阿尔茨海默病：发病和进展的数学模型
Alzheimer's disease: a mathematical model for onset and progression
摘要：
在本文中，我们提出了一个数学模型的基础上的运输和扩散方程的阿尔茨海默病的发病和发展。我们把大脑神经元看作是一个连续的介质，并根据它们的故障程度来构造它们。假设两种不同的机制与疾病的时间演变相关：i）由受损神经元产生的淀粉样蛋白的可溶性聚合物的扩散和聚集; ii）神经元至神经元朊病毒样传递。我们模拟这两个过程的淀粉样蛋白浓度的Smoluchowski方程的系统，再加上一个动力学类型的运输方程的神经元故障的程度的分布函数。第二个方程包含一个积分项，它将疾病的随机发作描述为一个跳跃过程，该过程局限于大脑的特别敏感区域。我们的数值模拟与从早期到晚期不同的疾病在大脑中分布的临床图像具有良好的定性一致性。
英文摘要：
In this paper we propose a mathematical model for the onset and progression of Alzheimer's disease based on transport and diffusion equations. We regard brain neurons as a continuous medium, and structure them by their degree of malfunctioning. Two different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble polymers of amyloid, produced by damaged neurons; ii) neuron-to-neuron prion-like transmission. We model these two processes by a system of Smoluchowski equations for the amyloid concentration, coupled to a kinetic-type transport equation for the distribution function of the degree of malfunctioning of neurons. The second equation contains an integral term describing the random onset of the disease as a jump process localised in particularly sensitive areas of the brain. Our numerical simulations are in good qualitative agreement with clinical images of the disease distribution in the brain which vary from early to advanced stages.
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