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 $\alpha$-synuclein and Amyloid-$\beta$, 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 $\alpha$-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-$\beta$ 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.

翻译：许多神经退行性疾病与错误折叠的朊蛋白传播有关。本文分析了与帕金森病和阿尔茨海默病分别相关的α-突触核蛋白及β-淀粉样蛋白的错误折叠与传播过程。我们引入并分析了一种保正数值方法，用于离散化描述朊蛋白累积和传播的Fisher-Kolmogorov方程。所提出的近似方法基于多边形和多面体网格上的间断伽辽金方法进行空间离散，并采用θ-方法时间积分格式。我们证明了离散解的存在性，以及采用隐式欧拉格式进行时间积分时的收敛性结果。研究表明该方法具有结构保持特性，即能保证离散解非负，这一特性在实际应用中至关重要。通过制造解法和二维多边形网格中的波前传播验证了数值模型。随后，我们模拟了冠状面二维脑切片中α-突触核蛋白的传播过程。该模拟中利用多面体间断伽辽金方法在网格构建上的灵活性，通过聚合方法生成多边形网格，同时保持白质与灰质的区分。最后，基于磁共振图像重建的三维几何结构及正电子发射断层扫描重建的初始条件，在患者特异性场景中模拟了β-淀粉样蛋白的传播。数值模拟证实，该方法能够有效捕捉帕金森病和阿尔茨海默病的病理演变过程。