Alzheimer's disease: a mathematical model for onset and progression

In this paper 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; 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 localised 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.