Discontinuous Galerkin methods for Fisher-Kolmogorov equation with application to α-synuclein spreading in Parkinson's disease

This spreading of prion proteins is at the basis of brain neurodegeneration. This paper deals with the numerical modelling of the misfolding process of α-synuclein in Parkinson’s disease. We introduce and analyse 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 α-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.