论文标题
对阿尔茨海默氏病动力学的淀粉样β和TAU协同相互作用的数学模型的灵敏度分析
A sensitivity analysis of a mathematical model for the synergistic interplay of Amyloid beta and tau on the dynamics of Alzheimer's disease
论文作者
论文摘要
我们提出了一个基于运输和扩散方程的阿尔茨海默氏病的发作和进展的数学模型。我们将脑神经元视为连续培养基，并通过其故障程度来构造它们。假定三种不同的机制与该疾病的时间演变有关：i）可溶性淀粉样蛋白β的扩散和凝聚，ii）磷酸化的tau蛋白和iii）疾病的神经元对神经元的prion元的疾病的影响。我们通过Smoluchowski方程系统的淀粉样β浓度，Tau蛋白动力学的进化方程和动力学型传输方程来对这些过程进行建模，以实现神经元故障程度的分布功能。后一个方程式包含一个不可或缺的术语，该术语描述了该疾病的随机发作是局部在大脑特别敏感区域中的跳跃过程。我们特别有兴趣研究淀粉样β和TAU的协同相互作用对阿尔茨海默氏病动力学的影响。我们的数值模拟的输出，尽管在2D中具有过度简化的几何形状，但与临床发现符合有关大脑中疾病分布的临床发现，该发现从早期到晚期变化，以及TAU对疾病动力学的影响。
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.
