Models of cardiac mechanics are increasingly used to investigate cardiac physiology. These models are characterized by a high level of complexity, including the particular anisotropic material properties of biological tissue and the actively contracting material. A large number of independent simulation codes have been developed, but a consistent way of verifying the accuracy and replicability of simulations is lacking. To aid in the verification of current and future cardiac mechanics solvers, this study provides three benchmark problems for cardiac mechanics. These benchmark problems test the ability to accurately simulate pressure-type forces that depend on the deformed objects geometry, anisotropic and spatially varying material properties similar to those seen in the left ventricle and active contractile forces. The benchmark was solved by 11 different groups to generate consensus solutions, with typical differences in higher-resolution solutions at approximately 0.5%, and consistent results between linear, quadratic and cubic finite elements as well as different approaches to simulating incompressible materials. Online tools and solutions are made available to allow these tests to be effectively used in verification of future cardiac mechanics software.

Reaction-diffusion mechanics (RDM) systems describe a wide range of practically important phenomena where deformation substantially affects wave and vortex dynamics. Here, we develop the first theory to describe the dynamics of rotating spiral waves in RDM systems, combining response function theory with a mechanical Green's function. This theory explains the mechanically-induced drift of spiral waves as a resonance phenomenon, and it can predict the drift trajectories and the final attractors from measurable characteristics of the system. Theoretical predictions are confirmed by numerical simulations. The results can be applied to cardiac tissue, where the drift of spiral waves is an important factor in determining different types of cardiac arrhythmias.In this work we present an analytical approach to study spiral wave dynamics in RDM systems. We combine response function theory [23] with a Green's function formalism to account for mechanical influence, and we derive an equation for spiral wave drift. The MEF-induced drift is reduced to a resonant forcing problem in the rotating frame of the spiral: during its rotation, the spiral wave perceives a time-varying perturbation since the domain boundaries are not stationary in the spiral's frame of reference. The resonant component of this boundary-induced forcing yields the net spiral drift.Our theoretical predictions are compared to numerical simulations. We find the relative angle and magnitude of the spiral wave's drift and identify the spatial attractors of the system. Some of them, such as the center of the medium, have already been reported in numerical simulations [19]. Using this theory, we also find new regimes and attractors that also are confirmed by simulations. For example, our theory predicts that the center of the domain can also be repulsive, and that multiple stable dynamical attractors may coexist. Although different from previous findings, these predictions are confirmed by numerical simulations.The developed analytical approach allows us to generalize the numerical results, as our analytical findings are based on an Archimedean description of spiral wave geometry, which is common for all types of excitable media. ModelThe reaction and MEF parts of our model for cardiac tissue are as in [18,20,24,25], supplemented here with the Navier-Cauchy equilibrium equations from linear elasticity to facilitate analytical calculations. In a twodimensional medium with Cartesian material coordinates (x, y), the transmembrane voltage u and recovery variable v are evolved according to modified Aliev-Panfilov kinetics [26], as used in [18]:

Cardiac contraction is coordinated by a wave of electrical excitation which propagates through the heart. Combined modeling of electrical and mechanical function of the heart provides the most comprehensive description of cardiac function and is one of the latest trends in cardiac research. The effective numerical modeling of cardiac electromechanics remains a challenge, due to the stiffness of the electrical equations and the global coupling in the mechanical problem. Here we present a short review of the inherent assumptions made when deriving the electromechanical equations, including a general representation for deformation-dependent conduction tensors obeying orthotropic symmetry, and then present an implicit-explicit time-stepping approach that is tailored to solving the cardiac mono- or bidomain equations coupled to electromechanics of the cardiac wall. Our approach allows to find numerical solutions of the electromechanics equations using stable and higher order time integration. Our methods are implemented in a monolithic finite element code GEMS (Ghent Electromechanics Solver) using the PETSc library that is inherently parallelized for use on high-performance computing infrastructure. We tested GEMS on standard benchmark computations and discuss further development of our software.

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