Reaction–diffusion model of atherosclerosis development

• DOI: 10.1007/s00285-011-0461-1
• Publication Date: 2011-08-20T04:38:18Z
• Authors: N. El Khatib; S. Genieys; B. Kazmierczak; V. Volpert
• Keywords: 0301 basic medicine; 610; Monocytes; 510; numerical simulations; Diffusion; Nonlinear boundary conditions; 03 medical and health sciences; Reaction–diffusion equations; Modelling and Simulation; Numerical simulations; Reaction Time; [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]; Humans; [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]; [NLIN.NLIN-AO]Nonlinear Sciences [physics]/Adaptation and Self-Organizing Systems [nlin.AO]; Inflammation; Applied Mathematics; Macrophages; existence; Models, Cardiovascular; Numerical Analysis, Computer-Assisted; Atherosclerosis; Agricultural and Biological Sciences (miscellaneous); Plaque, Atherosclerotic; AMS Class. 35k57, 92C50; Lipoproteins, LDL; Cholesterol; nonlinear boundary conditions; atherosclerosis; reaction-diffusion waves; Tunica Intima; Existence of travelling waves; Foam Cells

Atherosclerosis begins as an inflammation in blood vessel walls (intima). The inflammatory response of the organism leads to the recruitment of monocytes. Trapped in the intima, they differentiate into macrophages and foam cells leading to the production of inflammatory cytokines and further recruitment of white blood cells. This self-accelerating process, strongly influenced by low-density lipoproteins (cholesterol), results in a dramatic increase of the width of blood vessel walls, formation of an atherosclerotic plaque and, possibly, of its rupture. We suggest a 2D mathematical model of the initiation and development of atherosclerosis which takes into account the concentration of blood cells inside the intima and of pro- and anti-inflammatory cytokines. The model represents a reaction-diffusion system in a strip with nonlinear boundary conditions which describe the recruitment of monocytes as a function of the concentration of inflammatory cytokines. We prove the existence of travelling waves described by this system and confirm our previous results which suggest that atherosclerosis develops as a reaction-diffusion wave. The theoretical results are confirmed by the results of numerical simulations.