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.

翻译：错误折叠蛋白质的扩散是某些神经退行性疾病（称为蛋白病）的已知标志。一个重要的例子是与阿尔茨海默病等多种病理相关的tau蛋白。本文讨论并比较了两种不同的蛋白质错误折叠数学模型，即异二聚体模型和Fisher-Kolmogorov模型，以及它们的数值离散化方法。我们引入了一种基于多边形和多面体网格的不连续伽辽金方法进行空间离散，以精确模拟朊病毒传播中通常观察到的波阵面。基于半离散公式，我们采用Crank-Nicolson格式进行时间推进。最后，我们在一个使用多边形聚合网格的矢状面二维脑切片中模拟了错误折叠tau蛋白的扩散。模拟分别采用上述两种模型进行，我们比较了结果以及因建模选择导致的差异。