A comprehensive mathematical model of the multiphysics flow of blood and Cerebrospinal Fluid (CSF) in the brain can be expressed as the coupling of a poromechanics system and Stokes' equations: the first describes fluids filtration through the cerebral tissue and the tissue's elastic response, while the latter models the flow of the CSF in the brain ventricles. This model describes the functioning of the brain's waste clearance mechanism, which has been recently discovered to play an essential role in the progress of neurodegenerative diseases. To model the interactions between different scales in the porous medium, we propose a physically consistent coupling between Multi-compartment Poroelasticity (MPE) equations and Stokes' equations. In this work, we introduce a numerical scheme for the discretization of such coupled MPE-Stokes system, employing a high-order discontinuous Galerkin method on polytopal grids to efficiently account for the geometric complexity of the domain. We analyze the stability and convergence of the space semidiscretized formulation, we prove a-priori error estimates, and we present a temporal discretization based on a combination of Newmark's $\beta$-method for the elastic wave equation and the $\theta$-method for the other equations of the model. Numerical simulations carried out on test cases with manufactured solutions validate the theoretical error estimates. We also present numerical results on a two-dimensional slice of a patient-specific brain geometry reconstructed from diagnostic images, to test in practice the advantages of the proposed approach.

翻译：大脑中血液与脑脊液的多物理场流动的全面数学模型可表述为孔隙力学系统与斯托克斯方程的耦合：前者描述脑组织中的流体过滤及组织的弹性响应，后者则模拟脑室中脑脊液的流动。该模型刻画了大脑废物清除机制的功能，而近期研究发现这一机制在神经退行性疾病进展中起着关键作用。为模拟多孔介质中不同尺度的相互作用，我们提出了多隔室孔隙弹性方程与斯托克斯方程之间物理一致的耦合方案。本研究针对此类耦合MPE-Stokes系统引入了一种数值离散格式，采用多面体网格上的高阶间断伽辽金方法以有效处理域几何复杂性。我们分析了空间半离散格式的稳定性与收敛性，证明了先验误差估计，并基于弹性波动方程的纽马克β法与其他模型方程的θ法组合提出了时间离散方案。通过采用制造解法的测试案例进行数值模拟，验证了理论误差估计。我们还将该方法应用于从诊断图像重建的患者特异性脑几何二维切片，通过数值结果实际测试了所提方法的优势。