  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.
