A Mathematical Model on the Feedback Between Wall Shear Stress and Intimal Hyperplasia

Abstract

In this paper, we present a mathematical model linking blood flow, shear-dependent endothelium permeability and intimal thickening (hyperplasia) of the arterial wall, which is an initial stage in the development of atherosclerosis. The key concepts are that the intimal layer swells in response to the presence of excess oxidised LDL (OxLDL) in foam cells. The hyperplasia disturbs blood flow, affecting endothelial permeability via the wall shear stress (WSS). These changes produce a feedback mechanism. LDL is transported through the arterial wall by advection and diffusion, and the concentration of LDL at each time step is assumed to be quasi-steady since it equilibrates on a fast time scale. The process is controlled by the slow timescale of the increase in concentration of OxLDL. We consider a section of uniform axisymmetric artery, and impose an initial local injury or ‘hotspot’ of relatively high permeability that enhances the influx of LDL, triggering the development of a bump-shaped lesion. In the absence of further inflammatory processes, the lesion eventually decays back to the homeostatic state. The model is used to explore how the shape of the lesion changes over time, its effect on WSS, influx rates of LDL and the sensitivity of these processes to oxidation parameters. The lesion is shown to propagate downstream driven by regions of high and low WSS on either side of the bump, and it persists for some time after the hotspot has vanished, leaving ample time for further pro-atherogenic processes to develop.