Discontinuous Galerkin methods for Fisher–Kolmogorov equation with application to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si556.svg" display="inline" id="d1e2166"><mml:mi>α</mml:mi></mml:math>-synuclein spreading in Parkinson’s disease
Mattia Corti, Francesca Bonizzoni, Luca Dede’, Alfio Quarteroni, Paola F. Antonietti
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 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.