Perdikaris Paris, Grinberg Leopold, Karniadakis George Em
Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA,
Ann Biomed Eng. 2015 Jun;43(6):1432-42. doi: 10.1007/s10439-014-1221-3. Epub 2014 Dec 16.
The aim of the present work is to address the closure problem for hemodynamic simulations by developing a flexible and effective model that accurately distributes flow in the downstream vasculature and can stably provide a physiological pressure outflow boundary condition. To achieve this goal, we model blood flow in the sub-pixel vasculature by using a non-linear 1D model in self-similar networks of compliant arteries that mimic the structure and hierarchy of vessels in the meso-vascular regime (radii [Formula: see text]). We introduce a variable vessel length-to-radius ratio for small arteries and arterioles, while also addressing non-Newtonian blood rheology and arterial wall viscoelasticity effects in small arteries and arterioles. This methodology aims to overcome substantial cut-off radius sensitivities, typically arising in structured tree and linearized impedance models. The proposed model is not sensitive to outflow boundary conditions applied at the end points of the fractal network, and thus does not require calibration of resistance/capacitance parameters typically required for outflow conditions. The proposed model convergences to a periodic state in two cardiac cycles even when started from zero-flow initial conditions. The resulting fractal-trees typically consist of thousands to millions of arteries, posing the need for efficient parallel algorithms. To this end, we have scaled up a Discontinuous Galerkin solver that utilizes the MPI/OpenMP hybrid programming paradigm to thousands of computer cores, and can simulate blood flow in networks of millions of arterial segments at the rate of one cycle per 5 min. The proposed model has been extensively tested on a large and complex cranial network with 50 parent, patient-specific arteries and 21 outlets to which fractal trees where attached, resulting to a network of up to 4,392,484 vessels in total, and a detailed network of the arm with 276 parent arteries and 103 outlets (a total of 702,188 vessels after attaching the fractal trees), returning physiological flow and pressure wave predictions without requiring any parameter estimation or calibration procedures. We present a novel methodology to overcome substantial cut-off radius sensitivities.
本工作的目的是通过开发一个灵活有效的模型来解决血流动力学模拟中的封闭问题，该模型能准确地在下游脉管系统中分配血流，并能稳定地提供生理压力流出边界条件。为实现这一目标，我们在顺应性动脉的自相似网络中使用非线性一维模型对亚像素脉管系统中的血流进行建模，该网络模仿中尺度血管系统（半径[公式：见原文]）中血管的结构和层次。我们引入了小动脉和小动脉的可变血管长度与半径比，同时还考虑了小动脉和小动脉中的非牛顿血液流变学和动脉壁粘弹性效应。这种方法旨在克服通常出现在结构化树和线性化阻抗模型中的显著截止半径敏感性。所提出的模型对分形网络端点处应用的流出边界条件不敏感，因此不需要通常用于流出条件的电阻/电容参数校准。即使从零流量初始条件开始，所提出的模型在两个心动周期内也会收敛到周期性状态。由此产生的分形树通常由数千到数百万条动脉组成，这就需要高效的并行算法。为此，我们将一个利用MPI/OpenMP混合编程范式的间断伽辽金求解器扩展到数千个计算机核心，并能以每5分钟一个周期的速度模拟数百万个动脉段网络中的血流。所提出的模型已在一个大型复杂的颅骨网络上进行了广泛测试，该网络有50条源自患者的母动脉和21个连接分形树的出口，总共形成了一个多达4392484条血管的网络，以及一个详细的手臂网络，有276条母动脉和103个出口（连接分形树后共有702188条血管），无需任何参数估计或校准程序即可返回生理血流和压力波预测结果。我们提出了一种新颖的方法来克服显著的截止半径敏感性。
