This 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.

本文基于外科医生 Axel Haverich 博士教授提出的新假设（Circulation 135(3):205–207, 2017），讨论了动脉粥样硬化的数学模型。动脉粥样硬化是指动脉壁增厚。目前，有两种观点可以解释这种现象的根源：物质沉积导致的增厚和炎症过度生长导致的增厚。这里支持的假设是第二个范式，指出动脉粥样硬化只不过是由于壁营养破坏而导致的壁组织的炎症反应。众所周知，除了通过管腔的血液中的营养物自然扩散之外，称为血管滋养管（VV）的毛细血管网络还负责管壁的营养。由于病毒、细菌和非常细的灰尘颗粒（例如称为 PM 2.5 的空气污染物）堵塞血管滋养管，可能会导致营养物质流向壁组织的中断。它们首先通过呼吸系统进入体内，然后到达循环系统。因此，在新的假设中，动脉粥样硬化血管的根源被认为是滋养血管的微血管的功能障碍。大量临床观察支持这一假设。最近，与这项工作高度相关的是，在 COVID-19 大流行之后，肺部最常见的疾病之一归因于肺动脉粥样硬化，参见 Boyle 和 Haverich (Eur J Cardio Thorac Surg 58(6):1109– 1110, 2020）。在这项工作中，基于多物理场数学模型开发了一个通用框架，以捕获壁变形、营养可用性和炎症反应。 对于机械响应，调用各向异性本构关系来解释动脉壁中胶原纤维的存在。扩散反应方程控制着营养物在壁内的传输。使用具有双井电势的相场型方程描述炎症（过度生长），该方程捕获组织的两个区域（即健康部分和过度生长部分）之间的尖锐界面。生长的运动学通过梯度变形的经典乘法分解来处理。炎症通过相场变量来表示。提出了一种新颖的相场驱动机制来模拟病理学的进展。该模型是 3D 的并且完全基于问题的连续描述。数值实施是使用FEM 进行的。将模型的预测与临床观察结果进行比较。模型和数值工具具有通用性和适用性。