Mathematics

We consider the mathematical and numerical modeling of the human heart function, with focus on cardiac electromechanics. We proceed by integrating state-of-the art models for the electrophysiology of the tissue, mechanical activation at the cellular level, and the passive mechanical response of the muscle, thus yielding a coupled electromechanical problem within the active strain paradigm. We consider the spatial approximation of the Partial Differential Equations therein involved by means of the Finite Element method and the time discretization by Backward Differentiation Formulas. We numerically solve the coupledelectromechanics problem by exploiting both monolithic and staggered approaches, for which we verify, compare, and critically discuss their accuracy properties and computational efficiency in simulating the whole cardiac cycle. In addition, we develop a multiscale model for cardiac electromechanics that accounts formiscroscopic active force generation at the cellular level within the active stress paradigm; with this aim, we exploit model order reduction techniques based on Machine Learning algorithms to enable efficient numerical simulations of multiscale electromechanics. Finally, we present several numerical results of the electromechanics problem in the human left ventricle obtained in the high performance computing framework.