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A Cellular Scale Study of Low Density Lipoprotein Concentration Polarisation in Arteries

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A Cellular Scale Study of Low Density Lipoprotein Concentration Polarisation in Arteries 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
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