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

Keywords

Advanced Numerical Methods for Scientific Computing
Living Systems and Precision Medicine
Code:
51/2024
Title:
A discontinuous Galerkin method for the three-dimensional heterodimer model with application to prion-like proteins’ dynamics
Date:
Tuesday 23rd July 2024
Author(s):
Antonietti, P.F.; Corti, M.; Lorenzon, G.
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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.