Mathematical Analysis and Numerical Simulations for a Model of Atherosclerosis
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.
KeywordsAtherosclerosis Reaction-diffusion equations Nonlinear boundary conditions Upper and lower solutions Monotone sequences Existence-comparison theorem
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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