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的传播有关。本文分析了与帕金森病和阿尔茨海默病分别相关的$\alpha$-突触核蛋白和淀粉样蛋白-$\beta$的错误折叠与传播过程。我们引入并分析了一种保持正性的数值方法，用于离散化描述朊蛋白积累和传播的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不连续伽辽金方法进行空间离散，并采用$\vartheta$方法时间积分格式。我们证明了离散解的存在性，以及采用隐式欧拉方案进行时间积分时的收敛性结果。我们表明所提出的方法是结构保持的，即保证离散解非负，这一特性在实际应用中至关重要。数值验证通过解析解法和二维多边形网格中的波前传播实现。随后，我们展示了矢状面二维脑切片中$\alpha$-突触核蛋白传播的模拟。该模拟的多边形网格通过保持白质和灰质区分进行聚合，充分利用了PolyDG方法在网格构建中的灵活性。最后，我们利用磁共振图像重建的三维几何结构和正电子发射断层扫描重建的初始条件，在患者特异性设置中模拟了淀粉样蛋白-$\beta$的传播。数值模拟证实，所提出的方法能够捕捉帕金森病和阿尔茨海默病的演变过程。