Discontinuous Galerkin Methods for Fisher-Kolmogorov Equation with Application to Alpha-Synuclein Spreading in Parkinson’s Disease

10/2023

Discontinuous Galerkin Methods for Fisher-Kolmogorov Equation with Application to Alpha-Synuclein Spreading in Parkinson’s Disease

Tuesday 14th February 2023

Corti, M.; Antonietti, P.F.; Bonizzoni, F.; Dede', L., Quarteroni, A.

The spreading of prion proteins is at the basis of brain neurodegeneration. The 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, which allows us to accurately simulate the wavefronts typically observed in the prionic spreading. We prove stability and a priori error estimates for the semi-discrete formulation. 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 patient-specific brain geometry reconstructed from magnetic resonance images.

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Computer Methods in Applied Mechanics and Engineering 417(Part A) (2023), 116450