A mathematical macroscopic model for the onset and progression of Alzheimer's disease

In this work we deal with the study of a macroscopic mathematical model which describes the onset and progression of Alzheimer’s disease (AD) in the human brain. In particular, we analyze in detail the well-posedness of the model itself. The model is based on the so-called “amyloid cascade hypothesis” together with the “prionoid hypothesis”, which represents the spreading of the disease through neuron-to-neuron transmission. In particular, from a mathematical point of view, the model consists of a transport equation for a probability measure, coupled with a nonlinear Smoluchowski-type system with a diffusion term. In view of the features of the biological phenomena we are dealing with, the main hallmark of such a system is that the two groups of equations cannot be separated and they must be considered together. The main tools to prove existence and uniqueness of the solution of the problem coupled with appropriate initial and boundary conditions are: the definition of an appropriate metric space obtained by endowing with the Wasserstein distance the space of Borel probability measures supported on a bounded interval; the transport of measures along the characteristics of the transport equation; an ad hoc formulation of the classical contraction theorem and a priori estimates for solutions of the reaction diffusion equations.