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arXiv 2006.01749physics.bio-ph
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β淀粉样蛋白和tau蛋白协同作用对阿尔茨海默病动力学的数学模型的敏感性分析

A sensitivity analysis of a mathematical model for the synergistic interplay of Amyloid beta and tau on the dynamics of Alzheimer's disease

Michiel Bertsch, Bruno Franchi, Valentina Meschini, Maria Carla Tesi, Andrea Tosin

摘要:

我们提出了一个数学模型的基础上传输和扩散方程的阿尔茨海默病的发病和进展。我们将大脑神经元视为一个连续的介质,并根据它们的故障程度来构造它们。假设三种不同的机制与疾病的时间演变相关:i)可溶性淀粉样蛋白β的扩散和聚集,ii)磷酸化tau蛋白的作用和iii)疾病的神经元至神经元朊病毒样传播。我们模拟这些过程的Smoluchowski方程系统的淀粉样蛋白β浓度,tau蛋白的动力学的演化方程和动力学类型的运输方程的神经元故障的程度的分布函数。后一个方程包含一个积分项,它将疾病的随机发作描述为位于大脑特别敏感区域的跳跃过程。我们特别感兴趣的是研究淀粉样蛋白β和tau蛋白的协同相互作用对阿尔茨海默病动力学的影响。我们的数值模拟的输出,虽然在2D中具有过度简化的几何形状,但与关于大脑中的疾病分布(从早期到晚期各不相同)以及tau对疾病动力学的影响的临床发现具有良好的定性一致性。

英文摘要:

We propose a mathematical model for the onset and progression of Alzheimer's disease based on transport and diffusion equations. We treat brain neurons as a continuous medium and structure them by their degree of malfunctioning. Three different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble Amyloid beta, ii) effects of phosphorylated tau protein and iii) neuron-to-neuron prion-like transmission of the disease. We model these processes by a system of Smoluchowski equations for the Amyloid beta concentration, an evolution equation for the dynamics of tau protein and a kinetic-type transport equation for the distribution function of the degree of malfunctioning of neurons. The latter equation contains an integral term describing the random onset of the disease as a jump process localized in particularly sensitive areas of the brain. We are particularly interested in investigating the effects of the synergistic interplay of Amyloid beta and tau on the dynamics of Alzheimer's disease. The output of our numerical simulations, although in 2D with an over-simplified geometry, is in good qualitative agreement with clinical findings concerning both the disease distribution in the brain, which varies from early to advanced stages, and the effects of tau on the dynamics of the disease.

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