AbstractThis paper deals with the mathematical modeling of atherosclerosis based on a novel hypothesis proposed by a surgeon, Prof. Dr. Axel Haverich (Circulation 135(3):205–207, 2017). Atherosclerosis is referred as the thickening of the artery walls. Currently, there are two schools of thoughts for explaining the root of such phenomenon: thickening due to substance deposition and thickening as a result of inflammatory overgrowth. The hypothesis favored here is the second paradigm stating that the atherosclerosis is nothing else than the inflammatory response of of the wall tissues as a result of disruption in wall nourishment. It is known that a network of capillaries called vasa vasorum (VV) accounts for the nourishment of the wall in addition to the natural diffusion of nutrient from the blood passing through the lumen. Disruption of nutrient flow to the wall tissues may take place due to the occlusion of vasa vasorums with viruses, bacteria and very fine dust particles such as air pollutants referred to as PM 2.5. They can enter the body through the respiratory system at the first place and then reach the circulatory system. Hence in the new hypothesis, the root of atherosclerotic vessel is perceived as the malfunction of microvessels that nourish the vessel. A large number of clinical observation support this hypothesis. Recently and highly related to this work, and after the COVID-19 pandemic, one of the most prevalent disease in the lungs are attributed to the atherosclerotic pulmonary arteries, see Boyle and Haverich (Eur J Cardio Thorac Surg 58(6):1109–1110, 2020). In this work, a general framework is developed based on a multiphysics mathematical model to capture the wall deformation, nutrient availability and the inflammatory response. For the mechanical response an anisotropic constitutive relation is invoked in order to account for the presence of collagen fibers in the artery wall. A diffusion–reaction equation governs the transport of the nutrient within the wall. The inflammation (overgrowth) is described using a phase-field type equation with a double well potential which captures a sharp interface between two regions of the tissues, namely the healthy and the overgrowing part. The kinematics of the growth is treated by classical multiplicative decomposition of the gradient deformation. The inflammation is represented by means of a phase-field variable. A novel driving mechanism for the phase field is proposed for modeling the progression of the pathology. The model is 3D and fully based on the continuum description of the problem. The numerical implementation is carried out using FEM. Predictions of the model are compared with the clinical observations. The versatility and applicability of the model and the numerical tool allow.
Diffusion–reaction compartmental models formulated in a continuum mechanics framework: application to COVID-19, mathematical analysis, and numerical study