Polytopal discontinuous Galerkin methods for the prionic proteins' spreading in neurodegeneration
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Neurodegenerative diseases have a significant global impact affecting millions of individuals worldwide. Some of them, known as proteinopathies (for example Alzheimer's and Parkinson's diseases), are characterized by the accumulation and propagation of toxic proteins, known as prions. Mathematical models of prion dynamics play a crucial role in understanding disease progression. For this reason, several different models have been proposed in the literature to describe the misfolding process, with various levels of detail. In this context, we introduce and analyze a discontinuous Galerkin method on polytopal grids (PolyDG) for the semi-discrete approximation of different nonlinear models of population dynamics, such as the Fisher-Kolmogorov (FK) [1] and the heterodimer model [2]. We employ a PolyDG method for space discretization, which allows us to simulate the wavefronts typically observed in the prionic spreading. We prove stability and a-priori error estimates for the semi-discrete formulation. Finally, we carry out realistic simulations in three-dimensional and two-dimensional patient-specific brain geometries reconstructed from magnetic resonance images.
