Reaction-Diffusion Models of Protein Propagation in Neurodegenerative Diseases: The Case of Alzheimer's

The study of neurodegenerative diseases represents an open challenge for modern medicine today. The aim of this thesis is to study the progression of these conditions inside the brain - in particular Alzheimer's disease - using reaction-diffusion models. The present research links the continuous diffusion model with the brain connectome, which is described by a discrete network model. Using COMSOL Multiphysics software, a three-dimensional geometry of the right brain hemisphere and its connectome structure was constructed. The study initially implemented the Fisher-Kolmogorov model to investigate the role of tau protein in initiating the disease. Subsequently, the focus shifted to the interaction between tau and β -amyloid proteins, as recent studies suggest that the spread of β-amyloid is the key driver of early cognitive decline and highly interacts with tau. A tailored model was then developed to apply the findings to a cohort of patients at different stages of the disease. All numerical simulations were followed by a parametric study investigating the rates of conversion of healthy proteins into misfolded forms, the interaction parameter between pathological species, and the time required for complete invasion of the cerebral cortex. In the analysis of the patient cohort, simulation results were compared to clinical data to assess the correspondence between modelled predictions and observed disease progression. The results demonstrate that the implemented models effectively reproduce the spatial and temporal patterns of misfolded protein propagation, offering a realistic representation of disease dynamics. Future work could extend this framework by incorporating more complex structures to describe the anatomy of the cerebral cortex. From a modelling perspective, one could introduce patterns that take into account additional variables, such as the interaction with cerebral atrophy.