Bertsch Michiel, Franchi Bruno, Marcello Norina, Tesi Maria Carla, Tosin Andrea
Dipartimento di Matematica, Università di Roma 'Tor Vergata', Via della Ricerca Scientifica 1, 00133 Roma, Italy and Istituto per le Applicazoni del Calcolo 'M. Picone', Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185 Roma, Italy.
Department of Mathematics, University of Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy.
Math Med Biol. 2017 Jun 1;34(2):193-214. doi: 10.1093/imammb/dqw003.
In this article 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 and 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 localized 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.
在本文中，我们基于输运和扩散方程提出了一个关于阿尔茨海默病发病和进展的数学模型。我们将脑神经元视为连续介质，并根据其功能失调程度对其进行结构划分。假设两种不同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可溶性聚合物的扩散和聚集，以及ii）神经元间的朊病毒样传播。我们通过一个针对淀粉样蛋白浓度的斯莫卢霍夫斯基方程组来模拟这两个过程，该方程组与一个关于神经元功能失调程度分布函数的动力学型输运方程相耦合。第二个方程包含一个积分项，将疾病的随机发病描述为定位于大脑特别敏感区域的跳跃过程。我们的数值模拟在定性上与大脑中疾病分布的临床图像高度一致，这些图像涵盖了从早期到晚期的不同阶段。
