Mathematical Models for the Role of Toxic Proteins in Alzheimer's Disease

Alzheimer’s disease (AD) is a neurodegenerative disease leading to dementia, through a progressive decline in memory and other cognitive functions. Presently there are more than 50 million people suffering of AD and related disorders and by 2050 this figure is expected to increase to 150 million. Despite a rapidly growing amount of clinical data, there is no effective medical treatment to stop or slow down AD and many questions on the causes of AD remain unanswered. In this context macroscopic mathematical modeling and numerical simulation (so-called in silico research) are natural tools to provide additional insight, for example by simulating specific therapies or modeling hypotheses. AD is one of the neurodegenerative diseases involving more than one neurotoxic protein: beta-amyloid (Aβ) and pathological tau (τ). In his famous experimental post-mortem observations in 1907, Alois Alzheimer discovered extracellular plaques and intracellular neurofibrillary tangles (NFTs). By now we know that plaques contain Aβ and NFTs consist of pathological τ . There has been (and there still is) a lot of scientific debate on the role of the two proteins in AD and up to date Aβ and τ remain the major therapeutic targets for the treatment of the disease. Recent literature suggests that the interplay between the two proteins, as well as its timing, might be crucial for the development of the disease and should be taken into account when developing new therapies. In this talk we present the contributions of several authors carried on the last few years that can be distinguished as follows:

• Models focused on the role of Aβ both at a microscopic and at macroscopic level (the so-called Amyloid Cascate Hypothesis). Aggregation and diffusion of Aβ are described by means of a system of Smoluchowski equations.
• Models for simultaneous aggregation and diffusion of both Aβ and τ and their interaction, as well as for the evolution of the disease. Again the model relies on a double system of Smoluchowski equations in R 3 , where a transport equation describes the evolution of the disease.
• Relying on the so-called notion of connectome, the previous model is formulated in terms of analogous equations on a double graph.