Stability Analysis of a Model of Atherogenesis: An Energy Estimate Approach II

DYNAMICAL …, 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.

Computational and Mathematical Methods in Medicine, 2008

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 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 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 antioxidants upon transition to the inflammatory spiral.

SIAM Journal on Applied Mathematics, 2010

This paper presents a linear, asymptotic stability analysis for a reaction-diffusionconvection system modeling 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 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 a general framework is developed for constructing perturbations from a healthy state that exhibit blow-up which are interpreted as corresponding to disease initiation. The analysis reveals key features that arterial geometry, anti-oxident levels and the source of inflammatory components (through coupled third kind boundary conditions or through body sources) play in disease initiation.

Atherosclerosis is a chronic inflammatory disease occurs due to plaque accumulation in the inner artery wall. In atherosclerotic plaque formation monocytes and macrophages play a significant role in controlling the disease dynamics. In the present article, the entire biochemical process of atherosclerotic plaque formation is presented in terms of an autonomous system of nonlinear ordinary differential equations involving concentrations of oxidized low-density lipoprotein (LDL), monocytes, macrophages, and foam cells as the key dependent variables. To observe the capacity of monocytes and macrophages the model has been reduced to a two-dimensional temporal model using quasi steady state approximation theory. Linear stability analysis of the two-dimensional ordinary differential equations (ODEs) model has revealed the stability of the equilibrium points in the system. We have considered both one- and two- parameter bifurcation analysis with respect to parameters associated to the rate...

Mathematical Medicine and Biology, 2005

We construct a mathematical model of the early formation of an atherosclerotic lesion based on a simplification of Russell Ross' paradigm of atherosclerosis as a chronic inflammatory response. Atherosclerosis is a disease characterized by the accumulation of lipid-laden cells in the arterial wall. This disease results in lesions within the artery that may grow into the lumen restricting blood flow and, in critical cases, can rupture causing complete, sudden occlusion of the artery resulting in heart attack, stroke and possibly death. It is now understood that when chemically modified low-density lipoproteins (LDL cholesterol) enter into the wall of the human artery, they can trigger an immune response mediated by biochemical signals sent and received by immune and other cells indigenous to the vasculature. The presence of modified LDL can also corrupt the normal immune function triggering further immune response and ultimately chronic inflammation. In the construction of our mathematical model, we focus on the inflammatory component of the pathogenesis of cardiovascular disease (CVD). Because this study centres on the interplay between chemical and cellular species in the human artery and bloodstream, we employ a model of chemotaxis first given by E. F. Keller and Lee Segel in 1970 and present our model as a coupled system of non-linear reaction diffusion equations describing the state of the various species involved in the disease process. We perform numerical simulations demonstrating that our model captures certain observed features of CVD such as the localization of immune cells, the build-up of lipids and debris and the isolation of a lesion by smooth muscle cells.

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2009

Atherosclerosis begins as an inflammation in blood vessel walls (intima). The inflammatory response of the organism leads to the recruitment of monocytes. Trapped in the intima, they differentiate into macrophages and foam cells leading to the production of inflammatory cytokines and further recruitment of white blood cells. This self-accelerating process, strongly influenced by low-density lipoproteins (cholesterol), results in a dramatic increase of the width of blood vessel walls, formation of an atherosclerotic plaque and, possibly, of its rupture. We suggest a 2D mathematical model of the initiation and development of atherosclerosis which takes into account the concentration of blood cells inside the intima and of pro-and antiinflammatory cytokines. The model represents a reaction-diffusion system in a strip with nonlinear boundary conditions which describe the recruitment of monocytes as a function of the concentration of inflammatory cytokines. We prove the existence of travelling waves described by this system and confirm our previous results which suggest that atherosclerosis develops as a reaction-diffusion wave. The theoretical results are confirmed by the results of numerical simulations.

Journal of Mathematical Biology, 2012

Atherosclerosis begins as an inflammation in blood vessels walls (intima). Antiinflammatory response of the organism leads to the recruitment of monocytes. Trapped in the intima, they differentiate into macrophages and foam cells leading to the production of inflammatory cytokines and further recruitment of white blood cells. This self-accelerating process, strongly influenced by low-density lipoproteins (cholesterol), results in a dramatic increase of the width of blood vessel walls, formation of an atherosclerotic plaque and, possibly, of its rupture. We suggest a 2D mathematical model of the initiation and development of atherosclerosis which takes into account the concentration of blood cells inside the intima and of pro-and anti-inflammatory cytokines. The model represents a reaction-diffusion system in a strip with nonlinear boundary conditions which describe the recruitment of monocytes as a function of the concentration of inflammatory cytokines. We prove the existence of travelling waves described by this system and confirm our previous results which show that atherosclerosis develops as a reaction-diffusion wave. The theoretical results are confirmed by the results of numerical simulations.

International Journal of Biomathematics, 2018

A mathematical model, which takes into account new experimental results about diverse roles of macrophages in the atherosclerosis development, is proposed. Using technic of upper and lower solutions, the existence and uniqueness of its positive solution are justified. After the nondimensionalization, small parameters are found and the multiscale analysis of the corresponding perturbed problem is performed when those parameters tend to zero. In particular, the limit two-dimensional problem, which is a coupled system of reaction–diffusion equations and ordinary differential equations, is derived; the asymptotic approximation is constructed; the uniform pointwise estimate for the difference between the solution of the original problem and the solution of the limit problem as well as the respective [Formula: see text]-estimates for the fluxes are proved.

Journal of Mathematical Biology, 2014

We study a reaction-diffusion mathematical model for the evolution of atherosclerosis as an infiammation process by combining analytical tools with computer-intensive numerical calculations. The computational work involved the calculation of more than sixty thousand solutions of the full reaction-diffusion system and lead to the complete characterisation of the w-limit for every initial condition. Qualitative properties of the solution are rigorously proved, sorne of them hinted at by the numerical study.

Dyn. Contin. Discrete …, 2007

Atherosclerosis is a disease of the vasculature that is characterized by chronic inflammation and the accumulation of lipids and apoptic 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 United States, much of Europe, and parts of Asia. Here we discuss a dynamic model of the biochemical aspects of atherosclerosis. In particular, we consider the interaction between immune response cells in the presence of chemically modified low density lipoprotein which are known to interfere with normal immune function. The general model consists of a system of nonlinear evolution equations governing the interaction of chemical and cellular species leading to the disease initiation and progression.