American Institute of Mathematical Sciences

Advanced
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

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
[1]

Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063
[2]

Telma Silva, Adélia Sequeira, Rafael F. Santos, Jorge Tiago. Existence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosis. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 343-362. doi: 10.3934/dcdss.2016.9.343
[3]

Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control & Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014
[4]

Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic & Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883
[5]

Boris P. Belinskiy, Peter Caithamer. Energy estimate for the wave equation driven by a fractional Gaussian noise. Conference Publications, 2007, 2007 (Special) : 92-101. doi: 10.3934/proc.2007.2007.92
[6]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424
[7]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020292
[8]

Messoud Efendiev, Etsushi Nakaguchi, Wolfgang L. Wendland. Uniform estimate of dimension of the global attractor for a semi-discretized chemotaxis-growth system. Conference Publications, 2007, 2007 (Special) : 334-343. doi: 10.3934/proc.2007.2007.334
[9]

Xin Lai, Xinfu Chen, Mingxin Wang, Cong Qin, Yajing Zhang. Existence, uniqueness, and stability of bubble solutions of a chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 805-832. doi: 10.3934/dcds.2016.36.805
[10]

Chjan C. Lim, Da Zhu. Variational analysis of energy-enstrophy theories on the sphere. Conference Publications, 2005, 2005 (Special) : 611-620. doi: 10.3934/proc.2005.2005.611
[11]

Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064
[12]

Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014
[13]

Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 203-209. doi: 10.3934/dcdss.2020011
[14]

Qian Xu. The stability of bifurcating steady states of several classes of chemotaxis systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 231-248. doi: 10.3934/dcdsb.2015.20.231
[15]

Huanhuan Qiu, Shangjiang Guo. Global existence and stability in a two-species chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1569-1587. doi: 10.3934/dcdsb.2018220
[16]

Ismail Abdulrashid, Abdallah A. M. Alsammani, Xiaoying Han. Stability analysis of a chemotherapy model with delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 989-1005. doi: 10.3934/dcdsb.2019002
[17]

Shengji Li, Chunmei Liao, Minghua Li. Stability analysis of parametric variational systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 317-331. doi: 10.3934/naco.2011.1.317
[18]

Jean-Pierre Françoise, Hongjun Ji. The stability analysis of brain lactate kinetics. Discrete & Continuous Dynamical Systems - S, 2020, 13 (8) : 2135-2143. doi: 10.3934/dcdss.2020182
[19]

Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141
[20]

Shuai Zhang, L.R. Ritter, A.I. Ibragimov. Foam cell formation in atherosclerosis: HDL and macrophage reverse cholesterol transport. Conference Publications, 2013, 2013 (special) : 825-835. doi: 10.3934/proc.2013.2013.825

 Impact Factor: 

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences