与胆固醇比值相关的斑块形成自由边界模型的分歧分析
Bifurcation analysis of a free boundary model of plaque formation associated with the cholesterol ratio
摘要：
低密度脂蛋白（LDL）/高密度脂蛋白（HDL）-胆固醇比值已被证明与心血管风险评估高度相关。有没有可能用数学方法来量化这种相关性？本文对动脉粥样硬化早期斑块形成的自由边界数学模型进行了分岔分析。这种分叉分析，LDL/HDL的比例，是基于径向对称的稳态解的显式制定。通过对这些解进行扰动分析，我们建立了分支的存在性，并推导出了不同模式下发生分支的理论条件。然后我们还分析了径向对称稳态解的稳定性，并进行了数值模拟，以验证所有的理论结果。
英文摘要：
The low-density lipoprotein (LDL)/high-density lipoprotein (HDL)-cholesterol ratio has been shown a high correlation with the cardiovascular risk assessment. Is it possible to quantify the correlation mathematically? In this paper, we develop a bifurcation analysis for a mathematical model of the plaque formation with a free boundary in the early stage of atherosclerosis. This bifurcation analysis, to the ratio of LDL/HDL, is based on the explicit formulation of radially symmetric steady-state solutions. By performing the perturbation analysis to these solutions, we establish the existence of bifurcation branches and derive a theoretical condition that a bifurcation occurs for different modes. Then we also analyze the stability of radially symmetric steady-state solutions and conduct numerical simulations to verify all the theoretical results.
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