Structure preserving polytopal discontinuous Galerkin methods for the numerical modeling of neurodegenerative diseases (Q6571384)

This paper introduces a structure-preserving numerical approach for modeling neurodegenerative diseases through the propagation of misfolded proteins, specifically focusing on \(\alpha\)-synuclein and amyloid-\(\beta\) linked to Parkinson's and Alzheimer's diseases, respectively. The authors propose a positivity-preserving discretization for the Fisher-Kolmogorov equation, which governs the dynamics of protein accumulation and spread. Utilizing a DG method on polytopal grids for spatial discretization, combined with a \(\theta\)-method for time integration, they ensure stability and preservation of non-negativity in solutions, an essential feature for biological accuracy.\N\NThe proposed method's robustness is demonstrated through convergence analysis and simulation results. The authors highlight the method's effectiveness in handling heterogeneous tissue properties and confirm its computational efficiency and accuracy in approximating the spread of pathological proteins over decades.