Glioblastoma Multiforme is a malignant brain tumor with poor prognosis. There have been numerous attempts to model the invasion of tumorous glioma cells via partial differential equations in the form of advection–diffusion–reaction equations. The patient-wise parametrization of these models, and their validation via experimental data has been found to be difficult, as time sequence measurements are mostly missing. Also the clinical interest lies in the actual (invisible) tumor extent for a particular MRI/DTI scan and not in a predictive estimate. Therefore we propose a stationalized approach to estimate the extent of glioblastoma (GBM) invasion at the time of a given MRI/DTI scan. The underlying dynamics can be derived from an instationary GBM model, falling into the wide class of advection-diffusion-reaction equations. The stationalization is introduced via an analytic solution of the Fisher-KPP equation, the simplest model in the considered model class. We investigate the applicability in 1D and 2D, in the presence of inhomogeneous diffusion coefficients and on a real 3D DTI-dataset.

多形性胶质母细胞瘤是一种预后不良的恶性脑肿瘤。已经有许多尝试通过以对流-扩散-反应方程形式的偏微分方程来模拟肿瘤胶质瘤细胞的侵袭。已经发现这些模型的患者参数化以及通过实验数据对其进行验证很困难，因为时间序列测量大多缺失。此外，临床兴趣在于特定 MRI/DTI 扫描的实际（不可见）肿瘤范围，而不是预测估计。因此，我们提出了一种固定方法来估计给定 MRI/DTI 扫描时胶质母细胞瘤 (GBM) 的侵袭程度。潜在的动力学可以从一个不稳定的 GBM 模型中推导出来，属于广泛的对流-扩散-反应方程。平稳化是通过 Fisher-KPP 方程的解析解引入的，这是所考虑的模型类中最简单的模型。我们研究了在存在非均匀扩散系数和真实 3D DTI 数据集的情况下在 1D 和 2D 中的适用性。