A multi-order smoothed particle hydrodynamics method for cardiac electromechanics with the Purkinje network

In previous work, Zhang et al. (2021) \cite{zhang2021integrative} developed an integrated smoothed particle hydrodynamics (SPH) method to simulate the principle aspects of cardiac function, including electrophysiology, passive and active mechanical response of the myocardium. As the inclusion of the Purkinje network in electrocardiology is recognized as fundamental to accurately describing the electrical activation in the right and left ventricles, in this paper, we present a multi-order SPH method to handle the electrical propagation through the Purkinje system and in the myocardium with monodomain/monodomain coupling strategy. We first propose an efficient algorithm for network generation on arbitrarily complex surface by exploiting level-set geometry representation and cell-linked list neighbor search algorithm. Then, a reduced-order SPH method is developed to solve the one-dimensional monodomain equation to characterize the fast electrical activation through the Purkinje network. Finally, a multi-order coupling paradigm is introduced to capture the coupled nature of potential propagation arising from the interaction between the network and the myocardium. A set of numerical examples are studied to assess the computational performance, accuracy and versatility of the proposed methods. In particular, numerical study performed in realistic left ventricle demonstrates that the present method features all the physiological issues that characterize a heartbeat simulation, including the initiation of the signal in the Purkinje network and the systolic and diastolic phases. As expected, the results underlie the importance of using physiologically realistic Purkinje network for modeling cardiac functions.