Mathematical models for brain diseases: formation of senile plaques and neurofibrillary tangles in Alzheimer's disease.

Alzheimer’s disease (AD) is characterized by the presence of two specific structures in brain of patients: senile plaques (SP) and neurofibrillary tangles (NTFs). Oligomers of beta amyloid protein are the main component of SP, while NFTs are constituted by oligomers of hyperphosphorylated tau protein. The aim of this thesis is to present two studies concerning the process of diffusion and agglomeration of tau protein first into fibrils and eventually up to neurofibrillary tangles. The aforesaid researches share the use of Smoluchowski-type equations (including diffusion and monomers production) on a finite weighted graph, which is conceived as a theoretical approximation of the human brain, where each vertex represents a cerebral area of interest, while the connections between them are described by the edges. In the first chapter, we present a model for the aggregation and diffusion of hyperphosphorylated tau based on a finite dimensional Smoluchowski-type system on a finite graph, aimed at reproducing tau patterns obtained through suitable in vivo measurements. In such model, clusters whose length exceeds an a priori established critical threshold are defined to be tangles. Statistical analysis is performed, comparing the model-predicted tau concentrations (in monomeric, oligomeric form and in tangles) with cortical atrophy data and tau measurements in AD patients provided by in vivo neuroimaging. The second research considers the process of aggregation in tangles of hyperphosphorylated tau protein as a result of a coagulation process of this protein. The formation of neurofibrillary tangles is mathematically characterized in terms of a sol-gel phase transition for a polymerization problem, modelled by means of infinite dimensional Smoluchowski-type system on a finite graph. The process of formation of gel is a suitable tool for the description of the formation of neurofibrillary tangles, since it allows one to provide insights on the nature of this phenomenon in AD scenario.