Mathematical Modelling of Atherosclerosis

Abstract

In atherosclerosis, the arterial lining undergoes a specific sequence of inflammatory responses to an injury to the cells that line the blood vessel and to low density lipoprotein (LDL) particles from the blood stream that penetrate through this injury into the arterial wall. We model the events that take place inside the blood vessel wall that occur immediately after such an injury with a system of partial differential equations that involve the LDL particles, two proinflammatory cytokines, monocyte-derived macrophages and their lipid-filled counterparts, foam cells. The model includes the chemical and physical interactions with the endothelial cells that line the arterial wall. These interactions are formulated as boundary conditions. Through numerical simulations, we show that different LDL concentrations in the blood stream and different immune responses can qualitatively affect the development of a plaque. Numerical bifurcation analysis at the quasi-steady state through AUTO shows that there exists of a fold bifurcation when the flux of LDL into the plaque from the blood is high. An atherosclerotic plaque that develops within the intima, deforms the intima locally as macrophages and foam cells accumulate. We model the structure of the developing plaque by cell pressure and cell sorting models to account for the limited space within the intima. We do this by modelling cell movement in crowded tissue in a discrete space and extend this to a spatial domain where cells also moves due to cell pressure and chemotaxis. We model the mechanics of the physical interactions on the two bounding interfaces, (the lumen-intima boundary and the intima-media boundary) and of the tissue inside the domain and add advective terms to ensure that the mechanics of the cellular species is consistent with the underlying tissue deformation. Using a finite element solver, we produce numerical results in one dimension across the intima and in two dimensions as a cross section of an artery. With appropriate parameter values, this moving boundary problem produces results in agreement with the current theory on compensatory enlargement in atherosclerotic remodelling