Structure Preserving Polytopal Discontinuous Galerkin Methods for the Numerical Modeling of Neurodegenerative Diseases

TL;DR

This paper introduces a structure-preserving discontinuous Galerkin method on polygonal grids for simulating the spread of misfolded proteins in neurodegenerative diseases, ensuring non-negativity and accurate modeling.

Contribution

The paper presents a novel positivity-preserving numerical scheme for the Fisher-Kolmogorov equation using polygonal grids and $ heta$-method time integration, tailored for neurodegenerative disease modeling.

Findings

The method guarantees non-negative solutions, crucial for biological accuracy.

Numerical tests validate convergence and structure preservation.

Simulations successfully model protein spreading in realistic brain geometries.

Abstract

Many neurodegenerative diseases are connected to the spreading of misfolded prionic proteins. In this paper, we analyse the process of misfolding and spreading of both $\alpha$-synuclein and Amyloid-$\beta$, related to Parkinson's and Alzheimer's diseases, respectively. We introduce and analyze a positivity-preserving numerical method for the discretization of the Fisher-Kolmogorov equation, modelling accumulation and spreading of prionic proteins. The proposed approximation method is based on the discontinuous Galerkin method on polygonal and polyhedral grids for space discretization and on $\vartheta-$method time integration scheme. We prove the existence of the discrete solution and a convergence result where the Implicit Euler scheme is employed for time integration. We show that the proposed approach is structure-preserving, in the sense that it guaranteed that the discrete solution is non-negative, a feature that is of paramount importance in practical application. The numerical verification of our numerical model is performed both using a manufactured solution and considering wavefront propagation in two-dimensional polygonal grids. Next, we present a simulation of $\alpha$-synuclein spreading in a two-dimensional brain slice in the sagittal plane. The polygonal mesh for this simulation is agglomerated maintaining the distinction of white and grey matter, taking advantage of the flexibility of PolyDG methods in the mesh construction. Finally, we simulate the spreading of Amyloid-$\beta$ in a patient-specific setting by using a three-dimensional geometry reconstructed from magnetic resonance images and an initial condition reconstructed from positron emission tomography. Our numerical simulations confirm that the proposed method is able to capture the evolution of Parkinson's and Alzheimer's diseases.