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Reaction–diffusion model of atherosclerosis development

Abstract

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.

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Acknowledgments

The authors are grateful to John McGregor for discussions about the inflammatory aspect of atherosclerosis and clarifying some important details about the risk factors. B. Kazmierczak has been supported by the MNiSW grant N N201548738 and by the FNP project TEAM/2009-3/6.

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This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Correspondence to B. Kazmierczak.

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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El Khatib, N., Genieys, S., Kazmierczak, B. et al. Reaction–diffusion model of atherosclerosis development. J. Math. Biol. 65, 349–374 (2012). https://doi.org/10.1007/s00285-011-0461-1

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Keywords

  • Atherosclerosis
  • Reaction–diffusion equations
  • Nonlinear boundary conditions
  • Existence of travelling waves
  • Numerical simulations

Mathematics Subject Classification (2000)

  • 35K57
  • 92C50