The paper presents a conceptual mathematical model for Alzheimer’s disease (AD). According to the so-called amyloid cascade hypothesis, we assume that the progression of AD is associated with the presence of soluble toxic oligomers of beta-amyloid. Monomers of this protein are produced normally throughout life, but a change in the metabolism may increase their total production and, through aggregation, ultimately results in a large quantity of highly toxic polymers. The evolution from monomeric amyloid produced by the neurons to senile plaques (long and insoluble polymeric amyloid chains) is modelled by a system of ordinary differential equations (ODEs), in the spirit of the Smoluchowski equation. The basic assumptions of the model are that, at the scale of suitably small representative elementary volumes (REVs) of the brain, the production of monomers depends on the average degradation of the neurons and in turn, at a much slower timescale, the degradation is caused by the number of toxic oligomers. To mimic prion-like diffusion of the disease in the brain, we introduce an interaction among adjacent REVs that can be assumed to be isotropic or to follow given preferential patterns. We display the results of numerical simulations which are obtained under some simplifying assumptions. For instance, the amyloid cascade is modelled by just three ordinary differential equations (ODEs), and the simulations refer to abstract 2D domains, simplifications which can be easily avoided at the price of some additional computational costs. Since the model is suitably flexible to incorporate other mechanisms and geometries, we believe that it can be generalised to describe more realistic situations.

中文翻译：

---

淀粉样蛋白级联假说和阿尔茨海默病：一个数学模型

本文提出了阿尔茨海默病 (AD) 的概念数学模型。根据所谓的淀粉样蛋白级联假说，我们假设 AD 的进展与 β-淀粉样蛋白的可溶性毒性低聚物的存在有关。这种蛋白质的单体在整个生命过程中都会正常产生，但新陈代谢的变化可能会增加它们的总产量，并通过聚集最终导致大量剧毒聚合物。本着 Smoluchowski 方程的精神，从神经元产生的单体淀粉样蛋白到老年斑（长且不溶的聚合淀粉样蛋白链）的演变由常微分方程 (ODE) 系统建模。该模型的基本假设是，在大脑的适当小的代表性基本体积（REV）的范围内，单体的产生取决于神经元的平均降解，反过来，在慢得多的时间尺度上，降解是由有毒低聚物的数量引起的。为了模拟这种疾病在大脑中的朊病毒样扩散，我们引入了相邻 REV 之间的相互作用，这种相互作用可以假定为各向同性或遵循给定的优先模式。我们展示了在一些简化假设下获得的数值模拟结果。例如，淀粉样蛋白级联仅由三个常微分方程 (ODE) 建模，并且模拟涉及抽象的 2D 域，可以很容易地避免以一些额外的计算成本为代价的简化。由于该模型具有适当的灵活性，可以结合其他机制和几何形状，