Novel mathematical and computational approaches for modelling biological systems

This work presents the development, analysis and subsequent simulations of mathematical models aimed at providing a basis for modelling atherosclerosis. This cardiovascular disease is characterized by the growth of plaque in artery walls, forming lesions that protrude into the lumen. The rupture of these lesions contributes greatly to the number of cases of stroke and myocardial infarction. These are two of the main causes of death in the UK. Any work to understand the processes by which the disease initiates and progresses has the ultimate aim of limiting the disease through either its prevention or medical treatment and thus contributes a relevant addition to the growing body of research. The literature supports the view that the cause of atherosclerotic lesions is an in inflammatory process-succinctly put, excess amounts of certain biochemical species fed into the artery wall via the bloodstream spur the focal accumulation of extraneous cells. Therefore, suitable components of a mathematical model would include descriptions of the interactions of the various biochemical species and their movement in space and time. The models considered here are in the form of partial differential equations. Specifically, the following models are examined: first, a system of reaction-diffusion equations with coupling between surface and bulk species; second, a problem of optimisation to identify an unknown boundary; and finally, a system of advection-reaction-diffusion equations to model the assembly of keratin networks inside cells. These equations are approximated and solved computationally using the finite element method. The methods and algorithms shown aim to provide more accurate and efficient means to obtain solutions to such equations. Each model in this work is extensible and with elements from each model combined, they have scope to be a platform to give a fuller model of atherosclerosis.