Gradient flows and variational principles for cardiac electrophysiology: Toward efficient and robust numerical simulations of the electrical activity of the heart
Gradient flows and variational principles for cardiac electrophysiology: Toward efficient and robust numerical simulations of the electrical activity of the heart
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DOI：
10.1016/j.cma.2014.02.002
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发表时间：
2014-05-01
影响因子：
7.2
通讯作者：
Henao, Duvan
中科院分区：
文献类型：
--
作者：
Hurtado, Daniel E.;Henao, Duvan
心脏电活动的计算机模拟在过去十年中取得了巨大的进步。然而，计算方法在医学界的接受程度在很大程度上取决于其可靠性，效率和鲁棒性。在这项工作中，我们提出了一个梯度流重新制定的心脏电生理方程，并提出了一个极小极大变分原理的时间离散的电生理问题。基于变分分析的结果，我们推导出的时间步长上的界限，保证鞍点的存在性和唯一性，并反过来的电生理增量问题的弱解。我们还表明条件下，极大极小问题是等效的一个有效的最小化原则，这是服从瑞利-里兹有限元分析。推导出的时间步长范围保证了空间离散化后的目标函数的严格凸性，从而保证了梯度下降法的收敛性。所提出的理论被应用到广泛采用的FitzHugh-Nagumo模型，这是符合在这项工作中提出的变分框架。该方法的适用性和它的影响的鲁棒性的时间积分证明通过数值模拟的电行为在一个单一的细胞和三维楔形和双心室的几何形状。我们设想，所提出的框架将打开大门，强大的和有效的电生理模型和模拟的发展。(C)2014爱思唯尔有限公司版权所有。
The computer simulation of the electrical activity of the heart has experienced tremendous advances in the last decade. However, the acceptance of computational methods in the medical community will largely depend on their reliability, efficiency and robustness. In this work, we present a gradient-flow reformulation of the cardiac electrophysiology equations, and propose a minimax variational principle for the time-discretized electrophysiology problem. Based on results from variational analysis, we derive bounds on the time-step size that guarantee the existence and uniqueness of the saddle point, and in turn of the weak solution of the electrophysiology incremental problem. We also show conditions under which the minimax problem is equivalent to an effective minimization principle, which is amenable to a Rayleigh-Ritz finite-element analysis. The derived time-step bounds guarantee the strict convexity of the objective function resulting from spatial discretization, thus ensuring the convergence of gradient-descent methods. The proposed theory is applied to the widely employed FitzHugh-Nagumo model, which is shown to conform to the variational framework proposed in this work. The applicability of the method and its implications on the robustness of time integration are demonstrated by way of numerical simulations of the electrical behavior in a single-cell and 3D wedge and biventricular geometries. We envision that the proposed framework will open the door to the development of robust and efficient electrophysiology models and simulations. (C) 2014 Elsevier B.V. All rights reserved.