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In previous work, Zhang et al. (2021) \cite{zhang2021integrative} developedan integrated smoothed particle hydrodynamics (SPH) method to simulate theprinciple aspects of cardiac function, including electrophysiology, passive andactive mechanical response of the myocardium. As the inclusion of the Purkinjenetwork in electrocardiology is recognized as fundamental to accuratelydescribing the electrical activation in the right and left ventricles, in thispaper, we present a multi-order SPH method to handle the electrical propagationthrough the Purkinje system and in the myocardium with monodomain/monodomaincoupling strategy. We first propose an efficient algorithm for networkgeneration on arbitrarily complex surface by exploiting level-set geometryrepresentation and cell-linked list neighbor search algorithm. Then, areduced-order SPH method is developed to solve the one-dimensional monodomainequation to characterize the fast electrical activation through the Purkinjenetwork. Finally, a multi-order coupling paradigm is introduced to capture thecoupled nature of potential propagation arising from the interaction betweenthe network and the myocardium. A set of numerical examples are studied toassess the computational performance, accuracy and versatility of the proposedmethods. In particular, numerical study performed in realistic left ventricledemonstrates that the present method features all the physiological issues thatcharacterize a heartbeat simulation, including the initiation of the signal inthe Purkinje network and the systolic and diastolic phases. As expected, theresults underlie the importance of using physiologically realistic Purkinjenetwork for modeling cardiac functions.

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