Numerical modelling of protein misfolding in neurodegenerative diseases: a computational study

TL;DR

The paper addresses prion-like tau misfolding in neurodegenerative diseases by comparing two mathematical models, the two-species heterodimer system for healthy $p(\boldsymbol{x},t)$ and misfolded $q(\boldsymbol{x},t)$ proteins, and a Fisher–Kolmogorov reduction for the relative concentration $c(\boldsymbol{x},t)$. It advances a polytopal discontinuous Galerkin (PolyDG) framework with Crank–Nicolson time stepping to discretize these models on polygonal/polyhedral brain meshes, explicitly incorporating anisotropic diffusion via $\mathbf{D}$ and brain tissue heterogeneity. The study provides semi-discrete (space) and fully discrete (space-time) formulations, performs 2D brain-slice simulations in the sagittal plane, and compares front propagation, activation times, and computational costs between the two modelling approaches. The findings show that the FK model accelerates front propagation and reduces degrees of freedom, but the full heterodimer model is needed when the healthy protein concentration is not effectively constant, highlighting important trade-offs for model selection and numerical strategies in brain-scale tauopathy simulations.

Abstract

The spreading of misfolded proteins is a known hallmark in some neurodegenerative diseases, known as proteinopathies. A significant example is the tau protein, associated with many pathologies, such as Alzheimer's. In this work, we discuss and compare two different models for the mathematical modelling of protein misfolding, namely the heterodimer model and the Fisher-Kolmogorov model, as well as their numerical discretizations. We introduce a discontinuous Galerkin method on polygonal and polyhedral grids for space discretization to accurately simulate the wavefronts typically observed in the prionic spreading. Starting from the semidiscrete formulations, we use a Crank-Nicolson scheme to advance in time. Finally, we simulate the spreading of the misfolded tau protein in a two-dimensional brain slice in the sagittal plane with a polygonal agglomerated grid. The simulation is performed using both the presented models, and we compare the results and the differences deriving from the modelling choices.

Figures (6)

• Figure 1: Agglomerated mesh with the distinction of white matter (red) and grey matter (blue). The initial fine triangular mesh is visible in the left image with a zoom of a particular region.
• Figure 2: The initial configuration corresponds to the unstable healthy state, with a protein concentration of $p_h=1.2$ uniformly distributed all over the brain. The triggering of the system is due to the presence of misfolded proteins.
• Figure 3: Misfolded protein $q_h$, initially concentrated into the entorhinal cortex, spreads through the entire brain section over 25 years.
• Figure 4: Relative misfolded protein $c_h$ concentration, initially concentrated into the entorhinal cortex, spreads through the entire brain section.
• Figure 5: Activation time computed starting from the simulations of the two different models: heterodimer (a) and FK (b).

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3