This spreading of prion proteins is at the basis of brain neurodegeneration. This 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, to accurately simulate the wavefronts typically observed in the prionic spreading and we prove stability and a priori error estimates. 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 geometry reconstructed from magnetic resonance images of a patient's brain.

翻译：朊蛋白的扩散是脑神经退行性病变的基础。本文针对帕金森病中$\alpha$-突触核蛋白错误折叠过程的数值建模展开研究。我们引入并分析了一种适用于Fisher-Kolmogorov (FK)方程半离散近似的间断Galerkin方法，该方法可用于模拟该过程。采用基于多边形和多面体网格的间断Galerkin方法（PolyDG）进行空间离散化，以精确模拟朊病毒扩散中典型的波前传播现象，并证明其稳定性和先验误差估计。随后，我们使用Crank-Nicolson格式进行时间推进。数值验证阶段，首先采用人造解进行验证，其次考虑二维多边形网格中波前传播的案例。继而，利用PolyDG近似灵活性的优势，在基于聚合多边形网格的二维矢状面脑切片上模拟$\alpha$-突触核蛋白的扩散。最后，我们展示了基于患者脑部磁共振图像重建的三维几何模型中的模拟结果。