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Mathematical Fluid Dynamics, Present and Future pp 577–595Cite as

Mathematical Analysis and Numerical Simulations for a Model of Atherosclerosis

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 183)

Abstract

Atherosclerosis is a chronic inflammatory disease that occurs mainly in large and medium-sized elastic and muscular arteries. This pathology is essentially caused by the high concentration of low-density-lipoprotein (LDL) in the blood. It can lead to coronary heart disease and stroke, which are the cause of around 17.3 million deaths per year in the world. Mathematical modeling and numerical simulations are important tools for a better understanding of atherosclerosis and subsequent development of more effective treatment and prevention strategies. The atherosclerosis inflammatory process can be described by a model consisting of a system of three reaction-diffusion equations (representing the concentrations of oxidized LDL, macrophages and cytokines inside the arterial wall) with non-linear Neumann boundary conditions. In this work we prove the existence, uniqueness and boundedness of global solutions, using the monotone iterative method. Numerical simulations are performed in a rectangle representing the intima, to illustrate the mathematical results and the atherosclerosis inflammatory process.

Keywords

  • Atherosclerosis
  • Reaction-diffusion equations
  • Nonlinear boundary conditions
  • Upper and lower solutions
  • Monotone sequences
  • Existence-comparison theorem

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Notes

  1. 1.

    The wall shear stress function, WSS, is computed using the solution of the blood flow model (for instance a generalized Navier-Stokes model).

  2. 2.

    Let K be an arbitrary compact set and U an open subset of \(\varGamma _{end}\), taken as a very small neighbourhood of K, containing K. There exists a bump function \(\psi (x)\) which is equal to 1 on K and falls off rapidly to 0 outside of K, while still being smooth.

References

  1. Mitchell, M.E., Sidawy, A.N.: The pathophysiology of atherosclerosis. Semin. Vasc. Surg. 3(11), 134–141 (1998)

    Google Scholar 

  2. Ross, R.: Atherosclerosis—an inflammatory disease. Mass. Med. Soc. 340(2), 115–126 (1999)

    Google Scholar 

  3. Mitrovska, S., Jovanova, S., Matthiesen, I., Libermans, C.: Atherosclerosis: Understanding Pathogenesis and Challenge for Treatment. Nova Science Publishers Inc., New York (2009)

    Google Scholar 

  4. Guretzki, H.J., Gerbitza, K.D., Olgemöller, B., Schleicher, E.: Atherogenic levels of low density lipoprotein alter the permeability and composition of the endothelial barrier. Elsevier 107(1), 15–24 (1994)

    Google Scholar 

  5. Mendis, S., Puska, P., Norrving, B.: Global Atlas on Cardiovascular Disease Prevention and Control. World Health Organization, Geneva (2011)

    Google Scholar 

  6. El Khatib, N., Génieys, S., Kazmierczak, B., Volpert, V.: Reaction-diffusion model of atherosclerosis development. J. Math. Biol. 65, 349–374 (2012)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Calvez, V., Ebde, A., Meunier, N., Raoult, A.: Mathematical and numerical modeling of the atherosclerotic plaque formation. ESAIM Proc. 28, 1–12 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Calvez, V., Houot, J., Meunier, N., Raoult, A., Rusnakova, G.: Mathematical and numerical modeling of early atherosclerotic lesions. ESAIM Proc. 30, 1–14 (2010)

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Hao, W., Friedman, A.: The LDL-HDL profile determines the risk of atherosclerosis- a mathematical model. PLoS ONE 9(3), 1–15 (2014)

    Google Scholar 

  10. Liu, B., Tang, D.: Computer simulations of atherosclerosis plaque growth in coronary arteries. Mol. Cell. Biomech. 7(4), 193–202 (2010)

    Google Scholar 

  11. Filipovic, N., Nikolic, D., Saveljic, I., Milosevic, Z., Exarchos, T., Pelosi, G., Parodi, O.: Computer simulation of three-dimensional plaque formation and progression in the coronary artery. Comput. Fluids 88, 826–833 (2013)

    CrossRef  Google Scholar 

  12. Silva, T., Sequeira, A., Santos, R., Tiago, J.: Mathematical modeling of atherosclerotic plaque formation coupled with a non-Newtonian model of blood flow. In: Hindawi Publishing Corporation Conference Papers in Mathematics (2013)

    Google Scholar 

  13. El Khatib, N., Génieys, S., Kazmierczak, B., Volpert, V.: Mathematical modeling of atherosclerosis as an inflammatory disease. Philos. Trans. R. Soc. A 367, 4877–4886 (2009)

    CrossRef  MATH  Google Scholar 

  14. Silva, T., Sequeira, A., Santos, R., Tiago, J.: Existence, uniqueness, stability and asymptotic behavior of solutions for a model of atherosclerosis, DCDS-S, AIMS, 9(1), 343–362 (2016)

    Google Scholar 

  15. Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992)

    MATH  Google Scholar 

  16. Fife, P.C.: Mathematical Aspects of Reacting and Diffusing Systems. Springer-Verlag, Berlin Heidelberg (1979)

    CrossRef  MATH  Google Scholar 

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Acknowledgements

FCT (Fundação para a Ciência e a Tecnologia, Portugal) through the grant SFRH/BPD/66638/2009, the project EXCL/MAT-NAN/0114/2012 and the Research Center CEMAT-IST are gratefully acknowledged.

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Authors and Affiliations

  1. Department of Mathematics and CEMAT, IST, ULisboa, 1049-001, Lisbon, Portugal

    Telma Silva, Jorge Tiago & Adélia Sequeira

Authors
  1. Telma Silva
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  2. Jorge Tiago
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  3. Adélia Sequeira
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Correspondence to Adélia Sequeira .

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Editors and Affiliations

  1. Department of Mathematics, Waseda University, Tokyo, Japan

    Yoshihiro Shibata

  2. Department of Mathematics, Waseda University, Tokyo, Japan

    Yukihito Suzuki

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Silva, T., Tiago, J., Sequeira, A. (2016). Mathematical Analysis and Numerical Simulations for a Model of Atherosclerosis. In: Shibata, Y., Suzuki, Y. (eds) Mathematical Fluid Dynamics, Present and Future. Springer Proceedings in Mathematics & Statistics, vol 183. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56457-7_21

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