The spreading of prion proteins is at the basis of brain neurodegeneration. The paper deals with the numerical modelling of the misfolding process of $\alpha$-synuclein in Parkinson's disease. We introduce and analyze a discontinuous Galerkin method for the semi-discrete approximation of the Fisher-Kolmogorov (FK) equation that can be employed to model the process. We employ a discontinuous Galerkin method on polygonal and polyhedral grids (PolyDG) for space discretization, which allows us to accurately simulate the wavefronts typically observed in the prionic spreading. We prove stability and a priori error estimates for the semi-discrete formulation. Next, we use a Crank-Nicolson scheme to advance in time. For the numerical verification of our numerical model, we first consider a manufactured solution, and then we consider a case with wavefront propagation in two-dimensional polygonal grids. Next, we carry out a simulation of $\alpha$-synuclein spreading in a two-dimensional brain slice in the sagittal plane with a polygonal agglomerated grid that takes full advantage of the flexibility of PolyDG approximation. Finally, we present a simulation in a three-dimensional patient-specific brain geometry reconstructed from magnetic resonance images.

翻译：朊蛋白的传播是脑神经退行性疾病的病理基础。本文研究了帕金森病中α-突触核蛋白错误折叠过程的数值建模。我们提出并分析了一种用于Fisher-Kolmogorov (FK)方程半离散近似的间断伽辽金方法，该方法可有效模拟该病理过程。空间离散采用基于多边形和多面体网格的间断伽辽金方法（PolyDG），能精确模拟朊病毒传播中典型的波前现象。我们证明了半离散格式的稳定性和先验误差估计，并通过Crank-Nicolson格式进行时间推进。数值验证首先采用人为构造解进行测试，随后分析二维多边形网格上的波前传播案例。进一步地，通过充分利用PolyDG逼近灵活性的多边形聚合网格，在二维人脑矢状面切片中模拟了α-突触核蛋白的传播过程。最后，我们在基于磁共振影像重建的三维个体化脑几何结构上进行了仿真演示。