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

TL;DR

The paper addresses prion-like spreading in neurodegenerative disease by coupling native and misfolded protein dynamics through the heterodimer model in a 3D reaction-diffusion framework. It introduces a high-order polytopal discontinuous Galerkin (PolyDG) discretization on arbitrary polyhedral brain meshes and proves stability and a priori error estimates for the semi-discrete system, with fully discrete time stepping via a theta-method. Convergence tests in 2D and 3D, plus simulations on MRI-based brain geometries, validate the method and demonstrate realistic α-synuclein propagation patterns that align with Braak staging, including biomarker curves across regions. The work provides a robust, high-fidelity computational framework for analyzing prion-like disease progression and potential, region-specific therapeutic interventions, highlighting the importance of three-dimensional geometry and anisotropic diffusion in accurate modeling.

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.

Figures (12)

• Figure 1: Test case of Section \ref{['subsubsection:realistic_coefficients_trivial_domain']}. (a) Computed errors in the ${\left\vert\left\vert\left\vert \cdot \right\vert\right\vert\right\vert_{\epsilon}}$ norm versus the mesh size $h$ (loglog scale) for different polynomial approximation degrees $p=1,2,3,4$. (b) Computed errors in the $\|\cdot \|_{L^2(\Omega)}$ and $\|\cdot \|_{DG}$ norms, versus the mesh size $h$ (loglog scale) for different polynomial approximation degrees $p=1,2,3,4$. (c) Computed errors in the ${\left\vert\left\vert\left\vert \cdot \right\vert\right\vert\right\vert_{\epsilon}}$ norm versus the polynomial degree of approximation (semilog scale) for $h=1/4, \Delta t = 5\cdot 10^{-6}$.
• Figure 2: 3D domain partition.
• Figure 3: Test case of Section \ref{['subsubsection:realistic_coefficients_realistic_domain']}. Computed errors in the ${\left\vert\left\vert\left\vert \cdot \right\vert\right\vert\right\vert_{\epsilon}}$ norm versus the polynomial degree of approximation (semilog scale) for $\Delta t = 5\cdot 10^{-6}$.
• Figure 4: Test case of Section \ref{['sec:results']}. Snapshot of the computed solution $c_h(\boldsymbol{x},t)$ (blue) and $q_h(\boldsymbol{x},t)$ (red), for $t \in \{ 0, 5, 10, 15, 20, 25 \}$ years.
• Figure 5: Test case of Section \ref{['sec:results']}. Snapshot of the computed solution $c_h(\boldsymbol{x},t)$ (blue) and $q_h(\boldsymbol{x},t)$ (red), for $t \in \{ 0, 5, 10, 15, 20, 25 \}$ years in a sagittal medial section.

Theorems & Definitions (3)

• Remark 1
• Definition 1
• Theorem 1: Error bounds corti