Abstract:
Atherosclerosis is an inflammatory chronic illness characterized by the buildup of fats, cholesterol and other substances in and on the artery wall causing its hardening. This work presents a mathematical model that analyze the inflammatory stage of atherosclerosis. We present a system of four coupled type of partial differential equations. Four leading players are taken into consideration: LDL (L), oxidized LDL (Lox), immune cells (M ) and the pro-inflammatory cytokines (C). In addition, one characteristic parameter taken into account is the permeability of the endothelial layer.
A stability analysis and existence proof for fixed points of the kinetic system is presented leading to a biological interpretation. We are able to distinguish three main cases of the disease state that correlates with the permeability of the endothelial layer; where we can note that the permeability is classified according to its value based on the remaining parameters. In fact, when the permeability is low, this case is considered to be the disease free state since no chronic inflammatory reaction occurs due to the non-initiation of the auto-amplification process. With intermediate permeability, a wave propagation corresponding to a chronic inflammatory reaction might happen, whether the initial perturbation overcome a threshold or not. With high permeability, even a small perturbation of the disease free state drives to a chronic inflammatory reaction represented by a wave propagation.
