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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,
Chapter 1: Introduction 3
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.
4 Chapter 1: Introduction
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?
Chapter 1: Introduction 5
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
6 Chapter 1: Introduction
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.
Chapter 1: Introduction 7
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
8 Chapter 1: Introduction
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.
Chapter 1: Introduction 9
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
Chapter 2: Literature Review 13
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,
14 Chapter 2: Literature Review
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
Chapter 2: Literature Review 15
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
16 Chapter 2: Literature Review
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
Chapter 2: Literature Review 17
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
18 Chapter 2: Literature Review
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
Chapter 2: Literature Review 19
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
20 Chapter 2: Literature Review
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].
Chapter 2: Literature Review 21
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].
22 Chapter 2: Literature Review
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
Chapter 2: Literature Review 23
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].
24 Chapter 2: Literature Review
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
Chapter 2: Literature Review 25
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
26 Chapter 2: Literature Review
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
Chapter 2: Literature Review 27
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.
28 Chapter 2: Literature Review
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
Chapter 2: Literature Review 29
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
30 Chapter 2: Literature Review
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
Chapter 2: Literature Review 31
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].
32 Chapter 2: Literature Review
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.
Chapter 2: Literature Review 33
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
34 Chapter 2: Literature Review
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.
Chapter 2: Literature Review 35
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].
36 Chapter 2: Literature Review
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.
Chapter 2: Literature Review 37
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.
38 Chapter 2: Literature Review
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].
Chapter 2: Literature Review 39
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
40 Chapter 2: Literature Review
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
Chapter 2: Literature Review 41
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
42 Chapter 2: Literature Review
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
Chapter 2: Literature Review 43
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
44 Chapter 2: Literature Review
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
Chapter 3: New Agent and Rule 47
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.
48 Chapter 3: New Agent and Rule
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.
Chapter 3: New Agent and Rule 49
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
50 Chapter 3: New Agent and Rule
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.
Chapter 3: New Agent and Rule 51
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.
52 Chapter 3: New Agent and Rule
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.
Chapter 4: Using hybrid agents to create porous structures 55
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.
56 Chapter 4: Using hybrid agents to create porous structures
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
Chapter 4: Using hybrid agents to create porous structures 57
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
58 Chapter 4: Using hybrid agents to create porous structures
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.
Chapter 4: Using hybrid agents to create porous structures 59
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
60 Chapter 4: Using hybrid agents to create porous structures
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
Chapter 4: Using hybrid agents to create porous structures 61
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.
62 Chapter 4: Using hybrid agents to create porous structures
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.
Chapter 4: Using hybrid agents to create porous structures 63
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
Single-property region Single-property region
Uniform region B Transition region
Single-property region Single-property 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.
64 Chapter 4: Using hybrid agents to create porous structures
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
Chapter 4: Using hybrid agents to create porous structures 65
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
66 Chapter 4: Using hybrid agents to create porous structures
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
Chapter 4: Using hybrid agents to create porous structures 67
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
68 Chapter 4: Using hybrid agents to create porous structures
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
Chapter 4: Using hybrid agents to create porous structures 69
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.
70 Chapter 4: Using hybrid agents to create porous structures
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
Chapter 4: Using hybrid agents to create porous structures 71
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
72 Chapter 4: Using hybrid agents to create porous structures
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
Chapter 5: Diffusion throughout the articular cartilage 75
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.
76 Chapter 5: Diffusion throughout the articular cartilage
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).
Chapter 5: Diffusion throughout the articular cartilage 77
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.
78 Chapter 5: Diffusion throughout the articular cartilage
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.
Chapter 5: Diffusion throughout the articular cartilage 79
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
80 Chapter 5: Diffusion throughout the articular cartilage
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.
Chapter 5: Diffusion throughout the articular cartilage 81
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.
82 Chapter 5: Diffusion throughout the articular cartilage
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
Chapter 5: Diffusion throughout the articular cartilage 83
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
84 Chapter 5: Diffusion throughout the articular cartilage
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
Chapter 5: Diffusion throughout the articular cartilage 85
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
86 Chapter 5: Diffusion throughout the articular cartilage
(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
Chapter 5: Diffusion throughout the articular cartilage 87
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.
88 Chapter 5: Diffusion throughout the articular cartilage
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
Chapter 5: Diffusion throughout the articular cartilage 89
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.
90 Chapter 5: Diffusion throughout the articular cartilage
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).
Chapter 5: Diffusion throughout the articular cartilage 91
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)
92 Chapter 5: Diffusion throughout the articular cartilage
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
Chapter 5: Diffusion throughout the articular cartilage 93
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
94 Chapter 5: Diffusion throughout the articular cartilage
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
Chapter 5: Diffusion throughout the articular cartilage 95
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.
96 Chapter 5: Diffusion throughout the articular cartilage
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
Chapter 5: Diffusion throughout the articular cartilage 97
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.
98 Chapter 5: Diffusion throughout the articular cartilage
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].
Chapter 5: Diffusion throughout the articular cartilage 99
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
100 Chapter 5: Diffusion throughout the articular cartilage
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
Chapter 5: Diffusion throughout the articular cartilage 101
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
102 Chapter 5: Diffusion throughout the articular cartilage
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 MATERIAL AND METHODS
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).
Chapter 6: Deformation of the articular cartilage 105
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).
106 Chapter 6: Deformation of the articular cartilage
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
Chapter 6: Deformation of the articular cartilage 107
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.
108 Chapter 6: Deformation of the articular cartilage
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.
Chapter 6: Deformation of the articular cartilage 109
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
110 Chapter 6: Deformation of the articular cartilage
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.
Chapter 6: Deformation of the articular cartilage 111
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
112 Chapter 6: Deformation of the articular cartilage
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
Chapter 6: Deformation of the articular cartilage 113
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
114 Chapter 6: Deformation of the articular cartilage
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].
Chapter 6: Deformation of the articular cartilage 115
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
116 Chapter 6: Deformation of the articular cartilage
𝐹𝐿2 = 𝑘𝑐 𝑓𝑠2 𝑓𝑚2
𝐹𝐿3 = 𝑘𝑐 𝑓𝑠3 𝑓𝑚3
𝐹𝐿4 = 𝑘𝑐 𝑓𝑠4 𝑓𝑚4
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
𝐾𝐻𝑓𝑠𝑚2
+ 𝐾𝑉
𝑓𝑠𝑚3+
𝐾𝑂𝑓𝑠𝑚4
)
𝐹𝐺3 = 𝐹𝐿1 (
𝐾𝑉𝑓𝑠𝑚3
𝐾𝐻𝑓𝑠𝑚2
+ 𝐾𝑉
𝑓𝑠𝑚3+
𝐾𝑂𝑓𝑠𝑚4
)
𝐹𝐺4 = 𝐹𝐿1 (
𝐾𝑂𝑓𝑠𝑚4
𝐾𝐻𝑓𝑠𝑚2
+ 𝐾𝑉
𝑓𝑠𝑚3+
𝐾𝑂𝑓𝑠𝑚4
)
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,
Chapter 6: Deformation of the articular cartilage 117
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 (
𝐾𝐻𝑓𝑠𝑚1
𝐾𝐻𝑓𝑠𝑚1
+ 𝐾𝑂
𝑓𝑠𝑚3+
𝐾𝑉𝑓𝑠𝑚4
) + 𝐹𝐿3 (
𝐾𝑉𝑓𝑠𝑚1
𝐾𝑉𝑓𝑠𝑚1
+ 𝐾𝑂
𝑓𝑠𝑚2+
𝐾𝐻𝑓𝑠𝑚4
) + 𝐹𝐿4 (
𝐾𝑂𝑓𝑠𝑚1
𝐾𝑂𝑓𝑠𝑚1
+ 𝐾𝑉
𝑓𝑠𝑚2+
𝐾𝐻𝑓𝑠𝑚3
)
𝐹𝑛2 = 𝐹𝑖2 − 𝐹𝐿2 + 𝐹𝐿1 (
𝐾𝐻𝑓𝑠𝑚2
𝐾𝐻𝑓𝑠𝑚2
+ 𝐾𝑉
𝑓𝑠𝑚3+
𝐾𝑂𝑓𝑠𝑚4
) + 𝐹𝐿3 (
𝐾𝑂𝑓𝑠𝑚2
𝐾𝑉𝑓𝑠𝑚1
+ 𝐾𝑂
𝑓𝑠𝑚2+
𝐾𝐻𝑓𝑠𝑚4
) + 𝐹𝐿4 (
𝐾𝑉𝑓𝑠𝑚2
𝐾𝑂𝑓𝑠𝑚1
+ 𝐾𝑉
𝑓𝑠𝑚2+
𝐾𝐻𝑓𝑠𝑚3
)
𝐹𝑛3 = 𝐹𝑖3 − 𝐹𝐿3 + 𝐹𝐿1 (
𝐾𝑉𝑓𝑠𝑚3
𝐾𝐻𝑓𝑠𝑚2
+ 𝐾𝑉
𝑓𝑠𝑚3+
𝐾𝑂𝑓𝑠𝑚4
) + 𝐹𝐿2 (
𝐾𝑂𝑓𝑠𝑚3
𝐾𝐻𝑓𝑠𝑚1
+ 𝐾𝑂
𝑓𝑠𝑚3+
𝐾𝑉𝑓𝑠𝑚4
) + 𝐹𝐿4 (
𝐾𝐻𝑓𝑠𝑚3
𝐾𝑂𝑓𝑠𝑚1
+ 𝐾𝑉
𝑓𝑠𝑚2+
𝐾𝐻𝑓𝑠𝑚3
)
𝐹𝑛4 = 𝐹𝑖4 − 𝐹𝐿4 + 𝐹𝐿1 (
𝐾𝑂𝑓𝑠𝑚4
𝐾𝐻𝑓𝑠𝑚2
+ 𝐾𝑉
𝑓𝑠𝑚3+
𝐾𝑂𝑓𝑠𝑚4
) + 𝐹𝐿2 (
𝐾𝑉𝑓𝑠𝑚4
𝐾𝐻𝑓𝑠𝑚1
+ 𝐾𝑂
𝑓𝑠𝑚3+
𝐾𝑉𝑓𝑠𝑚4
) + 𝐹𝐿3 (
𝐾𝐻𝑓𝑠𝑚4
𝐾𝑉𝑓𝑠𝑚1
+ 𝐾𝑂
𝑓𝑠𝑚2+
𝐾𝐻𝑓𝑠𝑚4
)
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.
118 Chapter 6: Deformation of the articular cartilage
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
Chapter 6: Deformation of the articular cartilage 119
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.
120 Chapter 6: Deformation of the articular cartilage
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
Chapter 6: Deformation of the articular cartilage 121
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].
122 Chapter 6: Deformation of the articular cartilage
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.
Chapter 6: Deformation of the articular cartilage 123
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.
124 Chapter 6: Deformation of the articular cartilage
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.
Chapter 6: Deformation of the articular cartilage 125
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
126 Chapter 6: Deformation of the articular cartilage
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
Chapter 6: Deformation of the articular cartilage 127
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
128 Chapter 6: Deformation of the articular cartilage
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
Chapter 6: Deformation of the articular cartilage 129
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
130 Chapter 6: Deformation of the articular cartilage
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).
INDENTER
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Chapter 6: Deformation of the articular cartilage 131
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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132 Chapter 6: Deformation of the articular cartilage
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).
Chapter 6: Deformation of the articular cartilage 133
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].
134 Chapter 6: Deformation of the articular cartilage
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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Chapter 6: Deformation of the articular cartilage 135
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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136 Chapter 6: Deformation of the articular cartilage
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%
Chapter 6: Deformation of the articular cartilage 137
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
138 Chapter 6: Deformation of the articular cartilage
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 MATERIAL AND 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
Chapter 7: Dissolution of wet salt 141
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
142 Chapter 7: Dissolution of wet salt
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.
Chapter 7: Dissolution of wet salt 143
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
144 Chapter 7: Dissolution of wet salt
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.
Chapter 7: Dissolution of wet salt 145
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
146 Chapter 7: Dissolution of wet salt
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,
Chapter 7: Dissolution of wet salt 147
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].
148 Chapter 7: Dissolution of wet salt
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.
Chapter 7: Dissolution of wet salt 149
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,
150 Chapter 7: Dissolution of wet salt
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.
Chapter 7: Dissolution of wet salt 151
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
152 Chapter 7: Dissolution of wet salt
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.
Chapter 7: Dissolution of wet salt 153
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
154 Chapter 7: Dissolution of wet salt
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
Chapter 7: Dissolution of wet salt 155
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.
156 Chapter 7: Dissolution of wet salt
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.
Chapter 7: Dissolution of wet salt 157
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
158 Chapter 7: Dissolution of wet salt
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.
Chapter 7: Dissolution of wet salt 159
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
160 Chapter 7: Dissolution of wet 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.
Chapter 7: Dissolution of wet salt 161
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.
162 Chapter 7: Dissolution of wet salt
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],
Chapter 7: Dissolution of wet salt 163
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.
164 Chapter 7: Dissolution of wet salt
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);
cellS2 = zeros(TL,NCL);
cellF2 = zeros(TL,NCL);
cellS3 = zeros(TL,NCL);
cellF3 = zeros(TL,NCL);
cellS4 = zeros(TL,NCL);
cellF4 = zeros(TL,NCL);
cellS5 = zeros(TL,NCL);
cellF5 = zeros(TL,NCL);
cellS6 = zeros(TL,NCL);
cellF6 = zeros(TL,NCL);
cellS7 = zeros(TL,NCL);
cellF7 = zeros(TL,NCL);
cellS8 = zeros(TL,NCL);
cellF8 = 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
cellS2(LN+1,j) = 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
cellS3(LN+1,j) = 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
cellS5(LN+1,j) = 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))) ;
cellF6(LN+1,j) = 0 ;
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))) ;
cellF7(LN+1,j) = 0 ;
end ;
if cellF(LN,j-1) ~= 0 & cellF(LN,j) ~= 0 & cellF(LN,j+1) ~= 0
cellS8(LN+1,j) = 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 ;
XCent = round (size(FSL,1)/2) ;
YCent = round (size(FSL,2)/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 ;
XCent = round (size(FSL,1)/2) ;
YCent = round (size(FSL,2)/2) ;
FSL(XD1:XD2, YD1:YD2) = (FSL(XD1:XD2, YD1:YD2) + Beta) ./ (1- Beta) ;
XD1 = 2 ;
YD1 = 2 ;
Beta = 0;
XCent = round (size(FSL,1)/4) ;
YCent = round (size(FSL,2)/3) ;
FSL(XCent-XD1:XCent+XD1, YCent-YD1:YCent+YD1) = (FSL(XCent-XD1:XCent+XD1, YCent-
YD1:YCent+YD1) + Beta) ./ (1- Beta) ;
XD1 = 2 ;
YD1 = 2 ;
Beta = 0;
XCent = round (size(FSL,1)/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 = 2 ;
YD1 = 2 ;
Beta = 0;
XCent = round (size(FSL,1)/3.5) ;
YCent = round (size(FSL,2)/1.25) ;
FSL(XCent-XD1:XCent+XD1, YCent-YD1:YCent+YD1) = (FSL(XCent-XD1:XCent+XD1, YCent-
YD1:YCent+YD1) + Beta) ./ (1- Beta) ;
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-
1)+CPopulation(i,j-1)))*APopulation(i,j-1) ;
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-
1)+CPopulation(i,j-1)))*CPopulation(i,j-1) ;
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+1)+CPopulation(i,j+1)))*APopulation(i,j+1) ;
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+1)+CPopulation(i,j+1)))*CPopulation(i,j+1) ;
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) ;
APopulationNew(i,j) = APopulationNew(i,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) ;
CPopulationNew(i,j) = CPopulationNew(i,j) +
(TotalMFOUT/(APopulation(i+1,j)+CPopulation(i+1,j)))*CPopulation(i+1,j) ;
end ;
i = xg-5 ;
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) - 3 * TotalFAout ;
CPopulationNew(i,j) = CPopulationNew(i,j) - 3 * 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-
1)+CPopulation(i,j-1)))*APopulation(i,j-1) ;
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) ;
196 Appendices
CPopulationNew(i,j) = CPopulationNew(i,j) + (TotalMFOUT/(APopulation(i,j-
1)+CPopulation(i,j-1)))*CPopulation(i,j-1) ;
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+1)+CPopulation(i,j+1)))*APopulation(i,j+1) ;
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+1)+CPopulation(i,j+1)))*CPopulation(i,j+1) ;
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) ;
APopulationNew(i,j) = APopulationNew(i,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) ;
CPopulationNew(i,j) = CPopulationNew(i,j) + (TotalMFOUT/(APopulation(i-
1,j)+CPopulation(i-1,j)))*CPopulation(i-1,j) ;
end ;
j = 6 ;
for i = 7 : xg-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) = APopulationNew(i,j) + (TotalMFOUT/(APopulation(i,j-
1)+CPopulation(i,j-1)))*APopulation(i,j-1) ;
CPopulationNew(i,j) = CPopulationNew(i,j) + (TotalMFOUT/(APopulation(i,j-
1)+CPopulation(i,j-1)))*CPopulation(i,j-1) ;
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+1)+CPopulation(i,j+1)))*APopulation(i,j+1) ;
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+1)+CPopulation(i,j+1)))*CPopulation(i,j+1) ;
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) ;
APopulationNew(i,j) = APopulationNew(i,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) ;
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) ;
APopulationNew(i,j) = APopulationNew(i,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) ;
CPopulationNew(i,j) = CPopulationNew(i,j) +
(TotalMFOUT/(APopulation(i+1,j)+CPopulation(i+1,j)))*CPopulation(i+1,j) ;
end ;
j = yg-5 ;
for i = 7 : xg-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 ;
Appendices 197
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-
1)+CPopulation(i,j-1)))*APopulation(i,j-1) ;
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-
1)+CPopulation(i,j-1)))*CPopulation(i,j-1) ;
APopulationNew(i,j) = APopulationNew(i,j) +
(TotalMFOUT/(APopulation(i,j+1)+CPopulation(i,j+1)))*APopulation(i,j+1) ;
CPopulationNew(i,j) = CPopulationNew(i,j) +
(TotalMFOUT/(APopulation(i,j+1)+CPopulation(i,j+1)))*CPopulation(i,j+1) ;
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) ;
APopulationNew(i,j) = APopulationNew(i,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) ;
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) ;
APopulationNew(i,j) = APopulationNew(i,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) ;
CPopulationNew(i,j) = CPopulationNew(i,j) +
(TotalMFOUT/(APopulation(i+1,j)+CPopulation(i+1,j)))*CPopulation(i+1,j) ;
end ;
i = 6 ;
j = 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) = APopulationNew(i,j) + (TotalMFOUT/(APopulation(i,j-
1)+CPopulation(i,j-1)))*APopulation(i,j-1) ;
CPopulationNew(i,j) = CPopulationNew(i,j) + (TotalMFOUT/(APopulation(i,j-
1)+CPopulation(i,j-1)))*CPopulation(i,j-1) ;
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+1)+CPopulation(i,j+1)))*APopulation(i,j+1) ;
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+1)+CPopulation(i,j+1)))*CPopulation(i,j+1) ;
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) ;
APopulationNew(i,j) = APopulationNew(i,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) ;
CPopulationNew(i,j) = CPopulationNew(i,j) +
(TotalMFOUT/(APopulation(i+1,j)+CPopulation(i+1,j)))*CPopulation(i+1,j) ;
i = 6 ;
j = yg-5 ;
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 ;
198 Appendices
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-
1)+CPopulation(i,j-1)))*APopulation(i,j-1) ;
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-
1)+CPopulation(i,j-1)))*CPopulation(i,j-1) ;
APopulationNew(i,j) = APopulationNew(i,j) +
(TotalMFOUT/(APopulation(i,j+1)+CPopulation(i,j+1)))*APopulation(i,j+1) ;
CPopulationNew(i,j) = CPopulationNew(i,j) +
(TotalMFOUT/(APopulation(i,j+1)+CPopulation(i,j+1)))*CPopulation(i,j+1) ;
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) ;
APopulationNew(i,j) = APopulationNew(i,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) ;
CPopulationNew(i,j) = CPopulationNew(i,j) +
(TotalMFOUT/(APopulation(i+1,j)+CPopulation(i+1,j)))*CPopulation(i+1,j) ;
i = xg-5 ;
j = 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) - 3 * TotalFAout ;
CPopulationNew(i,j) = CPopulationNew(i,j) - 3 * TotalFCout ;
APopulationNew(i,j) = APopulationNew(i,j) + (TotalMFOUT/(APopulation(i,j-
1)+CPopulation(i,j-1)))*APopulation(i,j-1) ;
CPopulationNew(i,j) = CPopulationNew(i,j) + (TotalMFOUT/(APopulation(i,j-
1)+CPopulation(i,j-1)))*CPopulation(i,j-1) ;
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+1)+CPopulation(i,j+1)))*APopulation(i,j+1) ;
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+1)+CPopulation(i,j+1)))*CPopulation(i,j+1) ;
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) ;
APopulationNew(i,j) = APopulationNew(i,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) ;
CPopulationNew(i,j) = CPopulationNew(i,j) + (TotalMFOUT/(APopulation(i-
1,j)+CPopulation(i-1,j)))*CPopulation(i-1,j) ;
i = xg-5 ;
j = yg-5 ;
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) - 3 * TotalFAout ;
CPopulationNew(i,j) = CPopulationNew(i,j) - 3 * 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-
1)+CPopulation(i,j-1)))*APopulation(i,j-1) ;
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-
1)+CPopulation(i,j-1)))*CPopulation(i,j-1) ;
Appendices 199
APopulationNew(i,j) = APopulationNew(i,j) +
(TotalMFOUT/(APopulation(i,j+1)+CPopulation(i,j+1)))*APopulation(i,j+1) ;
CPopulationNew(i,j) = CPopulationNew(i,j) +
(TotalMFOUT/(APopulation(i,j+1)+CPopulation(i,j+1)))*CPopulation(i,j+1) ;
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) ;
APopulationNew(i,j) = APopulationNew(i,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) ;
CPopulationNew(i,j) = CPopulationNew(i,j) + (TotalMFOUT/(APopulation(i-
1,j)+CPopulation(i-1,j)))*CPopulation(i-1,j) ;
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 ;
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-
1)+CPopulation(i,j-1)))*APopulation(i,j-1) ;
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-
1)+CPopulation(i,j-1)))*CPopulation(i,j-1) ;
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+1)+CPopulation(i,j+1)))*APopulation(i,j+1) ;
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+1)+CPopulation(i,j+1)))*CPopulation(i,j+1) ;
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) ;
APopulationNew(i,j) = APopulationNew(i,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) ;
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) ;
APopulationNew(i,j) = APopulationNew(i,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) ;
CPopulationNew(i,j) = CPopulationNew(i,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
NAPos1 = NAPos1 + 1 ;
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 ;
PosCell2 = zeros(1,24) ;
clear('NCell2') ;
for i1 = -2 : 2 ;
for j1 = -2 : 2 ;
if i1 ~=0 | j1 ~= 0
NAPos2 = NAPos2 + 1 ;
NCell2(NAPos2) = APopulation(i+i1,j+j1) ;
if APopulation(i+i1,j+j1) > 0
PosCell2(NAPos2) = 1 ;
end ;
end ;
end ;
end ;
TotalNeighbourA2 = sum(NCell2 .* PosCell2 ) ;
if sum(TotalNeighbourA2) >= ExtraA
KK2=KK2+1 ;
TS2(KK2) = TimeStep ;
NAPos2 = 0 ;
for i1 = -2 : 2 ;
for j1 = -2 : 2 ;
if i1 ~=0 | j1 ~= 0
NAPos2 = NAPos2 + 1 ;
if PosCell2(NAPos2) == 1
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 ;
PosCell3 = zeros(1,48) ;
clear('NCell3') ;
for i1 = -3 : 3 ;
for j1 = -3 : 3 ;
if i1 ~=0 | j1 ~= 0
NAPos3 = NAPos3 + 1 ;
NCell3(NAPos3) = APopulation(i+i1,j+j1) ;
if APopulation(i+i1,j+j1) > 0
PosCell3(NAPos3) = 1 ;
end ;
end ;
end ;
end ;
TotalNeighbourA3 = sum(NCell3 .* PosCell3 ) ;
if sum(TotalNeighbourA3) >= ExtraA
KK3=KK3+1 ;
TS3(KK3) = TimeStep ;
NAPos3 = 0 ;
for i1 = -3 : 3 ;
for j1 = -3 : 3 ;
if i1 ~=0 | j1 ~= 0
NAPos3 = NAPos3 + 1 ;
if PosCell3(NAPos3) == 1
APopulation(i+i1,j+j1) =
APopulation(i+i1,j+j1) - (APopulation(i+i1,j+j1) / TotalNeighbourA3) * ExtraA ;
CPopulation(i+i1,j+j1) =
CPopulation(i+i1,j+j1) + (APopulation(i+i1,j+j1) / TotalNeighbourA3) * ExtraA ;
end ;
end ;
end ;
end ;
else
Neighbour3 = 0 ;
end ;
end ;
if Neighbour1 == 0 & Neighbour2 == 0 & Neighbour3 == 0
NAPos4 = 0 ;
PosCell4 = zeros(1,80) ;
clear('NCell4') ;
for i1 = -4 : 4 ;
for j1 = -4 : 4 ;
if i1 ~=0 | j1 ~= 0
NAPos4 = NAPos4 + 1 ;
NCell4(NAPos4) = APopulation(i+i1,j+j1) ;
if APopulation(i+i1,j+j1) > 0
PosCell4(NAPos4) = 1 ;
end ;
end ;
end ;
end ;
TotalNeighbourA4 = sum(NCell4 .* PosCell4 ) ;
if sum(TotalNeighbourA4) >= ExtraA
KK4=KK4+1 ;
TS4(KK4) = TimeStep ;
NAPos4 = 0 ;
for i1 = -4 : 4 ;
for j1 = -4 : 4 ;
if i1 ~=0 | j1 ~= 0
NAPos4 = NAPos4 + 1 ;
if PosCell4(NAPos4) == 1
APopulation(i+i1,j+j1) =
APopulation(i+i1,j+j1) - (APopulation(i+i1,j+j1) / TotalNeighbourA4) * ExtraA ;
CPopulation(i+i1,j+j1) =
CPopulation(i+i1,j+j1) + (APopulation(i+i1,j+j1) / TotalNeighbourA4) * ExtraA ;
end ;
end ;
end ;
end ;
else
Neighbour4 = 0 ;
end ;
end ;
if Neighbour1 == 0 & Neighbour2 == 0 & Neighbour3 == 0 & Neighbour4
== 0
202 Appendices
NAPos5 = 0 ;
PosCell5 = zeros(1,120) ;
clear('NCell5') ;
for i1 = -5 : 5 ;
for j1 = -5 : 5 ;
if i1 ~=0 | j1 ~= 0
NAPos5 = NAPos5 + 1 ;
NCell5(NAPos5) = APopulation(i+i1,j+j1) ;
if APopulation(i+i1,j+j1) > 0
PosCell5(NAPos5) = 1 ;
end ;
end ;
end ;
end ;
TotalNeighbourA5 = sum(NCell5 .* PosCell5 ) ;
if sum(TotalNeighbourA5) >= ExtraA
KK5=KK5+1 ;
TS5(KK5) = TimeStep ;
NAPos5 = 0 ;
for i1 = -5 : 5 ;
for j1 = -5 : 5 ;
if i1 ~=0 | j1 ~= 0
NAPos5 = NAPos5 + 1 ;
if PosCell5(NAPos5) == 1
APopulation(i+i1,j+j1) =
APopulation(i+i1,j+j1) - (APopulation(i+i1,j+j1) / TotalNeighbourA5) * ExtraA ;
CPopulation(i+i1,j+j1) =
CPopulation(i+i1,j+j1) + (APopulation(i+i1,j+j1) / TotalNeighbourA5) * ExtraA ;
end ;
end ;
end ;
end ;
else
Neighbour5 = 0 ;
end ;
end ;
if Neighbour1 == 0 & Neighbour2 == 0 & Neighbour3 == 0 & Neighbour4
== 0 & Neighbour5 == 0
NAPos6 = 0 ;
KK6=KK6+1 ;
TS6(KK6) = TimeStep ;
PosCell6 = zeros(1,(xg-10)*(yg-10)) ;
clear('NCell6') ;
for i1 = 6 : xg-5 ;
for j1 = 6 : yg-5 ;
NAPos6 = NAPos6 + 1 ;
NCell6(NAPos6) = APopulation(i1,j1) ;
if APopulation(i1,j1) > 0
PosCell6(NAPos6) = 1 ;
end ;
end ;
end ;
TotalNeighbourA6 = sum(NCell6 .* PosCell6 ) ;
if sum(TotalNeighbourA6) >= ExtraA
NAPos6 = 0 ;
for i1 = 6 : xg-5 ;
for j1 = 6 : yg-5 ;
NAPos6 = NAPos6 + 1 ;
if PosCell6(NAPos6) == 1
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 ;
if Neighbour1 == 0 & Neighbour2 == 0 & Neighbour3 == 0 & Neighbour4
== 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) ;
ErrorLR (i,3) = sum ((Exp4h - CNLR(:,i)) .^2) / size(Exp4h,1) ;
ErrorLR (i,4) = sum ((Exp6h - CNLR(:,i)) .^2) / size(Exp6h,1) ;
ErrorLR (i,5) = sum ((Exp8h - CNLR(:,i)) .^2) / size(Exp8h,1) ;
CVLR (i,1) = i ;
CVLR (i,2) = (sum ((Exp2h - CNLR(:,i)) .^2) / size(Exp2h,1))^0.5 /
mean(CNLR(:,i)) ;
CVLR (i,3) = (sum ((Exp4h - CNLR(:,i)) .^2) / size(Exp4h,1))^0.5 /
mean(CNLR(:,i)) ;
CVLR (i,4) = (sum ((Exp6h - CNLR(:,i)) .^2) / size(Exp6h,1))^0.5 /
mean(CNLR(:,i)) ;
CVLR (i,5) = (sum ((Exp8h - CNLR(:,i)) .^2) / size(Exp8h,1))^0.5 /
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))
;
MAELR (i,3) = ( sum (abs(Exp4h - CNLR(:,i))) ./ size(Exp4h,1) ) / mean(CNLR(:,i))
;
MAELR (i,4) = ( sum (abs(Exp6h - CNLR(:,i))) ./ size(Exp6h,1) ) / mean(CNLR(:,i))
;
MAELR (i,5) = ( sum (abs(Exp8h - CNLR(:,i))) ./ size(Exp8h,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) ;
ErrorD (i,3) = sum ((ExpD4h - CNLL(:,i)) .^2) / size(ExpD2h,1) ;
ErrorD (i,4) = sum ((ExpD6h - 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)) ;
CVD (i,3) = (sum ((ExpD4h - CNLL(:,i)) .^2) / size(ExpD4h,1))^0.5 /
mean(CNLL(:,i)) ;
CVD (i,4) = (sum ((ExpD6h - CNLL(:,i)) .^2) / size(ExpD6h,1))^0.5 /
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)) ;
MAED (i,3) = ( sum (abs(ExpD4h - CNLL(:,i))) ./ size(ExpD4h,1) ) /
mean(CNLL(:,i)) ;
MAED (i,4) = ( sum (abs(ExpD6h - CNLL(:,i))) ./ size(ExpD6h,1) ) /
mean(CNLL(:,i)) ;
MAED (i,5) = ( sum (abs(ExpD8h - CNLL(:,i))) ./ size(ExpD8h,1) ) /
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
TimeStepCLR(:,1)=b ;
[a,b] = min(CVLR(:,2:4)) ;
TimeStepCLR(:,4)=b ;
b(:,2) = round (b(:,2) ./ 2) ;
b(:,3) = round ( b(:,3) ./ 3) ;
minLR = min(b) ;
maxLR = max(b) ;
ErrorCLR(:,2)=a;
TimeStepCLR(:,2)=b ;
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(:,3) = Exp4h ;
LRfit(:,4) = CNLR(:,2*Int2h) ;
LRfit(:,5) = Exp6h ;
LRfit(:,6) = CNLR(:,3*Int2h) ;
Dfit(:,1) = ExpD2h ;
Dfit(:,2) = CNLL(:,Int2h) ;
Dfit(:,3) = ExpD4h ;
Dfit(:,4) = CNLL(:,2*Int2h) ;
Dfit(:,5) = ExpD6h ;
Dfit(:,6) = CNLL(:,3*Int2h) ;
[a,b] = min(MAED(:,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;
TimeStepCLR(:,1)=b ;
[a,b] = min(MAELR(:,2:4)) ;
TimeStepCLR(:,4)=b ;
b(:,2) = round (b(:,2) ./ 2) ;
b(:,3) = round ( b(:,3) ./ 3) ;
minLR = min(b) ;
maxLR = max(b) ;
ErrorCLR(:,2)=a;
TimeStepCLR(:,2)=b ;
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)) ;
CVLRfit (1,3) = (sum ((Exp6h - CNLR(:,3*Int2h)) .^2) / size(Exp6h,1))^0.5 /
mean(CNLR(:,3*Int2h)) ;
CVLRfit (1,4) = (sum ((Exp8h - CNLR(:,4*Int2h)) .^2) / size(Exp8h,1))^0.5 /
mean(CNLR(:,4*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)) ;
CVDfit (1,3) = (sum ((ExpD6h - CNLL(:,3*Int2h)) .^2) / size(ExpD6h,1))^0.5 /
mean(CNLL(:,3*Int2h)) ;
CVDfit (1,4) = (sum ((ExpD8h - CNLL(:,4*Int2h)) .^2) / size(ExpD8h,1))^0.5 /
mean(CNLL(:,4*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)) ;
MAELRfit (1,3) = ( sum (abs(Exp6h - CNLR(:,3*MInt2h))) ./ size(Exp6h,1) ) /
mean(CNLR(:,3*MInt2h)) ;
MAELRfit (1,4) = ( sum (abs(Exp8h - CNLR(:,4*MInt2h))) ./ size(Exp8h,1) ) /
mean(CNLR(:,4*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)) ;
MAEDfit (1,3) = ( sum (abs(ExpD6h - CNLL(:,3*MInt2h))) ./ size(ExpD6h,1) ) /
mean(CNLL(:,3*MInt2h)) ;
MAEDfit (1,4) = ( sum (abs(ExpD8h - CNLL(:,4*MInt2h))) ./ size(ExpD8h,1) ) /
mean(CNLL(:,4*MInt2h)) ;
[a,b] = min(MAPED(:,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;
TimeStepCLR(:,1)=b ;
[a,b] = min(MAPELR(:,2:4)) ;
TimeStepCLR(:,4)=b ;
b(:,2) = round (b(:,2) ./ 2) ;
b(:,3) = round ( b(:,3) ./ 3) ;
minLR = min(b) ;
maxLR = max(b) ;
ErrorCLR(:,2)=a;
TimeStepCLR(:,2)=b ;
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 ;
APopulation(isnan(APopulation)) = 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
;
Alpha(isnan(Alpha)) = 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 ;
WG2(isnan(WG2)) = 0 ;
WL2(isnan(WL2)) = 0 ;
WG3(isnan(WG3)) = 0 ;
WL3(isnan(WL3)) = 0 ;
WG4(isnan(WG4)) = 0 ;
WL4(isnan(WL4)) = 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 ;
APopulation(isnan(APopulation)) = 0 ;
Alpha = CP .* ( (APopulationRemain + APopulation) ./ (KC .* BPopulation) ) .^ 1.0
;
Alpha(isnan(Alpha)) = 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))) ;
cell(i+1,j+1) = (SolidC + ALPHAF * (Solid(2) - SolidC)) / (FluidC
- ALPHAF * (Solid(2) - SolidC)) ;
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))) /
(Fluid(3) - ALPHAF * (SolidC - Solid(3))) ;
cell(i+1,j+1) = (SolidC + ALPHAF * (Solid(3) - SolidC)) / (FluidC
- ALPHAF * (Solid(3) - SolidC)) ;
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))) /
(Fluid(4) - ALPHAF * (SolidC - Solid(4))) ;
cell(i+1,j+1) = (SolidC + ALPHAF * (Solid(4) - SolidC)) / (FluidC
- ALPHAF * (Solid(4) - SolidC)) ;
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))) /
(Fluid(5) - ALPHAF * (SolidC - Solid(5))) ;
cell(i+1,j+1) = (SolidC + ALPHAF * (Solid(5) - SolidC)) / (FluidC
- ALPHAF * (Solid(5) - SolidC)) ;
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))) /
(Fluid(6) - ALPHAF * (SolidC - Solid(6))) ;
214 Appendices
cell(i+1,j+1) = (SolidC + ALPHAF * (Solid(6) - SolidC)) / (FluidC
- ALPHAF * (Solid(6) - SolidC)) ;
elseif b(1) == 7
if cell(i+2,j+1) < LT & cell(i+1,j+1) < LT
ALPHAF = ALPHAL ;
else
ALPHAF = ALPHA ;
end ;
cell(i+2,j+1) = (Solid(7) + ALPHAF * (SolidC - Solid(7))) /
(Fluid(7) - ALPHAF * (SolidC - Solid(7))) ;
cell(i+1,j+1) = (SolidC + ALPHAF * (Solid(7) - SolidC)) / (FluidC
- ALPHAF * (Solid(7) - SolidC)) ;
elseif b(1) == 8
if cell(i+2,j+2) < LT & cell(i+1,j+1) < LT
ALPHAF = ALPHAL ;
else
ALPHAF = ALPHA ;
end ;
cell(i+2,j+2) = (Solid(8) + ALPHAF * (SolidC - Solid(8))) /
(Fluid(8) - ALPHAF * (SolidC - Solid(8))) ;
cell(i+1,j+1) = (SolidC + ALPHAF * (Solid(8) - SolidC)) / (FluidC
- ALPHAF * (Solid(8) - SolidC)) ;
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