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Mathematical Analysis and Numerical Simulations for a Model of Atherosclerosis

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


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.


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

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  • DOI: 10.1007/978-4-431-56457-7_21
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  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.


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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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Correspondence to Adélia Sequeira .

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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.

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