Macroscopic modelling of Alzheimer’s disease: difficulties and challenges

In the context of Alzheimer’s disease (AD), in silico research aims at giving complement- ary and better insight into the complex mechanisms which determine the development of AD. One of its important aspects is the construction of macroscopic mathematical models which are the basis for numerical simulations. In this paper we discuss some of the general and fundamental difficulties of macroscopic modelling of AD. In addition we formulate a mathem- atical model in the case of a specific problem in an early stage of AD, namely the propagation of pathological τ protein from the entorhinal cortex to the hippocampal region. The main feature of this model consists in the representation of the brain through two superposed finite graphs, which have the same vertices (that, roughly speaking, can be thought as parcels of a brain atlas), but different edges. We call these graphs “proximity graph” and “connectiv- ity graph”, respectively. The edges of the first graph take into account the distances of the vertices and the heterogeneity of the cerebral parenchyma, whereas the edges of the second graph represent the connections by white-matter fiber pathways between different structures. The diffusion of the proteins Aβ and τ are described through the Laplace operators on the graphs, whereas the phenomenon of aggregation of the proteins leading ultimately to senile plaques and neuro-fibrillar tangles (as already observed by A. Alzheimer in 1907) is modelled by means of the classical Smoluchowski aggregation system.