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

TL;DR

The paper tackles modeling the brain-wide spread of misfolded proteins in neurodegenerative diseases by the Fisher-Kolmogorov equation and proposes a structure-preserving, positivity-preserving PolyDG discretization on polygonal grids with a $\vartheta$-time integrator. It establishes discrete existence via the Leray-Schauder theorem, develops a coercive and continuous nonlinear diffusion form $\mathcal{A}$, and analyzes a fully discrete scheme with an $\varepsilon$-regularization, providing entropy-decay results and convergence insights for $\vartheta=1$. The method is applied to 2D brain slices and 3D patient-specific geometries with data-derived initial conditions, yielding wavefronts and activation patterns consistent with clinical knowledge of Parkinson's and Alzheimer's diseases. The work demonstrates non-negativity of the discrete solution, flexibility in mesh construction, and potential for patient-specific simulations in neurodegenerative disease modeling, offering a robust tool for clinical-assisted computational studies.

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 $α$-synuclein and Amyloid-$β$, 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 $α$-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-$β$ 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.

Figures (8)

• Figure 1: Test case 1: computed errors and convergence rates in DG-norm (left) and $L^2$-norm (right), $\Delta t = 10^{-6}$.
• Figure 2: Test case 1: computed errors and convergence rates in convergence concerning timestep (left) and polynomial degree (right) with $N_\mathrm{el} = 30$.
• Figure 3: Test case 2: Comparison of the numerical solutions computed both using the proposed positivity-preserving DG scheme and the DG method presented in corti:FK, with the exact solution at time $T=10$.
• Figure 4: Brain section with the specification of the white-gray matter subdivision (a), the agglomerated mesh (b), and the axonal directions (c).
• Figure 5: Patterns of $\alpha$-synuclein concentration at different stages of the pathology.

Theorems & Definitions (12)

• Remark 1
• Remark 2
• Proposition 1: Coercivity of $\mathcal{A}$
• proof
• Proposition 2: Continuity of $\mathcal{A}$
• proof
• Proposition 3: Existence of a solution
• proof
• Remark 3
• Proposition 4