Journal of Mathematical Biology

, Volume 65, Issue 2, pp 349–374 | Cite as

Reaction–diffusion model of atherosclerosis development

  • N. El Khatib
  • S. Genieys
  • B. Kazmierczak
  • V. Volpert
Open Access


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.


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

Mathematics Subject Classification (2000)

35K57 92C50 


  1. Chow CC, Clermont G, Kumar R, Lagoa C, Tawadrous Z, Gallo D, Betten B, Bartels J, Constantine G, Fink MP, Billiar TR, Vodovotz Y (2005) The acute inflammatory response in diverse shock states. Shock 24(1): 74–84CrossRefGoogle Scholar
  2. Edelstein-Keshet L, Spiros A (2002) Exploring the formation of Alzheimer’s disease senile plaques in silico. J Theor Biol 216(3): 301–326CrossRefGoogle Scholar
  3. El Khatib N, Genieys S, Volpert V (2007) Atherosclerosis initiation modeled as an inflammatory process. Math Model Nat Phenom 2(2): 126–141MathSciNetCrossRefGoogle Scholar
  4. Fan J, Watanabe T (2003) Inflammatory reactions in the pathogenesis of atherosclerosis. J Atheroscler Thromb 10: 63–71CrossRefGoogle Scholar
  5. Kazmierczak B (2009) Existence of global solutions to a model of chondrogenesis. Math Meth Appl Sci 32(3): 264–283MathSciNetMATHCrossRefGoogle Scholar
  6. Krasnoselskii MA, Zabreiko PP (1984) Geometrical methods of nonlinear analysis. Springer, New YorkCrossRefGoogle Scholar
  7. Ladyzhenskaya OA, Solonnikov VA, Uraltseva NN (1967) Linear and quasi-linear equations of parabolic type. Moskva, NaukaGoogle Scholar
  8. LaRosa JC, He J, Vupputuri S (1999) Effect of statins on risk of coronary disease: a meta-analysis of randomized controlled trials. J Am Med Assoc 282: 2340–2346CrossRefGoogle Scholar
  9. Leiderman KM, Miller LA, Fogelson AL (2008) The effects of spatial inhomogeneities on flow through the endothelial surface layer. J Theor Biol 252: 313–325CrossRefGoogle Scholar
  10. Li Z-Y, Howarth SP, Tang T, Gillard JH (2006a) How critical is fibrous cap thickness to carotid plaque stability? A flow plaque interaction model. Stroke 37: 1195–1196CrossRefGoogle Scholar
  11. Li Z-Y, Howarth SP, Trivedi RA, U-King-Im JM, Graves MJ, Brown A, Wang L, Gillard JH (2006b) Stress analysis of carotid plaque rupture based on in vivo high resolution MRI. J Biomech 39(14): 2611–2612CrossRefGoogle Scholar
  12. Lieberman GM (1996) Second order parabolic differential equations. World Scientific, SingaporeMATHGoogle Scholar
  13. Østerud B, Bjørklid E (2003) Role of monocytes in atherogenesis. Physiol Rev 83: 1070–1086Google Scholar
  14. Poston RN, Poston DRM (2007) A typical atherosclerotic plaque morphology produced in silico by an atherogenesis model based on self-perpetuating propagating macrophage recruitment. Math Model Nat Phenom 2(2): 142–149MathSciNetCrossRefGoogle Scholar
  15. Protter MH, Weinberger HF (1967) Maximum principles in differential equations. Springer, BerlinGoogle Scholar
  16. Ross R (1999) Atherosclerosis: an inflammatory disease. Mass Med Soc 340: 115–120Google Scholar
  17. Vincent PE, Sherwin SJ, Weinberg PD (2009) The effect of a spatially heterogeneous transmural water flux on concentration polarization of low density lipoprotein in arteries. Biophys J 96(8): 3102–3115CrossRefGoogle Scholar
  18. Volpert AI, Volpert VA, Volpert VA (2000) Traveling wave solutions of parabolic systems. American Mathematical Society, ProvidenceGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  • N. El Khatib
    • 1
  • S. Genieys
    • 1
  • B. Kazmierczak
    • 2
  • V. Volpert
    • 1
  1. 1.Institute of MathematicsUniversité Lyon 1, UMR 5208 CNRSVilleurbanneFrance
  2. 2.Institute of Fundamental Technological Research of PASWarsawPoland

Personalised recommendations