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：朊蛋白扩散是脑神经退行性病变的基础。本文针对帕金森病中α-突触核蛋白错误折叠过程进行数值建模。我们引入并分析了一种间断伽辽金方法，用于Fisher-Kolmogorov（FK）方程的半离散逼近，该方程可用于模拟该过程。我们采用多边形和多面体网格上的间断伽辽金方法（PolyDG）进行空间离散化，从而能够精确模拟朊蛋白扩散中通常观察到的波前。我们证明了半离散格式的稳定性和先验误差估计。接着，我们采用Crank-Nicolson格式进行时间推进。在数值验证环节，我们首先考虑了一个制造解案例，随后分析了二维多边形网格中的波前传播案例。进而，我们利用多边凝聚网格，在二维矢状面脑切片上开展了α-突触核蛋白扩散的模拟，充分发挥了PolyDG近似灵活性的优势。最后，我们展示了基于磁共振图像重建的三维患者特异性脑几何结构上的模拟结果。