Mathematical modelling of the atherosclerotic plaque formation

Abstract

Abstract. This article is devoted to the construction of a mathematical model describing the early formation of atherosclerotic lesions. Following the work of El Khatib, Genieys and Volpert [2], we model atherosclerosis as an inflammatory disease. We consider that the inflammatory process starts with the penetration of Low Density Lipoproteins cholesterol in the intima. This phenomenon is related to the local blood flow dynamics. Using a system of reaction-diffusion equations, we first provide a one-dimensional model of lesion growth. Then we perform numerical simulations on a two-dimensional geometry mimicking the carotid artery. We couple the previous mathematical model with blood flow and we provide a model in which the lesion appears in the area of lower shear stress.Résumé. Cet article est consacréà la construction d'un modèle mathématique décrivant la formation de lésions précoces dans l'athérosclérose. Suite au travail de El Khatib, Genieys and Volpert [2], nous modélisons l'athérosclérose comme une maladie inflammatoire. Nous considérons que le procesus inflammatoire débute avec la pénétration de "Low Density Lipoproteins cholesterol" dans l'intima. Ce phénomène est reliéà la dynamique locale de l'écoulement du sang. En utilisant un système d'équations de réaction-diffusion, nous donnons tout d'abord un modèle monodimensionel de croissance de lésion. Puis, dans le cas d'une bifurcation bidimensionnelle, nous couplons le modèle mathématique précédent avec l'écoulement du sang et nous fournissons un modèle dans lequel la lésion apparaît dans la zone de plus faible cisaillement.

Mathematical modelling of atheroma plaque formation and development in coronary arteries

“…More complex studies were presented by Siogkas et al [33], who included in their model oxidized LDL, macrophages and cytokines, considering that all the LDL molecules and the monocytes were oxidized and differentiated, respectively, at the instant in which these agents pass through the endothelium. A similar study was presented by Calvez et al [34] from a mathematical point of view, but their study also included the foam cell. Ougrinovskaia et al [25] explored the uptake of cholesterol by different scavenger receptors of macrophages during earlystage atherosclerosis using an ordinary differential equation (ODE) model.…”

“…Owing to the size of the monocytes, the convection term has been disregarded [34,35]. Equation (3.16) has several terms.…”

“…Finally, as the volume of the atheroma plaque is mainly due to the foam cells, SMCs and collagen, the volume contribution of the rest of the species has been neglected. It has been assumed that monocyte and macrophage sizes (before ingesting oxidized LDL) as well as cytokine, LDL and oxidized LDL sizes are small in comparison with the sizes of foam cells and SMCs [34]. Thus, the isotropic growth of the atheroma plaque can be estimated as…”

3D Modeling of Plaque Progression in the Human Coronary Artery

“…Three additional partial differential equations were used for solving the inflammatory process [8,9]:…”

Non-Gaussianity and dynamical trapping in locally activated random walks

“…1) [16,17]. As the cell grows, its ability to move decreases and the ultimate result is the formation of an atherosclerotic plaque [18].…”