A sensitivity analysis of a mathematical model for the synergistic interplay of amyloid beta and tau on the dynamics of Alzheimer’s disease

Abstract

We propose a mathematical model for the onset and progression of Alzheimer’s disease based on transport and diffusion equations. We treat brain neurons as a continuous medium and structure them by their degree of malfunctioning. Three different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble Amyloid beta, ii) effects of phosphorylated tau protein and iii) neuron-to-neuron prion-like transmission of the disease. We model these processes by a system of Smoluchowski equations for the Amyloid beta concentration, an evolution equation for the dynamics of tau protein and a kinetic-type transport equation for the distribution function of the degree of malfunctioning of neurons. The latter equation contains an integral term describing the random onset of the disease as a jump process localized in particularly sensitive areas of the brain. We are particularly interested in investigating the effects of the synergistic interplay of Amyloid beta and tau on the dynamics of Alzheimer’s disease. The output of our numerical simulations, although in 2D with an over-simplified geometry, is in good qualitative agreement with clinical findings concerning both the disease distribution in the brain, which varies from early to advanced stages, and the effects of tau on the dynamics of the disease.

Statement of Significance

We propose an in silico study of the onset and progression of Alzheimer’s disease (AD) in the brain by means of a mathematical model formulated in terms of kinetic and macroscopic integro-differential equations. From the biological side, our model takes into account the synergistic effect of Amiloid beta and phosphorylated tau protein and investigates the impact of their interplay on AD dynamics. From the mathematical side, unlike several other models present in the literature, our model does not focus on the detailed description of specific intra-cellular biochemical processes. It takes instead an aggregate point of view and, thanks to a multiscale approach inspired by statistical mechanics, describes the spatio-temporal patterns of the degree of neuronal malfunctioning due to AD in macroscopic portions of the brain tissue.