Stability analysis using an energy estimate approach of a reaction-diffusion model of atherogenesis

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2009, 2009(Special): 630-639. doi: 10.3934/proc.2009.2009.630

Stability analysis using an energy estimate approach of a reaction-diffusion model of atherogenesis

L.R. Ritter ^1, , Akif Ibragimov ^2, , Jay R. Walton ^3, and Catherine J. McNeal ^4,

1.	Southern Polytechnic State University, Marietta, GA 30060-2896, United States
2.	Department of Mathematics and Statistics, Texas Tech University, Lubbock TX, 79409-1042
3.	Texas A & M Unviersity, College Station, TX 77843, United States
4.	Scott & White Hospital, Temple, TX 76508, United States

Received June 2008 Revised July 2009 Published September 2009

Abstract
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This paper considers modeling the initiation of atherosclerosis, as an inflammatory instability. Motivated by the disease paradigm articulated by Russell Ross, atherogenesis is viewed as an inflammatory spiral with positive feedback loop involving key cellular and chemical species interacting and reacting within the intimal layer of muscular arteries. The inflammation is modeled through a system of nonlinear reaction/diffusion/convection partial differential equations. The inflammatory spiral is initiated as an instability from a healthy state which is defined to be an equilib- rium state devoid of certain key inflammatory markers. Disease initiation is studied through a linear, asymptotic stability analysis of a healthy equilibrium state. Conditions on system parameters guaranteeing stability of the health state and conditions on system parameters leading to instability are given. Among the questions addressed in the analysis is the possible mitigating effect of anti-oxidants upon transition to the inflammatory spiral.

Keywords: atherosclerosis, chemotaxis, energy estimate, atherogenesis, stability analysis.

Mathematics Subject Classification: 35K55, 92C17, 92C50.

Citation: L.R. Ritter, Akif Ibragimov, Jay R. Walton, Catherine J. McNeal. Stability analysis using an energy estimate approach of a reaction-diffusion model of atherogenesis. Conference Publications, 2009, 2009 (Special) : 630-639. doi: 10.3934/proc.2009.2009.630

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