1) * (1 - EXU * Alph2)

;

E(i+1,j+1) = APopulation(i+1,j+1) .* Alpha(i+1,j+1) ;

end;

end ;

end ;

end ;

Alpha = CP .* ( (APopulationRemain + APopulation) ./ (KC .* BPopulation) ) .^ 1.0

;


AlphaS = ((BPopulation./BPopulation) ./ APopulation).^1.0 ;

AlphaS(isnan(AlphaS)) = 0 ;

AlphaS(isinf(AlphaS)) = 1000 ;

AlphaDis = (APopulationRemain + APopulation) ./ (BPopulation) ;

Appendices 209

AlphaDis(isnan(AlphaDis)) = 0 ;

KV= (AlphaDis ./ Alphaini) .^ 1.0 ; %0.5;

KV(isinf(KV)) = 1 ;

KV(isnan(KV)) = 0 ;

KO= (KH + KV) ./ 2 ;

KO(isnan(KO)) = 0 ;

for i = 1+CoreX : 2 : xg-1

for j = 1+CoreX : 2 : yg-1

SAS1 = AlphaS(i,j+1) + AlphaS(i+1,j) + AlphaS(i+1,j+1) ;

SAS2 = AlphaS(i,j) + AlphaS(i+1,j) + AlphaS(i+1,j+1) ;

SAS3 = AlphaS(i,j) + AlphaS(i,j+1) + AlphaS(i+1,j+1) ;

SAS4 = AlphaS(i,j) + AlphaS(i,j+1) + AlphaS(i+1,j) ;

WL1 = E(i,j) * (KH(i,j) * AlphaS(i,j+1) + KV(i,j) * AlphaS(i+1,j) +

KO(i,j) * AlphaS(i+1,j+1)) / SAS1 ; %

WG1 = AlphaS(i,j) * (E(i,j+1) * KH(i,j+1)/SAS2 + E(i+1,j) *

KV(i+1,j)/SAS3 + E(i+1,j+1) * KO(i+1,j+1)/SAS4) ; %

WL2 = E(i,j+1) * (KH(i,j+1) * AlphaS(i,j) + KV(i,j+1) *

AlphaS(i+1,j+1) + KO(i,j+1) * AlphaS(i+1,j)) / SAS2 ;

WG2 = AlphaS(i,j+1) * (E(i,j) * KH(i,j)/SAS1 + E(i+1,j+1) *

KV(i+1,j+1)/SAS4 + E(i+1,j) * KO(i+1,j)/SAS3) ; %

WL3 = E(i+1,j) * (KH(i+1,j) * AlphaS(i+1,j+1) + KV(i+1,j) *

AlphaS(i,j) + KO(i+1,j) * AlphaS(i,j+1)) / SAS3 ;

WG3 = AlphaS(i+1,j) * (E(i+1,j+1) * KH(i+1,j+1)/SAS4 + E(i,j) *

KV(i,j)/SAS1 + E(i,j+1) * KO(i,j+1)/SAS2) ;

WL4 = E(i+1,j+1) * (KH(i+1,j+1) * AlphaS(i+1,j) + KV(i+1,j+1) *

AlphaS(i,j+1) + KO(i+1,j+1) * AlphaS(i,j)) / SAS4 ;

WG4 = AlphaS(i+1,j+1) * (E(i+1,j) * KH(i+1,j)/SAS3 + E(i,j+1) *

KV(i,j+1)/SAS2 + E(i,j) * KO(i,j)/SAS1) ;

WG1(isnan(WG1)) = 0 ;

WL1(isnan(WL1)) = 0 ;







APopulation(i,j) = APopulation(i,j) + WG1 - WL1 ;

APopulation(i,j+1) = APopulation(i,j+1) + WG2 - WL2 ;

APopulation(i+1,j) = APopulation(i+1,j) + WG3 - WL3 ;

APopulation(i+1,j+1) = APopulation(i+1,j+1) + WG4 - WL4 ;

end ;

end ;

for i = xg-1 : -1 : 2

for j = 2 : yg-1

Totalmass = APopulation(i,j) + APopulationRemain(i,j) + BPopulation(i,j)

;

if Totalmass > 1

ALF = (APopulation(i,j) + APopulationRemain(i,j)) / BPopulation(i,j)

;

PAP = APopulation(i,j) / (APopulation(i,j) + APopulationRemain(i,j))

;

PAR = APopulationRemain(i,j) / (APopulation(i,j) +

APopulationRemain(i,j)) ;

ALF(isnan(ALF))=0 ;

PAP(isnan(PAP))=0 ;

PAR(isnan(PAR))=0 ;

FTransfer = (Totalmass - 1) * ALF / (1+ALF) ;

STransfer = (Totalmass - 1) * 1 / (1+ALF) ;

if BPopulation(i,j)==0

STransfer = 0 ;

end ;

FTransfer(isnan(FTransfer))=0 ;

STransfer(isnan(STransfer))=0 ;

APopulation(i,j) = APopulation(i,j) - PAP * FTransfer ;

APopulationRemain(i,j) = APopulationRemain(i,j) - PAR * FTransfer ;

BPopulation(i,j) = BPopulation(i,j) - STransfer ;

APopulation(i-1,j) = APopulation(i-1,j) + PAP * FTransfer ;

APopulationRemain(i-1,j) = APopulationRemain(i-1,j) + PAR * FTransfer

;

BPopulation(i-1,j) = BPopulation(i-1,j) + STransfer ;

elseif Totalmass < 1

ALF = (APopulation(i-1,j) + APopulationRemain(i-1,j)) /

BPopulation(i-1,j) ;

PAP = APopulation(i-1,j) / (APopulation(i-1,j) + APopulationRemain(i-

1,j)) ;

210 Appendices

PAR = APopulationRemain(i-1,j) / (APopulation(i-1,j) +

APopulationRemain(i-1,j)) ;

ALF(isnan(ALF))=0 ;

PAP(isnan(PAP))=0 ;

PAR(isnan(PAR))=0 ;

FTransfer = (1 - Totalmass) * ALF / (1+ALF) ;

STransfer = (1 - Totalmass) * 1 / (1+ALF) ;

if BPopulation(i-1,j)==0

STransfer = 0 ;

end ;

FTransfer(isnan(FTransfer))=0 ;

STransfer(isnan(STransfer))=0 ;

if APopulation(i-1,j) + APopulationRemain(i-1,j) >= FTransfer

APopulation(i-1,j) = APopulation(i-1,j) - PAP * FTransfer ;

APopulationRemain(i-1,j) = APopulationRemain(i-1,j) - PAR *

FTransfer ;

BPopulation(i-1,j) = BPopulation(i-1,j) - STransfer ;

APopulation(i,j) = APopulation(i,j) + PAP * FTransfer ;

APopulationRemain(i,j) = APopulationRemain(i,j) + PAR * FTransfer

;

BPopulation(i,j) = BPopulation(i,j) + STransfer ;

elseif APopulation(i-1,j) + APopulationRemain(i-1,j) < FTransfer

APopulation(i,j) = APopulation(i,j) + APopulation(i-1,j) ;

APopulationRemain(i,j) = APopulationRemain(i,j) +

APopulationRemain(i-1,j) ;

BPopulation(i,j) = BPopulation(i,j) + BPopulation(i-1,j) ;

APopulation(i-1,j) = 0 ;

APopulationRemain(i-1,j) = 0 ;

BPopulation(i-1,j) = 0 ;

end ;

end;

end;

end ;

for j = 2:yg-1

for i=1:xg-4

if (APopulation(i,j) + APopulationRemain(i,j)) < 0.02 & BPopulation(i,j)

< 0.02 & (APopulation(i+1,j) + APopulationRemain(i+1,j)) > 0.02 & BPopulation(i+1,j)

> 0.02

XL(j,1) = i ;

XL(j,2) = j ;

APopulation(i+1,j) = APopulation(i+1,j) + APopulation(i,j) ;

APopulationRemain(i+1,j) = APopulationRemain(i+1,j) +

APopulationRemain(i,j) ;

BPopulation(i+1,j) = BPopulation(i+1,j) + BPopulation(i,j) ;

APopulation(i,j) = 0 ;

APopulationRemain(i,j) = 0 ;

BPopulation(i,j) = 0 ;

end ;

end ;

end ;


Alpha = CP .* ( (APopulationRemain + APopulation) ./ (KC .* BPopulation) ) .^ 1.0

;


TotalAlpha = sum(sum(Alpha)) ;

AlphaDis = (APopulationRemain + APopulation) ./ (BPopulation) ;

AlphaDis(isnan(AlphaDis)) = 0 ;

KV= (AlphaDis ./ Alphaini) .^ 1.0 ; %0.5;

KV(isinf(KV)) = 1 ;

KV(isnan(KV)) = 0 ;

KO= (KH + KV) ./ 2 ;

KO(isnan(KO)) = 0 ;

AAA=KH./KV ;

BulkKHKVchange(TimeStep,1) = sum(sum(AAA(2:xg-1,2:yg-1))) / ((xg-2)*(yg-2)) ;

Time(TimeStep,1) = toc ;

end;

ExpTime = [1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 20 25 30 35 40 45 50 55 60 65 70 75 80

85 90] ;

ExpTime = ExpTime' ;

Experimental = [5.95 8.86 11.36 13.41 14.8 16.25 17.45 18.65 19.9 20.7 21.6 22.5 23.2

23.9 24.4 27.1 29.1 30.45 31.5 32.3 33.07 33.75 34.12 34.55 34.77 35 35.35 35.57

35.68 35.8] ;

Experimental = Experimental' ;

Sign(1:size(Experimental,1),1) = 1 ;

k=1 ;

for i = 1: TimeStep

Appendices 211

if k<= size(Experimental,1) & (round (StrainT(i,1).*100)./100 ==

Experimental(k,1) ) & Sign(k,1)==1

Sign(k,1)=0 ;

CorTimestep(k,1) = ExpTime (k,1) ;

CorTimestep(k,2) = i ;

CorTimestep(k,3) = round(i/ExpTime (k,1)) ;

k=k+1 ;

end ;

end ;

LB = min(CorTimestep(:,3)) ;

UB = max(CorTimestep(:,3)) ;

k=0 ;

k1=0 ;

for i=LB:UB+1

k=0 ;

for j=1:15

k=k+1;

Simulation(k,1) = StrainT(j*i,1) ;

end;

for j=20:5:90

k=k+1;

Simulation(k,1) = StrainT(j*i,1) ;

end;

k1=k1+1 ;

Error (k1,2) = ( ( sum ((Experimental - Simulation(:,1)) .^2) /

size(Experimental,1) ) ^ 0.5 ) / mean(Simulation(:,1)) ;

Error (k1,1) = i ;

[ErB,Interval] = min(Error (:,2)) ;

IntB = Error(Interval,1) ;

Bestfit(:,1) = Experimental(:,1) ;

SumErrorNL = 0 ;

for is = 1: size(Experimental,1)-1

A= (Experimental(is+1) - Experimental(is)) / (ExpTime (is) - ExpTime (is+1))

;

B=1 ;

C = (Experimental(is) * ExpTime (is+1) - Experimental(is+1) * ExpTime (is))

/ (ExpTime (is) - ExpTime (is+1)) ;

SumErrorNL = SumErrorNL + ((abs(A*ExpTime(is) + B*Simulation(is) + C) /

(A^2+B^2)^0.5)).^2 ;

end

is = size(Experimental,1) ;

A= (Experimental(is) - Experimental(is-1)) / (ExpTime (is-1) - ExpTime (is)) ;

B=1 ;

C = (Experimental(is-1) * ExpTime (is) - Experimental(is) * ExpTime (is-1)) /

(ExpTime (is-1) - ExpTime (is)) ;

SumErrorNL = SumErrorNL + ((abs(A*ExpTime(is) + B*Simulation(is) + C) /

(A^2+B^2)^0.5))^2 ;

ErrorNL(k1,2) = SumErrorNL ;

ErrorNL (k1,1) = i ;

end ;

[ErBNL,IntervalNL] = min(ErrorNL (:,2)) ;

IntBNL = Error(IntervalNL,1) ;

BestfitNL(:,1) = Experimental(:,1) ;

k1=0 ;

k=0 ;

for j=1:15

k=k+1;

Bestfit(k,2) = StrainT(j*IntB,1) ;

Bestfit(k,3) = IntB * j ;

BestfitNL(k,2) = StrainT(j*IntBNL,1) ;

BestfitNL(k,3) = IntBNL * j ;

end;

for j=20:5:90

k=k+1;

Bestfit(k,2) = StrainT(j*IntB,1) ;

Bestfit(k,3) = IntB * j ;

BestfitNL(k,2) = StrainT(j*IntBNL,1) ;

BestfitNL(k,3) = IntBNL * j ;

end;

212 Appendices

Appendix D

Matlab program for simulating dissolution of wet salt.

clear;clc;

profile on

profile clear

Xg = 80 ; % No Layers

Yg = 80 ; % No columns

TL = 20000 ; % Total time steps

TNR = 1 ; % Number of repeat

SignCell = 0 ; % Interaction sign

CeC = 100 ; % Capacity of the cell

ALPHA = 0.2 ; % percentage of change

ALPHAL = 0.5 ; % percentage of change fluid

LT1 = 0.5 ; % Threshold for salt

LT = LT1 / (1 - LT1) ; % SW based on LT1

MA = 30 ;

%%% area of solid initially

Xc1 = round(Xg/2)- (fix(MA/2) - 1) ; %29 ;

Xc2 = round(Xg/2)+ round(MA/2) ; %30 ;

Yc1 = round(Yg/2)-(fix(MA/2) - 1) ; %29 ;

Yc2 = round(Yg/2)+ round(MA/2) ; %30 ;

%%%%%%%%%%%%%%%%%%%%%%%%

for NR=1:TNR;%00

k = 1 ;

for i=Xc1 : Xc2

for j= Yc1 : Yc2

if i < Xc1+round((Xc2-Xc1)/4) ;

ICC(k) = 9 ;

elseif i >= Xc1+round((Xc2-Xc1)/4) & i < Xc1+round((Xc2-Xc1)/2) ;

ICC(k) = 9 ;

elseif i >= Xc1+round((Xc2-Xc1)/2) & i <= Xc1+round(3*(Xc2-Xc1)/4) ;

ICC(k) = 9 ;

elseif i > Xc1+round(3*(Xc2-Xc1)/4) ;

ICC(k) = 9 ;

end ;

k=k+1 ;

end ;

end ;

%%%%% Initial conditions%%% 1:Fluid, 0: Solid, 0.5:combined %%%%%%

cell(1:Xg,1:Yg) = 0.25 ; %0 ; % 0.25 ;

k= 1 ;

for i=Xc1 : Xc2

for j= Yc1 : Yc2

cell(i,j) = ICC(k) ;

k=k+1 ;

end ;

end ;

for k=1:TL

CELL(:,:,k) = cell ;

CSolid = cell ./ (1 + cell) ;

CFluid = 1 ./ (1 + cell) ;

CC=cell;

imagesc(CC)

pause(0.00001)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

SN = 0;

for i=1 : Xg

for j= 1 : Yg

if cell(i,j) > LT

SolidCell(i,j) = 1 ;

SN = SN + 1 ;

else

SolidCell(i,j) = 0 ;

end ;

end ;

end ;

SCN(k,1) = SN ;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Appendices 213

for i=1+mod(k,3):3:Xg-2;

for j=1+mod(k,3):3:Yg-2;

Agents=[cell(i,j) cell(i,j+1) cell(i,j+2) cell(i+1,j) cell(i+1,j+2)

cell(i+2,j) cell(i+2,j+1) cell(i+2,j+2)] ;

Solid = Agents ./ (1 + Agents) ;

Fluid = 1 ./ (1 + Agents) ;

MS= mean(Solid) ;

SolidC = cell(i+1,j+1) / (1 + cell(i+1,j+1)) ;

FluidC = 1 / (1 + cell(i+1,j+1)) ;

c = [1 2 3 4 5 6 7 8] ;

[a,b] = sort(rand(size(c))) ;

if b(1) == 1

if cell(i,j) < LT & cell(i+1,j+1) < LT

ALPHAF = ALPHAL ;

else

ALPHAF = ALPHA ;

end ;

cell(i,j) = (Solid(1) + ALPHAF * (SolidC - Solid(1))) / (Fluid(1)

- ALPHAF * (SolidC - Solid(1))) ;

cell(i+1,j+1) = (SolidC + ALPHAF * (Solid(1) - SolidC)) / (FluidC

- ALPHAF * (Solid(1) - SolidC)) ;

elseif b(1) == 2

if cell(i,j+1) < LT & cell(i+1,j+1) < LT

ALPHAF = ALPHAL ;

else

ALPHAF = ALPHA ;

end ;

cell(i,j+1) = (Solid(2) + ALPHAF * (SolidC - Solid(2))) /

(Fluid(2) - ALPHAF * (SolidC - Solid(2))) ;



elseif b(1) == 3

if cell(i,j+2) < LT & cell(i+1,j+1) < LT

ALPHAF = ALPHAL ;

else

ALPHAF = ALPHA ;

end ;

cell(i,j+2) = (Solid(3) + ALPHAF * (SolidC - Solid(3))) /




elseif b(1) == 4

if cell(i+1,j) < LT & cell(i+1,j+1) < LT

ALPHAF = ALPHAL ;

else

ALPHAF = ALPHA ;

end ;

cell(i+1,j) = (Solid(4) + ALPHAF * (SolidC - Solid(4))) /




elseif b(1) == 5

if cell(i+1,j+2) < LT & cell(i+1,j+1) < LT

ALPHAF = ALPHAL ;

else

ALPHAF = ALPHA ;

end ;

cell(i+1,j+2) = (Solid(5) + ALPHAF * (SolidC - Solid(5))) /




elseif b(1) == 6

if cell(i+2,j) < LT & cell(i+1,j+1) < LT

ALPHAF = ALPHAL ;

else

ALPHAF = ALPHA ;

end ;

cell(i+2,j) = (Solid(6) + ALPHAF * (SolidC - Solid(6))) /


214 Appendices



elseif b(1) == 7


ALPHAF = ALPHAL ;

else

ALPHAF = ALPHA ;

end ;





elseif b(1) == 8


ALPHAF = ALPHAL ;

else

ALPHAF = ALPHA ;

end ;





end ;

end ;

end ;

end ;

if NR == 1

CCA = CELL ;

CCA(isnan(CCA))=0 ;

else

CCA = ((NR - 1) .* (CCA ./ (1 + CCA)) + CELL ./ (1 + CELL)) ./ ((NR - 1) .* (1

./ (1 + CCA)) + 1 ./ (1 + CELL) ) ;

CCA(isnan(CCA))=0 ;

end ;

end ; %%% NR end

CCA=SALT1./WATER1;

CCA(isnan(CCA))=0 ;

Appendices 215

Appendix E

216 Appendices

Appendix F

an innovative agent based cellular automata … · an innovative agent-based cellular automata...

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