A Cellular Scale Study of Low Density Lipoprotein Concentration Polarisation in Arteries
by
P. E. Vincent
Department of Aeronautics Imperial College London Prince Consort Road London SW7 2BY
This thesis is submitted for the degree of Doctor of Philosophy of the University of London
2009 Declaration
This is to certify that the work presented in this thesis has been carried out at Imperial College London and has not been previously submitted to any other university or technical institution for a degree or award. I further certify that all material in this thesis which is not my own work has been properly acknowledged.
P. E. Vincent Abstract
Uptake of Low Density Lipoprotein (LDL) by the arterial wall is likely to play a key role in the process of atherogenesis, which occurs non-uniformly within the ar- terial vasculature. A particular process that may cause vascular scale heterogeneity in the rate of transendothelial LDL transport is the formation of a flow-dependent LDL concentration polarisation layer adjacent to the luminal surface of the arte- rial endothelium. In this thesis the effects of cellular scale endothelial features on such LDL concentration polarisation are investigated using an idealised theoretical model. Specifically, the effect of a spatially heterogeneous transmural water flux is considered (flowing only through intercellular clefts), as well as the effect of the endothelial glycocalyx layer (EGL). The idealised model is implemented using both analytical techniques and the spectral/hp element method. A range of scenarios are considered, including those were no EGL is present, those where an EGL is present but LDL cannot penetrate into it, and finally those where an EGL is present and LDL can penetrate into it.
For cases where no EGL is present, particular attention is paid to the spatially averaged LDL concentration adjacent to various regions of the endothelial surface, as such measures may be relevant to the rate of transendothelial LDL transport. It is demonstrated, in principle, that a heterogeneous transmural water flux alone can act to enhance such measures, and cause them to develop a shear dependence (in addition to that caused by vascular scale flow features affecting the overall degree of LDL concentration polarisation). However, it is shown that this enhancement and additional shear dependence are likely to be negligible for a physiologically realistic transmural flux velocity of 0.0439µms−1 and an LDL diffusivity in blood plasma of 28.67µm2s−1.
For cases where an EGL is present, measures of LDL concentration polarisation relevant to the rate of transendothelial LDL transport can also be defined. It is demonstrated that an EGL is unlikely to cause any additional shear dependence of such measures directly, irrespective of whether or not LDL can penetrate into the EGL. However, it is found that such measures depend significantly on the nature of the interaction between LDL and the EGL (parameterised by the height of the EGL, the depth to which LDL penetrates into the EGL, and the diffusivity of LDL within the EGL). Various processes may regulate the interaction of LDL with the EGL, possibly in a flow dependent and hence spatially non-uniform fashion. It is concluded that any such processes may be as important as vascular scale flow features in terms of spatially modulating transendothelial LDL transport via an LDL concentration polarisation mechanism.
3 Acknowledgments
I would like to begin by thanking Prof. Colin Caro and Prof. Spencer Sherwin for giving me the opportunity to do my PhD in such a diverse and interesting field of research. I would like to thank Prof. Spencer Sherwin and Prof. Peter Weinberg for their excellent supervision. I would like to thank Prof. Kim Parker, Prof. Charles Michel, Prof. John Tarbell, Prof. Sheldon Weinbaum, Prof. David Rumschitzki, Prof. Darren Crowdy, Prof. Aaron Fogelson and Dr. Leopold Grinberg for their wisdom and assistance. I would like to thank all of my office mates, especially Donal and Andy, for all the useful discussions. I would like to thank Hakan and Marcus for all of their sensible thoughts. Finally, I would like to thank Ana for all of her love and support (especially the besitos and paella), and my parents for their unwavering support and understanding throughout the last 25 years.
This work was funded by the Engineering and Physical Sciences Research Council (UK). Nomenclature
b∗ Distance LDL can penetrate into the EGL b Non-dimensional distance LDL can penetrate into the EGL ∗ C Profile of generic macroscale LDL concentration polarisation layer
∗ CB LDL concentration in bulk blood flow ∗ CE LDL concentration adjacent to luminal surface of endothelium ∗ CW LDL concentration within arterial wall ∗ Cα LDL concentration field within ΩαC
Cα Non-dimensional LDL concentration field within ΩαC ∗ Cα Profile of macroscale LDL concentration polarisation layer when no EGL is considered
Cα Non-dimensional profile of macroscale LDL concentration polarisation layer when no EGL is considered
CαN Spatially averaged non-dimensional LDL concentration to which the endothelium is exposed (in vicinity of cleft entrances when no EGL is considered)
∗ Cαt Time dependent LDL concentration field within ΩαC
Cαt Non-dimensional time dependent LDL concentration field within ΩαC
CαtN Temporally and spatially averaged non-dimensional LDL concentration to which the endothelium is exposed (in vicinity of cleft entrances when no EGL is considered)
CαtU Temporally and spatially averaged non-dimensional LDL concentration to which the endothelium is exposed (when no EGL is considered) CαU Spatially averaged non-dimensional LDL concentration to which the endothelium is exposed (when no EGL is considered)
∗ Cβ LDL concentration field within ΩβC
Cβ Non-dimensional LDL concentration field within ΩβC ∗ Cβ Profile of macroscale LDL concentration polarisation layer when EGL is considered
Cβ Non-dimensional profile of macroscale LDL concentration polarisation layer when EGL is considered
CβN Spatially averaged non-dimensional LDL concentration to which the endothelium is exposed (in vicinity of cleft entrances when EGL is considered)
CβU Spatially averaged non-dimensional LDL concentration to which the endothelium is exposed (when EGL is considered) Da Non-dimensional inverse Darcy permeability tensor within the EGL
∗ DG LDL diffusivity within the EGL ∗ DL LDL diffusivity within the lumen
Dr Ratio of LDL diffusivity in the EGL to LDL diffusivity in the lumen
Dax Inverse of first eigenvalue of Da
Day Inverse of second eigenvalue of Da ∗ ∗ ∗ f Half-distance over which LDL diffusivity varies between DL and DG f Non-dimensional half-distance over which LDL diffusivity varies
∗ ∗ between DL and DG h∗ Height of EGL h Non-dimensional height of EGL
∗ JB LDL flux from bulk flow to endothelial surface ∗ JE LDL flux across endothelium K∗ Inverse Darcy permeability tensor within the EGL
∗ kB Blood side mass transfer coefficient ∗ kE Permeability of endothelium to LDL ∗ ∗ Kx Inverse of first eigenvalue of K
6 ∗ ∗ Ky Inverse of second eigenvalue of K ∗ Lα Height of domain ΩαC
Lα Non-dimensional height of domain ΩαC ∗ Lβ Height of domain ΩβC
Lβ Non-dimensional height of domain ΩβC
P eG Peclet number associated with heterogeneous transmural flux within EGL
P eL Peclet number associated with heterogeneous transmural flux within lumen
∗ pα Pressure field within ΩαV pα Non-dimensional pressure field within ΩαV ∗ pβG Pressure field within ΩβV G pβG Non-dimensional pressure field within ΩβV G ∗ pβL Pressure field within ΩβV L pβL Non-dimensional pressure field within ΩβV L
T Non-dimensional period of γt uα x component of non-dimensional water velocity field vα uαt x component of non-dimensional water velocity field vαt uβG x component of non-dimensional water velocity field vβG uβL x component of non-dimensional water velocity field vβL ∗ V Average transmural water flux velocity vα y component of non-dimensional water velocity field vα ∗ vα Water velocity field within ΩαV vα Non-dimensional water velocity field within ΩαV vαt y component of non-dimensional water velocity field vαt ∗ vαt Time dependent water velocity field within ΩαV vαt Non-dimensional time dependent water velocity field within ΩαV ∗ vβ Water velocity field within ΩβV vβ Non-dimensional water velocity field within ΩβV vβG y component of non-dimensional water velocity field vβG
7 ∗ vβG Water velocity field within ΩβV G vβG Non-dimensional water velocity field within ΩβV G vβL y component of non-dimensional water velocity field vβL ∗ vβL Water velocity field within ΩβV L vβL Non-dimensional water velocity field within ΩβV L γ∗ Shear rate applied orthogonal to length-wise extent of idealised clefts γ Non-dimensional shear rate applied orthogonal to length-wise extent of idealised clefts
∗ ∗ γa Amplitude of γt
γa Non-dimensional amplitude of γt ∗ ∗ γo Offset of γt
γo Non-dimensional offset of γt ∗ γt Time dependent shear rate applied orthogonal to length-wise extent of idealised clefts
γt Non-dimensional time dependent shear rate applied orthogonal to length-wise extent of idealised clefts
∗ γT Shear rate applied to region of interest δ∗ Cleft half-width δ Non-dimensional cleft half-width ∆∗ Cleft half-spacing ζ∗ Thickness of macroscale LDL concentration polarisation layer ζ Non-dimensional thickness of macroscale LDL concentration polarisation layer θ∗ Angle between applied shear and idealised intercellular clefts
κα Non-dimensional distance from endothelium at which Cα becomes approximately one-dimensional
κβ Non-dimensional distance from endothelium at which Cβ becomes approximately one-dimensional µ∗ Dynamic viscosity of water
σα Non-dimensional stress tensor within ΩαV
8 σβG Non-dimensional stress tensor within ΩβV G
σβL Non-dimensional stress tensor within ΩβV L
ψα Stream function within ΩαV
ψβG Stream function within ΩβV G
ψβL Stream function within ΩβV L ∗ ∗ ω Frequency of γt
ω Non-dimensional frequency of γt
ΩαC Domain in which LDL transport is modelled when no EGL is considered
ΩαV Domain in which water motion is modelled when no EGL is considered
ΩβC Domain in which LDL transport is modelled when EGL is considered
ΩβCG Sub-domain of ΩβC that covers EGL
ΩβCL Sub-domain of ΩβC that covers lumen
ΩβV Domain in which water motion is modelled when EGL is considered
ΩβV G Sub-domain of ΩβV that covers EGL
ΩβV L Sub-domain of ΩβV that covers lumen
9 Contents
Declaration 1
Abstract 2
Acknowledgements 4
Nomenclature 5
Table of Contents 10
List of Figures 18
List of Tables 21
1 Introduction 23
1.1 Atherosclerosis ...... 23
1.1.1 Overview ...... 23
1.1.2 Initiation of Atherosclerosis ...... 24
1.1.3 Inter-Human Variation ...... 25
1.1.4 Intra-Human Variation ...... 25 1.2 Blood Flow and Atherosclerosis ...... 26
1.2.1 Overview ...... 26
1.2.2 Regulation by Mechanical Forces ...... 28
1.2.3 Regulation by Mass Transport Mechanisms ...... 28
1.3 Flow Dependent LDL Concentration Polarisation within Arteries . . . 33
1.3.1 Concentration Polarisation ...... 33
1.3.2 LDL Concentration Polarisation within Arteries ...... 34
1.3.3 Flow Dependent LDL Concentration Polarisation within Ar- teries ...... 35
1.3.4 Flow Dependent Regulation of Atherogenesis ...... 37
1.3.5 Previous Studies ...... 37
1.4 Relevant Cellular Scale Features of the Endothelial Surface ...... 38
1.4.1 Spatially Heterogeneous Transmural Water Flux ...... 38
1.4.2 Endothelial Glycocalyx Layer ...... 39
1.5 Objective of Thesis ...... 41
1.6 Overview of Thesis ...... 41
2 Spectral/hp Element Method 42
2.1 Overview ...... 42
2.2 One-Dimensional Spectral/hp Element Method ...... 44
2.2.1 Example Problem ...... 44
2.2.2 Weak Formulation ...... 44
11 2.2.3 An Approximate Solution ...... 45
2.2.4 A Spectral/hp Element Basis ...... 46
2.2.5 Galerkin Approximation ...... 51
2.2.6 Global Matrix Equation ...... 51
2.2.7 Matrix Structure ...... 52
2.2.8 Global Assembly ...... 55
2.2.9 Other Elemental Bases ...... 58
2.3 Extension to Two-Dimensions ...... 58
2.3.1 Overview ...... 58
2.3.2 Basis Functions Within Quadrangular Elements ...... 59
2.3.3 Basis Functions Within Triangular Elements ...... 60
2.4 Advection Diffusion Equation ...... 62
2.4.1 Steady State Two-Dimensional Advection Diffusion Equation . 62
2.4.1.1 Definition ...... 62
2.4.1.2 Weak Form ...... 62
2.4.1.3 Obtaining an Approximate Solution ...... 63
2.4.2 Unsteady Two-Dimensional Advection Diffusion Equation . . 64
2.4.2.1 Definition ...... 64
2.4.2.2 Temporal Discretisation ...... 64
2.4.2.3 Weak Form ...... 65
2.4.2.4 Obtaining an Approximate Solution ...... 66
12 2.5 Summary ...... 67
3 The Effect of a Spatially Heterogeneous Transmural Water Flux 68
3.1 Overview ...... 68
3.2 Idealised Model of Water Flow ...... 69
3.2.1 Heterogeneous Transmural Flux ...... 69
3.2.2 Applied Shear ...... 71
3.2.3 Domain ...... 72
3.2.4 Governing Equations ...... 73
3.2.5 Non-Dimensionalisation ...... 74
3.2.6 Non-Dimensional Boundary Conditions ...... 74
3.2.6.1 Periodicity in x ...... 74
3.2.6.2 Conditions at y =0 ...... 75
3.2.6.3 Large y Behavior ...... 76
3.3 Idealised Model of LDL Transport ...... 76
3.3.1 Domain and Governing Equation ...... 76
3.3.2 Suitable Definition of Macroscale LDL Concentration Polari- sation Layer ...... 76
3.3.3 Non-Dimensionalisation ...... 77
3.3.4 Non-Dimensional Boundary Conditions ...... 79
3.3.4.1 Periodicity in x ...... 79
3.3.4.2 Condition at y =0...... 79
13 3.3.4.3 Condition at y = Lα ...... 80
3.4 Implementation ...... 82
3.4.1 Water Velocity Field ...... 82
3.4.1.1 Formulating in Terms of a Stream Function . . . . . 82
3.4.1.2 Trial Solution for ψα ...... 82
3.4.1.3 Application of Boundary Conditions ...... 82
3.4.1.4 Velocity Field Solution ...... 83
3.4.2 LDL Concentration Field ...... 83
3.4.3 Parameter Values ...... 84
3.4.3.1 Value for the Non-Dimensional Cleft Half-Width δ . 84
3.4.3.2 Values for the Non-Dimensional Shear Rate γ .... 85
3.4.3.3 Values for the Peclet Number P eL ...... 85
3.4.3.4 Value for the Non-Dimensional Height Lα of ΩαC .. 86
3.5 Results and Analysis ...... 86
3.5.1 Water Velocity Field ...... 86
3.5.2 LDL Concentration Field ...... 87
3.5.3 Transendothelial LDL Transport ...... 91
3.5.3.1 Definitions ...... 91
3.5.3.2 Dependence of CαU and CαN on P eL ...... 92
3.5.3.3 Dependence of CαU and CαN on γ ...... 94
3.5.4 Extending the Parameter Space ...... 96
14 3.5.5 Implications of Neglecting Tight Junction Strands ...... 100
3.5.6 Implications of Neglecting Pulsatile Blood Flow ...... 100
3.6 Conclusions ...... 101
4 The Effect of an Endothelial Glycocalyx Layer 102
4.1 Overview of Idealised Model ...... 102
4.2 Idealised Model of Water Flow ...... 103
4.2.1 Heterogeneous Transmural Flux and Applied Shear ...... 103
4.2.2 Domain and Governing Equations ...... 103
4.2.3 Non-Dimensionalisation ...... 106
4.2.4 Non-Dimensional Boundary Conditions ...... 108
4.2.4.1 EGL/Lumen Interface ...... 108
4.2.4.2 Periodicity in x ...... 109
4.2.4.3 Conditions at y =0 ...... 109
4.2.4.4 Large y Behavior ...... 110
4.3 Idealised Model of LDL Transport ...... 110
4.3.1 Domain and Governing Equation ...... 110
4.3.2 Suitable Definition of Macroscale LDL Concentration Polari- sation Layer ...... 112
4.3.3 Non-Dimensionalisation ...... 113
4.3.4 Non-Dimensional Boundary Conditions ...... 115
4.3.4.1 Periodicity in x ...... 115
15 4.3.4.2 Condition at y = h − b ...... 115
4.3.4.3 Condition at y = h − b + L ...... 115
4.4 Implementation ...... 117
4.4.1 Water Velocity Field ...... 117
4.4.1.1 Formulating in Terms of Stream Functions ...... 117
4.4.1.2 Trial Solution for ψβL ...... 118
4.4.1.3 Trial Solution for ψβG when Dax ≤ Day ...... 118
4.4.1.4 Trial Solution for ψβG when Dax > Day ...... 119
4.4.1.5 Application of Boundary Conditions ...... 120
4.4.1.6 Velocity Field Solution ...... 123
4.4.2 LDL Concentration Field ...... 124
4.4.3 Parameter Values ...... 125
4.4.3.1 Value for the Non-Dimensional Cleft Half-Width δ . 125
4.4.3.2 Values for the Non-Dimensional Shear Rate γ .... 125
4.4.3.3 Values for Cleft Peclet Number P eL ...... 125
4.4.3.4 Values for the Non-Dimensional EGL Height h ... 125
4.4.3.5 Values for the Non-Dimensional Darcy Permeabili-
ties Dax and Day ...... 126
4.4.3.6 Values for the Ratio of EGL to Lumen LDL Diffu-
sivity Dr ...... 126
4.4.3.7 Values for the Non-Dimensional Distance b that LDL Penetrates into the EGL ...... 126
16 4.4.3.8 Value for the Non-Dimensional Height Lβ of ΩβC .. 127
4.5 Results and Analysis ...... 127
4.5.1 Water Velocity Field ...... 127
4.5.1.1 Streamlines ...... 127
4.5.1.2 Wall Normal Velocity ...... 129
4.5.2 LDL Concentration Field (No LDL Penetration into the EGL) 130
4.5.3 Transendothelial LDL Transport (No LDL Penetration into the EGL) ...... 133
4.5.3.1 Definitions ...... 133
4.5.3.2 Dependence of CβU and CβN on Applied Shear γ .. 135
4.5.3.3 Dependence of CβU and CβN on EGL Properties h,
Dax and Day ...... 135
4.5.4 LDL Concentration Field (LDL Penetration into the EGL) . . 135
4.5.5 Transendothelial LDL Transport (LDL Penetration into the EGL) ...... 141
4.5.5.1 Dependence of CβU and CβN on Applied Shear γ .. 141
4.5.5.2 Dependence of CβU and CβN on LDL Interaction
with the EGL (h, b and Dr)...... 141
4.6 Conclusions ...... 145
5 Conclusions and Future Research 146
5.1 Conclusions ...... 146
5.2 Future Research ...... 147
17 5.2.1 Future Theoretical Research ...... 147
5.2.2 Future Experimental Research ...... 149
Bibliography 150
A Applicability 160
B Convergence 162
C Time Dependent Simulations 164
C.1 Water Velocity Field ...... 164
C.2 LDL Concentration Field ...... 165
C.3 Transendothelial LDL Transport ...... 166
C.4 Results ...... 167
D Trial Solution for ψβG when Dax > Day 168
ˆ ˆ ˆ ˆ ˆ ˆ E Explicit Expressions for Aβn, Bβn, Cβn, Dβn, Eβn and Fβn 171
F Resulting Publications and Presentations 179
F.1 Journal Articles ...... 179
F.2 Conference Contributions ...... 180
F.3 Seminars ...... 182
18 List of Figures
1.1 Schematic illustration of the interface between the arterial wall and the lumen...... 30
1.2 Diagram illustrating how concentration polarisation of a solute can occur adjacent to a generic membrane...... 34
1.3 Schematic illustration of LDL concentration polarisation adjacent to the luminal surface of the arterial endothelium...... 36
1.4 Schematic illustration of a tight junction strand within an intercellu- lar cleft...... 38
1.5 Illustration of EGL structure...... 39
1.6 Image of the EGL within a rat ventricular myocardial capillary ob- tained using electron microscopy...... 40
2.1 Graphical illustration of global spectral/hp basis functions...... 50
2.2 A quadrangular standard element...... 59
2.3 A triangular standard element...... 60
3.1 Idealisation of intercellular cleft pattern...... 70
3.2 Idealised periodically repeating domains ΩαV and ΩαC adjacent to the endothelial surface...... 73 ∗ 3.3 Schematic illustration of Cα...... 77
3.4 Computational mesh of the domain ΩαC ...... 84
3.5 Velocity streamlines in the vicinity of an intercellular cleft...... 87
3.6 Contour plots of Cα...... 88
3.7 Plots illustrating how the LDL distribution adjacent to the endothe- lium varies with γ...... 89
3.8 Plots of CαU and CαN against P eL...... 92
3.9 Plots of (CαU − 1) against P eL...... 94
3.10 Plots of CαU and CαN against γ...... 95
3.11 Plots of (CαU − 1) and (CαN − 1) against δ...... 97
3.12 Plots of CαU against γ for various values of δ...... 98
3.13 Plots of CαU against P eL and (CαU -1) against P eL...... 99
4.1 Idealised periodically repeating domain ΩβV ...... 104
4.2 Idealised periodically repeating domain ΩβC ...... 111
∗ 4.3 Schematic illustration of Cβ...... 113
4.4 Computational mesh of the domain ΩβC ...... 124
4.5 Velocity streamlines in the vicinity of an intercellular cleft...... 128
4.6 Plots of the wall normal velocity at the luminal surface of the EGL. . 129
4.7 Plots illustrating how the LDL distribution adjacent to the luminal surface of the EGL varies with EGL height h for cases where LDL cannot penetrate into the EGL...... 131
20 4.8 Plots illustrating how the LDL distribution adjacent to the luminal
surface of the EGL varies with Dax and Day for cases where no LDL can penetrate into the EGL...... 132
4.9 Contour plots of Cβ...... 137
4.10 Plots illustrating how the LDL distribution evaluated at y = 0.05 − b varies with b for cases where LDL can penetrate a distance b into an EGL of height h = 0.05...... 139
4.11 Plots illustrating how the LDL distribution evaluated at y = 0.1 − b varies with b for cases where LDL can penetrate a distance b into an EGL of height h = 0.1...... 140
4.12 Plots of CβU and CβN against b for the case where LDL can penetrate a distance b into an EGL of height h = 0.05...... 142
4.13 Plots of CβU and CβN against b for the case where LDL can penetrate a distance b into an EGL of height h = 0.1...... 143
∗ ∗ A.1 Plot of CEm/CB against γ...... 161
B.1 Plots of EM against M...... 163
21 List of Tables
4.1 Ranges of CβU and CβN as LDL penetration distance varies over the range b = 0.01 − 0.05...... 144 Chapter 1
Introduction
1.1 Atherosclerosis
1.1.1 Overview
Atherosclerosis is a common disease of the human vascular system, characterised by the formation of lipid-rich lesions within the walls of large arteries. After several decades of growth such lesions may act to stenose their native vessel, leading to symptoms such as angina (caused by coronary stenoses restricting blood flow to the heart). Further to causing stenoses, lesions may also become unstable and hence prone to rupture. When an atherosclerotic lesion ruptures a mass of thrombogenic material is released into the lumen, causing a blood clot to form. Such a clot may cause sudden and complete arterial occlusion, and hence severe disruption of the oxygen supply to organs within the body. Two well known consequences of such occlusion are heart attacks (caused by occlusion of a coronary artery), and strokes (caused by occlusion of an artery that supplies the brain). In addition to its severe consequences, atherosclerosis is also striking in its worldwide prevalence. According to the World Health Organisation (WHO) cardiovascular disease, and in particular atherosclerosis, is now the leading cause of death globally each year [1]. Furthermore, by 2020 the WHO predict that atherosclerosis will become the single most burdensome disease on the health systems of the world [2].
1.1.2 Initiation of Atherosclerosis
This thesis is primarily concerned with the initiation of atherosclerosis, as opposed to processes involved in the later stages of the disease. At a simplistic level the most fundamental steps involved in the early stages of atherosclerosis development can be outlined as follows:
• Low Density Lipoproteins (LDL), which are the primary transporter of choles- terol within the blood, begin to accumulate in the intima of the arterial wall. Accumulated LDL becomes modified (oxidised for example), and re- leases chemicals that attract monocytes to the regions of LDL build up.
• Monocytes enter the vascular wall at sites of LDL accumulation and differ- entiate into macrophages. These macrophages ingest the accumulated LDL, creating lipid-rich foam cells.
• Mechanisms involving High Density Lipoproteins (HDL), for example, can act to remove accumulated lipids from foam cells.
• If lipid influx into the arterial wall is consistently greater than lipid efflux from the wall, then a net accumulation of lipid-rich foam cells will begin to occur.
• This unchecked net accumulation of lipid-rich foam cells is a marker of an early stage atherosclerotic lesion.
The outline of the atherosclerosis initiation process presented above is generally accepted in the literature [3, 4]. What remains contentious, however, are the mech- anisms by which the process is regulated i.e. what promotes and what limits its occurrence.
24 1.1.3 Inter-Human Variation
The fact that not all humans exhibit pathological atherosclerosis indicates that its development is regulated at an inter-human level (i.e. between humans). It is observed that the disease develops preferentially in individuals exhibiting a range of risk factors. The most important of these risk factors are listed below:
• High levels of LDL in the blood.
• Low levels of HDL in the blood.
• High blood pressure.
• Diabetes.
• Smoking.
• Gender (being male).
• Genetic predisposition.
• Excess stress.
• Lack of exercise.
Generally, the factors in the above list have been identified empirically. They offer some useful insight into the process of atherosclerosis initiation. However, they do not allow detailed elucidation of any processes that regulate onset of the disease.
1.1.4 Intra-Human Variation
As well as varying between individuals, the prevalence of atherosclerotic lesions is also seen to vary at the intra-human level (i.e. within the arterial vasculature), with lesions developing preferentially in regions of arterial branching and high curvature. Since these particular regions are associated with complex blood flow patterns, it
25 has been postulated that blood flow, which is also spatially heterogeneous at the vascular scale, may play an important role in regulating atherogenesis [5, 6]. Sev- eral mechanisms that could lead to a flow dependence of atherogenesis have been suggested. Such mechanisms will be discussed in more detail shortly.
A fuller understanding of the mechanisms that cause atherosclerosis to be spatially heterogeneous could lead to a fuller understanding of the mechanisms that regulate onset of atherosclerosis per se. Such enhanced understanding could potentially lead to the development of improved therapies to avert or treat the disease.
1.2 Blood Flow and Atherosclerosis
1.2.1 Overview
Correlations between blood flow patterns and sites of arterial disease have been suggested for over a century. One of the earliest proponents of such correlations was Rindfleisch, who in 1872 stated that atherosclerotic lesions formed in locations ‘exposed to the full stress and impact of the blood’ [7]. Based on the observations of Rindfleisch, although almost 100 years after they were initially made, a variety of mechanisms to explain correlations between flow and disease were proposed during the 1960’s. Generally, such mechanisms relied on flow induced wall damage ren- dering particular regions of the vasculature more permeable to atherogenic species such as LDL, and thus more susceptible to atherosclerosis [8]. As a consequence of these initial theories, a general consensus developed by the end of the 1960’s that regions exposed to high levels of wall shear stress were predisposed to developing atherosclerosis [9].
In the late 1960’s and early 1970’s, however, two papers by Caro et al. overturned the previously held consensus that high levels of wall shear stress were atherogenic. Caro et al. instead suggested that regions of arterial disease were colocated with areas exposed to low levels of wall shear stress [10, 11]. Furthermore, Caro et al. also
26 proposed a mechanism by which such colocation could occur. This mechanism was more sophisticated than any previously suggested, and relied on the shear modulated efflux of species (such as LDL) from the arterial wall into the lumen.
After its conception in the early 1970’s, the low shear hypothesis of Caro et al. became widely accepted. To a large extent, the idea that lesions develop at sites of low wall shear stress is still accepted today (although the original mechanism proposed by Caro et al. to cause such colocation is now considered unfeasible). In recent decades, however, it has become increasingly evident that a direct correla- tion between disease patterns and (solely) wall shear stress magnitude may be too simplistic. A wide range of more complex relationships have been suggested, with a tentative consensus now developing that a combination of low and temporally oscillating wall shear stress predisposes regions of arteries to disease [12, 13, 14]. An array of mechanisms attempting to explain these more complex relationships have also been suggested. Such mechanisms generally, although not always, fall into one of two categories. The first category contains mechanisms whereby mechani- cal forces due to blood flow directly influence properties of the arterial wall [15], whereas the second category contains mechanisms reliant on the flow dependent transport of species to, into, and out of the arterial wall [16, 17]. Mechanisms from either category often have interlinking components, and the implications of individ- ual mechanisms often overlap. The current picture is thus highly complex. Brief overviews of each type of mechanism are given below.
Before proceeding with discussions, it is important to highlight the following salient point. It is not being suggested that atherosclerosis occurs solely because individuals develop atherogenic blood flow patterns. Rather it is being suggested that blood flow predisposes certain regions of the vasculature to atherosclerosis (when the disease is induced by the risk factors mentioned in section 1.1.3). This point was also noted over 100 years ago by Rindfleisch, who stated that ‘mechanical irritation must not be regarded as the sole cause of the disorder, though it determines its localisation. Among other predisposing causes are advanced age and free living’ [7].
27 1.2.2 Regulation by Mechanical Forces
The luminal surface of the arterial wall is lined with a monolayer of endothelial cells, known collectively as the endothelium. The endothelium is likely to play a key role in the process of atherogenesis, since it acts as a barrier [18] between the luminal blood (containing LDL and monocytes) and the intima (where atherosclerotic lesions begin to develop).
It has been demonstrated that the endothelium is able to sense applied flow forces, and respond in various ways. One well know example of such a response is the observed alignment of vascular endothelial cells with the direction of blood flow [19, 20]. There is currently much debate regarding exactly how endothelial cells sense and respond to applied forces. A wide range of mechano-chemical transducers have been implicated in the process, including flow activated ion channels, tyrosine kinase receptors, G proteins, cell-cell junction molecules and the endothelial gly- cocalyx layer. These transducers are currently associated with a vast array of flow dependent endothelial responses [15, 21, 22], a detailed review of which is beyond the scope of this thesis. Suffice it to say that many of the responses may affect arterial susceptibility to atherosclerosis (by varying endothelial permeability to atherogenic species for example). Therefore, endothelial mechano-chemical transduction mech- anisms constitute a direct way in which mechanical forces, and hence blood flow patterns, could regulate the onset of arterial disease.
1.2.3 Regulation by Mass Transport Mechanisms
Several cells/species are thought to play key roles in the process of atherosclerosis initiation. LDL, HDL, and monocytes have already been mentioned, since they are likely to play a fundamental role. Other important species include oxygen (O2), ni- tric oxide (NO), adenosine triphosphate (ATP), and adenosine diphosphate (ADP). It has been postulated that blood flow patterns may regulate transport of several of the aforementioned species to, into and out of the arterial wall. Such flow depen-
28 dent species transport could offer an explanation for the observed flow dependence of atherosclerosis.
There have been a range of experimental and numerical studies investigating O2 [23], ATP, ADP [24] and LDL [25, 26, 27, 28] transport within arteries. A review of computational modelling techniques applied to arterial mass transport is given by Ethier [16], and a review of arterial mass transport in general (with a particular focus on its relation to the localisation of atherosclerosis) is given by Tarbell [17]. The review of Tarbell splits arterial mass transport processes into two distinct groups, namely those termed ‘fluid-phase-limited’ and those termed ‘wall-limited’. Broadly speaking these two types of process can be described as follows:
• Fluid-phase-limited processes are those in which the rate of species transport into the arterial wall is limited by the rate at which species are transported to the wall from the flowing blood. It has been suggested that low molecular
weight species that react quickly with the arterial wall (such as O2 and ATP) enter the arterial wall in a fluid-phase-limited fashion [17].
• Wall-limited processes are those in which the rate of species transport into the arterial wall is limited by the wall itself, not by a fluid-phase-limited trans- port process. It has been suggested that high molecular weight species that are offered significant resistance by the arterial wall (such as LDL) enter the arterial wall in a wall-limited fashion [17].
For a fluid-phase-limited transport process, it is clear that blood flow patterns may control the rate of species uptake by the arterial wall (potentially causing flow de- pendent regulation of atherogenesis). For wall-limited transport processes, however, the situation is less clear. It is often implied that wall-limited processes will be flow-independent (simply because they are independent of fluid-phase transport). Indeed the terms wall-limited and flow-independent are often used synonymously. For a variety of reasons, however, such synonymous use of these terms is not nec- essarily correct. As an illustrative example of why wall-limited does not necessarily
29 mean flow-independent, consider specifically the process of transendothelial LDL transport. It has been suggested that this process is independent of fluid-phase transport and hence wall-limited [17], with the argument as to why proceeding in the following manner [17, 29].
Consider LDL transport within the locality of an arbitrary point on the arterial wall (as illustrated in Fig. 1.1).
Endothelium
Arterial Lumen Wall Artery
* * * CW CE CB
* * kE kB
Figure 1.1: Schematic illustration of the interface between the arterial wall and ∗ the lumen. LDL concentration in the arterial wall is denoted CW , LDL concentra- ∗ tion adjacent to the luminal surface of the endothelium is denoted CE, and LDL ∗ concentration in the bulk flow is denoted CB (all within the locality of an arbitrary point on the arterial wall). The endothelium is considered to have a permeability ∗ kE to LDL. It is assumed that fluid-phase LDL transfer from the bulk flow to the ∗ endothelial surface is characterised by a mass transfer coefficient kB.
Further, consider that the flux of LDL across the endothelium into the arterial wall
∗ JE is given by
∗ ∗ ∗ ∗ JE = kE(CE − CW ), (1.1)
∗ where CE is the LDL concentration adjacent to the luminal surface of the endothe-
30 ∗ lium, CW is the LDL concentration on the abluminal surface of the endothelium ∗ (in the arterial wall), and kE is the permeability of the endothelium to LDL. Also, consider that the flux of LDL from the flowing blood to the luminal surface of the
∗ endothelium JB is given by
∗ ∗ ∗ ∗ JB = kB(CB − CE), (1.2)
∗ ∗ where CB is the LDL concentration in the bulk blood flow, and kB is a blood side mass transfer coefficient. The review of Tarbell [17] (from which the present analysis
∗ is taken) states that kB characterises a convective-diffusive LDL transport mecha- nism. This statement is true, although slightly misleading, since due to the form of
∗ ∗ Eq. (1.2) JB can only explicitly model diffusional LDL transport. Therefore, kB only accounts for convection insomuch as it accounts for its effect on the thickness and form of the LDL concentration boundary layer (and thus on the diffusional transport
∗ of LDL to the endothelial surface). The mass transfer coefficient kB, and hence the present analysis, cannot directly account for any convective LDL transport towards the endothelium. The implications of this limitation will be discussed shortly. For now, however, we return to the analysis and assume that in steady state
∗ ∗ JB = JE. (1.3)
Further, it is assumed that since the endothelium offers a significant resistance to LDL
∗ ∗ CW CE. (1.4)
∗ Substituting Eq. (1.1) and Eq. (1.2) into Eq. (1.3), and setting CW = 0 due to Eq. (1.4), one obtains
∗ ∗ ∗ ∗ ∗ kB(CB − CE) = kECE, and hence
31 ∗ ∗ CB CE = ∗ ∗ . (1.5) 1 + (kE/kB)
∗ ∗ Finally, it is assumed that for LDL kE kB, hence Eq. (1.5) can be re-written as
∗ ∗ CE = CB. (1.6)
Eq. (1.6) indicates that LDL concentration adjacent to the luminal surface of the
∗ ∗ endothelium CE is likely to be independent of kB, and hence independent of fluid- ∗ phase transport processes. The implication of this result is that JE (which is given ∗ ∗ by CBkE under the above assumptions) is also likely to be independent of fluid-phase transport. Transendothelial LDL transport is therefore deemed to be a wall-limited process.
Although the above analysis indicates that transendothelial LDL transport is inde- pendent of fluid-phase LDL transport, it has not shown the process to be completely flow-independent. There are in fact three ways in which blood flow could control the rate of LDL transport across the endothelium. The first two mechanisms can be understood within the context of the above analysis, and can be described as follows:
• Flow could regulate endothelial permeability to LDL via a mechanochemical transduction mechanism (as described in Section 1.2.2). Such flow-dependent modulation of endothelial permeability would clearly cause the transendothe- lial LDL flux to become flow dependent.
• A fluid-phase-limited mass transfer process (of a species such as O2) could regulate endothelial permeability to LDL. Such flow-dependent modulation of endothelial permeability would clearly cause the transendothelial LDL flux to become flow dependent.
The third and final mechanism, however, cannot be understood within the context of the above analysis, since it relies on the existence of a wall-normal convective LDL
32 flux, which (as has been discussed) cannot be explicitly represented by Eq. (1.2). It has been postulated that such a wall normal convective flux will cause flow- dependent LDL concentration polarisation to occur adjacent to the luminal surface of the endothelium. Details of this final mechanism are given in the following section.
1.3 Flow Dependent LDL Concentration Polari- sation within Arteries
1.3.1 Concentration Polarisation
Consider a fluid containing a dissolved solute flowing through a membrane. If the membrane offers a higher resistance to the solute than the fluid, then solute will be rejected by the membrane and accumulate on its upstream surface. It is this process of solute accumulation that is referred to as concentration polarisation. The layer of rejected solute is often referred to as a concentration polarisation layer (see Fig. 1.2). As concentration polarisation occurs a concentration gradient will develop, causing solute to diffuse back away from the membrane (in the opposite direction to the approaching fluid flow). A steady state solute distribution is reached when solute convection towards the membrane balances with solute diffusion away from the membrane and solute transport across it.
33
Convection
Diffusion Fluid
Solute
Concentration Polarisation Layer
Membrane Transport Across Membrane
Figure 1.2: Diagram illustrating how concentration polarisation of a dissolved solute can occur adjacent to a generic membrane from which the solute is fully or partially rejected. A steady state is reached when convection towards the membrane (solid arrow) balances with diffusion away from the membrane (checkered arrow) and transport across the membrane (striped arrow).
The phenomenon of concentration polarisation is well known within various fields of science and engineering, in particular those involving filtration processes.
1.3.2 LDL Concentration Polarisation within Arteries
LDL concentration polarisation is postulated to occur in arteries due to the trans- mural water flux that flows radially outwards (from the lumen) through the arte- rial wall. There is an imbalance between the measured velocity of this water flux (∼ 4 × 10−2 µms−1 [30]) and the measured permeability of the endothelium to LDL (∼ 2×10−4 µms−1 [31]). Due to this imbalance LDL convection towards the luminal surface of the endothelium is likely to be far greater that the rate of transendothelial LDL transport, resulting in the formation of an LDL-rich (concentration polarisa- tion) layer adjacent to the endothelial surface.
34 1.3.3 Flow Dependent LDL Concentration Polarisation within Arteries
Consider the formation of an LDL concentration polarisation layer within the local- ity of an arbitrary point adjacent to the luminal surface of the arterial wall. Further, consider defining a wall normal coordinate y∗ at this arbitrary point of interest (see Fig. 1.3). When modelling LDL transport above the endothelium it can be assumed that LDL is completely rejected from the endothelial surface (since endothelial per- meability to LDL is likely negligible c.f. the transmural water flux). If it is assumed that LDL is rejected from the endothelium at y∗ = R∗ (where R∗ is arbitrary and introduced for later convenience), then lumen side convection and diffusion must balance at y∗ = R∗, and hence
∗ ∂C ∗ ∗ D∗ (R∗) = −V C (R∗), (1.7) L ∂y∗
∗ ∗ where DL is the diffusivity of LDL in the lumen and V is the magnitude of the transmural flux velocity (assumed to flow in the negative y∗ direction towards the endothelial surface).
Given the condition defined by Eq. (1.7), and assuming that the concentration po- larisation layer can be modelled locally as a stagnant film [32, 33], convection and diffusion must balance along the y∗ coordinate within the concentration polarisa- ∗ tion layer. Hence, the y∗ dependent LDL distribution C within the concentration polarisation layer is a solution of
∗ ∂C ∗ ∗ D∗ = −V C . (1.8) L ∂y∗
In order to solve Eq. (1.8) a single boundary condition is required. To obtain such a condition it is assumed that the concentration polarisation layer has a finite thickness ζ∗ within the locality of the arbitrary point of interest (see Fig. 1.3), where ζ∗ is dependent upon the local haemodynamics. If (as before) LDL concentration
∗ in the bulk flow is denoted CB, then the above assumption requires that
35 ∗ ∗ ∗ ∗ C (R + ζ ) = CB. (1.9)
On solving Eq. (1.8) subject to the boundary condition defined by Eq. (1.9), one obtains the following revised expression for the LDL concentration adjacent to the
∗ luminal surface of the endothelium CE (at the arbitrary point of interest)
∗ ∗ ∗ ∗ ∗ (V ζ /DL) CE = CBe , (1.10)
∗ along with the following expression for the LDL distribution C within the concen- tration polarisation layer (i.e. within the region R∗ < y∗ < R∗ +ζ∗) at the arbitrary point of interest
∗ ∗ ∗ ∗ ∗ ∗ −V (y −R )/DL C = CEe . (1.11)
The concentration polarisation layer defined by Eq. (1.11) will henceforth be referred to as the ‘macroscale LDL concentration polarisation layer’ (at the arbitrary point of interest).
C*
* CE Lumen Artery
y*
* CB !* * 0 R* R*+!* y
Figure 1.3: Schematic illustration of LDL concentration polarisation adjacent to the luminal surface of the arterial endothelium. It is assumed that the concentration polarisation layer has a flow dependent thickness ζ∗. Further, it is assumed that LDL is rejected from the endothelium at y∗ = R∗, where R∗ is arbitrary and introduced for later convenience.
36 Eq. (1.10) can be compared directly with Eq. (1.6) (which was obtained via a more simplistic analysis that ignored the effect of the transmural water flux). The
∗ important point to note is that CE given by Eq. (1.10) is dependent on local ∗ ∗ haemodynamics (via its dependence on ζ ), whereas CE given by Eq. (1.6) does not depend on flow. Consideration of the transmural water flux has revealed a mechanism by which flow can control the degree of endothelial exposure to LDL.
1.3.4 Flow Dependent Regulation of Atherogenesis
The route by which LDL crosses the endothelium and enters the arterial wall is not known definitively. It is observed, however, that increasing plasma LDL concentra- tion (and hence endothelial exposure to LDL) increases the rate of transendothelial LDL transport [34]. Such a finding implies that the degree of LDL concentration polarisation could act to modulate the rate of LDL transport into the intima, and thus the likelihood of atherosclerosis occurring. If (as suggested above) the local degree of LDL concentration polarisation is dependent on vascular scale flow fea- tures, then the above mechanism provides a viable explanation for the observed flow dependence of atherogenesis [35, 36].
1.3.5 Previous Studies
Experiments suggest that a flow dependent LDL concentration polarisation layer does form within the vasculature [25, 26]. These experimental results are supported by vascular scale theoretical studies [27, 28], which also quantify how the degree of concentration polarisation might depend on vascular scale flow features. It is the case, however, that these previous studies have generally overlooked several cellular scale features of the endothelial surface. In particular they have not accounted for the fact that the transmural water flux is likely to be spatially heterogeneous as it crosses the endothelium [37]. Also, they have not accounted for the fact that an endothelial glycocalyx layer covers the luminal surface of endothelial cells [38, 39].
37 1.4 Relevant Cellular Scale Features of the En- dothelial Surface
1.4.1 Spatially Heterogeneous Transmural Water Flux
Blood plasma consists of approximately 92% water. Studies of capillaries suggest that the water component of blood plasma traverses the endothelium predominantly via intercellular clefts (between endothelial cells) [37] rather than through the en- dothelial cells themselves. Therefore the transmural water flux is likely to be spa- tially heterogeneous at the cellular scale as it crosses the endothelium. Further studies of capillaries reveal that within intercellular clefts there exists a complex structure of connections referred to as tight junction strands [40]. These are thought to form a disjointed barrier around almost the entire cell periphery (see Fig. 1.4), with the passage of water occurring preferentially at locations where the junction strands are broken. Such structures within the depth of each intercellular cleft may result in further spatial localisation of the transmural water flux as it crosses the endothelium [41].
Tight Junction Strand Break in Tight Junction Strand
Intercellular Cleft Entrance
Endothelial Cell
Endothelial Cell
Figure 1.4: Schematic illustration of a tight junction strand within an intercellular cleft. Note that the strands are broken at particular locations.
38 Previous review articles [16, 17] have stated that a heterogeneous transmural water flux is likely to affect LDL concentration polarisation within the vasculature. How- ever, the precise nature of such an effect and its influence on the flow-dependent entry of LDL into the arterial wall have not been established, and are not intu- itively obvious. The problem has been addressed by one earlier numerical study [42]. This study, however, did not use realistic intercellular cleft dimensions, nor did it fully investigate a realistic range of LDL diffusivities.
1.4.2 Endothelial Glycocalyx Layer
Attached to the luminal surface of vascular endothelial cells, and likely covering the intercellular clefts, is a structure known as the endothelial glycocalyx layer (EGL). The EGL is currently understood to be formed of two distinct yet closely interacting regions [38, 39] as illustrated in Fig. 1.5.
Lumen
(b) ~ 0.5 µm EGL
0.05-0.4 µm (a)
Endothelial Cell
Figure 1.5: Illustration of EGL structure (not to scale). It is thought that the EGL is composed of a well adhered layer of long chain macromolecules extending ∼ 0.05−0.4µm from the endothelial surface (a) and a more dynamic loosely attached layer of endothelial and plasma derived macromolecules (b), which act to extend the EGL ∼ 0.5 − 1.0µm into the lumen.
The first layer (Fig. 1.5 label (a)) is composed of long chain macromolecules (pre- dominantly proteoglycans and glycoproteins) firmly attached to the luminal surface of the underlying endothelial cells. This region of the EGL is often observed in
39 ex-vivo electron microscope studies (see Fig. 1.6), and extends ∼ 0.05 − 0.4µm [43] from the cell surface. The second layer (Fig. 1.5 label (b)) is thought to be more dynamic in nature, composed of loosely attached plasma and endothelial derived proteins which act to extend the EGL ∼ 0.3−0.5µm [44, 45, 46] (or possibly further [47]) into the lumen.
Figure 1.6: Image of the EGL within a rat ventricular myocardial capillary ob- tained using electron microscopy. Reused with permission from B. M. van den Berg et al. [48]. Copyright 2003, Wolters Kluwer Health.
The EGL is postulated to play an important role in several processes, including the prevention of red blood cell interaction with the endothelium [49] and the transduc- tion of mechanical forces to the surface of endothelial cells [50]. It is the role of the EGL as a macro-molecular sieve [51], however, that is most likely to have an effect on LDL concentration polarisation within arteries. Specifically, the distance that LDL can penetrate into the EGL will determine the local water velocity profile and LDL diffusivity close to the endothelium, both of which are important determinants of the structure and degree of any LDL concentration polarisation that occurs.
It can be noted that the EGL (as described above) is sometimes referred to simply as the ‘glycocalyx’ or alternatively the ‘endothelial surface layer’ [52]. To add fur- ther confusion, the well adhered layer of macromolecules at the base of the EGL is
40 sometimes referred to alone as the glycocalyx (since this region was the first to be regularly observed). To avoid such confusions the term EGL is adopted consistently throughout this thesis to refer to the entire structure described above. Individual regions of the EGL are referred to explicitly when necessary.
Overviews of the structure and function of the EGL are given in recent review articles by Weinbaum et al. [38] and Reitsma et. al. [39].
1.5 Objective of Thesis
The objective of this thesis is to assess the effect of a heterogeneous transmural water flux and an EGL on LDL concentration polarisation in arteries. Particular attention is paid to measures of LDL concentration polarisation relevant to the rate of transendothelial LDL transport, since this rate has implications for the likelihood of atherosclerosis onset.
1.6 Overview of Thesis
In Chapter 2 an overview is given of the numerical scheme that will be used through- out this thesis (namely the spectral/hp element method). Particular attention is focused on obtaining numerical solutions to the two-dimensional (2D) steady and unsteady advection diffusion equations. In Chapter 3 an idealised model is devel- oped and implemented in order to assess the effect of a heterogeneous transmural water flux on LDL concentration polarisation in arteries (published in Biophysical Journal [53]). In Chapter 4 this model is extended to include the effect of the EGL (published in Physics of Fluids [54], and submitted for publication in Journal of Theoretical Biology). Finally, in Chapter 5 conclusions and ideas for future research are presented.
41 Chapter 2
Spectral/hp Element Method
2.1 Overview
It is often difficult, and in some cases impossible, to derive analytical solutions to a differential equation. The applicability of analytical techniques is therefore somewhat limited. As a consequence, a range of numerical methods have been developed in order to obtain approximate solutions to differential equations. Such numerical techniques have a wide range of applicability, and are hence currently employed in numerous fields of science and engineering.
There are several well known types of numerical method. These include finite dif- ference methods [55], finite volume methods [56], finite element methods [57] and spectral methods [58], each of which has its own advantages and disadvantages when applied to particular problems. Numerous implementations and variations of the aforementioned methods have been developed. A spectral/hp element method [59] can be viewed as an implementation of a finite element scheme in which the principles of spectral methods are also employed.
When implementing a finite element method, the domain of interest is decomposed spatially into separate regions known as elements. The solution is then represented within each element of the domain using a polynomial function (which is often sim- ply linear). Such decomposition is often referred to as h-type discretisation, and increasing the number of elements is often referred to as h-type refinement. The spatial discretisation employed when using a finite element method does not have to be uniform, hence finite element techniques lend themselves well to problems in which the geometry is complex. Also, due to the flexibility of the spatial discretisa- tion, elements (and hence degrees of freedom) can be localised in particular regions of the domain. The ability to spatially localise degrees of freedom can be extremely useful if one has a priori knowledge of where complex features of the solution may develop.
When implementing a spectral method the solution is decomposed in frequency space into modes, which are defined globally within the domain of interest. Such decomposition is often referred to as p-type discretisation, and increasing the num- ber of modes is often referred to as p-type refinement. Spectral methods lack the geometrical flexibility of finite element methods, since it is often difficult to define continuous global modes within complex geometries. However, in geometries where spectral methods can be applied they are known to exhibit excellent convergence properties. Specifically, if the solution is smooth then spectral methods will converge exponentially as the number of modes (degrees of freedom) is increased. This can be compared with the algebraic convergence exhibited by finite element methods as the number of elements (degrees of freedom) is increased.
The principle of spectral/hp element methods is to combine the geometrical flexi- bility of finite element methods with the superior convergence properties of spectral methods. This is achieved by decomposing the domain of interest spatially into ele- ments (h-type discretisation), and then representing the solution within each element using a summation of modes (p-type discretisation). A particular implementation of the spectral/hp element method will be employed in Chapters 3 and 4 to find approximate solutions of the 2D steady and unsteady advection diffusion equations. The main principles of the method are outlined below.
43 2.2 One-Dimensional Spectral/hp Element Method
2.2.1 Example Problem
For illustrative purposes, consider the inhomogeneous one-dimensional (1D) Helmholtz equation d2u − λu = f, (2.1) dx2 where u = u(x) is a scalar field, f = f(x) is a forcing function, and λ is a real positive constant. Further, consider finding a solution of Eq. (2.1) within the domain Ω = {x | x0 ≤ x ≤ x4}, that satisfies the following boundary conditions
u(x0) = gD, (2.2)
du (x ) = g , (2.3) dx 4 N where gD and gN are known constants. Eq. (2.1), along with the boundary con- ditions defined by Eq. (2.2) and Eq. (2.3), are said to pose the problem in its strong form (as opposed to in its weak form which will be discussed shortly). It can be noted that in the above example the boundary condition applied at x = x0 is a condition on the solution (often referred to as a Dirichlet boundary condition), whereas the boundary condition applied at x = x4 is a condition on the derivative of the solution (often referred to as a Neumann boundary condition).
2.2.2 Weak Formulation
If Eq. (2.1) is multiplied through by an arbitrary test function v = v(x) (defined to be zero on all Dirichlet boundaries), and then integrated over the whole domain, the following expression is obtained
Z x4 d2u Z x4 Z x4 v 2 dx − λ vu dx = vf dx. x0 dx x0 x0
44 Integration by parts leads to
x4 Z x4 Z x4 Z x4 du dv du v − dx − λ vu dx = vf dx, dx dx dx x0 x0 x0 x0 and hence
Z x4 dv du Z x4 du du Z x4 dx + λ vu dx = v(x4) (x4) − v(x0) (x0) − vf dx. x0 dx dx x0 dx dx x0
Therefore, using Eq. (2.3) along with the fact that v is zero at x = x0, one obtains
Z x4 dv du Z x4 Z x4 dx + λ vu dx = v(x4)gN − vf dx. (2.4) x0 dx dx x0 x0
Eq. (2.4) is a weak (or variational) form of Eq. (2.1). It should be noted that the Neumann boundary condition defined by Eq. (2.3) is naturally included within this weak formulation. Solutions to Eq. (2.4) are known as weak solutions to Eq. (2.1).
2.2.3 An Approximate Solution
Consider a generic approximation uδ = uδ(x) to the solution u. Further, consider writing this approximate solution as the sum of two components, namely uD and δ uH , such that
δ δ u = uD + uH , (2.5)
where uD is a known function that satisfies the Dirichlet boundary condition at δ x = x0, and uH is an unknown function (to be determined) that is zero on the
Dirichlet boundary at x = x0. Note that the superscript δ is omitted from uD since δ uD is a known function, not an approximation. The approximate nature of u is δ contained within uH . Also note that due to this decomposition, the approximate δ solution u will always satisfy the Dirichlet boundary condition at x = x0 exactly.
If u is replaced by uδ in Eq. (2.4) (and also if v is replaced with a generic approxima- tion vδ which is exactly zero on Dirichlet boundaries) then the following expression
45 is obtained
Z x4 δ δ Z x4 Z x4 dv du δ δ δ δ dx + λ v u dx = v (x4)gN − v f dx. (2.6) x0 dx dx x0 x0
Substitution of Eq. (2.5) into Eq. (2.6) leads to
Z x4 δ δ Z x4 dv duD duH δ δ + dx + λ v (uD + uH ) dx = x0 dx dx dx x0
Z x4 δ δ v (x4)gN − v f dx, x0 and hence
Z x4 δ δ Z x4 dv duH δ δ dx + λ v uH dx = x0 dx dx x0
Z x4 Z x4 δ Z x4 δ δ dv duD δ v (x4)gN − v f dx − dx, −λ v uD dx. (2.7) x0 x0 dx dx x0
The problem as formulated by Eq. (2.7) includes all boundary conditions. Note that all terms on the right hand side of Eq. (2.7) are either known or arbitrary.
2.2.4 A Spectral/hp Element Basis
δ δ The approximations uH and v are as yet undefined. In order to proceed with a description of the spectral/hp element method, it is necessary to cast these approx- imations in terms of a global spectral/hp basis. There are two important steps in the formation of a spectral/hp basis. The first is to partition the solution domain into elements (an h-type discretisation). This step is often referred to as defining a mesh on the domain. The second step is to define a spectral basis locally within each element of the domain (a p-type discretisation). The mesh, along with the spectral basis within each element, can then be used to define a global spectral/hp basis within the entire domain.
46 Consider constructing a mesh on the domain Ω. In this example Ω will be split into four non-overlapping elements defined as
Ω0 = {x | x0 ≤ x ≤ x1}, Ω1 = {x | x1 ≤ x ≤ x2},
Ω2 = {x | x2 ≤ x ≤ x3}, Ω3 = {x | x3 ≤ x ≤ x4}, where
x0 < x1 < x2 < x3 < x4, and hence
3 3 [ \ Ω = Ωi Ωi = ∅. i=0 i=0
Further to this definition of a mesh, consider constructing a spectral basis within each element Ωi. To facilitate such a process it is useful to introduce the concept of a standard element ΩS = {ρ | − 1 ≤ ρ ≤ 1}. For each element Ωi bounded between xi and xi+1 (where i = 0, 1, 2, 3), a mapping Si between ρ within ΩS and x within Ω can be defined as
1 − ρ 1 + ρ x = S (ρ) = x + x ρ ∈ Ω . i 2 i 2 i+1 S
Hence, for each element Ωi bounded between xi and xi+1 (where i = 0, 1, 2, 3), an −1 inverse mapping Si between x within Ω and ρ within ΩS can also be defined as
−1 x − xi ρ = Si (x) = 2 − 1 x ∈ Ωi. xi+1 − xi
rd S S Within the standard element ΩS a 3 order modal polynomial basis φj = φj (ρ), containing four modes, can be defined as 1 − ρ j = 0 2 1 − ρ1 + ρ S 1,1 φj = P (ρ) 0 < j < 3 (2.8) 2 2 j−1 1 + ρ j = 3, 2
47 1,1 where Pj−1(ρ) are Jacobi polynomials. Note that a variety of spectral bases could have been used within the standard element. Other types of basis will be discussed shortly in Section 2.2.9.
S The basis φj within the standard element ΩS, along with the mappings between ΩS and each element Ωi of Ω, can be used to define a global spectral/hp basis within Ω. Here it will be required that the global basis is C0 continuous i.e. it will be required that the global basis functions are continuous everywhere (but the first and further derivatives of the functions need not be). With this constraint in mind, a global spectral/hp basis Φk = Φk(x) within Ω can be defined as
S −1 S −1 φ0 (S0 (x)) x ∈ Ω0 φ1 (S0 (x)) x ∈ Ω0 Φ0(x) = Φ1(x) = 0 x∈ / Ω0 0 x∈ / Ω0
S −1 φ3 (S0 (x)) x ∈ Ω0 S −1 φ2 (S0 (x)) x ∈ Ω0 Φ (x) = Φ (x) = S −1 2 3 φ0 (S1 (x)) x ∈ Ω1 0 x∈ / Ω 0 0 x∈ / Ω0 ∪ Ω1
S −1 S −1 φ1 (S1 (x)) x ∈ Ω1 φ2 (S1 (x)) x ∈ Ω1 Φ4(x) = Φ5(x) = 0 x∈ / Ω1 0 x∈ / Ω1
S −1 φ3 (S1 (x)) x ∈ Ω1 S −1 φ1 (S2 (x)) x ∈ Ω2 Φ (x) = S −1 Φ (x) = 6 φ0 (S2 (x)) x ∈ Ω2 7 0 x∈ / Ω 2 0 x∈ / Ω1 ∪ Ω2
48 S −1 φ3 (S2 (x)) x ∈ Ω2 S −1 φ2 (S2 (x)) x ∈ Ω2 Φ (x) = Φ (x) = S −1 8 9 φ0 (S3 (x)) x ∈ Ω3 0 x∈ / Ω 2 0 x∈ / Ω2 ∪ Ω3
S −1 S −1 φ1 (S3 (x)) x ∈ Ω3 φ2 (S3 (x)) x ∈ Ω3 Φ10(x) = Φ11(x) = 0 x∈ / Ω3 0 x∈ / Ω3
S −1 φ3 (S3 (x)) x ∈ Ω3 Φ12(x) = 0 x∈ / Ω3
Since the global basis Φk is constructed from a mesh containing four elements, and each element is spectrally decomposed into four modes, one may expect the basis to be formed of 16 = 4 × 4 global modes. However, due to the requirement of C0 continuity at the three inter-element interfaces there are three constraints, and hence only 13 = (4 × 4) − 3 global modes in the basis. It can be noted that only five of the 13 modes have values at elemental boundaries, these modes are known as global boundary modes. The remaining modes, which are zero at elemental boundaries, are known as global interior modes. The basis functions Φk are illustrated graphically in Fig. 2.1.
49 1 1 1
0.8 0.8 0.8
0.6 0.6 0.6
0 0.4 1 0.4 2 0.4 ! ! !
0.2 0.2 0.2
0 0 0
-0.2 -0.2 -0.2 x x x x x x x x x x x x x x x 0 1 x2 3 4 0 1 x2 3 4 0 1 x2 3 4
1 1 1
0.8 0.8 0.8
0.6 0.6 0.6
3 0.4 4 0.4 5 0.4 ! ! !
0.2 0.2 0.2
0 0 0
-0.2 -0.2 -0.2 x x x x x x x x x x x x x x x 0 1 x2 3 4 0 1 x2 3 4 0 1 x2 3 4
1 1 1
0.8 0.8 0.8
0.6 0.6 0.6
6 0.4 7 0.4 8 0.4 ! ! !
0.2 0.2 0.2
0 0 0
-0.2 -0.2 -0.2 x x x x x x x x x x x x x x x 0 1 x2 3 4 0 1 x2 3 4 0 1 x2 3 4
1 1 1
0.8 0.8 0.8
0.6 0.6 0.6 0 1 9 0.4 1 0.4 1 0.4 ! ! ! 0.2 0.2 0.2
0 0 0
-0.2 -0.2 -0.2 x x x x x x x x x x x x x x x 0 1 x2 3 4 0 1 x2 3 4 0 1 x2 3 4
1
0.8
0.6 2
1 0.4 ! 0.2
0
-0.2 x x x x x 0 1 x2 3 4
Figure 2.1: Graphical illustration of global spectral/hp basis functions Φk. 50 2.2.5 Galerkin Approximation
δ δ δ It is now possible to represent u = uD +uH and v in terms of the global spectral/hp basis Φk. Here it will be chosen that
uD = gDΦ0, (2.9)
12 δ X uH = uˆkΦk, (2.10) k=1
12 δ X v = vˆlΦl, (2.11) l=1
δ whereu ˆk are coefficients to be determined, andv ˆl are arbitrary coefficients since v is arbitrary. It can be noted that uD is written in terms of the known function Φ0, since this is the only basis function which is non-zero on the Dirichlet boundary at
δ δ x = x0. It can also be noted that uH and v are both represented by the same basis functions, all of which are zero on Dirichlet boundaries. A implementation such as
δ δ this, where identical bases are used to represent both uH and v , is often referred to as a Galerkin type implementation.
2.2.6 Global Matrix Equation
On substituting Eq. (2.9), Eq. (2.10) and Eq. (2.11) into Eq. (2.7), one obtains
12 12 12 12 Z x4 Z x4 X X dΦk dΦl X X uˆ vˆ dx + λ uˆ vˆ Φ Φ dx = k l dx dx k l k l k=1 l=1 x0 k=1 l=1 x0
12 12 Z x4 Z x4 X X dΦl dΦ0 vˆ g − vˆ Φ f dx − vˆ g dx− 12 N l l l D dx dx l=1 x0 l=1 x0
12 X Z x4 λ vˆlgD ΦlΦ0 dx. (2.12) l=1 x0
51 Defining the matrices
Z x4 Z x4 dΦk dΦl L[k][l] = dx, M[k][l] = ΦkΦl dx, x0 dx dx x0 and the vectors
uˆ[k] =u ˆk, vˆ[l] =v ˆl,
0 l 6= 12 Z x4 n[l] = , f[l] = Φlf dx, x0 gN l = 12
Z x4 Z x4 dΦl dΦ0 d[l] = gD dx − λ ΦlΦ0 dx , x0 dx dx x0 where L[1][1] and M[1][1] denote the top left entries in L and M respectively, and uˆ[1][1], vˆ[1][1], n[1], f[1] and d[1] denote the first entries in uˆ, vˆ, n, f and d respec- tively, one can write Eq. (2.12) succinctly in matrix form as
vˆ>(L + λM)uˆ = vˆ>(n − f − d). (2.13)
Since vδ is arbitrary, vˆ is also arbitrary, and thus Eq. (2.13) is only satisfied if
(L + λM)uˆ = (n − f − d). (2.14)
On pre-multiplication of Eq. (2.14) by (L + λM)−1 one obtains the following ex- pression for uˆ (and hence the approximate weak solution uδ)
uˆ = (L + λM)−1(n − f − d). (2.15)
2.2.7 Matrix Structure
The matrix L is often referred to as the global Laplacian matrix of the system, and the matrix M is often referred to as the global mass matrix of the system. It can be
52 noted that both L and M have some degree of structure. Implementations of the spectral/hp element method seek to use such structure in order to facilitate matrix creation, storage and inversion.
The structure of the global mass matrix M can be illustrated in the following schematic fashion
∗ 0 0 0 0 0 0 0 0 0 ∗ 0 0 0 0 0 0 0 0 0 ∗ ∗ N ∗ ∗ N 0 0 0 0 0 0 0 0 ∗ ∗ 0 0 0 0 0 0 0 0 ∗ ∗ 0 0 0 0 0 0 0 0 N ∗ ∗ N ∗ ∗ N 0 0 0 M = , 0 0 0 0 0 ∗ ∗ 0 0 0 0 0 0 0 0 ∗ ∗ 0 0 0 0 0 0 0 0 N ∗ ∗ N ∗ ∗ N 0 0 0 0 0 0 0 0 ∗ ∗ 0 0 0 0 0 0 0 0 ∗ ∗ 0 0 0 0 0 0 0 0 N ∗ ∗ N where the solid black triangles represent non-zero entries due to coupling between global boundary modes, the asterisks represent non-zero entries due to coupling between global boundary modes and global interior modes, and the solid black squares represent non-zero entries due to coupling between global interior modes. It can be noted that M is symmetric due to the form of the Helmholtz operator in Eq. (2.1) and the nature of the weak form derived in Eq. (2.4). Also, many of the entries in M are zero due to orthogonality between various global modes.
Additional structure of M is revealed if entries within the solution vector uˆ and the right hand side vectors n, f and d are re-ordered. Specifically, consider defining a re-ordering matrix R as
53 001000000000 000001000000 000000001000 000000000001 100000000000 010000000000 R = . 000100000000 000010000000 000000100000 000000010000 000000000100 000000000010
On multiplication of uˆ, n, f and d by R one obtains the vectors uˆR = Ruˆ, nR = Rn, fR = Rf and dR = Rd, which are all ordered such that global boundary modes appear first, followed by global interior modes. Eq. (2.15) can be rewritten in terms of uˆR, nR, fR and dR as
−1 uˆR = (LR + λMR) (nR − fR − dR), (2.16) where
−1 −1 LR = RLR , MR = RMR .
Eq. (2.16) encodes the same information as Eq. (2.15). However, entries within LR and MR are rearranged c.f. entries in L and M respectively. Specifically, MR has the modified form
54 NN 0 0 ∗ ∗ ∗ ∗ 0 0 0 0 NNN 0 0 0 ∗ ∗ ∗ ∗ 0 0 0 NNN 0 0 0 0 ∗ ∗ ∗ ∗ 0 0 NN 0 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 ∗ 0 0 0 0 0 0 0 0 0 MR = ∗ ∗ 0 0 0 0 0 0 0 0 ∗ ∗ 0 0 0 0 0 0 0 0 0 ∗ ∗ 0 0 0 0 0 0 0 0 ∗ ∗ 0 0 0 0 0 0 0 0 0 ∗ ∗ 0 0 0 0 0 0 0 0 ∗ ∗ 0 0 0 0 0 0 where the solid black triangles, asterisks, and solid black squares represent various non-zero entries as previously described.
The structure of MR is apparent, with boundary-boundary interactions grouped in the top left quadrant, boundary-interior interactions grouped in the top right and bottom left quadrants, and interior-interior interactions grouped in block diagonal form in the bottom right quadrant. The structure of MR lends its self to inversion via a static condensation method, which avoids explicit construction, storage and inversion of the entire matrix. For further details see Karniadakis and Sherwin [59].
2.2.8 Global Assembly
Due to the elemental structure of a spectral/hp element basis, it is possible to assemble global matrices (such as M) and global vectors (such as d) from their elemental contributions. The ability to assemble global matrices and global vectors from elemental contributions is advantageous for several reasons. In particular, it allows operations such as integration to be carried out solely at an elemental level
55 within a standard element, rather than throughout the entire domain.
To illustrate the procedure of global matrix assembly consider defining a matrix m as
m0 0 0 0 0 m1 0 0 m = , 0 0 m2 0 0 0 0 m3 where
Z 1 S S dSi mi[n][m] = φn−1(ρ)φm−1(ρ) dρ (i = 0, 1, 2, 3), −1 dρ
with each mi[1][1] denoting the top left entry of each elemental mass matrix mi. The global mass matrix M can be assembled from the elemental mass matrices held within m via the following matrix manipulation
M = A>mA, where A is an assembly matrix defined as
56 000000000000 100000000000 010000000000 001000000000 001000000000 000100000000 000010000000 000001000000 A = . 000001000000 000000100000 000000010000 000000001000 000000001000 000000000100 000000000010 000000000001
It can be noted that columns three, six and nine of the assembly matrix A contain two non-zero entries (whereas all other columns contain only one). This is because the global modes Φ3,Φ6 and Φ9 are constructed from two local modes, whereas all other global modes are constructed from only one. Also, it can be noted that the top row of A contains all zero entries. This is because none of the global modes
δ S used in the expansion of uH contain the local mode φ0 mapped to the element Ω0. Finally, it can be noted that A is rectangular, not square, since there are more local degrees of freedom than global degrees of freedom.
In practice (since A is very sparse) M is not assembled via matrix manipulations; instead it is assembled using a series of mappings. For further details of the afore- mentioned global matrix assembly procedure see Karniadakis and Sherwin [59].
57 2.2.9 Other Elemental Bases
The elemental basis defined by Eq. (2.8) is an example of a modal basis. Modal bases are heirarchical in nature, that is to say lower order modal expansions are subsets of higher order modal expansions. Other types of expansion basis can also be used to represent the solution within a standard element, including those that are nodal in nature (which are defined in terms of a set of nodal points situated within the standard element). Such nodal bases are not heirarchical in nature, that is to say higher order nodal expansions cannot be formed by adding further modes to lower order nodal expansions.
When choosing an appropriate elemental basis one should take several factors into consideration. In particular it is preferable that the basis functions are orthogonal (or at least near to being orthogonal), so as to reduce coupling between elemental modes. Also, it is preferable that the basis can be decomposed into boundary and interior modes, since this facilitates the enforcement of C0 continuity between adjacent elements. Other considerations include how linearly independent the basis functions are, since this will impact the conditioning of any matrices that are formed (which in turn affects the ease with which matrices can be numerically inverted). For further details regarding the choice of suitable basis functions see Karniadakis and Sherwin [59].
2.3 Extension to Two-Dimensions
2.3.1 Overview
The basic principles of the spectral/hp element method previously described within a 1D context also apply in multiple dimensions. It is the case, however, that extension of the method to multiple dimensions gives rise to several additional complications. The most notable of these complications is the fact that a standard element (which is simply a line segment in 1D) can take various forms in multiple dimensions. In
58 2D, standard elements can take the form of quadrangles or triangles. An overview of suitable basis functions within such quadrangular and triangular standard elements is given below.
2.3.2 Basis Functions Within Quadrangular Elements
A quadrangular standard element ΩSQ is illustrated in Fig. 2.2, and can be defined mathematically in the following fashion
ΩSQ = (ρ1, ρ2 | − 1 ≤ ρ1 ≤ 1, −1 ≤ ρ2 ≤ 1).
2 (-1,1) (1,1)
(0,0)
1
(-1,-1) (1,-1)
Figure 2.2: A quadrangular standard element.
A suitable basis can be defined within the quadrangular standard element ΩSQ by SQ taking the tensor product of 1D bases. For example, a basis φpq can be defined rd S within ΩSQ in terms of the 3 order 1D basis φi as
SQ S S φpq (ρ1, ρ2) = φp (ρ1)φq (ρ2).
59 SQ S The basis φpq exhibits favorable properties of the 1D basis φi from which it is SQ constructed. In particular, the 2D basis φpq can be decomposed into boundary and interior modes.
2.3.3 Basis Functions Within Triangular Elements
A triangular standard element ΩST is illustrated in Fig. 2.3, and can be defined mathematically in the following fashion
ΩST = (ρ1, ρ2 | − 1 ≤ ρ1, −1 ≤ ρ2, 0 ≥ ρ1 + ρ2).
2 (-1,1)
(0,0)
1
(-1,-1) (1,-1)
Figure 2.3: A triangular standard element.
S It is the case that 1D basis functions such as φi run between constant limits. How- ever, the bounds of the triangular standard element ΩST defined in terms of ρ1 and
ρ2 are not constant. For example, when ρ2 = 0, ρ1 is bounded between values of
ρ1 = −1 and ρ1 = 0. However, when ρ2 = 0.5, ρ1 is bounded between values of
ρ1 = −1 and ρ1 = −0.5. It is therefore not possible to construct a 2D basis within
ΩST (defined in terms of ρ1 and ρ2) simply by taking a tensor product of 1D bases
60 S such as φi .
In order to construct a tensorial-type basis within a triangular standard element it is necessary to develop a coordinate system in which the standard element has constant bounds. Such a coordinate system can be defined by the following transformation
1 + ρ1 ρ1 = 2 − 1 1 − ρ2
ρ2 = ρ2.
The triangular standard element ΩST can be defined in terms of the new coordinates
ρ1 and ρ2 as
ΩST = (ρ1, ρ2 | − 1 ≤ ρ1 ≤ 1, −1 ≤ ρ2 ≤ 1).
ST and a tensorial-type basis φpq can be defined within the triangular standard element as
ST S S φpq (ρ1, ρ2) = φp (ρ1)φpq(ρ2) where
φS(ρ ) p = 0, 0 ≤ q ≤ 3 q 2 p+1 1 − ρ2 1 ≤ p < 3, p = 0 S 2 φpq(ρ2) = 1 − ρ p+11 + ρ 2 2 2p+1,1 Pq−1 (ρ2) 1 ≤ p < 3, 1 ≤ q < 3 2 2 S φq (ρ2) p = 3, 0 ≤ q ≤ 3
2p+1,1 and Pq−1 (ρ2) are Jacobi polynomials. For further details regarding bases within triangular standard elements see Karniadakis and Sherwin [59].
61 2.4 Advection Diffusion Equation
2.4.1 Steady State Two-Dimensional Advection Diffusion Equation
2.4.1.1 Definition
Consider solving the following steady state 2D advection diffusion equation for a scalar field C within a 2D domain ΩC bounded by an edge σC
v · ∇C − ∇2C = 0, (2.17) where v is a convective velocity field. Further, consider that a Dirichlet boundary condition of the following form is applied at various locations on σC
C = gD, and a Neumann boundary condition of the following form is applied at all other locations on σC
∇C · nˆ = gN ,
where nˆ is a unit vector normal to σC .
2.4.1.2 Weak Form
If one multiplies Eq. (2.17) by an arbitrary test function B (defined to be zero on all
Dirichlet boundaries), and then integrates over the whole domain ΩC , the following expression is obtained
Z Z 2 B v · ∇C dΩC − B ∇ C dΩC = 0. ΩC ΩC
62 Integration by parts leads to
Z Z Z B v · ∇C dΩC − B ∇C · nˆ dσC + ∇B · ∇C dΩC = 0. ΩC σC ΩC
Hence, since B = 0 on all Dirichlet boundaries, and ∇C · nˆ = gN on all Neumann boundaries, the following weak from of Eq. (2.17) can be obtained
Z Z B v · ∇C + ∇B · ∇C dΩC = B gN dσC . (2.18) ΩC σC
2.4.1.3 Obtaining an Approximate Solution
δ δ Consider defining C = CD + CH as an approximation to C, where CD is a known δ function that satisfies all Dirichlet boundary conditions, and CH is zero on all Dirich- let boundaries (but is otherwise as yet undetermined). If C is replaced by Cδ in Eq. (2.18) (and also if B is replaced with an approximate form Bδ, which is exactly zero on all Dirichlet boundaries) then the following expression is obtained
Z δ δ δ δ B v · ∇CH + ∇B · ∇CH dΩC = ΩC Z Z Z δ δ δ B gN dσC − B v · ∇CD dΩ − ∇B · ∇CD dΩ. (2.19) σC ΩC ΩC
δ δ All terms on the right hand side of Eq. (2.19) are known. Writing B and CH in terms of a finite number of spectral/hp basis functions allows Eq. (2.19) to be formu-
δ lated as a matrix equation, which can be solved computationally to obtain CH and hence the approximate solution Cδ. It can be noted that the matrix representation
δ δ of the advection term B v · ∇CH will be non-symmetric.
63 2.4.2 Unsteady Two-Dimensional Advection Diffusion Equa- tion
2.4.2.1 Definition
Consider solving the following unsteady 2D advection diffusion equation for a time dependent scalar field C within a 2D domain ΩC bounded by an edge σC
∂C v · ∇C − ∇2C = , (2.20) ∂t where v is a (possibly time dependent) convective velocity field, and t is time. Further, consider that a Dirichlet boundary condition of the following form is applied at various locations on σC
C = gD, and a Neumann boundary condition of the following form is applied at all other locations on σC
∇C · nˆ = gN ,
0 where nˆ is a unit vector normal to σC . Finally, consider that at time t = t the 0 scaler field C = C within ΩC .
2.4.2.2 Temporal Discretisation
Consider temporally discretising Eq. (2.20) using the following first-order scheme
Cn+1 − Cn vn · ∇Cn − ∇2Cn+1 = , (2.21) ∆t where vn is the convective velocity field v at time t = tn, Cn is the scalar field C at time t = tn, Cn+1 is the scalar field C at time t = tn+1, and ∆t = tn+1 − tn. The
64 above scheme treats the diffusion term implicitly. However, the advection term is treated explicitly (to avoid subsequent inversion of a non-symmetric matrix), and hence ∆t must be restricted by a suitable Courant-Friedrichs-Lewy (CFL) condition to ensure stability [59].
Rearranging Eq. (2.21) results in the following expression
1 ∇2Cn+1 + Cn+1 = Cn/∆t + vn · ∇Cn, ∆t which is a Helmholtz equation for Cn+1 (where the forcing function depends on Cn).
2.4.2.3 Weak Form
If one multiplies Eq. (2.21) by an arbitrary test function B (defined to be zero on all
Dirichlet boundaries), and then integrates over the whole domain ΩC , the following expression is obtained
Z Z Z 2 n+1 1 n+1 n n n B ∇ C dΩC + BC dΩC = B (C /∆t + v · ∇C ) dΩC . ΩC ∆t ΩC ΩC
Integration by parts leads to
Z Z Z n+1 n+1 1 n+1 B ∇C · nˆ dσC − ∇B · ∇C dΩC + BC dΩC = σC ΩC ∆t ΩC Z n n n B (C /∆t + v · ∇C ) dΩC . ΩC
n+1 Hence, since B = 0 on all Dirichlet boundaries, and ∇C ·nˆ = gN on all Neumann boundaries, the following weak from of Eq. (2.21) can be obtained
65 Z Z n+1 1 n+1 ∇B · ∇C dΩC − BC dΩC = ΩC ∆t ΩC Z Z n n n B gN dσC − B (C /∆t + v · ∇C ) dΩC . σC ΩC
2.4.2.4 Obtaining an Approximate Solution
δn+1 n+1 δn+1 n+1 n+1 Consider defining C = CD + CH as an approximation to C , where CD δn+1 is a known function that satisfies all Dirichlet boundary conditions, and CH is zero on all Dirichlet boundaries (but is otherwise as yet undetermined). If Cn+1 is replaced by Cδn+1 in Eq. (2.18) (and also if B is replaced with an approximate form Bδ, which is exactly zero on all Dirichlet boundaries) then the following expression is obtained
Z Z δ δn+1 1 δ δn+1 ∇B · ∇CH dΩC − B CH dΩC = ΩC ∆t ΩC Z Z δ δ n n n B gN dσC − B (C /∆t + v · ∇C ) dΩC σC ΩC Z Z δ n+1 δ n+1 − ∇B · ∇CD dΩC + B CD /∆t dΩC . (2.22) ΩC ΩC
δ δn+1 Writing B and CH in terms of a finite number of spectral/hp basis functions allows Eq. (2.22) to be formulated as a matrix equation. Assuming the solution Cn is known (at time t = tn), one can use such a matrix formulation to obtain
δn+1 δn+1 n+1 CH and hence the approximate solution C (at time t = t ). Eq. (2.22) can therefore be used to advance the solution in time from the initial condition C = C0 at time t = t0.
66 2.5 Summary
An overview of the spectral/hp element method has been presented. In the following chapters a in-house C/C++ implementation of the method is employed to solve both the steady and unsteady 2D advection diffusion equations.
The 2D steady advection diffusion equation is solved using the approach detailed in Section 2.4.1. For all steady state cases the 2D domain of interest is tessellated with both triangular and quadrangular elements. A modal tensor product basis
SQ is used within each quadrangular element (of the form described by φpq but ex- tended to variable polynomial order), and a modal generalised tensor product basis
ST is used within each triangular element (of the form described by φpq but extended to variable polynomial order).
The 2D unsteady advection diffusion equation is solved using the approach detailed in Section 2.4.2. Once again the 2D domain of interest is tessellated with both
SQ triangular and quadrangular elements. Modal bases of the form described by φpq ST and φpq (extended to variable polynomial order) are used within the quadrangular and triangular elements respectively. A suitable time-step ∆t is chosen to ensure that the CFL stability condition is satisfied.
67 Chapter 3
The Effect of a Spatially Heterogeneous Transmural Water Flux
3.1 Overview
In this chapter an idealised model is developed in order to assess the effect of a heterogeneous transmural water flux on LDL concentration polarisation adjacent to the luminal surface of the endothelium. Specifically, the objective of the model is to determine cellular scale modifications to the macroscale LDL concentration polarisation layer introduced in Section 1.3.3, within the cellular scale locality of an arbitrary point on the arterial wall. Before specific details of the model are given, it is useful to highlight five general assumptions that are made:
• It is assumed that the macroscale LDL concentration polarisation layer is spatially uniform at the cellular scale.
• Only transport of water and LDL are considered; all other constituents of the blood are ignored. • It is assumed that LDL has no effect on the motion of the water.
• The EGL will be ignored. Such an approach follows that adopted in pre- vious studies of mass transport above the endothelium [60]. A model that incorporates the EGL will be developed in Chapter 4.
• For the majority of the analysis it will be assumed that dynamics within the domain are at steady state. The implications of such an assumption will be assessed at the end of the chapter via a series of time dependent simulations.
3.2 Idealised Model of Water Flow
3.2.1 Heterogeneous Transmural Flux
It is assumed that water traverses the endothelium only via intercellular clefts. Fig. 3.1(a) depicts an en face view of the endothelium illustrating a typical pattern of interendothelial cell cleft entrances. As a first approximation it is reasonable to represent the intercellular clefts by a repeating diamond pattern (Fig. 3.1(b)). Here, however, a further simplification is made and the clefts are modelled as an infinite series of parallel outflow slits of width 2δ∗ and inter-slit spacing 2∆∗ (Fig. 3.1(c)). The domain is considered sufficiently small that the curvature of the arterial wall can be ignored. Further, it is also assumed that the surfaces of the endothelial cells are flat. Therefore, the infinitely long parallel outflow slits in Fig. 3.1(c) are assumed to reside in a flat plane. The objective of such simplifications is to create a model that is straightforward to implement, yet also still captures the key physics of the problem.
69 (a) (b)
(c)
Figure 3.1: Mouse endothelial cells shown in (a) (personal communication from A. R. Bond, Bristol Royal Infirmary, UK) appear approximately diamond shaped (cleft entrances are stained dark using silver nitrate). Based on this observation the cleft entrance structure can be approximated by (b), or even further simplified to an infinitely repeating array of infinitely long parallel outflow slits as shown in (c). The simplified geometry (c) is characterised by only three parameters, namely the cleft half width δ∗, the cleft half spacing ∆∗ , and the angle θ∗ between the clefts and the applied flow.
70 ∗ A transmural water flux with an average velocity V is drawn towards the endothe- lium via the application of a parabolic outflow velocity profile (with peak magnitude ∗ 3V ∆∗/2δ∗) across the width of each cleft entrance. This profile is maintained uni- formly along the length of each cleft. Applying such an entrance velocity profile is an approximation for two reasons. Firstly, it assumes the width-wise flow profile develops instantaneously to become parabolic, and secondly it neglects any further localisation of the flow caused by tight junction strands [40] blocking portions of the clefts. The former of these two approximations is unlikely to have a significant effect on the results. However, the implications of the latter approximation are less clear, and will be discussed when the results are presented. Both of the approximations could be avoided by adding an idealisation of the cleft’s internal structure to the model.
The formulation described above allows (and indeed requires) the direct enforcement of a measured transmural flux magnitude; it is not necessary to apply a pressure drop in order to drive the flow. Such an approach simplifies calculations, as it avoids having to couple equations governing the water velocity with those governing species concentration (which would be necessary to account for the effects of osmotic pressure). It is clearly the case, however, that such a formulation cannot predict the magnitude of the transmural water flux.
3.2.2 Applied Shear
Consider the diffusion of a species within a fluid. An associated Schmidt number (Sc) can be defined as the ratio of the kinematic viscosity of the fluid (the momentum diffusivity) to the diffusivity of the species within the fluid. For LDL in blood Sc 1, implying that the rate of diffusional momentum transport is far greater than the rate of diffusional LDL transport. Based on this imbalance in transport rates, it can be assumed that the thickness of the macroscale LDL concentration polarisation layer is substantially less than the thickness of the momentum boundary layer (at any arbitrary point on the arterial wall). Therefore, in the present model it
71 is assumed that the domain of interest resides well within the momentum boundary layer.
The parallel outflow slits illustrated in Fig. 3.1(c) (which represent the intercellular clefts) are aligned at an angle θ∗ to the boundary layer flow, which applies a constant
∗ ∗ shear rate γT to the endothelium. Such alignment results in a shear rate of γ = ∗ ∗ γT sin θ being applied orthogonal to the length-wise extent of the clefts (in the x∗ direction). The shear rate applied to the endothelium is considered spatially constant, since the cellular scale domain is assumed to be small c.f. the scale of spatial variations in the momentum boundary layer.
3.2.3 Domain
As a result of the simplifications outlined above, the problem becomes 2D in nature.
The periodically repeating 2D domain of interest ΩαV within which water motion is considered is illustrated in Fig. 3.2. Note that the wall normal y∗ coordinate in Fig. 3.2 is the same wall normal y∗ coordinate used to define the macroscale LDL concentration polarisation layer in Section 1.3.3.
72
Macroscale Shearing Flow !* = !* sin" * T y* Lumen
* % $V Lumen L$ % $C
(0,0) * !#* #* x Endothelial Cell Intercellular Cleft !"* "*
Figure 3.2: Idealised periodically repeating domains ΩαV and ΩαC adjacent to the endothelial surface. The domain ΩαV (within which water motion is modelled) is semi-infinite. However, the domain ΩαC ⊂ ΩαV (within which LDL transport is modelled) is of finite extent in the y∗ direction, bounded between y∗ = 0 and ∗ ∗ y = Lα. The depth of the intercellular cleft is not included in either domain.
3.2.4 Governing Equations
The continuum approximation for water holds even at the scale of the intercellular clefts [61]. Therefore water is modelled within ΩαV as a free fluid. For any flowing fluid a Reynolds number (Re) can be defined (thought of as the ratio between inertial and viscous forces within the flow). For water flow within ΩαV Re 1 and hence ∗ flow is dominated by viscous forces. Therefore the water velocity field vα within
ΩαV is a solution of the continuity equation
∗ ∗ ∇ · vα = 0, (3.1) and Stokes equation
∗2 ∗ ∗ ∗ ∗ ∇ vα = (1/µ )∇ pα, (3.2)
∗ ∗ where pα is the hydrodynamic pressure in ΩαV and µ is the dynamic viscosity of water.
73 3.2.5 Non-Dimensionalisation
Consider defining x∗ y∗ x = , y = , (3.3) ∆∗ ∆∗ and
∗ ∗ ∗ vα pα∆ vα = uαˆex + vαˆey = ∗ , pα = ∗ , (3.4) V µ∗V where ˆex and ˆey are unit vectors in the x and y directions respectively. On sub- stituting Eq. (3.3) and Eq. (3.4) into Eq. (3.1) and Eq. (3.2), one obtains the following non-dimensional governing equations
∇ · vα = 0, (3.5)
2 ∇ vα = ∇pα. (3.6)
Consistent with these conventions, a non-dimensional cleft half-width δ can be de- fined as δ∗ δ = , (3.7) ∆∗ and a non-dimensional shear rate γ applied orthogonal to the length-wise extent of the clefts can be defined as γ∗∆∗ γ = ∗ . (3.8) V
3.2.6 Non-Dimensional Boundary Conditions
3.2.6.1 Periodicity in x
Since the domain ΩαV is periodic in x, it is required on physical grounds that the velocity fields at x = ±1 and the stress traction vectors acting on the inter-domain interfaces at x = ±1 are continuous. The stress traction vector acting on a surface
74 is given by the product of the stress tensor at the surface with an outward facing unit vector normal to the surface. Therefore, defining the non-dimensional stress tensor σα within ΩαV as
∂uα ∂uα ∂vα 2 − pα + ∂x ∂y ∂x σα = ∂v ∂u ∂v α + α 2 α − p ∂x ∂y ∂y α these interface conditions can be written as
vα(−1, y) = vα(1, y) (3.9) and
σα(−1, y) · −ˆex = −σα(1, y) · ˆex (3.10) where the negative sign on the right hand side of Eq. (3.10) is to account for the fact that the traction vectors are obtained from surface normals of opposite sense.
3.2.6.2 Conditions at y = 0
A no-slip boundary condition is applied at the endothelial cell surface [62], hence
uα(x, 0) = 0. (3.11)
Also a parabolic outflow velocity profile with peak (non-dimensional) magnitude 3/(2δ) is applied at the entrance to the cleft, hence
0 |x| ≥ δ vα(x, 0) = 2 2 (3.12) 3(x − δ ) |x| < δ. 2δ3
75 3.2.6.3 Large y Behavior
As y → ∞ the x component of the velocity field should tend to γy, hence
uα(x, y → ∞) → γy. (3.13)
3.3 Idealised Model of LDL Transport
3.3.1 Domain and Governing Equation
∗ Fig. 3.2 illustrates the domain ΩαC ⊂ ΩαV within which the LDL distribution Cα ∗ ∗ is obtained. ΩαC is periodic in x . However, unlike ΩαV , it is finite in y , bounded ∗ ∗ ∗ between y = 0 and y = Lα. The choice of a suitable value for the height of the ∗ ∗ domain Lα will be discussed shortly. Within ΩαC it is assumed that Cα is a solution of the steady state advection diffusion equation
∗ ∗ ∗ ∗ ∗2 ∗ vα · ∇ Cα − DL∇ Cα = 0, (3.14)
∗ where, as previously defined, DL is the diffusivity of LDL in the lumen.
3.3.2 Suitable Definition of Macroscale LDL Concentration Polarisation Layer
The objective of the present model is to determine modifications to the macroscale LDL concentration polarisation layer caused by a heterogeneous transmural water flux. It has already been noted that the y∗ coordinate illustrated in Fig. 3.2 is the same y∗ coordinate used to define the macroscale LDL concentration polarisation layer in Section 1.3.3. Further, it will shortly be established that LDL is rejected
∗ at the base of ΩαC (at y = 0) in the present model. Therefore, a specific form of the macroscale LDL concentration polarisation layer suitable for comparison with
76 results from the present model can be obtained by setting R∗ = 0 in Eq. (1.11). ∗ The profile Cα within such a layer is given by
∗ ∗ ∗ ∗ ∗ −V y /DL Cα = CEe .
A schematic illustration of such a profile is shown in Fig. 3.3.
* C"
* CE Lumen Artery
y*
* CB !* * 0 !* y
Figure 3.3: Schematic illustration of the macroscale LDL concentration polarisa- ∗ tion layer Cα.
The above procedure ensures that the y∗ coordinate at which LDL is rejected in the
∗ cellular scale domain ΩαC is coincident with the y coordinate at which LDL is re- jected when considering formation of the macroscale LDL concentration polarisation ∗ layer Cα.
3.3.3 Non-Dimensionalisation
∗ Since CE (the LDL concentration predicted to occur adjacent to the luminal surface of the endothelium within the cellular scale locality of ΩαC due to a homogeneous transmural flux) is assumed to be spatially constant within ΩαC , a non-dimensional
77 concentration distribution Cα can be defined as
∗ Cα Cα = ∗ . (3.15) CE
Using Eq. (3.15) and relevant relations from Eq. (3.3) and Eq. (3.4), Eq. (3.14) can be non-dimensionalised to give
2 P eL vα · ∇Cα − ∇ Cα = 0 (3.16) where ∗ V ∆∗ P eL = ∗ (3.17) DL is a Peclet number associated with the spatially heterogeneous convection towards the intercellular clefts. Consistent with the above conventions, a non-dimensional macroscale concentration polarisation layer thickness ζ can be defined as
ζ∗ ζ = , (3.18) ∆∗
a non-dimensional domain height Lα can be defined as
L∗ L = α , α ∆∗
and a non-dimensional concentration profile Cα within the macroscale concentration polarisation layer can be defined as
∗ C α −P eLy Cα = ∗ = e . CE
78 3.3.4 Non-Dimensional Boundary Conditions
3.3.4.1 Periodicity in x
The domain ΩαC is periodic in x, hence
Cα(−1, y) = Cα(1, y), and
∂C ∂C α (−1, y) = α (1, y). ∂x ∂x
3.3.4.2 Condition at y = 0
LDL can cross the vascular endothelium via both paracellular [63] and transcellular [64, 65, 66] routes. However, in-vivo it remains unclear as to which of these two routes is dominant (if either) [67].
Paracellular transport of LDL through ‘normal’ intercellular clefts is considered un- likely given that they contain tight junctional strands. A small fraction of clefts, however, are thought to be ‘leaky’. Such leaky clefts provide a viable route for the paracellular transport of LDL [63, 68]. The periodicity of ΩαC implies that all inter- cellular clefts considered in the present model must be identical i.e. either all normal or all leaky. In this study it is assumed that all the clefts are normal, and therefore that LDL is rejected from the cleft entrances. This is a reasonable assumption since the large majority of clefts are normal. However, it obviously precludes this study from investigating the effect of leaky clefts on LDL concentration polarisation, and also precludes this study from assessing the effect of LDL concentration polarisation on paracellular LDL transport.
Transcellular transport of LDL is thought to occur via vesicular pathways (transcy- tosis) or transcellular pores. Macroscale measurements of endothelial permeability
79 to LDL [31] indicate that the rate of transcellular LDL transport is negligible com- pared to the bulk rate at which LDL is convected towards the endothelium by the transmural water flux [30]. The effect of transcellular LDL transport can therefore be ignored when investigating LDL concentration polarisation adjacent to the lu- minal surface of the endothelium. In this study it will be assumed that the rate of transcellular LDL transport is zero (i.e. that LDL is completely rejected from the surface of the endothelium). Note that although this assumption precludes tran- scellular LDL transport from having an effect on LDL concentration polarisation, it does not preclude the degree of concentration polarisation from having an ef- fect on the rate of transcellular LDL transport; the latter can still be estimated by post-processing the obtained concentration polarisation results.
In summary, it is assumed that LDL is completely rejected from the endothelial surface. Convection must therefore balance diffusion at y = 0 and hence
∂C α (x, 0) = P e v (x, 0)C (x, 0). ∂y L α α
3.3.4.3 Condition at y = Lα
The following three assumptions are made regarding the nature of modifications (caused by a heterogeneous transmural water flux) to the macroscale LDL concen- tration polarisation layer.
• It is assumed, a priori, that modifications do not affect the overall thickness ζ of the macroscale LDL concentration polarisation layer. This is equivalent to stating that ζ is determined only by vascular scale flow features.
• It is assumed, a priori, that beyond a non-dimensional distance y = κα from
the endothelium the non-dimensional concentration field solutions Cα obtained
within ΩαC are approximately 1D and given by
80 −P eLy Cα ≈ Zαe = ZαCα ∀ y > κα,
where the constant Zα will depend on the boundary condition applied at y =
Lα. Such behavior is to be expected for two reasons. Firstly, 2D perturbations to the velocity field decay with y causing the problem to become 1D in nature as y increases, and secondly, it is assumed that LDL is completely rejected from the endothelium at y = 0.
• It is assumed, a priori, that ζ > κα i.e. that the macroscale LDL concentration polarisation layer is not completely destroyed by modifications arising from a heterogeneous water flux. The validity of this assumption and any restrictions that it places on the results are assessed in Appendix A.
The first assumption can be enforced by requiring that for y > κα the solution Cα within ΩαC tends towards the macroscale solution Cα. Via the second and third assumptions this trivially implies that Zα = 1, which can be enforced by applying
Cα(x, Lα) = Cα(Lα),
where Lα can be chosen arbitrarily but must satisfy Lα > κα.
Although the above procedure may seem overly involved, it is in fact necessary in order to ensure that the results are independent of Lα, the arbitrary y extent of
ΩαC .
81 3.4 Implementation
3.4.1 Water Velocity Field
3.4.1.1 Formulating in Terms of a Stream Function
The water velocity field vα within ΩαV can be obtained analytically. Consider defining a stream function ψα such that
∂ψ ∂ψ u = α , v = − α , α ∂y α ∂x
where uα and vα are the x and y components of vα respectively. The non-dimensional governing equations Eq. (3.5) and Eq. (3.6) can be reformulated in terms of ψα as
4 ∇ ψα = 0.
3.4.1.2 Trial Solution for ψα
A suitable form for ψα that is able to satisfy all relevant boundary conditions can be written as
∞ 2 X −λαny ψα = Gαx + Hαy + sin(λαnx)[Aαn + Bαny]e , n=1
where Aαn, Bαn and λαn (which can depend on n) and Gα and Hα are all constants to be determined via the application of the boundary conditions defined by Eq. (3.9), Eq. (3.10), Eq. (3.11), Eq. (3.12) and Eq. (3.13).
3.4.1.3 Application of Boundary Conditions
Application of relevant boundary conditions results in
82 Gα = 1,Hα = γ/2,
+ λαn = nπ n ∈ Z ,
6[sin(λαnδ) − λαnδ cos(λαnδ)] Aαn = 4 3 , λαnδ and
Bαn = λαnAαn.
3.4.1.4 Velocity Field Solution
As a result of the above analysis, expressions for uα and vα can be written as
∞ X 2 −λαny uα = γy − Aαnλαn sin(λαnx)ye , n=1
∞ X −λαny vα = −1 − Aαnλαn cos(λαnx)[1 + λαny]e . n=1
Note that vα depends on only two parameters, namely the non-dimensional cleft half-width δ and the non-dimensional shear rate γ.
3.4.2 LDL Concentration Field
Solutions to Eq. (3.16) within ΩαC are obtained numerically using an in-house C/C++ implementation of the spectral/hp element method previously described in
Chapter 2. The domain ΩαC is meshed in an unstructured fashion (with triangular and quadrangular elements) using the commercial mesh generator Gambit 2.4.6 (ANSYS, Inc). Within each element the solution is represented using 2D modal basis
83 functions, which are constructed from generalised tensor products of 1D polynomial bases of 9th order. An example mesh is shown in Fig. 3.4(a). Smaller elements are grouped near the intercellular cleft as shown in Fig. 3.4(b), since localised features of the solution are expected to develop in this region. Convergence of the numerical solutions is assessed in Appendix B.
1.5 0.06
1 0.04 y y
0.5 0.02
0 0 -1 -0.5 0 0.5 1 -0.04 -0.02 0 0.02 0.04 x x
(a) (b)
Figure 3.4: An example computational mesh of the entire domain ΩαC (a). Smaller elements are concentrated near to the intercellular cleft (b).
Note that the concentration field solutions Cα depend on vα, and hence on δ and
γ, as well as on the Peclet number P eL associated with convection towards the intercellular clefts.
3.4.3 Parameter Values
3.4.3.1 Value for the Non-Dimensional Cleft Half-Width δ
Values of ∆∗ = 10µm and δ∗ = 0.01µm [37, 60] are considered to be physiologically realistic, resulting in a fixed value of δ = 0.001.
84 3.4.3.2 Values for the Non-Dimensional Shear Rate γ
Shear rates in the range γ∗ = 0 − 1000s−1 are considered, along with transmural ∗ flux velocities in the range V = 0.02 − 0.08µms−1. The transmural flux velocity of 0.0439µms−1 obtained by Lever et al. [30] sits within this range. Based on these values (and using ∆∗ = 10µm) a range of γ = 0−5×105 is used in the present model. ∗ For the physiologically realistic value of V = 0.0439µms−1 (and using ∆∗ = 10µm), only the limited range of γ = 0 − 2.28 × 105 is relevant.
3.4.3.3 Values for the Peclet Number P eL
∗ The choice of a suitable value for LDL diffusivity within the lumen, DL, is con- ∗ 2 −1 tentious. In several vascular scale studies of LDL transport a value of DL = 5µm s has been used [28, 69]. However, it has been suggested that this value is unreal- istically low and may therefore artificially enhance calculated LDL concentration
∗ 2 −1 polarisation [16]. Cellular scale studies have also used a value of DL = 5µm s [42], where the value has been obtained via the Stokes-Einstein equation using the viscosity of whole blood. Such an approach is, however, unjustified since at the cellular scale red blood cells should be regarded as discrete objects, and thus LDL should be considered as residing and diffusing within the blood plasma. A range of
∗ 2 −1 DL = 1 − 100µm s is considered in the following analysis. This range spans an ∗ 2 −1 order of magnitude either side of DL = 28.67µm s , the diffusivity of LDL in blood ∗ ∗ ∗ plasma [70]. Based on this range for DL, and the value for ∆ and the range for V given above, a range of P eL = 0.002 − 0.8 is considered in the present model. When ∗ ∗ −1 ∗ 2 −1 using values of ∆ = 10µm, V = 0.0439µms and DL = 28.67µm s (suggested here to be physiologically realistic) a value of P eL = 0.015 is obtained. When using ∗ ∗ −1 ∗ 2 −1 values of ∆ = 10µm, V = 0.04µms and DL = 5µm s (the transmural flux velocity and LDL diffusivity often used in previous studies [27, 28, 42]) an increased value of P eL = 0.08 is obtained.
It should be noted that certain combinations of γ and P eL are unattainable within
85 the limits of the dimensional parameter ranges given above. Specifically, for the dimensional parameter ranges used here, if P eL > 0.2 then it is required that γ < 5 (1 × 10 /P eL). Such a limitation is taken into account when the non-dimensional parameters γ and P eL are varied.
3.4.3.4 Value for the Non-Dimensional Height Lα of ΩαC
For all cases a value of Lα = 1.5 is used. This value is large enough to ensure that
Lα > κα is always satisfied.
3.5 Results and Analysis
3.5.1 Water Velocity Field
Fig. 3.5 shows streamlines of the water velocity field in the vicinity of an intercellular cleft for two values of the non-dimensional shear rate γ, with a fixed non-dimensional cleft half-width δ = 0.001. As γ is increased, the perturbation to the velocity field caused by flow into the intercellular cleft extends less far into the lumen.
86 0.03 0.03
0.02 0.02 y y
0.01 0.01
0 0 -0.02 -0.01 0 0.01 0.02 -0.02 -0.01 0 0.01 0.02 x x
(a) (b)
0.03 0.03
0.02 0.02 y y
0.01 0.01
0 0 -0.02 -0.01 0 0.01 0.02 -0.02 -0.01 0 0.01 0.02 x x
(c) (d)
Figure 3.5: Velocity streamlines in the vicinity of an intercellular cleft obtained using γ = 0 (a), γ = 3 × 103 (b), γ = 3 × 104 (c) and γ = 1.5 × 105 (d) with fixed δ = 0.001. Streamlines are seen to enter the intercellular cleft which is located at the origin. As γ is increased, the perturbation to the velocity field caused by flow into the intercellular cleft extends less far into the lumen.
3.5.2 LDL Concentration Field
Fig. 3.6 shows contour plots of the non-dimensional LDL concentration field Cα for various values of the non-dimensional shear rate γ and the Peclet number P eL, with a fixed non-dimensional cleft half-width δ = 0.001.
87 0.06 0.06 1.011 1.016
0.04 0.04 1.015 1.02 y y 1.024 1.02 0.02 0.02 1.025 1.03 1.035 1.035 1.045 1.045 0 0 -0.04 -0.02 0 0.02 0.04 -0.04 -0.02 0 0.02 0.04 x x 0.6 0.6 0.992 0.4 0.992 0.994 0.4 0.998 0.994 y y 0.998 0.2 1.002 0.2 0.995 1.001 1.01 1.01 0 0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x x
(a) (b)
0.06 0.06
1.09
1.04
0.04 1.11 0.04 1.06 y y
1.14 1.08 0.02 0.02 1.10 1.17 1.12
1.22 1.16 1.3 1.22 0 0 -0.04 -0.02 0 0.02 0.04 -0.04 -0.02 0 0.02 0.04 x x 0.6 0.6 0.96 0.96 0.4 0.97 0.98 0.4 0.97
y 1 y 0.98 0.2 1.02 0.2 0.985 0.99 1 1.07 1.03 0 0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x x
(c) (d)
Figure 3.6: Contour plots of Cα obtained using values of γ = 0 and P eL = 0.015 3 3 (a), γ = 3 × 10 and P eL = 0.015 (b), γ = 0 and P eL = 0.08 (c), and γ = 3 × 10 and P eL = 0.08 (d), all with fixed δ = 0.001. Within each subfigure the upper plot is an enlargement of the region in the dotted box marked on the lower plot. Note that a spatially homogeneous transmural water flux (with the same average velocity as the heterogeneous flux used here) would result in a constant non-dimensional concentration of unity adjacent to the endothelium.
88 Fig. 3.7 shows plots of the non-dimensional LDL concentration adjacent to the endothelium (at y = 0). Again, various values of the non-dimensional shear rate γ and the Peclet number P eL are considered, with fixed non-dimensional cleft half- width δ = 0.001. Note that the scale of the Cα(x, 0) axis in Fig. 3.7 varies between plots.
" = 0 " = 0 " = 3×103 " = 3 103 1.008 1.06 1.008 = 3 104 = 3 104 " × 1.06 " " = 5×105 " = 5 105 1.006 1.04 1.006 1.04 0) 0) 1.02
(x, 1.004 (x, ! -0.005 0 0.005 ! -0.005 0 0.005 C C
1.02 1.002
1 1
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x x
(a) (b)
" = 0 = 0 3 30 3 1.4 1.4 " = 3 10 = 3 10 " = 3 104 30 = 3 104 5 5 " = 5 10 20 = 1.25 10 1.3 1.3 10 1.2 20 0) 0)
(x, 1.2 (x, ! -0.005 0 0.005 -0.005 0 0.005 C C
1.1 10
1 0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x x
(c) (d)
Figure 3.7: Plots illustrating how the LDL distribution adjacent to the endothe- lium varies with γ for P eL = 0.002 (a), P eL = 0.015 (b), P eL = 0.08 (c) and P eL = 0.8 (d) with fixed δ = 0.001. Intercellular clefts are centered at x = 0. Note that a spatially homogeneous transmural water flux (with the same average velocity as the heterogeneous flux used here) would result in a constant non-dimensional concentration of unity adjacent to the endothelium. Also note that the scale of the
Cα(x, 0) axis varies between plots.
89 Shear-dependent cellular scale heterogeneity in the LDL concentration polarisation layer is observed for all values of the Peclet number P eL, with peaks in LDL concen- tration localised above intercellular clefts (at x = 0). Such localisation is expected since LDL is convected directly toward the clefts by the heterogeneous transmural water flux, but cannot pass through them. As the non-dimensional shear rate γ is raised, the peaks in LDL concentration become increasingly skewed (swept down- stream), spreading the LDL distribution more evenly over the endothelial surface and reducing the peak LDL concentration above the clefts.
For the case of P eL = 0.015 illustrated in Fig. 3.7(b) (considered here to be the most physiologically realistic) the degree of heterogeneity in the LDL concentration polarisation layer is relatively small. Specifically, peaks in the LDL concentration profile above the intercellular clefts are only 7.2% (2 s.f.) greater than the aver- age LDL concentration adjacent to the endothelium predicted by a vascular scale simulation using a homogeneous transmural water flux. This relatively small de- gree of heterogeneity is due to the fact that P eL 1 in this realistic case, and hence diffusional transport of LDL dominates over convective transport towards the clefts. For the case of P eL = 0.08 illustrated in Fig. 3.7(c) (obtained using values for the transmural water velocity and LDL diffusivity suggested in previous stud- ies [27, 28, 42]) a more significant degree of heterogeneity is observed. Specifically, peaks in the LDL concentration profile adjacent to the endothelium are 43% (2 s.f.) greater than the average LDL concentration adjacent to the endothelium predicted by a vascular scale simulation using a homogeneous transmural water flux. For the case of P eL = 0.8 illustrated in Fig. 3.7(d) convective transport of LDL by the heterogeneous transmural flux becomes significant, and the LDL concentration po- larisation layer develops very significant spatial heterogeneity. Specifically, peaks in LDL concentration adjacent to the endothelium are 3300% (2 s.f.) greater than the average LDL concentration adjacent to the endothelium predicted by a vascular scale simulation using a homogeneous transmural water flux. It should be noted, however, that a value of P eL = 0.8 is unlikely to occur physiologically.
90 3.5.3 Transendothelial LDL Transport
3.5.3.1 Definitions
Leaky junctions are not represented in the current model. Therefore the implications of the results for the rate of paracellular LDL transport cannot be assessed. It is still possible, however, to consider measures of the concentration polarisation layer that may be relevant to the rate of transcellular LDL transport across the endothelium. Two such measures are considered. The first is simply the average non-dimensional
LDL concentration adjacent to the entire endothelium. This is denoted CαU and defined as 1 Z 1 CαU = Cα(x, 0) dx. 2 −1
CαU is a reasonable measure to use if the endothelium is considered uniformly perme- able to LDL at the cellular scale. The second measure is the average non-dimensional LDL concentration adjacent to the endothelium within a (non-dimensional) distance of 1/10 from each cleft centre. This is denoted CαN and defined as
1 Z 1/10 CαN = Cα(x, 0) dx. 2(1/10) −1/10
CαN may be more relevant to the rate of transcellular LDL transport than CαU since evidence suggests that caveolae, which can facilitate LDL transcytosis, are localised near cell borders [71, 72]. Further, endothelial cells are thinner at their edges and have a nucleus near their centre. These structural features may also increase the relative ease of transcellular LDL transport near the borders of endothelial cells.
It should be noted that both CαU and CαN are obtained from LDL concentration ∗ fields that have been non-dimensionalised by CE. Therefore, the measures are in- dependent of the effect of vascular scale flow features on the macroscale LDL con- centration polarisation layer. The measures only reflect additional (possibly flow dependent) modifications to the LDL concentration polarisation layer caused by the heterogeneous transmural water flux.
91 3.5.3.2 Dependence of CαU and CαN on P eL
Fig. 3.8 shows plots of CαU and CαN against the Peclet number P eL for various values of the non-dimensional shear rate γ with fixed non-dimensional cleft half- width δ = 0.001.
" = 0 3 1.2 " = 3×10 " = 3×104 " = 5×105 1.15 U !
C 1.1
1.05
1 0 0.2 0.4 0.6 0.8 PeL
(a)
" = 0 3.5 " = 3×103 " = 3×104 5 3 " = 5×10
2.5 N ! C 2
1.5
1 0 0.2 0.4 0.6 0.8 PeL
(b)
Figure 3.8: Plots of CαU (a) and CαN (b) against P eL for various values of γ with fixed δ = 0.001.
92 It can be seen that both CαU and CαN increase monotonically as P eL is increased
(up to a value of P eL = 0.8). It can also be seen that values of CαU and CαN are greater than unity in all cases. Finally, it can be seen that values of CαN are always greater than values of CαU obtained using the same parameters, reflecting the fact that LDL is convected towards the intercellular clefts.
The above results indicate that a heterogeneous transmural water flux acts to en- hance the measures CαU and CαN to values greater than unity (i.e. to values greater than those resulting from a homogeneous transmural flux with the same average ve- locity). However, for cases where P eL = 0.015 (considered here to be physiologically realistic), CαU exceeds unity by at most 0.0060% (2 s.f.) and CαN exceeds unity by at most 2.1% (2 s.f.). For physiologically realistic cases the aforementioned enhance- ment is therefore unlikely to be significant.
Fig. 3.9 shows a log-log plot of (CαU − 1) against the Peclet number P eL, with fixed non-dimensional cleft half-width δ = 0.001. The straight lines indicate that for a given value of the non-dimensional shear rate γ (and when P eL < 0.8) a relationship exists between CαU and P eL of the form
d CαU = 1 + c(P eL) ,
where both c and d depend on the value of γ considered.
93 " = 0 3 -1 " = 3×10 10 " = 3×104 " = 5×105
10-2 -1 U ! C 10-3
10-4
0.2 0.4 0.6 0.8 PeL
Figure 3.9: Plots of (CαU − 1) against P eL for various values of γ with fixed δ = 0.001. The plots are on a log-log scale and hence the straight lines indicate a power law relationship between (CαU − 1) and P eL.
3.5.3.3 Dependence of CαU and CαN on γ
Fig. 3.10 shows plots of CαU and CαN against the non-dimensional shear rate γ for various values of the Peclet number P eL with fixed non-dimensional cleft half-width δ = 0.001.
94 PeL = 0.002 1.2 PeL = 0.015 PeL = 0.08
PeL = 0.8 1.15 U "
C 1.1
1.05
1 0 0.5×105 1×105 !
(a)
Pe = 0.002 3.5 L PeL = 0.015
PeL = 0.08 Pe = 0.8 3 L
2.5 N " C 2
1.5
1 0 0.5×105 1×105 !
(b)
Figure 3.10: Plots of CαU (a) and CαN (b) against γ for various values of P eL with fixed δ = 0.001.
For the case of P eL = 0.015 (considered here to be physiologically realistic) CαU varies by only 0.0060% (2 s.f.) and CαN varies by only 2.1% (2 s.f.) as γ varies 5 between 0 and 5 × 10 . For the case of P eL = 0.08 (obtained using values for the transmural water velocity and LDL diffusivity suggested in previous studies
95 [28, 42, 27]) CαU varies by only 0.16% (2 s.f.) and CαN varies by 11% (2 s.f.) as γ 5 varies between 0 and 5 × 10 . For the case of P eL = 0.8 both CαU and CαN deviate significantly from unity and exhibit a pronounced shear dependence.
It has been shown in previous studies [27, 28] that vascular scale flow features may affect the overall degree of LDL concentration polarisation adjacent to the endothe- lium, potentially resulting in a flow dependent rate of transendothelial LDL trans- port. The results in Fig. 3.10 indicate that, in principle, a spatially heterogeneous transmural water flux can cause additional shear (and hence flow) dependence of transendothelial LDL transport if P eL ∼ 1. However, this additional shear depen- dence is observed to be insignificant for the physiological value of P eL = 0.015.
3.5.4 Extending the Parameter Space
The behavior of solutions within an extended parameter space is briefly investigated. Although such behavior is unlikely to have physiological relevance, it does put the physiological solutions into context.
The previous results were obtained using a fixed (physiologically reasonable) value for the non-dimensional cleft half-width of δ = 0.001. Fig. 3.11 illustrates the effect of varying δ on CαU and CαN for various values of the Peclet number P eL with fixed non-dimensional shear rate γ = 0. It can be seen that both CαU and CαN asymptote towards a constant value as δ decreases. Specifically, for the physiological value of
P eL = 0.015, CαU and CαN become independent of δ when δ < 0.01. Therefore,
CαU and CαN are likely to be independent of δ for all physiologically reasonable parameter values.
96 PeL = 0.015 100 PeL = 0.08
PeL = 0.8
10-1
-2
-1 10 U " C
10-3
10-4
10-3 10-2 10-1 !
(a)
1 10 PeL = 0.015 PeL = 0.08
PeL = 0.8
100 -1 N " C 10-1
10-2
10-3 10-2 10-1 !
(b)
Figure 3.11: Plots of (CαU − 1) (a) and (CαN − 1) (b) against δ for various values of P eL with fixed γ = 0. The plots are on a log-log scale.
It has been shown in Fig. 3.10(a) that CαU becomes shear dependent if the Peclet number P eL ∼ 1. Specifically, results for the case of P eL = 0.8 were presented.
Fig. 3.12 shows further plots of CαU against γ with fixed P eL = 0.8 for various values of δ. The shear dependence of CαU decreases significantly as δ is increased.
97 This observation supports the assertion that a heterogeneous transmural water flux facilitates the shear dependence of CαU .
# = 0.5 1.2 # = 0.1 # = 0.01 # = 0.001 # = 0.5 1.15 # = 0.5 # = 0.01 U
" # = 0.001 C 1.1
1.05
1 0 1×104 2×104 !
Figure 3.12: Plots of CαU against γ for various values of δ with fixed P eL = 0.8. The larger solid symbols at γ = 0 help to indicate where each line begins.
Fig. 3.13(a) shows a plot of CαU against the Peclet number P eL for fixed γ = 0 and δ = 0.001. An extended range of P eL = 0.002 − 13.3 is considered. It can be seen that a value of P eL exists that maximises CαU . Also, as P eL becomes large
CαU → 0.
Fig. 3.13(b) shows a log-log plot of (CαU − 1) against Peclet number P eL for fixed
γ = 0 and δ = 0.001. Beyond P eL ∼ 1 the power law relationship between (CαU −1) and P eL observed in Fig. 3.9 breaks down, indicating that the relationship is not valid when transport is dominated by convection.
98 6
4 U ! C
2
0 0 5 10 PeL
(a)
101
100
-1 -1 U
! 10 C
10-2
10-3
10-1 100 101 PeL
(b)
Figure 3.13: Plots of CαU against P eL (a) and (CαU -1) against P eL (b) for fixed γ = 0 and δ = 0.001. The plot in (b) is on a log-log scale. The solid straight line in (b) highlights the power law relationship between (CαU -1) and P eL previously observed in Fig. 3.9. This power law relationship is seen to break down when
P eL > 1.
99 3.5.5 Implications of Neglecting Tight Junction Strands
The effect of tight junction strands on the cleft entrance velocity profile has been ignored in this simple model. In reality, such strands (which form within the depth of the intercellular clefts) are likely to block water flow through certain portions of the cleft, further localising flow to a sub-area of each cleft entrance. The results obtained in this study indicate that CαU and CαN depend strongly on the Peclet number P eL (which can be thought of as the total water flux towards each cleft), but very weakly on the non-dimensional cleft half width δ (which determines how localised this total flux becomes at each cleft entrance). Increased localisation of flow at the cleft entrances due to the presence of tight junction strands is therefore unlikely to have a significant impact on the results. It should be noted, however, that such an assertion can only be confirmed by developing a model that explicitly includes tight junction strands (or at least their effect on the cleft entrance velocity profile).
3.5.6 Implications of Neglecting Pulsatile Blood Flow
Since blood flow is pulsatile the shear rate applied to ΩαV should vary with time. Time-dependent studies (the formulation of which is given in Appendix C) indicate that the LDL concentration field within ΩαC is not completely quasi-steady when exposed to pulsatile flow i.e. additional time-dependent dynamics do exist. It is found, however, that when a physiologically realistic value of P eL = 0.015 is considered, time-averaged values of CαU and CαN are almost exactly unity and independent of the applied flow wave form. The main findings of the steady state analysis presented in this chapter are therefore not affected by the application of a time-dependent shear rate.
100 3.6 Conclusions
The results presented in this chapter indicate that a spatially heterogeneous trans- mural water flux will cause spatially heterogeneous and shear dependent modifi- cations to any LDL concentration polarisation layer that develops adjacent to the endothelium. Measures of LDL concentration polarisation that may be relevant to the rate of transendothelial LDL transport have been defined and calculated. It has been demonstrated, in principle, that a spatially heterogeneous transmural water flux can act to enhance such measures, and cause them to develop a shear depen- dence (in addition to any shear dependence of LDL uptake caused by vascular scale flow features affecting the overall degree of LDL concentration polarisation). How- ever, it has been shown that this enhancement and additional shear dependence are unlikely to be significant for physiological values of the Peclet number P eL.
101 Chapter 4
The Effect of an Endothelial Glycocalyx Layer
4.1 Overview of Idealised Model
In this chapter the idealised model developed in Chapter 3 is extended in order to assess the effect of a heterogeneous transmural water flux and an EGL on LDL concentration polarisation adjacent to the arterial endothelium. As in Chapter 3, the objective of the present model is to determine modifications to the macroscale LDL concentration polarisation layer within the cellular scale locality of an arbitrary point adjacent to the arterial wall. Also, in line with the approach adopted in Chapter 3 the following four general assumptions are made:
• It is assumed that the macroscale LDL concentration polarisation layer is spatially uniform at the cellular scale.
• Only transport of water and LDL are considered; all other constituents of the blood are ignored.
• It is assumed that LDL has no effect on the motion of the water. • All dynamics within the domain are considered to be at steady state.
4.2 Idealised Model of Water Flow
4.2.1 Heterogeneous Transmural Flux and Applied Shear
The present model of water flow is based on the previous model outlined in Chapter 3. The intercellular clefts are represented by a repeating array of outflow slits as ∗ illustrated in Fig. 3.1(c). A transmural water flux with an average velocity V is drawn towards the endothelium via the application of a parabolic outflow velocity ∗ profile (with peak magnitude 3V ∆∗/2δ∗) across the width of each cleft entrance. It is assumed that the relevant domain of interest resides within the momentum
∗ boundary layer, which applies a constant shear rate γT to the endothelium. It is further assumed that the idealised intercellular clefts are aligned at an angle θ∗
∗ ∗ ∗ to the boundary layer flow, resulting in a shear rate γ = γT sin θ being applied perpendicular to the length-wise extent of the clefts (in the x∗ direction).
4.2.2 Domain and Governing Equations
The periodically repeating 2D domain of interest ΩβV = ΩβV L ∪ ΩβV G within which water motion is modelled is illustrated in Fig. 4.1. Note that the wall normal y∗ coordinate in Fig. 4.1 is the same wall normal y∗ coordinate used to define the macroscale LDL concentration polarisation layer in Section 1.3.3.
103
Macroscale Shearing Flow !* = !* sin" * T y*
! "VL Lumen
! "V h* EGL !"VG (0,0) * #%* %* x Endothelial Cell Intercellular Cleft #$* $*
Figure 4.1: Idealised periodically repeating domain ΩβV = ΩβV L ∪ ΩβV G within which water motion is considered. ΩβV L is a semi-infinite region of free viscous fluid representing the lumen. ΩβV G is a region of saturated anisotropic Brinkman medium representing the EGL. The depth of the intercellular cleft is not included in ΩβV .
Region ΩβV L represents the lumen. As in Section 3.2, water in the lumen is modelled ∗ as a free fluid within which viscous forces dominate. The water velocity field vβL within ΩβV L is therefore a solution of the continuity equation
∗ ∗ ∇ · vβL = 0, (4.1) and Stokes equation
∗2 ∗ ∗ ∗ ∗ ∇ vβL = (1/µ )∇ pβL, (4.2)
∗ ∗ where pβL is the hydrodynamic pressure in ΩβV L and, as previously defined, µ is the dynamic viscosity of water.
Region ΩβV G represents the EGL. As has been discussed, the EGL is believed to be formed of two distinct yet closely interacting layers. The first layer is thought to be composed of long chain macromolecules firmly attached to the luminal surface of the underlying endothelial cells, whereas the second layer is considered to be more dynamic in nature, composed of loosely attached plasma and endothelium derived proteins.
104 In the present model the entire EGL (both layers) is modelled as a saturated fibrous layer of height h∗. Based on measurements of hyaluronate gel [73] and other relevant biological tissues [74], it is likely that the EGL has a solid volume fraction ∼ 1% or less. In this study it will be assumed that the EGL has a vanishing solid volume fraction, and hence can be modelled from a fluid dynamics perspective as a Brinkman medium. Representing the EGL as a Brinkman medium significantly simplifies analysis c.f. modelling the structure of the EGL explicitly. This is because such a medium can be fully characterised by its Darcy permeability and the viscosity of the fluid residing within it. In the present model it will be assumed that the Darcy permeability of the glycocalyx is anisotropic, with the anisotropy modelled using the inverse Darcy permeability tensor
∗ ∗ (1/Kx) 0 K = . ∗ 0 (1/Ky )
∗ In line with the above assumptions, the velocity field vβG within ΩβV G is a solution of the continuity equation
∗ ∗ ∇ · vβG = 0, (4.3) and the anisotropic Brinkman equation
∗2 ∗ ∗ ∗ ∗ ∗ ∗ ∇ vβG − K · vβG = (1/µ )∇ pβG, (4.4)
∗ where pβG is the hydrodynamic pressure in ΩβV G.
There are several points that should be noted regarding this particular representation of the EGL:
• The inverse Darcy permeability tensor K∗ is not the most generic that can be formed. Specifically, since K∗ is diagonal, the model is limited to cases where the principle axes of the inverse Darcy permeability tensor (and hence also the Darcy permeability tensor) are aligned with the x∗ and y∗ axes of the adopted
105 coordinate system (i.e. parallel and normal to the endothelial cell surface at the base of the domain). This is a reasonable way to introduce anisotropy to the EGL given the limited information about its structure.
• One could potentially construct more complex representations of the EGL in
∗ order to model its dual layered structure. For example one could vary Kx ∗ ∗ and Ky with y (resulting in the EGL having an inhomogeneous as well as anisotropic Darcy permeability), or one could model the most luminal layer of the EGL as a highly viscous fluid rather than a porous layer. Given that there is currently limited information about the structure of the EGL, it would be unwise to consider models of such complexity at this time.
• A slightly modified form of Eq. (4.4) can be used to model cases where the solid volume fraction of the porous medium is appreciable. This modification is achieved through the introduction of an additional parameter know as the effective viscosity, which depends on solid volume fraction and is thus generally not equal to the viscosity of the free fluid [75]. To obtain the modified form of
∗2 ∗ Eq. (4.4) the Laplacian term ∇ vβG is multiplied by the ratio of the effective viscosity to the free fluid viscosity. Such a modification is not employed in this study, since it is assumed that the solid volume fraction of the EGL is vanishingly small.
∗ Finally, note that the entire velocity field vβ within ΩβV can be defined as
v∗ y∗ ≥ h∗ ∗ βL vβ = ∗ ∗ ∗ vβG y < h .
4.2.3 Non-Dimensionalisation
Consider defining
106 ∗ ∗ vβL vβG vβL = uβLˆex + vβLˆey = ∗ , vβG = uβGˆex + vβGˆey = ∗ , V V (4.5) ∗ ∗ ∗ ∗ pβL∆ pβG∆ pβL = ∗ , pβG = ∗ . µ∗V µ∗V
On substituting Eq. (4.5) and expressions for x and y given by Eq. (3.3) into Eq. (4.1), Eq. (4.2), Eq. (4.3) and Eq. (4.4) one obtains the following non-dimensional governing equations
∇ · vβL = 0, (4.6)
2 ∇ vβL = ∇pβL, (4.7)
∇ · vβG = 0, (4.8)
2 ∇ vβG − Da · vβG = ∇pβG, (4.9) where (1/Dax) 0 ∗2 ∗ Da = = ∆ K . 0 (1/Day)
Consistent with these conventions one can define a non-dimensional cleft half-width δ and a non-dimensional shear rate applied orthogonal to the length-wise extent of the clefts γ (previously defined by Eq. (3.7) and Eq. (3.8) respectively). Also, a non-dimensional EGL height h can be defined as
h∗ h = , ∆∗
and the entire non-dimensional velocity field vβ within ΩβV can be defined as
vβL y ≥ h vβ = vβG y < h.
107 4.2.4 Non-Dimensional Boundary Conditions
4.2.4.1 EGL/Lumen Interface
At the interface between the lumen (ΩβV L) and the EGL (ΩβV G) velocity field com- ponents are matched, as are the components of the stress traction vectors acting on the interface. Defining the non-dimensional stress tensor σβL within ΩβV L as
∂uβL ∂uβL ∂vβL 2 − pβL + ∂x ∂y ∂x σβL = , ∂v ∂u ∂v βL + βL 2 βL − p ∂x ∂y ∂y βL
and the non-dimensional stress tensor σβG within ΩβV G as
∂uβG ∂uβG ∂vβG 2 − pβG + ∂x ∂y ∂x σβG = , ∂v ∂u ∂v βG + βG 2 βG − p ∂x ∂y ∂y βG
the interface conditions between ΩβV L and ΩβV G be written as
vβL(x, h) = vβG(x, h), (4.10)
σβL(x, h) · −ˆey = −σβG(x, h) · ˆey, (4.11) where the negative sign on the right hand side of Eq. (4.11) is to account for the fact that the traction vectors are obtained from normals of opposite sense. Writing Eq. (4.11) explicitly implies
∂u ∂v ∂u ∂v βL (x, h) + βL (x, h) = βG (x, h) + βG (x, h), ∂y ∂x ∂y ∂x
∂v ∂v −p (x, h) + 2 βL (x, h) = −p (x, h) + 2 βG (x, h). βL ∂y βG ∂y
108 Conditions defined by Eq. (4.10) and Eq. (4.11) are the same as those imposed by Damiano et al. [76] and others [77, 78]. They assume that the interface cannot support any discontinuity in stress, and are only considered valid when the solid volume fraction of the porous material is small. Other interface conditions such as those involving a jump in tangential shear stress proportional to the tangential ve- locity at the interface have been suggested [79, 80]. However such conditions (which require an additional empirically determined parameter) will not be considered in this study.
4.2.4.2 Periodicity in x
Since the domain ΩβV is periodic in x the velocity fields at x = ±1 and the stress traction vectors acting on the inter-domain interfaces at x = ±1 should be continu- ous, hence
vβL(−1, y) = vβL(1, y),
vβG(−1, y) = vβG(1, y), and
σβL(−1, y) · −ˆex = −σβL(1, y) · ˆex, (4.12)
σβG(−1, y) · −ˆex = −σβG(1, y) · ˆex, (4.13) where the negative signs on the right hand sides of Eq. (4.12) and Eq. (4.13) are to account for the fact that the traction vectors are obtained from surface normals of opposite sense.
4.2.4.3 Conditions at y = 0
The boundary conditions imposed at y = 0 are physically identical to those imposed in the previous model presented in Chapter 3. Therefore
109 uβG(x, 0) = 0
and 0 |x| ≥ δ vβG(x, 0) = 2 2 3(x − δ ) |x| < δ. 2δ3
4.2.4.4 Large y Behavior
The boundary condition imposed at y = ∞ is physically identical to that imposed in the previous model presented in Chapter 3. Therefore
uβL(x, y → ∞) → γy.
4.3 Idealised Model of LDL Transport
4.3.1 Domain and Governing Equation
∗ Fig. 4.2 illustrates the domain ΩβC ⊂ ΩβV within which the LDL distribution Cβ ∗ is obtained. The y extent and location of ΩβC can be changed by varying the ∗ ∗ ∗ ∗ ∗ parameters b and Lβ. The parameter b (where 0 ≤ b ≤ h ) determines the extent to which ΩβC overlaps with ΩβV G (i.e. the extent to which ΩβC overlaps with the EGL). It will be established shortly that b∗ effectively determines the distance to
∗ which LDL can penetrate into the EGL. The parameter Lβ determines the overall ∗ ∗ extent of ΩβC in the y direction. The choice of a suitable value for Lβ will be discussed shortly. It can be noted that the region of overlap between ΩβC and the
EGL is denoted ΩβCG = ΩβC ∩ ΩβV G, and the region of overlap between ΩβC and the lumen is denoted ΩβCL = ΩβC ∩ ΩβV L.
110
Macroscale Shearing Flow !* = !* sin" * T y*
Lumen ! "CL * ! "C L"
* b * !"CG h EGL (0,0) * #%* %* x Endothelial Cell Intercellular Cleft #$* $*
Figure 4.2: Idealised periodically repeating domain ΩβC ⊂ ΩβV within which LDL transport is considered. ΩβC = ΩβCL ∪ ΩβCG, where ΩβCL = ΩβC ∩ ΩβV L and ΩβCG = ΩβC ∩ ΩβV G i.e. ΩβCL is in the lumen and ΩβCG is in the EGL. For the ∗ special case of b = 0, ΩβC ∩ ΩβV G = 0, hence ΩβCG = 0 and ΩβC = ΩβCL.
∗ Within ΩβC it is assumed that the LDL concentration field Cβ is a solution of the steady state advection diffusion equation
∗ ∗ ∗ ∗ ∗ ∗ ∗ v · ∇ Cβ − ∇ (D ∇ Cβ) = 0, (4.14)
∗ where D is the (potentially spatially dependent) LDL diffusivity within ΩβC . Defin- ∗ ∗ ing DG as the LDL diffusivity in the EGL (and as before defining DL as the LDL diffusivity in the lumen) two sets of cases can be considered. The first set of cases
∗ are those where b = 0 (hence ΩβC is solely within the lumen and ΩβCG = 0). In ∗ ∗ these cases it is assumed that D is spatially constant with a value DL. The second ∗ set of cases are those where b 6= 0 (hence the domain ΩβC spans both the lumen ∗ and the EGL, and ΩβCG 6= 0). In these cases it is assumed that D varies spatially, following a hyperbolic tangent function, between an approximately constant value of
∗ ∗ DL in the lumen and an approximately constant value of DG in the EGL. It should be noted that the choice a hyperbolic tangent function to represent variations in D∗ has no particular physical motivation. It is simply chosen as a convenient way to parameterise a step-type transition between two spatially constant values of LDL
111 diffusivity.
An explicit expression for D∗, encompassing all scenarios, can be written as
∗ DL ΩβCG = 0 D∗ = ∗ ∗ ∗ ∗ ∗ ∗ ∗ (1/2)[(DL − DG) tanh[(y − h )/f ] + (DL + DG)] ΩβCG 6= 0,
∗ ∗ ∗ ∗ where 2f is the approximate distance taken by D to vary between DL and DG when ΩβCG 6= 0.
The present model, as formulated, accounts for the effect of the EGL on the water velocity profile above the endothelium (which will affect convective LDL transport), as well as the effect of the EGL on LDL diffusivity near the luminal surface of the endothelium (which will affect diffusive LDL transport). However, it is important to note that the model does not account for several other potential effects of the EGL on LDL transport. For example it does not account for any possible reaction of LDL with the EGL, or any possible binding of LDL to sites within the EGL.
4.3.2 Suitable Definition of Macroscale LDL Concentration Polarisation Layer
The objective of the present model is to determine modifications to the macroscale LDL concentration polarisation layer caused by a heterogeneous transmural water flux and an EGL. It has already been noted that the y∗ coordinate illustrated in Fig. 4.1 and Fig. 4.2 is the same y∗ coordinate used to define the macroscale LDL concentration polarisation layer in Section 1.3.3. Further, it will shortly be
∗ established that in the present model LDL is rejected at the base of ΩβC (at y = h∗ −b∗). Therefore, a specific form of the macroscale LDL concentration polarisation layer suitable for comparison with results from the present model can be obtained
∗ ∗ ∗ ∗ by setting R = h − b in Eq. (1.11). The profile Cβ within such a layer is given by
112 ∗ ∗ ∗ ∗ ∗ ∗ ∗ −V (y −h +b )/DL Cβ = CEe .
A schematic illustration of such a profile is shown in Fig. 4.3.
* C
* CE Lumen Artery
y*
* CB * * 0 h*-b* h*-b*+ * y
Figure 4.3: Schematic illustration of the macroscale LDL concentration polarisa- ∗ tion layer Cβ.
The above procedure ensures that the y∗ coordinate at which LDL is rejected in the
∗ cellular scale domain ΩβC is coincident with the y coordinate at which LDL is re- jected when considering formation of the macroscale LDL concentration polarisation ∗ layer Cβ.
4.3.3 Non-Dimensionalisation
∗ Since CE is assumed to be spatially constant within ΩβC , a non-dimensional con- centration distribution Cβ can be defined as
∗ Cβ Cβ = ∗ . (4.15) CE
113 Using Eq. (4.15) and relevant relations from Eq. (3.3) and Eq. (4.5), Eq. (4.14) can be non-dimensionalised to give
P eLv · ∇Cβ − ∇(D∇Cβ) = 0, (4.16)
where P eL is a Peclet number, previously defined by Eq. (3.17), and
1 ΩβCG = 0 ∗ ∗ D = D /DL = (1/2)[(1 − Dr) tanh[(y − h)/f] + (1 + Dr)] ΩβCG 6= 0, where
∗ ∗ DG f Dr = ∗ , f = ∗ . DL ∆
Consistent with the above conventions the non-dimensional height Lβ of ΩβC can be defined as L∗ L = β , β ∆∗ the non-dimensional distance b that LDL penetrates into the EGL can be defined as b∗ b = , ∆∗
and the non-dimensional profile Cβ of the macroscale LDL concentration polarisa- tion layer can be defined as
∗ C β −P eL(y−h+b) Cβ = ∗ = e . CE
Once again, the non-dimensional thickness of the macroscale LDL concentration polarisation is denoted ζ, which has been previously defined by Eq. (3.18).
114 4.3.4 Non-Dimensional Boundary Conditions
4.3.4.1 Periodicity in x
The domain ΩβC is periodic in x, hence
Cβ(−1, y) = Cβ(1, y), and ∂C ∂C β (−1, y) = β (1, y). ∂x ∂x
4.3.4.2 Condition at y = h − b
Following the arguments presented in Chapter 3, it is assumed in the present model that LDL is completely rejected at the endothelial surface. Therefore, in steady state, convection must balance diffusion at the base of ΩβC (at y = h − b) and hence
P e v (x, h)C (x, h)Ω = 0 ∂C L βL β βCG β (x, h − b) = 2P e v (x, h − b)C (x, h − b) ∂y L βG β ΩβCG 6= 0. (1 − Dr) tanh(−b/f) + (1 + Dr)
The enforced rejection of LDL at y = h − b implies that b effectively determines the distance to which LDL can penetrate into the EGL.
4.3.4.3 Condition at y = h − b + L
In line with the procedure adopted in Chapter 3, the following assumptions are made regarding the nature of modifications (caused by a heterogeneous transmural water flux and an EGL) to the macroscale LDL concentration polarisation layer.
115 • It is assumed, a priori, that modifications do not affect the overall thickness ζ of the macroscale LDL concentration polarisation layer. This is equivalent to stating that ζ is determined only by vascular scale flow features.
• It is assumed, a priori, that beyond a non-dimensional distance κβ from the point of LDL rejection (y = h − b), the non-dimensional concentration field
solutions Cβ obtained within ΩβC are approximately 1D and given by
−P eL(y−h+b) Cβ ≈ Zβe = ZβCβ ∀ y > h − b + κβ,
where the constant Zβ will depend on the boundary condition applied at y = h − b + L. Such behavior is to be expected for three reasons. Firstly, 2D perturbations to the velocity field decay with y causing the problem to become 1D in nature as y increases. Secondly, the non-dimensional LDL diffusivity D tends towards unity as y increases. Thirdly, it is assumed that LDL is completely rejected from the endothelium at y = h − b.
• It is assumed, a priori, that ζ > κβ i.e. that the macroscale LDL concentration polarisation layer is not completely destroyed by modifications arising from a heterogeneous water flux and an EGL.
The first assumption can be enforced by requiring that for y > κβ the solution Cβ within ΩβC tends towards the macroscale solution Cβ. Via the second and third assumptions this trivially implies that Zβ = 1, which can be enforced by applying
Cβ(x, Lβ + h − b) = Cβ(Lβ),
where Lβ can be chosen arbitrarily but must satisfy Lβ > κβ.
116 4.4 Implementation
4.4.1 Water Velocity Field
4.4.1.1 Formulating in Terms of Stream Functions
The water velocity field vβ within ΩβV can be obtained analytically. Consider defining stream functions ψβL and ψβG such that
∂ψ ∂ψ u = βL , v = − βL (4.17) βL ∂y βL ∂x and ∂ψ ∂ψ u = βG , v = − βG . (4.18) βG ∂y βG ∂x where uβL and vβL are the x and y components of vβL respectively, and uβG and vβG are the x and y components of vβG respectively.
The non-dimensional governing equations Eq. (4.6) and Eq. (4.7) for flow within
ΩβV L can be reformulated in terms of the stream function ψβL as
4 ∇ ψβL = 0.
Similarly, the non-dimensional governing equations Eq. (4.8) and Eq. (4.9) for flow in ΩβV G can be reformulated in terms of the stream function ψβG as
2 2 4 1 ∂ ψβG 1 ∂ ψβG ∇ ψβG − 2 − 2 = 0. (4.19) Day ∂x Dax ∂y
Using the above formulation, the fundamental governing equations for flow within both ΩβV L and ΩβV G are reduced to single fourth order equations for the non- dimensional scalars ψβL and ψβG.
117 4.4.1.2 Trial Solution for ψβL
A suitable form for ψβL that is able to satisfy all relevant boundary conditions can be written as
∞ 2 X −λβny −λβny ψβL = Gβx + Hβy + Jβy + sin(λβnx)[Aβne + Bβnye ], n=1
where Aβn, Bβn and λβn (which can depend on n) and Gβ, Hβ and Jβ are all constants to be determined.
4.4.1.3 Trial Solution for ψβG when Dax ≤ Day
It will be established that ψβG takes different forms depending on whether Dax ≤
Day or Dax > Day. For cases where Dax ≤ Day (i.e. where the EGL is more permeable in the wall normal direction than in the wall parallel direction) consider letting ∞ ξβ y X χβny ψβG = Kβx + Pβe + sin(λβnx)[Cβne ], (4.20) n=1 where Cβn, λβn and χβn (which can depend on n) and Kβ, Pβ and ξβ are all constants to be determined.
The trial solution defined by Eq. (4.20) will only satisfy the stream function formu- lation of the anisotropic Brinkman equation Eq. (4.19) for certain values of ξβ and
χβn. Inserting Eq. (4.20) into Eq. (4.19) and collecting terms in Pβ and Cβn one obtains
4 2 ξβ − (1/Dax)ξβ = 0 (4.21) and
4 2 2 4 2 2 χβn − 2λβnχβn + λβn + (λβn/Day) − (χβn/Dax) = 0. (4.22)
The two non-trivial values of ξβ that satisfy Eq. (4.21) are given by
118 p ξβ = ±(1/ Dax),
and the four values of χβn that satisfy Eq. (4.22) are given by
r 0 q 00 χβn = ± χβn ± χβn, (4.23) where
0 2 1 00 2 1 1 1 χβn = λβn + and χβn = λβn − + 2 . (4.24) 2Dax Dax Day 4Dax
For cases where Dax ≤ Day it can be shown that
00 0 q 00 χβn > 0 and χβn − χβn > 0 ∀ λβn.
This implies that all values of χβn are real, and hence the following real trial solution for ψβG can be constructed
√ √ (y/ Dax) −(y/ Dax) ψβG = Kβx + Pβe + Qβe + ∞ X τβny −τβny ηβny −ηβny sin(λβnx)[Cβne + Dβne + Eβne + Fβne ], (4.25) n=1 where r r 0 q 00 0 q 00 τβn = χβn − χβn, ηβn = χβn + χβn,
and Cβn, Dβn, Eβn, Fβn and λβn (which can depend on n) and Kβ, Pβ and Qβ are all real constants to be determined.
4.4.1.4 Trial Solution for ψβG when Dax > Day
On examining Eq. (4.23) and Eq. (4.24) it becomes clear that if Dax > Day, then all four values of χβn will become complex for a particular range of λβn. For complex
χβn it is still possible to construct a real trial solution for ψβG, however the form
119 of the solution will be different to that of Eq. (4.25). It is also the case that if
Dax > Day the roots of Eq. (4.22) may be degenerate for a particular value of λβn.
This will result in only two distinct values of χβn, and again a form of trial solution different to Eq. (4.25) must be constructed to ensure that four degrees of freedom are retained. Suitable trial solutions for cases where Dax > Day are developed in Appendix D.
4.4.1.5 Application of Boundary Conditions
For brevity, only the application of boundary conditions to Eq. (4.25) is presented
(i.e. to cases where Dax ≤ Day). In order to apply the conditions outlined in Section
4.2.4 expressions for vβL, vβG, pβL and pβG are required. Expressions for vβL and vβG (omitted for brevity) can be obtained trivially from the stream functions ψβL and ψβG via Eq. (4.17) and Eq. (4.18). Using these expressions for vβL and vβG, expressions for pβL and pβG can then be obtained via Eq. (4.7) and Eq. (4.9). The resulting expressions for pβL and pβG can be written as
∞ X −λβny pβL =p ¯βL − cos(λβnx)[2λβnBβne ], n=1
Ky pβG =p ¯βG + Day ∞ X −τβny τβny −ηβny ηβny − cos(λβnx)[Γβn(Dβne − Cβne ) + Θβn(Fβne − Eβne )], n=1
wherep ¯βL andp ¯βG are constants and
2 Γβn = τβn(λβn − (τβn/λβn) + (1/Daxλβn)),
2 Θβn = ηβn(λβn − (ηβn/λβn) + (1/Daxλβn)).
After application of boundary conditions one obtains
120 Gβ = 1,Kβ = 1,Hβ = γ/2,
along with
ˆ Jβ = γJβ,
where √ √ ! −2(h/ Dax) ˆ Dax(1 − e ) Jβ = √ − h (1 + e−2(h/ Dax)) and
ˆ Pβ = Qβ = γPβ,
where ˆ Dax Pβ = √ √ . (e(h/ Dax) + e−(h/ Dax))
Finally, one finds
+ λβn = nπ n ∈ Z ,
as well as six equations for the six unknowns Aβn, Bβn, Cβn, Dβn, Eβn and Fβn, which can be written in matrix form as
−1 Υβn = MβnΛβn where
121 Aβn 0 Bβn βn Cβn 0 Υβn = , Λβn = , Dβn 0 Eβn 0 Fβn 0 0 0 τβn 0 0 1 −λβnh −λβnh τβnh λβne (λβnh − 1)e τβne Mβn = ... −λβnh −λβnh τβnh −e −he e 2 −λβnh −λβnh 2 2 τβnh −2λβne 2λβn(1 − λβnh)e (τβn + λβn)e 2 −λβnh 2 −λβnh τβnh −2λβne −2λβnhe −(Γβn + 2λβnτβn)e
−τβn ηβn −ηβn 1 1 1 −τβnh ηβnh −ηβnh −τβne ηβne −ηβne ... −τβnh ηβnh −ηβnh e e e 2 2 −τβnh 2 2 ηβnh 2 2 −ηβnh (τβn + λβn)e (ηβn + λβn)e (ηβn + λβn)e −τβnh ηβnh −ηβnh (Γβn + 2λβnτβn)e −(Θβn + 2λβnηβn)e (Θβn + 2λβnηβn)e
with
6(sin(λβnδ) − λβnδ cos(λβnδ)) βn = 4 3 . λβnδ
122 4.4.1.6 Velocity Field Solution
In order to make the final form of the solution amenable to computation, it is useful ˆ ˆ ˆ ˆ ˆ ˆ to define the terms An, Bn, Cn, Dn, En and Fn as
ˆ −λβnh ˆ −λβnh ˆ τβnh Aβn = Aβne , Bβn = Bβne , Cβn = Cβne , (4.26) ˆ ˆ ηβnh ˆ Dβn = Dβn, Eβn = Eβne , Fβn = Fβn.
The coefficients defined by Eq. (4.26) can be calculated without the evaluation any positive exponentials (which become very large for high values of n). Explicit ˆ ˆ ˆ ˆ ˆ ˆ expressions for An, Bn, Cn, Dn, En and Fn are given in Appendix E.
As a result of the above analysis, the non-dimensional water velocity field compo- nents uβL, vβL, uβG and vβG (for cases where Day ≥ Dax) can be written in terms ˆ ˆ ˆ ˆ ˆ ˆ of An, Bn, Cn, Dn, En and Fn as
∞ ˆ X ˆ ˆ ˆ λβn(h−y) uβL = γ[y + Jβ] + sin(λβnx)[Bβn − λβn(Aβn + Bβny)]e n=1
∞ X ˆ ˆ λβn(h−y) vβL = −1 − λβn cos(λβnx)[Aβn + Bβny]e n=1
ˆ √ √ Pβ (y/ Dax) −(y/ Dax) uβG = γ √ [e − e ] Dax ∞ X ˆ τβn(y−h) ˆ −τβny ˆ ηβn(y−h) ˆ −ηβny + sin(λβnx)[τβnCβne − τβnDβne + ηβnEβne − ηβnFβne ] n=1
∞ X ˆ τβn(y−h) ˆ −τβny ˆ ηβn(y−h) ˆ −ηβny vβG = −1 − λβn cos(λβnx)[Cβne + Dβne + Eβne + Fβne ]. n=1
Note that the velocity field vβ depends on five parameters, namely the non-dimensional cleft half-width δ, the non-dimensional shear rate γ, the non-dimensional EGL height h, and the non-dimensional inverse Darcy permeabilities Dax and Day.
123 4.4.2 LDL Concentration Field
Solutions to Eq. (4.16) within ΩβC are obtained numerically using an in-house C/C++ implementation of the spectral/hp element method described in Chapter 2. Following the approach adopted in Chapter 3, a hybrid mesh consisting of both triangular and quadrangular elements is used to tessellate the domain. Once again the mesh is obtained using the commercial mesh generator Gambit 2.4.6 (ANSYS, Inc). An example mesh, used for cases where h = 0.05 and b = 0.05, is shown in Fig. 4.4(a). Smaller elements are grouped in regions where localised features of the solution are expected to develop as shown in Fig 4.4(b). Within each element the solution is represented using 2D modal basis functions, which are constructed from generalised tensor products of 1D polynomial bases of 9th order.
1.5 0.06
1 0.04 y y
0.5 0.02
0 0 -1 -0.5 0 0.5 1 -0.04 -0.02 0 0.02 0.04 x x
(a) (b)
Figure 4.4: An example computational mesh of the entire domain ΩβC (a). Smaller elements are concentrated near to the intercellular cleft and at the interface between the lumen and the EGL (b).
Note that when b = 0 (ΩβCG = 0) the LDL concentration field Cβ depends on six additional parameters, namely δ, γ, h, Dax, Day and P eL. When b 6= 0 (ΩβCG 6= 0),
Cβ depends on eight additional parameters, namely δ, γ, h, Dax, Day, P eL, Dr and f.
124 4.4.3 Parameter Values
4.4.3.1 Value for the Non-Dimensional Cleft Half-Width δ
Once again, values of ∆∗ = 10µm and δ∗ = 0.01µm are considered to be physiologi- cally realistic, resulting in a fixed value of δ = 0.001.
4.4.3.2 Values for the Non-Dimensional Shear Rate γ
Once again, shear rates in the range γ∗ = 0 − 1000s−1 are considered, along with ∗ an average transmural flux velocity of V = 4.39 × 10−2µms−1 [30]. Based on these values (and using ∆∗ = 10µm) a range of γ = 0 − 2.28 × 105 is used in the present model.
4.4.3.3 Values for Cleft Peclet Number P eL
∗ 2 −1 Once again, a value of DL = 28.67µm s is considered here to be physiologically realistic. This is equal to the diffusivity of LDL in blood plasma [70]. Based on
∗ ∗ ∗ −2 −1 this value for DL (and using ∆ = 10µm and V = 4.39 × 10 µms ), a value of
P eL = 0.015 is obtained. A higher value of P eL = 0.8 will be used occasionally to illustrate behavior of the solutions when P eL ∼ 1. However, as mentioned in
Chapter 3, this increased value for P eL is not deemed to be physiologically realistic.
4.4.3.4 Values for the Non-Dimensional EGL Height h
An EGL height range of h∗ = 0.2 − 1.0µm is considered here, which results in a range of h = 0.02 − 0.1 (assuming a fixed value of ∆∗ = 10µm). This range of h spans results from both ex-vivo electron microscope studies [43] and in-vivo studies [44, 45, 46, 47] of the EGL.
125 4.4.3.5 Values for the Non-Dimensional Darcy Permeabilities Dax and
Day
∗ −6 −4 2 EGL Darcy permeability ranges of Kx = 2 × 10 − 1 × 10 µm [38, 81] and ∗ −6 −4 2 Ky = 1 × 10 − 2 × 10 µm are considered here, resulting in ranges of Dax = −8 −6 −8 −6 2 × 10 − 1 × 10 and Day = 1 × 10 − 2 × 10 (assuming a fixed value of ∆∗ = 10µm) [82]. It should be noted that for a medium with a low solid volume
p ∗ p ∗ fraction such as the EGL, fibre spacing is of order Kx (and/or Ky ) . Therefore, √ p the ratios Dax/δ and Day/δ are approximate ratios of the fibre spacing to the cleft half width. For the highest values of Dax and Day considered in this study, √ p the associated values of Dax/δ and Day/δ are ∼ 1, hence the continuum view of the EGL, which is necessary in order to apply the Brinkman equation, begins to break down at the scale of the intercellular clefts.
4.4.3.6 Values for the Ratio of EGL to Lumen LDL Diffusivity Dr
∗ There have been no direct measurements of DG, however the diffusivity of LDL in a hyaluronate gel (2.5% w/v at 20oC) has been measured as 1µm2s−1 [83]. This
∗ ∗ 2 −1 value provides an initial estimate for DG. A range of DG = 0.2867 − 28.67µm s will be considered here, which results in a range of Dr = 0.01 − 1.0 when using ∗ 2 −1 ∗ 2 −1 DL = 28.67µm s . It can be noted that when using values of DG = 1µm s ∗ 2 −1 (measured diffusivity of LDL in a hyaluronate gel) and DL = 28.67µm s (the diffusivity of LDL in blood plasma) a value of Dr = 0.0349 is obtained.
4.4.3.7 Values for the Non-Dimensional Distance b that LDL Penetrates into the EGL
The parameter b determines the distance that LDL can penetrate into the EGL before it is rejected. The actual extent to which LDL penetrates into the EGL is not known. Investigations suggests that macromolecules smaller than LDL are excluded from the EGL [84], indicating that LDL itself may also be excluded. However, these
126 same investigations [84] note that the EGL does not behave as a purely size selective filter, and further studies suggest that macromolecules such as albumin penetrate into, and interact with, a layer on the luminal surface of the endothelium [85, 86, 87]. These observations mean that the possibility of some LDL penetration into the EGL cannot be discounted.
LDL penetration distances of b∗ = 0 − 0.5µm will be considered here, resulting in a range of b = 0 − 0.05 (assuming a fixed value of ∆∗ = 10µm). For cases where b 6= 0 it is necessary to define f ∗ (a measure of the distance taken to transition between the lumen diffusivity and the EGL diffusivity). A value of f ∗ = 0.02µm will be used here, resulting in a value of f = 0.002 (assuming a fixed value of ∆∗ = 10µm). The parameter f is chosen to be significantly less than all non-zero values of b that are investigated.
4.4.3.8 Value for the Non-Dimensional Height Lβ of ΩβC
For all cases a value of Lβ = 1.5 is used. This value is large enough to ensure that
Lβ > κβ is always satisfied.
4.5 Results and Analysis
4.5.1 Water Velocity Field
4.5.1.1 Streamlines
Fig. 4.5 shows streamlines of the water velocity field in the vicinity of an intercellular cleft for four values of the non-dimensional shear rate γ with fixed δ = 0.001,
−8 −8 h = 0.05, Dax = 2 × 10 and Day = 2 × 10 .
127 0.2 0.2
0.15 0.15 y y 0.1 0.1
0.05 0.05
0 0 -0.15 -0.075 0 0.075 0.15 -0.15 -0.075 0 0.075 0.15 x x
(a) (b)
0.2 0.2
0.15 0.15 y y 0.1 0.1
0.05 0.05
0 0 -0.15 -0.075 0 0.075 0.15 -0.15 -0.075 0 0.075 0.15 x x
(c) (d)
Figure 4.5: Velocity streamlines in the vicinity of an intercellular cleft obtained using γ = 0 (a), γ = 300 (b), γ = 3 × 103 (c) and γ = 3 × 104 (d) with fixed −8 −8 δ = 0.001, h = 0.05, Dax = 2 × 10 and Day = 2 × 10 . The dashed lines mark the interface between the lumen and the EGL.
√ For the parameter values considered in this study Dax/h < 0.05 in all cases. The effects of shear therefore never penetrate significantly into the EGL, and flow within the EGL is seen to be almost completely independent of γ. Above the EGL, perturbations to the velocity field caused by flow into each intercellular cleft protrude less far into the lumen as γ is increased.
128 4.5.1.2 Wall Normal Velocity
Fig. 4.6 shows plots of the wall normal velocity at the luminal surface of the EGL vβL(x, h) in the vicinity of an intercellular cleft entrance. Results for various values of the non-dimensional EGL height h and non-dimensional inverse Darcy permeability
−8 Day are presented, with fixed values of δ = 0.001 and Dax = 2 × 10 . The effect of applied shear is not considered since the wall normal velocity is independent of γ.
0 0
-10 -20 L L
! -20 !
v v -40
-8 h = 0.02 -30 Day = 1×10 -8 h = 0.04 Day = 1.3×10 -60 -8 h = 0.06 Day = 2×10 -8 h = 0.08 Day = 3×10 -40 -8 h = 0.1 Day = 4×10 -80 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 x x
(a) (b)
Figure 4.6: Plots of the wall normal velocity at the luminal surface of the EGL vβL(x, h) in the vicinity of an intercellular cleft entrance. Results for various values −8 of Day with fixed h = 0.05 (a), and various values of h with fixed Day = 2 × 10 −8 (b) are presented, all with fixed values of δ = 0.001 and Dax = 2 × 10 .
As h is decreased, or as Day is increased, the peak in wall normal velocity magnitude becomes increasingly localised above the intercellular cleft entrance. Note that in all cases the peak in the velocity profile remains localised above the cleft c.f. the cleft spacing. This result implies that the transmural water velocity profile is likely to remain heterogeneous at the luminal surface of the EGL.
129 4.5.2 LDL Concentration Field (No LDL Penetration into the EGL)
Figs. 4.7 and 4.8 show plots of the non-dimensional LDL concentration field Cβ evaluated at the luminal surface of the EGL (at y = h) for cases where LDL cannot penetrate into the EGL (b = 0). Fig. 4.7 illustrates the effect of varying the non- dimensional EGL height h for various values of the non-dimensional shear rate γ
−8 and Peclet number P eL with fixed δ = 0.001, Dax = Day = 2×10 and b = 0. Fig. 4.8 illustrates the effect of varying the non-dimensional inverse Darcy permeabilities
Dax and Day, for various values of non-dimensional shear rate γ and Peclet number
P eL with fixed δ = 0.001, h = 0.05 and b = 0.
130 1.035 h = 0.02 h = 0.02 1.015 h = 0.04 1.015 h = 0.04 1.03 1.03 h = 0.06 h = 0.06 h = 0.08 1.01 h = 0.08 1.025 h = 0.1 h = 0.1 1.01 1.005 1.02 1.02 h) h) 1
(x, -0.05 0 0.05 (x, -0.05 0 0.05 ! ! C C 1.01 1.005
1 1
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x x
(a) (b)
6 6 h = 0.02 1.3 1.3 h = 0.02 h = 0.04 h = 0.04 5 h = 0.06 h = 0.06 5 1.2 h = 0.08 h = 0.08 4 h = 0.1 h = 0.1 1.2 1.1 4 3
h) h) 1
(x, -0.05 0 0.05 (x, -0.05 0 0.05 ! 3 ! C C 1.1
2
1 1
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x x
(c) (d)
Figure 4.7: Plots illustrating how the LDL distribution adjacent to the luminal surface of the EGL varies with EGL height h for cases where LDL cannot penetrate 5 into the EGL. Values of γ = 0, P eL = 0.015 (a) γ = 2.28 × 10 , P eL = 0.015 (b) 5 γ = 0, P eL = 0.8 (c) and γ = 2.28 × 10 , P eL = 0.8 (d) are considered. For all cases −8 δ = 0.001, Dax = Day = 2 × 10 and b = 0. Note that a spatially homogeneous transmural water flux (with the same average velocity as the heterogeneous flux used here) would result in a constant non-dimensional concentration of unity at y = h.
Also note that the scale of the Cβ(x, h) axis varies between plots.
131 eed togyon strongly depends above (at localised clefts concentration LDL intercellular in ob- the peaks with is here, layer considered polarisation cases all concentration for LDL served the of heterogeneity scale Sub-cellular γ noteEL ausof Values EGL. the with into varies EGL the of surface 4.8: Figure δ u wt h aeaeaevlct stehtrgnosflxue ee would here) at used unity the flux of of heterogeneous concentration scale the the non-dimensional that constant as a velocity average in result same the (with flux 0 = 0, = C!(x,0.05) C!(x,0.05) 1. 1. 1. 01 02 03 . 001, 1 2 3 4 5 1 -1 -1 e P L h 0 = 1 1 1 1 1 1 1 1 1 1 ...... 0 =
0 0 C(x,0 20) 0 0 0 0 C(x,0 20) 0 0 3 3 3 3 3 3 3 3 3 3 5 6 7 8 9 5 6 7 8 9 lt lutaighwteLLdsrbto daett h luminal the to adjacent distribution LDL the how illustrating Plots -0. -0. . c and (c) 8 5 5 . 5and 05 (a) (c) D D D D γ C a a a a x x x x β and = = = = x x 0 0 ( b 2 2 2 2 x, × × × × γ γ .Nt htasailyhmgnostasua water transmural homogeneous spatially a that Note 0. = 10 10 10 10 x 0 e P 0, = 2 = - - - - 8 8 8 8 . , , , , 5 xsvre ewe plots. between varies axis 05) D D D D ) h tutr n ereo n heterogeneity any of degree and structure The 0). = L a a a a y y y y . 0. 0. nasmlrfsint htse nCatr3when 3 Chapter in seen that to fashion similar a in , Da 28 = = = = 5 5 e P 4 2 4 2 × × × × × x 10 10 10 10 L - - - - and 10 8 8 8 8 0 = 5 , 1 1 132 Da e P . 1 (a) 015 y L C!(x,0.05) 1. C!(x,0.05)1. 1. 1. 1. 004 008 012 o ae hr oLLcnpenetrate can LDL no where cases for 1 1 0 = 05 15 . . 1 1 1 2 -1 -1 . d r osdrd o l cases all For considered. are (d) 8 γ 2 = 1 1 1 1 1 1 1 1 1 1 ...... 0 0 0 0 0 0 0 C(x,0 20) 0 0 C(x,20) 3 3 3 3 3 3 3 3 3 3 5 6 7 8 9 5 6 7 8 9 -0. -0. . 28 5 5 × (b) (d) D D D D 10 a a a a x x x x = = = = x x 5 0 0 y 2 2 2 2 , × × × × e P 10 10 10 10 = - - - - 8 8 8 8 , , , , L D D D D h a a a a lonote Also . 0 = y y y y 0. 0. = = = = 5 5 4 2 4 2 × × × × 10 10 10 10 . 1 (b) 015 - - - - 8 8 8 8 1 1 no EGL is present. Specifically, as γ is raised the peaks in LDL concentration be- come skewed (swept downstream), and as P eL is raised the degree of heterogeneity becomes increasingly significant.
As h is decreased (Fig. 4.7) and as Day becomes greater than Dax (Fig. 4.8), the peaks in LDL concentration above the intercellular clefts become more prominent. Such variation occurs because, as shown in Section 4.5.1.2, the water velocity pro-
file vβL(x, h) at the luminal surface of the EGL becomes increasingly localised as h decreases and as Day increases relative to Dax. For each value of γ and P eL −7 considered in Fig. 4.8, results for cases where Dax = Day = 1 × 10 and cases −6 where Dax = Day = 1 × 10 are almost identical to the presented isotropic cases −8 −7 where Dax = Day = 2×10 . Similarly, results for cases where Dax = 1×10 and −7 −6 −6 Day = 2×10 and cases where Dax = 1×10 and Day = 2×10 are almost iden- −8 −8 tical to the presented anisotropic cases where Dax = 2 × 10 and Day = 4 × 10 . These additional results are therefore omitted for brevity. It is clear from Figs. 4.7 and 4.8 that (when b = 0 and hence LDL cannot penetrate into the EGL) the effects of varying h, Dax and Day on LDL concentration polarisation are far more pronounced for the cases where P eL = 0.8 than for the more physiologically realistic cases where P eL = 0.015. This is to be expected since h, Dax and Day affect only the convective water flux, and for the physiologically realistic cases where P eL 1 transport is dominated by diffusion.
4.5.3 Transendothelial LDL Transport (No LDL Penetra- tion into the EGL)
4.5.3.1 Definitions
Evidence suggests that cationised ferritin (used as a marker of the EGL) enters vesicles on the endothelial cell surface [85]. The implication of this result is that vesicles are able to ingest the EGL, along with any LDL residing within it or on its luminal surface. Exclusion of LDL from the EGL does not, therefore, necessarily
133 preclude transcellular LDL transport from occurring. Hence, even when LDL is fully or partially rejected from the EGL, it is still reasonable to define measures of LDL concentration polarisation that may be relevant to the rate of transendothelial LDL transport.
Following the arguments adopted in Chapter 3, two such measured are considered in the present model. The first is simply the average non-dimensional LDL concen- tration adjacent to the point of LDL rejection from the endothelial surface. This is denoted CβU and defined as
1 Z 1 CβU = Cβ(x, h − b) dx. 2 −1
The second measure is the average non-dimensional LDL concentration adjacent to the point of LDL rejection from the endothelial surface within a (non-dimensional) distance of 1/10 from each cleft centre. This is denoted CβN and defined as
1 Z 1/10 CβN = Cβ(x, h − b) dx. 2(1/10) −1/10
The above definitions of CβU and CβN contain the parameter b, which is zero for the first set of cases considered here (where LDL cannot penetrate into the EGL). The parameter b is included to add generality to the definitions so they may be used for cases where b 6= 0 (which will be considered shortly).
It is important to note that both CβU and CβN are obtained from LDL concen- ∗ tration fields that have been non-dimensionalised by CE. Therefore, the measures are independent of the effect of vascular scale flow features on the macroscale LDL concentration polarisation layer. The measures only reflect additional (possibly flow dependent) modifications to the LDL concentration polarisation layer caused by the heterogeneous transmural water flux and the EGL.
134 4.5.3.2 Dependence of CβU and CβN on Applied Shear γ
For physiologically realistic cases where the Peclet number P eL = 0.015, CαU varies by at most only 0.0060% (2 s.f.) and CαN varies by at most only 2.1% (2 s.f.) as 5 γ varies between 0 and 5 × 10 . Only when P eL ∼ 1 do CβU and CβN develop a significant shear dependence. Such behavior is similar to that observed in Chapter 3 when no EGL is present.
4.5.3.3 Dependence of CβU and CβN on EGL Properties h, Dax and Day
For cases where the Peclet number P eL = 0.8, calculated values of CβU and CβN vary significantly as the non-dimensional EGL height h, and the non-dimensional inverse Darcy permeabilities Dax and Day are varied. Specifically, for fixed values of
δ = 0.001, γ = 0 and P eL = 0.8, ranges of CβU = 1.11 − 1.22 and CβN = 2.34 − 3.60 are obtained as h, Dax and Day are varied in the ranges h = 0.02 − 0.1, Dax = −8 −6 2 × 10 − 1 × 10 , Day = Dax − 2Dax. For the physiologically realistic cases where P eL = 0.015, however, varying h, Dax and Day has almost no effect on either
CβU or CβN . Specifically, for fixed values of δ = 0.001, γ = 0 and P eL = 0.015, ranges of CβU = 1.00004 − 1.00006 and CβN = 1.017 − 1.021 are obtained as h, −8 −6 Dax and Day are varied in the ranges h = 0.02 − 0.1, Dax = 2 × 10 − 1 × 10 ,
Day = Dax − 2Dax. These results show that for cases where an EGL exists but
LDL does not penetrate into it, there is likely to be almost no dependence of CβU and CβN on the structural properties of the EGL via the mechanisms considered here.
4.5.4 LDL Concentration Field (LDL Penetration into the EGL)
Fig. 4.9 shows contour plots of the non-dimensional LDL concentration field Cβ for cases where LDL can penetrate into the EGL (b 6= 0). Various values of non-
135 dimensional shear rate γ and non-dimensional EGL height h are considered, with
−8 fixed δ = 0.001, P eL = 0.015, Dax = Day = 2 × 10 , b = 0.05, Dr = 0.0349 and f = 0.002. It can be seen that the concentration field Cβ is heterogeneous at the cellular scale. The heterogeneity is similar in nature to that which develops when LDL cannot penetrate into the EGL, with peaks in LDL concentration localised above intercellular clefts. However, for cases of LDL penetration, the peaks occur predominantly within the EGL (as opposed to in the lumen). The contour plots in Fig. 4.9 also show that the LDL distribution within the EGL is almost completely independent of the non-dimensional shear rate γ (when it is varied over a physiolog- ical range). This is because the water velocity field vβG within the EGL is shielded from the applied shear, as illustrated in Fig. 4.5.
136 0.12 0.12 1.011
1.013 0.999
0.08 1.016 0.08 1.003 1.021 y y 1.024
1.070 0.04 0.04 1.024 1.070 1.206 1.206 1.409 1.409
2.093 2.093 0 0 -0.08 -0.04 0 0.04 0.08 -0.08 -0.04 0 0.04 0.08 x x 0.6 0.6 0.991 0.992 0.4 0.993 0.4 0.995
y 0.998 y 0.2 0.2 1.010 0.997 0.999 0 0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x x
(a) (b)
0.24 0.24 1.005 0.997 1.008 0.998 0.16 1.011 0.16 0.999
1.002 y 1.017 y
0.08 1.082 0.08 1.082 1.216 1.216
0 0 -0.16 -0.08 0 0.08 0.16 -0.16 -0.08 0 0.08 0.16 x x 0.6 0.6 0.991 0.993 0.993 0.4 0.998 0.4
y y 0.996 0.2 0.2 0.998 1.010 0.998 1.010 0 0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x x
(c) (d)
Figure 4.9: Contour plots of Cβ obtained using values of γ = 0 and h = 0.05 (a), γ = 2.28 × 105 and h = 0.05 (b), γ = 0 and h = 0.1 (c), γ = 2.28 × 105 −8 and h = 0.1 (d) all with fixed δ = 0.001, P eL = 0.015, Dax = Day = 2 × 10 , b = 0.05, Dr = 0.0349 and f = 0.002. The upper plot within each subfigure is an enlargement of the region in the dotted box marked on the lower plot within each corresponding subfigure. The dashed lines mark the interface between the EGL and the lumen. The solid black shading denotes regions of the EGL into which LDL cannot penetrate.
137 Figs. 4.10 and 4.11 show plots of the non-dimensional LDL concentration field Cβ evaluated at a distance h − b from the endothelial cell surface, for cases where LDL can penetrate a distance b into the EGL (i.e. evaluated at the point of LDL rejection within the EGL). Fig. 4.10 shows results for cases where the non-dimensional EGL height h = 0.05. Fig. 4.11 shows results for cases where the non-dimensional EGL height h = 0.1. Various values of the non-dimensional LDL penetration distance b and the EGL to lumen diffusivity ratio Dr are considered, with fixed values of −8 δ = 0.001, γ = 0, P eL = 0.015, Dax = Day = 2 × 10 and f = 0.002. Only results from cases where the applied shear γ = 0 are presented since the concentration profiles at y = h − b are almost completely independent of γ.
As the non-dimensional LDL penetration distance b increases and as the EGL to lumen diffusivity ratio Dr decreases, the peaks in LDL concentration above the inter- cellular clefts become more prominent. Enhancement of the peaks with decreasing
Dr is to be expected, since the peaks form predominantly within the EGL, and low- ering Dr raises the ratio P eG = P eL/Dr (where P eG is a Peclet Number analogous to P eL but based on the diffusivity of LDL in the EGL). Enhancement of the peaks with increasing b is also expected, for two reasons. Firstly, raising b increases the localisation of the water velocity profile vβG(x, h − b) at the point of LDL rejection, and secondly raising b increases the region of retarded diffusivity (if Dr < 1) into which LDL can penetrate.
138 b = 0.01 b = 0.01 b = 0.02 b = 0.02 b = 0.03 b = 0.03 1.06 b = 0.04 4 b = 0.04 b = 0.05 b = 0.05
05-b) 05-b) 3 0. 1.04 0. (x, (x, ! ! C C
2
1.02
1 -0.1 -0.05 0 0.05 0.1 -0.1 -0.05 0 0.05 0.1 x x
(a) (b)
b = 0.01 b = 0.02 b = 0.03 150 b = 0.04 b = 0.05
05-b) 100 0. (x, ! C
50
0 -0.1 -0.05 0 0.05 0.1 x
(c)
Figure 4.10: Plots illustrating how the LDL distribution evaluated at y = 0.05 − b varies with b for cases where LDL can penetrate a distance b into an EGL of height h = 0.05. Values of Dr = 1.0 (a), Dr = 0.0349 (b) and Dr = 0.01 (c) are considered, −8 with fixed values of δ = 0.001, γ = 0, P eL = 0.015, Dax = Day = 2 × 10 and f = 0.002.
139 b = 0.01 1.3 b = 0.01 b = 0.02 b = 0.02 1.02 b = 0.03 b = 0.03 b = 0.04 b = 0.04 b = 0.05 b = 0.05 1.2 1-b) 1-b) 0. 1.01 0. (x, (x, ! ! C C 1.1
1 1 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 x x
(a) (b)
b = 0.01 b = 0.02 b = 0.03 b = 0.04 2 b = 0.05 1-b) 0. (x, ! 1.5 C
1 -0.4 -0.2 0 0.2 0.4 x
(c)
Figure 4.11: Plots illustrating how the LDL distribution evaluated at y = 0.1 − b varies with b for cases where LDL can penetrate a distance b into an EGL of height h = 0.1. Values of Dr = 1.0 (a), Dr = 0.0349 (b) and Dr = 0.01 (c) are considered, −8 with fixed values of δ = 0.001, γ = 0, P eL = 0.015, Dax = Day = 2 × 10 and f = 0.002.
140 4.5.5 Transendothelial LDL Transport (LDL Penetration into the EGL)
4.5.5.1 Dependence of CβU and CβN on Applied Shear γ
For cases where an EGL is present but LDL cannot penetrate into it, and cases where no EGL is present, it has been shown that measures such as CβU and CβN only develop a shear dependence if the Peclet number P eL ∼ 1. Therefore, for cases where LDL can penetrate into the EGL, one may expect CβU and CβN to develop a shear dependence when P eG = P eL/Dr ∼ 1. Values of P eG = 0.015 − 1.5 are used in this study. However, even when P eG ∼ 1, both CβU and CβN remain almost completely independent of γ. Such shear independence occurs because the concentration profiles at y = h − b, from which CβU and CβN are calculated, are shielded from applied shear by the EGL.
4.5.5.2 Dependence of CβU and CβN on LDL Interaction with the EGL
(h, b and Dr)
Three parameters characterise the interaction of LDL with the EGL, namely the non-dimensional EGL height h, the non-dimensional distance that LDL penetrates into the EGL b, and the EGL to lumen diffusivity ratio Dr. Figs. 4.12 and 4.13 show plots of CβU and CβN against b. Fig. 4.12 shows results for cases where h = 0.05.
Fig. 4.13 shows results for cases where h = 0.1. Various values of Dr are considered, −8 with fixed values of δ = 0.001, γ = 0, P eL = 0.015, Dax = Day = 2 × 10 and f = 0.002. For all cases, CβU and CβN increase as b increases and as Dr decreases.
141 Dr = 0.01 1.3 Dr = 0.0349 Dr = 1.0
1.2 U ! C
1.1
1 0.01 0.02 0.03 0.04 0.05 b
(a)
Dr = 0.01 D = 0.0349 4 r Dr = 1.0
3 N ! C
2
1 0.01 0.02 0.03 0.04 0.05 b
(b)
Figure 4.12: Plots of CβU (a) and CβN (b) against b for the case where LDL can penetrate a distance b into an EGL of height h = 0.05. Various values of Dr are considered, with fixed values of δ = 0.001, γ = 0, P eL = 0.015, Dax = Day = 2 × 10−8 and f = 0.002.
142 Dr = 0.01
Dr = 0.0349 D = 1.0 1.08 r U ! C 1.04
1 0.01 0.02 0.03 0.04 0.05 b
(a)
1.8 Dr = 0.01
Dr = 0.0349
Dr = 1.0 N ! 1.4 C
1 0.01 0.02 0.03 0.04 0.05 b
(b)
Figure 4.13: Plots of CβU (a) and CβN (b) against b for the case where LDL can penetrate a distance b into an EGL of height h = 0.1. Various values of Dr are considered, with fixed values of δ = 0.001, γ = 0, P eL = 0.015, Dax = Day = 2 × 10−8 and f = 0.002.
Table 4.1 shows the ranges over which CβU and CβN vary as the non-dimensional LDL penetration distance is varied within the range b = 0.01 − 0.05. Various values
143 of the non-dimensional EGL height h and the EGL to lumen diffusivity ratio Dr are considered, with fixed values of δ = 0.001, γ = 0, P eL = 0.015, Dax = Day = 2 × 10−8 and β = 0.002.
The extent to which CβU and CβN vary with the non-dimensional LDL penetration distance b depends significantly on the EGL to lumen diffusivity ratio Dr. When using a value of Dr = 1.0, variations of CβU and CβN with b are negligible. When using a value of Dr = 0.0349, variations of CβU with b are relatively small. However, variations of CβN with b are of the same order as variations in LDL concentration polarisation (and hence it is assumed transendothelial LDL transport) caused by vascular scale flow features (typically ∼ 20%) [28]. Finally, when using a value of
Dr = 0.01, variations of both CβU and CβN with b are of the same order as (or greater than) variations in LDL concentration polarisation caused by vascular scale flow features [28].
It has been suggested that blood flow patterns may be able to control properties of the EGL, such as its hyaluronate content [88], and the thickness of the well adhered macromolecular layer at its abluminal base [43]. The above results indicate that such flow dependent regulation of EGL properties could be as important as vascular scale flow features in terms of spatially modulating transendothelial LDL transport via an LDL concentration polarisation mechanism.
Dr = 0.01 Dr = 0.0349 Dr = 1.0
CβU = 1.009 − 1.328 CβU = 1.003 − 1.027 CβU = 1.000 − 1.000 h = 0.05 CβN = 1.105 − 4.284 CβN = 1.049 − 1.288 CβN = 1.020 − 1.021
CβU = 1.009 − 1.091 CβU = 1.003 − 1.023 CβU = 1.000 − 1.000 h = 0.1 CβN = 1.082 − 1.764 CβN = 1.039 − 1.190 CβN = 1.017 − 1.018
Table 4.1: Ranges of CβU and CβN as the LDL penetration distance varies over the range b = 0.01 − 0.05. Various values of h and Dr are considered, with fixed values −8 −8 of δ = 0.001, γ = 0, P eL = 0.015, Dax = 2 × 10 , Day = 2 × 10 and f = 0.002.
144 4.6 Conclusions
The results indicate that for cases where LDL cannot penetrate into the EGL, spa- tially heterogeneous and shear dependent perturbations to the LDL concentration polarisation layer develop on the luminal surface of the EGL. For physiologically realistic cases where P eL = 0.015, however, only small perturbations are predicted to occur, and measures relevant to the rate of transendothelial LDL transport CβU and CβN are almost completely independent of applied shear γ. These results are similar to those obtained when no EGL is present.
For cases where LDL can penetrate into the EGL, results indicate that spatially heterogeneous perturbations to the LDL concentration polarisation layer will develop within the EGL. It is observed that such perturbations are shielded from the flowing blood, hence measures relevant to the rate of transendothelial LDL transport CβU and CβN are once again predicted to be independent of applied shear γ (at least in a direct sense). However, it is also observed that the measures CβU and CβN depend significantly on the nature of interaction between LDL with the EGL, which is parameterised by Dr (the ratio of LDL diffusivity in the EGL to LDL diffusivity in the lumen), b (the non-dimensional distance that LDL penetrates into the EGL) and h (the non-dimensional EGL height). Specifically, as Dr, b and h are varied within reasonable bounds, CβU and CβN vary in the ranges CβU = 1.000−1.328 and
CβN = 1.017 − 4.284.
Various processes may regulate the interaction of LDL with the EGL, possibly in a flow dependent and hence spatially non-uniform fashion. It is concluded that any such processes may be as important as vascular scale flow features in terms of spatially modulating transendothelial LDL transport via an LDL concentration polarisation mechanism.
145 Chapter 5
Conclusions and Future Research
5.1 Conclusions
In this thesis the effects of cellular scale endothelial features on LDL concentration polarisation have been assessed using an idealised theoretical model. In particular the effect of a spatially heterogeneous transmural water flux has been investigated (flowing through intercellular clefts only) along with the effect of an EGL. A range of scenarios have been considered, including cases were no EGL is present, cases where an EGL is present but LDL cannot penetrate into it, and finally cases where an EGL is present and LDL can penetrate into it.
For cases where no EGL is present, measures of LDL concentration relevant to the rate of transendothelial LDL transport have been defined as CαU and CαN (where
CαU is the spatially averaged non-dimensional LDL concentration to which the en- dothelium is exposed and CαN is the spatially averaged non-dimensional LDL con- centration to which the endothelium is exposed within the vicinity of the intercel- lular cleft entrances). It has been demonstrated, in principle, that a heterogeneous transmural water flux can act to enhance such measures, and cause them to develop a shear dependence (in addition to that caused by vascular scale flow features af- fecting the overall degree of LDL concentration polarisation). However, it has also been shown that this enhancement and additional shear dependence are likely to be negligible for physiologically realistic values of the Peclet number P eL.
For cases where an EGL is present, measures of LDL concentration relevant to the rate of transendothelial LDL transport have been defined as CβU and CβN (where
CβU is the spatially averaged non-dimensional LDL concentration to which the en- dothelium is exposed and CβN is the spatially averaged non-dimensional LDL con- centration to which the endothelium is exposed within the vicinity of the intercellular cleft entrances). It has been demonstrated that the presence of an EGL is unlikely to (directly) cause any additional shear dependence of such measures, irrespective of whether or not LDL can penetrate into the EGL. However, it has been found that both CβU and CβN depend significantly on the nature of the interaction between LDL and the EGL, which is parameterised by the height of the EGL, the depth to which LDL penetrates into the EGL, and the diffusivity of LDL in the EGL.
Various processes may regulate the interaction of LDL with the EGL, possibly in a flow dependent and hence spatially non-uniform fashion. It is concluded that any such processes may be as important as vascular scale flow features in terms of spatially modulating transendothelial LDL transport via an LDL concentration polarisation mechanism.
5.2 Future Research
5.2.1 Future Theoretical Research
This thesis offers significant insight into various aspects of arterial LDL concentra- tion polarisation at the cellular scale. However, there still remains much scope for future theoretical studies. Specifically, the idealised model presented in this thesis could be extended in the following ways:
• Model a third dimension running along the lengthwise extent of the idealised
147 clefts, so that the effect of heterogeneity in the transmural flux due to tight junction strands can be fully assessed.
• Model the depth of the intercellular clefts in a similar fashion to Phillips et al. [89] or Zeng et al. [90] so that the effect of tight junction strands on the cleft entrance velocity profile can be explicitly calculated.
• Model leaky intercellular clefts through which LDL can readily pass in a similar fashion to Tzeghai et al. [91]. Such an alteration necessitates modelling flow and mass transport within a region adjacent to the abluminal surface of the endothelium. However, it would allow one to assess the effect of leaky clefts on LDL concentration polarisation, as well as the effect of LDL concentration polarisation on paracellular LDL transport.
• The current model of the EGL is very simple, primarily due to a lack of detailed information about its structure, and how it is thought to interact with water and LDL. Studies presented in this thesis suggest that the nature of the interaction between LDL and the EGL may play an important role in regulating the degree of LDL concentration polarisation. It is therefore desirable to develop a more detailed (and hence realistic) model of the EGL, possibly incorporating non-continuum dynamics.
In addition to improving the model of LDL concentration polarisation presented in this thesis, it would also be interesting to investigate various physical phenom- ena revealed by the present analysis. The following phenomena warrant particular attention:
• The mechanism by which a heterogeneous transmural water flux causes an
overall increase in concentration polarisation when P eL ∼ 1.
• The time dependent dynamics of the system described in Appendix C when
P eL ∼ 1.
148 5.2.2 Future Experimental Research
The theoretical analysis presented in this thesis has identified several unknowns that may determine the degree and distribution of LDL concentration polarisation within arteries. The most critical of these unknowns are related to how LDL inter- acts with the EGL (in particular the depth to which LDL penetrates into the EGL and the diffusivity of LDL within the EGL). Experimentally obtaining values for these unknowns in-vivo currently poses a significant challenge. However, attempts to ascertain such values for the interaction of albumin with the EGL have been successfully undertaken in-vitro using Fluorescence Correlation Spectroscopy [87]. Extension of such experiments to investigate the interaction of LDL with the EGL could potentially determine values for the critical unknowns identified in the pre- sented theoretical analysis (i.e. the depth to which LDL penetrates into the EGL and the diffusivity of LDL within the EGL).
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159 Appendix A
Applicability
It has been assumed a priori that ζ > κα. Consider defining
Z 1 1/2 1 2 Eκα (y) = Cα(x, y) − Cα(y) dx 2 −1
as a measure of the y dependent deviation of Cα from the non-dimensional macroscale concentration polarisation layer Cα produced by a homogeneous transmural flux.
For a given non-dimensional concentration distribution Cα, κα can be quantitatively defined as the value of y such that
Eκα (y) = e,
where e is a tolerance taken here to be e = 0.005. The value of κα obtained for a given Cα is equal to the non-dimensional thickness ζ of the thinnest macroscale concentration polarisation layer to which the model can be applied. Employing Eq. (1.10) one can define a minimum LDL concentration adjacent to the endothelium,
∗ CEm, associated with this minimum layer thickness as
∗ ∗ P eLκα CEm = CBe . ∗ The value of κα and hence CEm will vary with δ, γ and P eL. Fig. A.1 shows plots ∗ ∗ of CEm/CB against γ for values of δ = 0.001 and P eL = 0.08. This particular value ∗ ∗ of P eL allows comparisons to be made with plots of CE/CB against wall shear stress produced by Wada et al. [28] for an artery with multiple bends.
When considering the dimensional parameter values used by Wada et al. [28] (along with a fixed value of ∆ = 10µm), the upper non-dimensional bound of γ in Fig. A.1 is equivalent to a dimensional shear stress of 7Pa. With this in mind, comparisons can be made between Fig. A.1 and the results of Wada et al. [28]. Such comparisons
∗ ∗ reveal that almost all values of CE/CB obtained for a given shear stress by Wada ∗ ∗ ∗ ∗ et al. [28] sit above the line plotted in Fig. A.1. Therefore CEm/CB < CE/CB for the majority of points adjacent to the luminal surface the arterial wall. This result indicates that the applicability of the model is not significantly limited by the a priori assumption that κα < ζ (i.e. the assumption that the macroscale con- centration polarisation layer is not completely destroyed by spatially heterogeneous modifications).
1.05
1.04
1.03 * B C / m * E
C 1.02
1.01
1 0 2×105 4×105 !
∗ ∗ Figure A.1: Plot of CEm/CB against γ with fixed δ = 0.001 and P eL = 0.08.
161 Appendix B
Convergence
Consider representing Cα within each element of the domain ΩαC using 2D modal basis functions generated from generalised tensor products of M th order 1D bases.
Based on the relevant measure of interest CαN , an M dependent measure of error
EM in the solution Cα can be defined as
|CαN (M) − CαN (14)| EM = , CαN (14)
where CαN (M) is the value of CαN obtained from a simulation using a 2D basis th within each element generated from M order 1D bases, and CαN (14) is the value of CαN obtained from a simulation using a 2D basis within each element generated from 14th order 1D bases (viewed as a definitive solution).
Convergence of EM with increasing M (for a given mesh) is assessed for cases where
P eL = 0.8. Within the range P eL = 0.002 − 0.8 such cases can be viewed as the most demanding from a convergence standpoint since they produce the most spatially heterogeneous solutions. Fig. B.1 shows plots of EM against M for various values of γ with fixed δ = 0.001 and P eL = 0.8. Note that when using P eL = 0.8 the maximum value of γ attainable given the dimensional parameter ranges is 1.25×105.
For all values of γ the error EM is seen to converge approximately exponentially with −4 M. For the value of M = 9 used here the relative error E9 < 1 × 10 for all values of γ.
! = 0 ! = 3×103 ! = 3×104 ! = 7.5×104 ! = 1.25×105
10-4 M E
10-5
4 6 8 10 12 M
Figure B.1: Plots of EM against M for various values of γ with fixed δ = 0.001 and P eL = 0.8.
163 Appendix C
Time Dependent Simulations
C.1 Water Velocity Field
Water flow within the cellular scale domain ΩαC is dominated by viscous forces.
Hence momentum transport within ΩαC is a predominantly diffusional process.
Timescales for momentum diffusion within ΩαC are found to be far less than the period at which blood flow pulsates (which is of order one second). Therefore, any time dependent water motion within ΩαC (caused by pulsating blood flow) can be considered highly quasi-steady.
Due to the quasi-steady nature of water flow within ΩαC an approximate time de- ∗ pendent velocity field vαt within ΩαC can be obtained simply by replacing the steady ∗ ∗ state shear rate γ (in an expression for the steady state velocity field vα) with a ∗ time dependent shear rate γt . Such an approach is adopted in this study.
∗ Here it will be assumed that γt has the following form
∗ ∗ ∗ ∗ ∗ γt = γo + γa sin(ω t ),
∗ ∗ ∗ where t is the dimensional time, γo is a constant dimensional offset, γa is a constant dimensional amplitude, and ω∗ is a constant dimensional frequency. Defining
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ t V γo ∆ γa∆ ω ∆ t = , γo = ∗ , γa = ∗ , ω = ∗ , (C.1) ∆∗ V V V one can write
∗ ∗ γt ∆ γt = ∗ = γo + γa sin(ωt). V
∗ ∗ Hence, the x and y components of the non-dimensional velocity field vαt = vαt/V
(denoted uαt and vαt respectively) can be written as
∞ X 2 −λαny uαt = [γo + γa sin(ωt)]y − Aαnλαn sin(λαnx)ye , n=1
vαt = vα.
C.2 LDL Concentration Field
∗ It is assumed that the time dependent LDL concentration field Cαt within ΩαC is a solution of the following time dependent advection diffusion equation
∂C∗ v∗ · ∇∗C∗ − D∗ ∇∗2C∗ = αt . (C.2) αt αt L αt ∂t∗
Defining
∗ Cαt Cαt = ∗ CE and using relevant relations from Eq. (3.3), Eq. (3.4) and Eq. (C.1), Eq. (C.2) can be non-dimensionalised to give
∂C P e v · ∇C − ∇2C = P e αt . (C.3) L αt αt αt L ∂t 165 In line with the steady state analysis presented in Chapter 3, the following boundary conditions are applied to Eq. (C.3)
Cαt(−1, y, t) = Cαt(1, y, t), (C.4)
∂C ∂C αt (−1, y, t) = αt (1, y, t), (C.5) ∂x ∂x ∂C αt (x, 0, t) = P e v (x, 0, t)C (x, 0, t), (C.6) ∂y L αt αt
Cαt(x, Lα, t) = Cα(Lα), (C.7)
where Lα can be chosen arbitrarily but must satisfy Lα > κ.
Eq. (C.3), subject to the boundary conditions defined by Eq. (C.4), Eq. (C.5), Eq. (C.6) and Eq. (C.7), is solved using the spectral/hp element method described in Chapter 2. The non-dimensional time t is advanced until the non-dimensional solution Cαt becomes temporally periodic.
C.3 Transendothelial LDL Transport
Temporally averaged measures of LDL concentration polarisation that may be rel- evant to the rate of transendothelial LDL transport can be defined as
1 Z T Z 1 CαtU = Cαt(x, 0, t) dxdt, 2T 0 −1
1 Z T Z 1/10 CαtN = Cαt(x, 0, t) dxdt, 2(1/10)T 0 −1/10 where T is the non-dimensional period of the applied shear, defined as
2π T = . ω
Such measures are analogous to the steady state measures CαU and CαN .
166 C.4 Results
Values of P eL = 0.015 and P eL = 0.8 were used in all studies, with fixed δ =
0.001. The effect of various non-dimensional applied shear rates γt was investigated, including those with a finite mean flow in the y direction (γo 6= 0) as well as those with zero mean flow in the y direction (γo = 0).
Results indicate that the LDL concentration field solutions exhibit additional time dependent dynamics i.e. the LDL concentration field is not quasi-steady when exposed to oscillatory shear. However, the measures CαtU and CαtN are observed to be almost exactly unity when P eL = 0.015 (which is considered here to be physiologically realistic). Such a result implies that the main findings of the steady state analysis presented in Chapter 3 are not affected by the application of a time- dependent shear rate to ΩαC . When P eL = 0.8, both CαtU and CαtN are seen to increase above unity, and become dependent on the form of γt. However, as has been previously noted, a value of P eL = 0.8 is unlikely to occur physiologically.
167 Appendix D
Trial Solution for ψβG when
Dax > Day
The analysis presented in Section 4.4.1 can be extended to consider cases where
Dax > Day (i.e. cases where the porous layer is more permeable in the wall parallel direction than in the wall normal direction). For these cases it becomes evident that Eq. (4.22) will no longer have four independent real roots for all values of λβn.
Specifically, when Dax > Day there are three possible scenarios:
• The first scenario occurs if
2 0 λβn < Da
where Da Da0 = y . 4Dax(Dax − Day)
00 In this case χβn > 0 and hence χβn has four distinct real values given by
r 0 q 00 χβn = ± χβn ± χβn. • The second scenario occurs if
2 0 λβn = Da
which is only possible if √ + Da0/π ∈ Z .
00 In this case χβn = 0 and there exists only two distinct values of χβn given by
q 0 χβn = ± χβn.
• The third scenario occurs if
2 0 λβn > Da .
00 In this case χβn < 0 and hence χβn has four distinct complex values (two complex conjugate pairs) given by
r r ! 1 q q χ = ± 2 χ0 2 + |χ00 | + 2χ0 ± i 2 χ0 2 + |χ00 | − 2χ0 . βn 2 βn βn βn βn βn βn
To account for variation in the form of the above roots when Dax > Day, one can define the coefficients τ βn and ηβn as
r q χ0 − χ00 λ2 < Da0 βn βn βn q 0 2 0 τ βn = χβn λβn = Da r 1 q 2 2 0 2 χ0 + |χ00 | − 2χ0 λ > Da 2 βn βn βn βn
r q χ0 + χ00 λ2 < Da0 βn βn βn η = βn r q 1 0 2 00 0 2 0 2 χβn + |χβn| + 2χβn λβ > Da . 2 n
169 After due consideration, a real form for ψβG that is suitable for cases where Dax >
Day can be written as
√ 0 + ψβG0 + ψβG1 + ψβG3 Da /π∈ / Z ψβG = √ 0 + ψβG0 + ψβG1 + ψβG2 + ψβG3 Da /π ∈ Z where
√ √ (y/ Dax) −(y/ Dax) ψβG0 = Kβx + Pβe + Qβe
n00 X τ βny −τ βny ηβny −ηβny ψβG1 = sin(λβnx)[Cβne + Dβne + Eβne + Fβne ] n=1
τ 0 y τ 0 y −τ 0 y −τ 0 y ψβG2 = sin(λβn0 x)[Cβn0 e βn + Dβn0 ye βn + Eβn0 e βn + Fβn0 ye βn ]
∞ X ηβny −ηβny ψβG3 = 2sin(λβnx)[sin(τ βny)(Cβne + Dβne )+ n=n0+1
ηβny −ηβny cos(τ βny)(Eβne + Fβne )] with √ n0 = b Da0/πc
√ n0 Da0/π∈ / + 00 Z n = √ n0 − 1 Da0/π ∈ Z+
and where Kβ, Pβ, Qβ, and Cβn, Dβn, Eβn and Fβn (which can depend on n), are all real coefficients to be determined.
170 Appendix E
Explicit Expressions for ˆ ˆ ˆ ˆ ˆ ˆ Aβn, Bβn, Cβn, Dβn, Eβn and Fβn
−τβn h −ηβn h −h(τβn+2 ηβn) −h(2 τβn+ηβn) βn −Ξn1e + Ξn2e + Ξn3e − Ξn4e ˆ Aβn = −2 Πn
−τβn h −ηβn h −h(τβn+2 ηβn) −h(2 τβn+ηβn) βn Ξn10e − Ξn11e − Ξn12e + Ξn13e ˆ Bβn = 2 Πn
−τβn h −ηβn h −h(τβn+2 ηβn) βn Ξn14e − 2 Ξn15e − Ξn16e ˆ Cβn = − Πn
−2 ηβn h −h(τβn+ηβn) βn Ξn17e − 2 Ξn15e − Ξn18 ˆ Dβn = Πn
−τβn h −ηβn h −h(2 τβn+ηβn) βn 2 Ξn19e − Ξn20e − Ξn21e ˆ Eβn = Πn
−2 τβn h −h(τβn+ηβn) βn −Ξn22e + 2 Ξn19e − Ξn23 ˆ Fβn = Πn where
−2 τβn h −2 ηβn h −2 h(τβn+ηβn) −h(τβn+ηβn) Πn = −Ξn5e − Ξn6e + Ξn7e + 4 Ξn8e + Ξn9
with
2 Ξn1 = ξn1λβn + ξn2λβn − ξn3
2 Ξn2 = ξn4λβn − ξn5λβn − ξn6
2 Ξn3 = ξn1λβn − ξn7λβn + ξn8
2 Ξn4 = ξn4λβn + ξn9λβn − ξn10
3 2 Ξn5 = 2 ξn11λβn + ξn12λβn − 2 ξn13λβn − ξn14
3 2 Ξn6 = 2 ξn11λβn − ξn15λβn − 2 ξn16λβn − ξn14
3 2 Ξn7 = 2 ξn17λβn − ξn18λβn + 2 ξn19λβn + ξn20
3 2 Ξn8 = 4 ξn21λβn − ξn22λβn − 2 ξn23λβn − ξn24
3 2 Ξn9 = 2 ξn17λβn + ξn25λβn + 2 ξn26λβn + ξn20
2 Ξn10 = ξn27λβn + ξn28λβn − ξn29
172 2 Ξn11 = ξn30λβn + ξn31λβn − ξn32
2 Ξn12 = ξn27λβn − ξn28λβn − ξn29
2 Ξn13 = ξn30λβn − ξn31λβn − ξn32
3 2 Ξn14 = 2 ξn33λβn + ξn34λβn − 2 ξn35λβn − ξn36
3 2 Ξn15 = 2 ξn21λβn − ξn37λβn − 2 ξn38λβn − ξn39
3 2 Ξn16 = 2 ξn40λβn − ξn41λβn + 2 ξn42λβn + ξn43
3 2 Ξn17 = 2 ξn33λβn − ξn44λβn − 2 ξn45λβn − ξn36
3 2 Ξn18 = 2 ξn40λβn + ξn46λβn + 2 ξn47λβn + ξn43
3 2 Ξn19 = 2 ξn21λβn − ξn48λβn − 2 ξn49λβn − ξn50
3 2 Ξn20 = 2 ξn51λβn − ξn52λβn − 2 ξn53λβn − ξn54
3 2 Ξn21 = 2 ξn55λβn − ξn56λβn + 2 ξn57λβn + ξn58
3 2 Ξn22 = 2 ξn51λβn + ξn59λβn − 2 ξn60λβn − ξn54
3 2 Ξn23 = 2 ξn55λβn + ξn61λβn + 2 ξn62λβn + ξn58
173 and
ξn1 = ηβn h (−τβn Θβn + Γβn ηβn)
2 2 ξn2 = ηβn (−2 τβn + hΓβn) ηβn − 2 Γβn ηβn − τβn −2 τβn + hΓβn τβn − 2 Θβn
3 2 2 ξn3 = ηβn −τβn hΘβn + Γβn (−1 + ηβn h) τβn + Γβn ηβn
ξn4 = τβn h (−τβn Θβn + Γβn ηβn)
3 2 2 ξn5 = −τβn −2 ηβn + ηβn hΘβn + 2 τβn − 2 Γβn ηβn − τβn Θβn (τβn h − 2)
3 2 2 ξn6 = τβn ηβn Γβn h + Θβn (1 − τβn h) ηβn − τβn Θβn
2 2 ξn7 = ηβn (−2 τβn + hΓβn) ηβn + 2 Γβn ηβn − τβn −2 τβn + hΓβn τβn + 2 Θβn
2 2 3 ξn8 = −ηβn (1 + ηβn h) τβn − ηβn Γβn − τβn hΘβn
3 2 2 ξn9 = −τβn −2 ηβn + ηβn hΘβn + 2 τβn + 2 Γβn ηβn − τβn Θβn (τβn h + 2)
3 2 2 ξn10 = τβn ηβn Γβn h + Θβn (−τβn h − 1) ηβn + τβn Θβn
2 ξn11 = (τβn + ηβn)
2 2 ξn12 = 2 ηβn − τβn − 1/2 Θβn − 1/2 Γβn (τβn + ηβn)
174 2 2 ξn13 = τβn ηβn + ηβn − Θβn τβn + Γβn ηβn (τβn + ηβn)
2 2 ξn14 = (τβn + ηβn) Γβn ηβn + τβn Θβn
2 2 ξn15 = 2 ηβn + 1/2 Θβn − τβn + 1/2 Γβn (τβn + ηβn)
2 2 ξn16 = τβn ηβn + ηβn + Θβn τβn − Γβn ηβn (τβn + ηβn)
2 ξn17 = (−τβn + ηβn)
2 2 ξn18 = 2 (−τβn + ηβn) ηβn − 1/2 Γβn − τβn + 1/2 Θβn
2 2 ξn19 = (−τβn + ηβn) −τβn ηβn + ηβn + Θβn τβn − Γβn ηβn
2 2 ξn20 = (−τβn + ηβn) Γβn ηβn − τβn Θβn
ξn21 = τβn ηβn
ξn22 = Γβn ηβn + τβn Θβn
3 3 ξn23 = τβn ηβn + τβn ηβn
2 2 ξn24 = τβn Γβn ηβn + τβn ηβn Θβn
2 2 ξn25 = 2 (−τβn + ηβn) ηβn + 1/2 Γβn − τβn − 1/2 Θβn
175 2 2 ξn26 = (−τβn + ηβn) −τβn ηβn + ηβn − Θβn τβn + Γβn ηβn
2 ξn27 = −τβn Θβn ηβn + Γβn ηβn
2 2 ξn28 = Γβn ηβn −τβn + ηβn
3 2 2 ξn29 = −ηβn τβn Θβn + ηβn τβn Γβn
2 ξn30 = −τβn Θβn + ηβn Γβn τβn
2 2 ξn31 = τβn Θβn −τβn + ηβn
2 2 3 ξn32 = −τβn ηβn Θβn + τβn Γβn ηβn
ξn33 = ηβn (τβn + ηβn)
2 2 ξn34 = ηβn −2 τβn − Θβn + 2 ηβn − Γβn
2 2 ξn35 = ηβn τβn ηβn + Γβn + τβn ηβn − τβn Θβn
3 2 ξn36 = Γβn ηβn + τβn Θβn ηβn
ξn37 = τβn Θβn
3 ξn38 = τβn ηβn
176 2 ξn39 = τβn ηβn Θβn
ξn40 = ηβn (−τβn + ηβn)
2 2 ξn41 = ηβn −2 τβn − Γβn + Θβn + 2 ηβn
2 2 ξn42 = τβn ηβn + −τβn − Γβn ηβn + τβn Θβn ηβn
3 2 ξn43 = Γβn ηβn − τβn Θβn ηβn
2 2 ξn44 = ηβn −2 τβn + 2 ηβn + Θβn + Γβn
2 2 ξn45 = τβn ηβn + −Γβn + τβn ηβn + τβn Θβn ηβn
2 2 ξn46 = ηβn −Θβn + 2 ηβn − 2 τβn + Γβn
2 2 ξn47 = ηβn τβn ηβn + Γβn − τβn ηβn − τβn Θβn
ξn48 = Γβn ηβn
3 ξn49 = τβn ηβn
2 ξn50 = τβn Γβn ηβn
ξn51 = τβn (τβn + ηβn)
177 2 2 ξn52 = τβn −2 τβn + 2 ηβn + Θβn + Γβn
2 2 ξn53 = τβn ηβn + ηβn + Θβn τβn − Γβn ηβn τβn
2 3 ξn54 = Γβn ηβn τβn + τβn Θβn
ξn55 = τβn (−τβn + ηβn)
2 2 ξn56 = τβn −2 τβn − Γβn + Θβn + 2 ηβn
2 2 ξn57 = τβn −τβn ηβn + ηβn + Θβn τβn − Γβn ηβn
2 3 ξn58 = Γβn ηβn τβn − τβn Θβn
2 2 ξn59 = τβn −2 τβn − Θβn + 2 ηβn − Γβn
2 2 ξn60 = τβn τβn ηβn + ηβn − Θβn τβn + Γβn ηβn
2 2 ξn61 = τβn −Θβn + 2 ηβn − 2 τβn + Γβn
2 2 ξn62 = τβn −τβn ηβn + ηβn − Θβn τβn + Γβn ηβn
178 Appendix F
Resulting Publications and Presentations
F.1 Journal Articles
The Effect of the Endothelial Glycocalyx Layer on Concentration Polari- sation of Low Density Lipoprotein in Arteries. P. E. Vincent, S. J. Sherwin, P. D. Weinberg. Submitted to Journal of Theoretical Biology, July 2009.
A Randomized Optical Coherence Tomography Study of Coronary Stent Strut Coverage and Luminal Protrusion With Rapamycin-Eluting Stents. P. Moore, P. Barlis, J. Spiro, G. Ghimire, M. Roughton, C. Di Mario, W. Wallis, C. Ilsley, A. Mitchell, M. Mason, R. Kharbanda, P. E. Vincent, S. J. Sherwin, M. Dalby. JACC Cardiovascular Interventions, Volume 2, May 2009, Pages 437-444.
The Effect of a Spatially Heterogeneous Transmural Water Flux on Con- centration Polarisation of Low Density Lipoprotein in Arteries. P. E. Vincent, S. J. Sherwin, P. D. Weinberg. Biophysical Journal, Volume 96, Issue 8, 22 April 2009, Pages 3102-3115. Viscous Flow Over Outflow Slits Covered by an Anisotropic Brinkman Medium: A Model of Flow Above Inter-Cellular Clefts. P. E. Vincent, S. J. Sherwin, P. D. Weinberg. Physics of Fluids, Volume 20, 063106 (2008).
F.2 Conference Contributions
Flow Features In a Realistic Representation of the Rabbit Aorta. A. Plata Garc´ıa, P. E. Vincent, A. A. E. Hunt, S. J. Sherwin, P. D. Weinberg. Oral pre- sentation, 12th International Symposium on Computer Simulation in Biomechanics, 2-4 July 2009. Cape Town, South Africa.
A Realistic Representation of the Rabbit Aorta for use in Computational Haemodynamic Studies. P. E. Vincent, A. A. E. Hunt, L. Grinberg, S. J. Sherwin, P. D. Weinberg. Poster Presentation, American Society of Mechanical Engineering, Summer Bioengineering Conference, 17-21 June 2009. Lake Tahoe, California, USA. (Finalist in poster prize competition).
Sub-Cellular Scale Features of Low Density Lipoprotein Concentration Polarisation Adjacent to the Arterial Endothelium. P. E. Vincent, S. J. Sherwin, P. D. Weinberg. Oral Presentation, Bioengineering 2008, 18-19 September 2008. London, UK.
The Effect of Sub-Cellular Scale Endothelial Features on Low Density Lipoprotein Concentration Polarisation in Arteries. P. E. Vincent, S. J. Sherwin, P. D. Weinberg. Oral Presentation, UK Focus for Biomedical Engineer- ing Futures Meeting Study and Treatment of Cardiovascular Disease: Devices and Fluidics, 15-17 September 2008. London, UK.
Sub-Cellular Scale Features of Low Density Lipoprotein Concentration Polarisation in Arteries. P. E. Vincent, S. J. Sherwin, P. D. Weinberg. Oral Presentation, 16th International Conference on Mechanics in Medicine and Biology, 23-25 July 2008. Pittsburgh, Pennsylvania, USA.
180 Effect of the Endothelial Glycocalyx Layer on Low Density Lipoprotein Concentration Polarisation Within Arteries. P. E. Vincent, S. J. Sherwin, P. D. Weinberg. Poster Presentation, 3rd International Symposium on Biomechan- ics in Vascular Biology and Cardiovascular Disease, 24-25 April 2008. Rotterdam, Netherlands.
Arterial Low Density Lipoprotein Concentration Polarisation at the Sub- Cellular Scale. P. E. Vincent, S. J. Sherwin, P. D. Weinberg. Oral Presenta- tion, 5th International Bio-Fluid Symposium and Workshop, 28-30 March 2008. Pasadena, Los Angeles, California, USA.
Arterial Low Density Lipoprotein Concentration Polarisation at the Sub- Cellular Scale. P. E. Vincent, S. J. Sherwin, P. D. Weinberg. Poster Presen- tation, 5th International Bio-Fluid Symposium and Workshop, 28-30 March 2008. Pasadena, Los Angeles, California, USA.
Sub-Cellular Scale Variations in Low Density Lipoprotein Concentration Adjacent to the Endothelium. P. E. Vincent, S. J. Sherwin, P. D. Weinberg. Poster Presentation, BMES Annual Fall Meeting, 26-29 September 2007. Holly- wood, Los Angeles, California, USA.
The Effect of Sub-Cellular Scale Variations in Transmural Water Flux on LDL Buildup Adjacent to the Endothelium. P. E. Vincent, S. J. Sherwin, P. D. Weinberg. Poster Presentation, Fifth Physiological Flow Meeting: Size, Sex and Sight, 3-4 September 2007. London, UK.
Sub-Cellular Scale Variations in LDL Concentration Polarisation on the Luminal Side of the Endothelium. P. E. Vincent, S. J. Sherwin, P. D. Wein- berg. Poster Presentation, Fourth Physiological Flow Meeting: Respiratory Biome- chanics and Physiological Fluid-Structure Interaction Problems, 2-3 April 2007. Manchester, UK. (Runner up in poster prize competition).
Computational Investigation of a Mechanism by Which Blood Flow Could Control Lipoprotein Uptake by the Arterial Wall. P. E. Vincent, S. J. Sher-
181 win, P. D. Weinberg. Oral Presentation, BAS and BSCR Joint Autumn Meeting, 21-22 September 2006. Cambridge, UK. Published in HEART, 2007, Vol: 93, ISSN: 1355-6037.
2D Computational Study of Cellular Scale Variations in LDL Concen- tration at an Endothelium with Physiologically Realistic Inter-Cellular Cleft Dimensions. P. E. Vincent, S. J. Sherwin, P. D. Weinberg. Poster Presen- tation, Third Physiological Flow Meeting: Imaging and Modelling for Interventional Planning, 18-19 April 2006. Oxford, UK.
F.3 Seminars
Computational Studies of Blood Flow and Lipid Transport Within Ar- teries. June 2009. Department of Aeronautics and Astronautics, Stanford Uni- versity, Palo Alto, CA, USA.
Computational Studies of Blood Flow and Lipid Transport Within Arter- ies. April 2009. Department of Physics, Harvard University, Cambridge, MA, USA.
Computational Studies of Blood Flow and Lipid Transport Within Ar- teries. April 2009. Department of Applied Mathematics, Brown University, Providence, RI, USA.
Computational Studies of Blood Flow and Lipid Transport Within Arter- ies. April 2009. Department of Visualisation, Harvard University, Cambridge, MA, USA.
Realistic Reconstructions of the Rabbit Aorta. January 2009. Department of Bioengineering, Imperial College London, London, UK.
Sub-Cellular Scale Features of Low Density Lipoprotein Concentration Polarisation in Arteries. November 2008. Department of Aeronautics, Imperial
182 College London, London, UK.
Flow and Mass Transport near the Luminal Surface of the Arterial Wall. June 2008. Department of Mathematics, Imperial College London, London, UK.
Sub-Cellular Scale Features of LDL Concentration Polarisation Adjacent to the Endothelium. December 2007. Department of Bioengineering, Imperial College London, London, UK.
Sub-Cellular Scale Variations in LDL Concentration Polarisation Adja- cent to the Endothelium. October 2007. Department of Mathematics, Univer- sity of Utah, Salt Lake City, UT, USA.
Sub-Cellular Scale Variations in LDL Concentration Polarisation Adja- cent to the Endothelium - Implications for Atherosclerosis Initiation. May 2007. Department of Aeronautics, Imperial College London, London, UK.
Sub-Cellular Scale Variations in LDL Concentration Polarisation on the Luminal Side of the Endothelium. January 2007. Department of Biomedical Engineering, City College New York, New York, NY, USA.
Computational Modelling of Atherosclerosis Initiation. May 2006. Depart- ment of Aeronautics, Imperial College London, London, UK.
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