Xie Xuming
Department of Mathematics, Morgan State University, Baltimore, MD 21251.
J Math Anal Appl. 2022 Jan 15;505(2). doi: 10.1016/j.jmaa.2021.125606. Epub 2021 Aug 21.
Atherosclerosis is a leading cause of death in the United States and worldwide; it emerges as a result of multiple dynamical cell processes including hemodynamics, endothelial damage, innate immunity and sterol biochemistry. Making matters worse, nearly 21 million Americans have diabetes, a disease where patients' cells cannot efficiently take in dietary sugar, causing it to build up in the blood. In part because diabetes increases atherosclerosis-related inflammation, diabetic patients are twice as likely to have a heart attack or stroke. Past work has shown that hyperglycemia and insulin resistance alter function of multiple cell types, including endothelium, smooth muscle cells and platelets, indicating the extent of vascular disarray in this disease. Although the pathophysiology of diabetic vascular disease is generally understood, there is no mathematical model to date that includes the effect of diabetes on plaque growth. In this paper, we propose a mathematical model for diabetic atherosclerosis; the model is given by a system of partial differential equations with a free boundary. We establish local existence and uniqueness of solution to the model. The methodology is to use Hanzawa transformation to reduce the free boundary to a fixed boundary and reduce the system of partial differential equations to an abstract evolution equation in Banach spaces, and apply the theory of analytic semigroup.
动脉粥样硬化是美国乃至全球的主要死因；它是多种动态细胞过程的结果，包括血液动力学、内皮损伤、先天免疫和甾醇生物化学。更糟糕的是，近2100万美国人患有糖尿病，这种疾病患者的细胞无法有效摄取膳食糖分，导致糖分在血液中积聚。部分由于糖尿病会增加与动脉粥样硬化相关的炎症，糖尿病患者心脏病发作或中风的可能性是正常人的两倍。过去的研究表明，高血糖和胰岛素抵抗会改变多种细胞类型的功能，包括内皮细胞、平滑肌细胞和血小板，这表明了这种疾病中血管紊乱的程度。尽管糖尿病血管疾病的病理生理学已被普遍了解，但迄今为止还没有一个数学模型能够包含糖尿病对斑块生长的影响。在本文中，我们提出了一个糖尿病动脉粥样硬化的数学模型；该模型由一个带有自由边界的偏微分方程组给出。我们建立了该模型解的局部存在性和唯一性。方法是利用半泽变换将自由边界化为固定边界，并将偏微分方程组化为巴拿赫空间中的一个抽象发展方程，然后应用解析半群理论。
