TU Kaiserslautern, Felix-Klein-Zentrum für Mathematik, Paul-Ehrlich-Street 31, 67663, Kaiserslautern, Germany.
College of Science, Minzu University of China, Beijing, 100081, People's Republic of China.
J Math Biol. 2021 Apr 12;82(6):49. doi: 10.1007/s00285-021-01599-x.
Gliomas are primary brain tumors with a high invasive potential and infiltrative spread. Among them, glioblastoma multiforme (GBM) exhibits microvascular hyperplasia and pronounced necrosis triggered by hypoxia. Histological samples showing garland-like hypercellular structures (so-called pseudopalisades) centered around the occlusion site of a capillary are typical for GBM and hint on poor prognosis of patient survival. We propose a multiscale modeling approach in the kinetic theory of active particles framework and deduce by an upscaling process a reaction-diffusion model with repellent pH-taxis. We prove existence of a unique global bounded classical solution for a version of the obtained macroscopic system and investigate the asymptotic behavior of the solution. Moreover, we study two different types of scaling and compare the behavior of the obtained macroscopic PDEs by way of simulations. These show that patterns (not necessarily of Turing type), including pseudopalisades, can be formed for some parameter ranges, in accordance with the tumor grade. This is true when the PDEs are obtained via parabolic scaling (undirected tissue), while no such patterns are observed for the PDEs arising by a hyperbolic limit (directed tissue). This suggests that brain tissue might be undirected - at least as far as glioma migration is concerned. We also investigate two different ways of including cell level descriptions of response to hypoxia and the way they are related .
神经胶质瘤是具有高侵袭性和浸润性扩散能力的原发性脑肿瘤。其中，多形性胶质母细胞瘤（GBM）表现为微脉管增生和缺氧触发的明显坏死。组织学样本显示出围绕毛细血管闭塞部位的花环状细胞增生结构（所谓的假栅格）是 GBM 的典型特征，并提示患者生存预后不良。我们在活性粒子动力学理论框架中提出了一种多尺度建模方法，并通过上采样过程推导出一个具有排斥 pH 趋性的反应扩散模型。我们证明了所得到的宏观系统的一个版本的唯一全局有界经典解的存在，并研究了解的渐近行为。此外，我们研究了两种不同类型的标度，并通过模拟比较了所得到的宏观 PDE 的行为。这些结果表明，对于某些参数范围，可以形成图案（不一定是图灵类型），包括假栅格，与肿瘤分级一致。这对于通过抛物型标度（无向组织）获得的 PDE 是正确的，而对于通过双曲极限（有向组织）获得的 PDE 则不会观察到这种模式。这表明脑组织结构可能是无向的——至少就胶质瘤迁移而言是这样。我们还研究了两种不同的方法来包含对缺氧的细胞水平描述以及它们之间的关系。
