A fractional integral ghost-structure method for a space-fractional anisotropic monodomain model in realistic bi-ventricular cardiac electrophysiology

This paper proposes a space-fractional anisotropic monodomain model for simulating action potential propagation in deforming cardiac tissue. The model incorporates an anisotropic conductivity tensor that depends on the local fiber orientation and evolves with tissue motion, thereby capturing both fiber anisotropy and tissue heterogeneity. To address the numerical challenge of solving this model on moving irregular domains, we develop a fractional integral ghost-structure (FIGS) method. In the FIGS framework, the deforming cardiac structure is immersed within a fixed regular domain (the ghost structure). This approach employs an Eulerian description for the transmembrane potential on the fixed grid and a Lagrangian description for the membrane dynamics on the moving material points, enabling the approximation of domain-dependent fractional derivatives using the finite difference method on a Cartesian grid. The transformation between Eulerian variables and Lagrangian variables is achieved by a variable interchange mechanism based on a local search algorithm and finite element interpolation basis functions. Extensive numerical experiments demonstrate the effectiveness of both the model and the method. Simulations in a static circular tissue analyze the influence of fractional orders on wavefront dynamics. The displacement fields for both the oscillating sphere and the realistic contracting bi-ventricular heart are obtained by solving a fluid-structure interaction model using the immersed boundary method. Tests on a moving oscillating sphere confirm the capability of the FIGS method to handle deforming geometries. Finally, application to a physiologically realistic contracting bi-ventricular heart shows that the model successfully reproduces normal potential propagation and, under arrhythmic conditions, captures the emergence of stable spiral waves leading to re-entrant activation patterns.