A discontinuous Galerkin method for the three-dimensional heterodimer model with application to prion-like proteins' dynamics

TL;DR

This paper develops a high-order discontinuous Galerkin numerical method for simulating three-dimensional heterodimer models of prion-like protein dynamics, providing insights into neurotoxic pattern formation in brain diseases.

Contribution

It introduces a novel DG discretization for complex 3D reaction-diffusion systems modeling protein misfolding, with theoretical error analysis and realistic brain simulations.

Findings

High-order accuracy achieved in 3D brain geometries

Validated convergence rates through numerical tests

Simulated realistic neurotoxic patterns in brain models

Abstract

Neurocognitive disorders, such as Alzheimer's and Parkinson's, have a wide social impact. These proteinopathies involve misfolded proteins accumulating into neurotoxic aggregates. Mathematical and computational models describing the prion-like dynamics offer an analytical basis to study the diseases' evolution and a computational framework for exploring potential therapies. This work focuses on the heterodimer model in a three-dimensional setting, a reactive-diffusive system of nonlinear partial differential equations describing the evolution of both healthy and misfolded proteins. We investigate traveling wave solutions and diffusion-driven instabilities as a mechanism of neurotoxic pattern formation. For the considered mathematical model, we propose a space discretization, relying on the Discontinuous Galerkin method on polytopal/polyhedral grids, allowing high-order accuracy and flexible handling of the complicated brain's geometry. Further, we present a priori error estimates for the semi-discrete formulation and we perform convergence tests to verify the theoretical results. Finally, we conduct simulations using realistic data on a three-dimensional brain mesh reconstructed from medical images.