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Seminario del 2021
2021
29 aprile
Maria Carla Tesi
nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI
Seminario di analisi matematica
There is currently a great deal of interest in the scientific community in investigating the effects of the synergistic interplay of Amyloid beta and tau on the dynamics of Alzheimer’s disease.
I will present a mathematical model for the onset and progression of Alzheimer’s disease based on transport and diffusion equations for the two proteins. In the model neurons are treated as a continuous medium and structured by their degree of mal- functioning. 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 misfolded tau protein and iii) neuron-to- neuron prion-like transmission of the disease. These processes are modelled 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. I will explain in detail the structure of the model and give a hint of the main results obtained and the techniques used for the purpose. Eventually I will also show the output of some numerical simulations, of some significance even if performed in an over-simplified 2D geometry.