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arXiv · 2302.07126

Discontinuous Galerkin Methods for Fisher-Kolmogorov Equation with Application to $\alpha$-Synuclein Spreading in Parkinson's Disease

Mattia Corti·Francesca Bonizzoni·Luca Dede'·Alfio Quarteroni·Paola F. Antonietti
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Abstract

This spreading of prion proteins is at the basis of brain neurodegeneration. This paper deals with the numerical modelling of the misfolding process of $\alpha$-synuclein in Parkinson's disease. We introduce and analyze a discontinuous Galerkin method for the semi-discrete approximation of the Fisher-Kolmogorov (FK) equation that can be employed to model the process. We employ a discontinuous Galerkin method on polygonal and polyhedral grids (PolyDG) for space discretization, to accurately simulate the wavefronts typically observed in the prionic spreading and we prove stability and a priori error estimates. Next, we use a Crank-Nicolson scheme to advance in time. For the numerical verification of our numerical model, we first consider a manufactured solution, and then we consider a case with wavefront propagation in two-dimensional polygonal grids. Next, we carry out a simulation of $\alpha$-synuclein spreading in a two-dimensional brain slice in the sagittal plane with a polygonal agglomerated grid that takes full advantage of the flexibility of PolyDG approximation. Finally, we present a simulation in a three-dimensional geometry reconstructed from magnetic resonance images of a patient's brain.

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Mattia Corti, Francesca Bonizzoni, Luca Dede', Alfio Quarteroni, Paola F. Antonietti. 2023-02-14. Discontinuous Galerkin Methods for Fisher-Kolmogorov Equation with Application to $\alpha$-Synuclein Spreading in Parkinson's Disease. https://doi.org/10.1016/j.cma.2023.116450

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