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

TL;DR

The paper advances meshless cardiac modeling by integrating Purkinje-network conduction with myocardial electrophysiology and mechanics through a multi-order SPH framework. It introduces an efficient level-set–based fractal-tree network generation on complex surfaces, a reduced-order SPH method for fast Purkinje conduction, and a multi-order MM coupling with a multi-time stepping scheme to couple network and myocardium efficiently. Key contributions include a network representation compatible with SPH via a cell-linked list, an anisotropic diffusion discretization in a reduced-dimensional space, and demonstration on both simplified geometries and a realistic left ventricle, revealing the critical impact of physiologically realistic Purkinje networks on activation and contraction. The results show substantial computational speedups and validate the method’s capability to resolve electromechanical behavior in a physiologically plausible LV, highlighting the potential for a fully meshless total-heart simulator with clinical relevance.

Abstract

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.

Figures (15)

• Figure 1: Schematic diagram for iteratively generation branches.
• Figure 2: Schematic diagram for one branch-to-grow branch grows into two child branches and segment growth inside a branch. Each branch-to-grow branch will grow into two child branches whose directions are given in Eq. \ref{['eq:branchgrowing']}. For the segment growth, the growth direction is determined by the previous one and the gradient defined in Eq. \ref{['eq:graddistance']}. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
• Figure 3: Schematic diagram for the CLL scheme for nearest node search. The neighbor search of a single node can be conducted by searching all other nodes (left panel)or by dividing the domain into cells with an length of at least the cutoff radius and searching the neighbors between the node and all nodes in the same (red) and in the adjacent (green) cells. Here, we consider the cutoff radius equals to $2.6 l_{seg}$ for consistency with the SPH framework. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
• Figure 4: Schematic diagram for the network-based neighboring particle search scheme. The searching domain for typical particles are represented by the arrow line with the same color. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
• Figure 5: Schematic diagram of a simple myocardial domain $\Phi_m$ with a generic network $\Phi_p$. Here, the generic network consists of the AV node and two terminal nodes $T_1$ and $T_2$ denoting the PKJs which have the influence of spherical region with a radius of $r$ centered in the node. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)