Dynamical Model of Mild Atherosclerosis: Applied Mathematical Aspects

TL;DR

This paper develops a mathematical model of atherosclerosis using nonlinear differential equations, analyzes stability and bifurcations, and explores wave-like plaque development through reaction-diffusion equations.

Contribution

It introduces a novel autonomous system model of plaque formation, including bifurcation analysis and a spatiotemporal PDE model with traveling wave solutions.

Findings

Model reveals stability conditions of plaque-related variables.

Bifurcation analysis identifies thresholds for uncontrollable plaque growth.

Reaction-diffusion model demonstrates traveling wave phenomena in plaque development.

Abstract

Atherosclerosis is a chronic inflammatory disease occurs due to plaque accumulation in the inner artery wall. In atherosclerotic plaque formation monocytes and macrophages play a significant role in controlling the disease dynamics. In the present article, the entire biochemical process of atherosclerotic plaque formation is presented in terms of an autonomous system of nonlinear ordinary differential equations involving concentrations of oxidized low-density lipoprotein (LDL), monocytes, macrophages, and foam cells as the key dependent variables. To observe the capacity of monocytes and macrophages the model has been reduced to a two-dimensional temporal model using quasi steady state approximation theory. Linear stability analysis of the two-dimensional ordinary differential equations (ODEs) model has revealed the stability of the equilibrium points in the system. We have considered both one- and two- parameter bifurcation analysis with respect to parameters associated to the rate at which macrophages phagocytose oxidised LDL and the rate at which LDL enters into the intima. The bifurcation diagrams reveal the oscillating nature of the curves representing concentration of monocytes and macrophages with respect to significant model parameters. We are able to find the threshold values at which the plaque accumulation accelerates in an uncontrollable way. Further to observe the impact of diffusion, a spatiotemporal model has been developed. Numerical investigation of the partial differential equations (PDEs) model reveals the existence of travelling wave in the system which ensures the fact that the development of atherosclerotic plaque formation follows reaction-diffusion wave.