Department of Biomedical Engineering, King's College London, King's Health Partners, St. Thomas' Hospital, London SE1 7EH, UK.
Int J Numer Method Biomed Eng. 2013 Feb;29(2):217-32. doi: 10.1002/cnm.2520. Epub 2012 Oct 18.
We present a method to efficiently simulate coronary perfusion in subject-specific models of the heart within clinically relevant time frames. Perfusion is modelled as a Darcy porous-media flow, where the permeability tensor is derived from homogenization of an explicit anatomical representation of the vasculature. To account for the disparity in length scales present in the vascular network, in this study, this approach is further refined through the implementation of a multi-compartment medium where each compartment encapsulates the spatial scales in a certain range by using an effective permeability tensor. Neighbouring compartments then communicate through distributed sources and sinks, acting as volume fluxes. Although elegant from a modelling perspective, the full multi-compartment Darcy system is computationally expensive to solve. We therefore enhance computational efficiency of this model by reducing the N-compartment system of Darcy equations to N pressure equations, and N subsequent projection problems to recover the Darcy velocity. The resulting 'reduced' Darcy formulation leads to a dramatic reduction in algebraic-system size and is therefore computationally cheaper to solve than the full multi-compartment Darcy system. A comparison of the reduced and the full formulation in terms of solution time and memory usage clearly highlights the superior performance of the reduced formulation. Moreover, the implementation of flux and, specifically, impermeable boundary conditions on arbitrarily curved boundaries such as epicardium and endocardium is straightforward in contrast to the full Darcy formulation. Finally, to demonstrate the applicability of our methodology to a personalized model and its solvability in clinically relevant time frames, we simulate perfusion in a subject-specific model of the left ventricle.
我们提出了一种在心脏的特定于个体的模型中高效模拟冠状动脉灌注的方法，该方法可在临床相关的时间范围内进行。灌注被建模为达西多孔介质流，其中渗透率张量是通过对血管结构的显式解剖表示进行均匀化而得出的。为了解决血管网络中存在的长度尺度差异，在本研究中，通过实现多腔介质进一步改进了该方法，其中每个腔室通过使用有效渗透率张量来封装特定范围内的空间尺度。然后，相邻的腔室通过分布式源和汇进行通信，充当体积通量。尽管从建模的角度来看很优雅，但完整的多腔达西系统求解计算成本很高。因此，我们通过将 N 腔达西方程组简化为 N 个压力方程组，并将 N 个后续投影问题恢复为达西速度，从而提高了该模型的计算效率。由此产生的“简化”达西公式导致代数系统大小的急剧减小，因此比完整的多腔达西系统求解成本更低。在求解时间和内存使用方面对简化公式和完整公式进行比较，清楚地突出了简化公式的优越性能。此外，与完整的达西公式相比，通量的实现，特别是在心脏外膜和心内膜等任意曲面上不可渗透的边界条件的实现非常简单。最后，为了证明我们的方法在个性化模型中的适用性及其在临床相关时间范围内的可解性，我们模拟了左心室特定于个体的模型中的灌注。
