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.
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, 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.
Frontal polymerization is a process of converting a monomer into a polymer by means of a self-propagating thermal reaction wave. We study initiation of polymerization waves by a high temperature heat source. A five species reaction model is considered with a focus on the evolution of two of these species and the temperature of the mixture. The temperature is tracked from the inert heating to the transition stage. Through an asymptotic analysis, the first correction to the temperature in transition is found as the solution to an integral equation. Two parameters govern the qualitative behavior of the solution to the integral equation. Depending on the magnitude of these parameters, the solution exhibits either bounded or unbounded behavior indicating the onset or inhibition of propagation of a polymerization wave.
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