Reduced order modeling for cardiac electrophysiology and mechanics: New methodologies, challenges and perspectives

Reduced-order modeling techniques enable a remarkable speed up in the solution of the parametrized electromechanical model for heart dynamics. Being able to rapidly approximate the solution of this problem allows to investigate the impact of significant model parameters querying the parameter-to-solution map in a very inexpensive way. The construction of reduced-order approximations for cardiac electromechanics faces several challenges from both modeling and computational viewpoints, because of the multiscale nature of the problem, the need of coupling different physics, and the nonlinearities involved. Our approach relies on the reduced basis method for parametrized PDEs. This technique performs a Galerkin projection onto low-dimensional spaces built from a set of snapshots of the high-fidelity problem by the Proper Orthogonal Decomposition technique. Snapshots are obtained for different values of the parameters and computed, e.g., by the finite element method. Then, suitable hyper-reduction techniques, in particular the Discrete Empirical Interpolation Method and its matrix version, are called into play to efficiently handle nonlinear and parameter-dependent terms. In this work we show how a fast and reliable approximation of both the electrical and the mechanical model can be achieved by developing two separate reduced order models where the interaction of the cardiac electrophysiology system with the contractile muscle tissue, as well as the sub-cellular activation-contraction mechanism, are included. Open challenges and possible perspectives are finally outlined.