Inflammation propagation modeled as a reaction-diffusion wave

Inflammation is a physiological process aimed to protect the organism in various diseases and injuries. This work presents a generic inflammation model based on the reaction-diffusion equations for the concentrations of uninflamed cells, inflamed cells, immune cells and the inflammatory cytokines. The analysis of the model shows the existence of three different regimes of inflammation progression depending on the value of a parameter R called the inflammation number. If R>1, then inflammation propagates in cell culture or tissue as a reaction-diffusion wave due to diffusion of inflammatory cytokines produced by inflamed cells. If 0 Collapse

Key Words

• Cytokine storm
• Inflammation
• Inflammation propagation
• Reaction–diffusion equations
• Wave speed

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Affiliation(s)

W El Hajj Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
N El Khatib Department of Computer Science and Mathematics, Lebanese American University, P.O. Box 36, Byblos, Lebanon.
V Volpert Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France; Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow 117198, Russian Federation

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Effect of permeability on the initiation of Atherosclerosis modeled as an inflammatory process

This work presents a mathematical model, based on partial differential equations, that analyzes the inflammatory stage of atherosclerosis. Four leading players are taken into consideration: Low Density Lipoproteins, oxidized Low Density Lipoproteins, immune cells and the inflammatory cytokines. In addition to this, the permeability of the endothelial layer is taken into account in the model. A stability analysis of the fixed points of the kinetic system is presented in details followed by the proof of existence of traveling wave solutions of the system of partial differential equations. The mathematical analysis leads to a biological interpretation. We distinguish three main cases of the disease state that correlate with the permeability of the endothelial layer. In fact, having a low permeability indicates the disease free state since no chronic inflammatory reaction occurs due to the non initiation of the inflammation. With intermediate permeability, a wave propagation corresponding to a chronic inflammatory reaction might occur whether the initial perturbation overcomes a threshold or not. With high permeability, even a small perturbation of the disease free state leads to a chronic inflammatory reaction represented by a wave propagation. We perform numerical simulations of the solutions to illustrate the biological results.