Computational and Mathematical Methods in Medicine
Volume 11 (2010), Issue 1, Pages 67-88
Stability Analysis of a Model of Atherogenesis: An Energy Estimate Approach II
1Department of Mathematics, Texas Tech University, Lubbock, TX 79409, USA
2Division of Cardiology, Department of Internal Medicine, Temple, TX 76508, USA
3Division of Endocrinology, Department of Pediatrics, Scott & White, Temple, TX 76508, USA
4Department of Mathematics, Southern Polytechnic State University, Marietta, GA 30060, USA
5Department of Mathematics, Texas A & M University, College Station, TX 77843-3368, USA
Received 21 July 2008; Accepted 12 December 2008
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This paper considers modelling atherogenesis, 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 modelled through a system of non-linear reaction–diffusion–convection partial differential equations. The inflammatory spiral is initiated as an instability from a healthy state which is defined to be an equilibrium state devoid of certain key inflammatory markers. Disease initiation is studied through a linear, asymptotic stability analysis of a healthy equilibrium state. Various theorems are proved giving conditions on system parameters guaranteeing stability of the health state and conditions on system parameters leading to instability. Among the questions addressed in the analysis is the possible mitigating effect of anti-oxidants upon transition to the inflammatory spiral.