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Stability analysis of a model of atherogenesis: an energy estimate approach. (English) Zbl 1140.92011
Summary: Atherosclerosis is a disease of the vasculature that is characterized by chronic inflammation and the accumulation of lipids and apoptotic cells in the walls of large arteries. This disease results in plaque growth in an infected artery typically leading to occlusion of the artery. Atherosclerosis is the leading cause of human mortality in the US, much of Europe, and parts of Asia. In a previous work [see A. I. Ibragimov et al., Math. Med. Biol. 22, No. 4, 305–333 (2005; Zbl 1080.92040)], we introduced a mathematical model of the biochemical aspects of the disease, in particular the inflammatory response of macrophages in the presence of chemoattractants and modified low density lipoproteins. Herein, we consider the onset of a lesion as resulting from an instability in an equilibrium configuration of cells and chemical species. We derive an appropriate norm by taking an energy estimate approach and present stability criteria. A bio-physical analysis of the mathematical results is presented.

MSC:
92C50 Medical applications (general)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35B35 Stability in context of PDEs
35K57 Reaction-diffusion equations
92C17 Cell movement (chemotaxis, etc.)
35B40 Asymptotic behavior of solutions to PDEs
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