In numerical simulations of cardiac mechanics, coupling the heart to a model of the circulatory system is essential for capturing physiological cardiac behavior. A popular and efficient technique is to use an electrical circuit analogy, known as a lumped parameter network or zero-dimensional (0D) fluid model, to represent blood flow throughout the cardiovascular system. Due to the strong physical interaction between the heart and the blood circulation, developing accurate and efficient numerical coupling methods remains an active area of research. In this work, we present a modular framework for implicitly coupling three-dimensional (3D) finite element simulations of cardiac mechanics to 0D models of blood circulation. The framework is modular in that the circulation model can be modified independently of the 3D finite element solver, and vice versa. The numerical scheme builds upon a previous work that combines 3D blood flow models with 0D circulation models (3D fluid - 0D fluid). Here, we extend it to couple 3D cardiac tissue mechanics models with 0D circulation models (3D structure - 0D fluid), showing that both mathematical problems can be solved within a unified coupling scheme. The effectiveness, temporal convergence, and computational cost of the algorithm are assessed through multiple examples relevant to the cardiovascular modeling community. Importantly, in an idealized left ventricle example, we show that the coupled model yields physiological pressure-volume loops and naturally recapitulates the isovolumic contraction and relaxation phases of the cardiac cycle without any additional numerical techniques. Furthermore, we provide a new derivation of the scheme inspired by the Approximate Newton Method of Chan (1985), explaining how the proposed numerical scheme combines the stability of monolithic approaches with the modularity and flexibility of partitioned approaches.

翻译：在心脏力学的数值模拟中，将心脏与循环系统模型耦合对于捕捉生理学心脏行为至关重要。一种流行且高效的技术是采用电路类比方法，即集总参数网络或零维（0D）流体模型，来表征心血管系统中的血流。由于心脏与血液循环之间存在强烈的物理相互作用，开发精确且高效的数值耦合方法仍是活跃的研究领域。本文提出一种模块化框架，用于将心脏力学的三维（3D）有限元模拟与血液循环的0D模型进行隐式耦合。该框架的模块化特性体现在：循环模型可独立于3D有限元求解器进行修改，反之亦然。数值方案基于先前将3D血流模型与0D循环模型耦合的工作（3D流体-0D流体）进行构建。我们将其扩展至耦合3D心脏组织力学模型与0D循环模型（3D结构-0D流体），证明两类数学问题可在统一耦合框架下求解。通过多个心血管建模领域的实例，评估了算法的有效性、时间收敛性及计算成本。重要的是，在理想化左心室示例中，该耦合模型可生成生理性压力-容积环，并在无需额外数值技术的情况下自然重现心动周期的等容收缩期和等容舒张期。此外，我们受Chan（1985）近似牛顿法启发，对该方案进行新推导，阐明当前数值方案如何融合整体法的稳定性与分区法的模块化及灵活性。