# Mathematical and numerical modeling of the cardiac electro-mechanical coupling

Aim. Our goal is to develop effective computational tools to push forward the understanding of the fundamental mechanisms underlying the origin of life-threatening arrhythmias and contractile disorders in the human heart and to provide theoretical support to cardiologists in developing more successful pharmacological and surgical treatments for these pathologies.

State of the heart. The spread of the electrical impulse in the cardiac muscle and the subsequent contraction-relaxation process are quantitatively described by the cardiac electro-mechanical coupling model (see e.g. [2]), which consists of the following four components:

• the quasi-static finite elasticity model of the deformation of cardiac tissue, derived from a strain energy function which characterizes the anisotropic mechanical properties of the myocardium;
• the active tension model, consisting of a non-linear ODE system, describing the intracellular calcium dynamics and the cross bridges binding;
• the electrical current flow model of the cardiac tissue, i.e. the Bidomain model (or its reduction called Monodomain model, see [1]), which is a degenerate parabolic system of two non-linear partial differential equations (PDEs) of reaction-diffusion type, describing the evolution in space and time of the intra- and extracellular electric potentials;
• the membrane model of the cardiac myocyte, i.e. a stiff system of ordinary differential equations (ODEs), describing the flow of the ionic currents through the cellular membrane.

This complex non-linear model poses great theoretical and numerical challenges. At the theoretical level, the well-posedness of the cardiac electro-mechanical coupling model is still an open problem, as well as the convergence of its finite element approximation. At the numerical level, the approximation and simulation of the cardiac electro-mechanical coupling model is a very demanding and expensive task, because of the very different space and time scales associated with the electrical and mechanical models, as well as their non-linear and multiphysics interactions.

Parallel solvers for cardiac electro-mechanics. Balancing Domain Decomposition by Constraints (BDDC) preconditioners are non-overlapping domain decomposition preconditioners first introduced in [C.R. Dohrmann. SIAM J. Sci. Comput., 2003 ] for scalar elliptic problems. BDDC can be regarded as an evolution of balancing Neumann-Neumann methods where all local and coarse problems are treated additively due to a choice of so-called primal continuity constraints across the interface of the subdomains. The primal constraints can be point constraints and/or averages or moments over edges or faces of the subdomains. The objectives of this project are: to construct BDDC preconditioners for finite element discretizations of the cardiac electro-mechanical coupling model; to analyze theoretically the scalability and quasi-optimality of the resulting algorithms; to test numerically their effectiveness and robustness on parallel computational platforms with O(105) processors, by considering both idealized and realistic patient-specific cardiac geometries.

Cardiac resynchronization therapy defibrillation. Defibrillation by strong electric shock is the only known procedure that reliably terminates ventricular fibrillation. Implantable Cardioverter Defibrillators (ICDs) are electrical devices placed under the skin that keep track the heart rate. Thin wires connect the ICD to patient’s heart. If an abnormal heart rhythm is detected, e.g. the heart beats much too fast or chaotically, the device will deliver an electric shock to restore the normal heartbeat, resynchronizing the whole muscle. Several simulation studies have been devoted to determine numerically the defibrillation thresholds associated with different ICD configurations. Due to the high computational costs required by cardiac electro-mechanical simulations, most of these studies have neglected the influence of deformation on the bioelectrical activity of the cardiac tissue, the so-called mechano-electric feedback (MEF). The objective of this project is to study, by means of realistic three-dimensional electro-mechanical simulations, the effects of MEF on defibrillation thresholds, in order to give theoretical insights aimed at determining optimal ICD configurations.

References

[1] P. Colli Franzone, L. F. Pavarino, S. Scacchi. Mathematical Cardiac Electrophysiology. Springer, 2014.

[2] P. Colli Franzone, L. F. Pavarino, S. Scacchi. Bioelectrical effects of mechanical feedbacks in a strongly coupled cardiac electro-mechanical model. Math. Mod. Meth. Appl. Sci., 26 (1): 27-57, 2016.

For further information please contact Luca Pavarino (luca.pavarino@unimi.it ) or Simone Scacchi (simone.scacchi@unimi.it ), Università degli Studi di Milano.

Joint work with Piero Colli-Franzone, Lara Charawi (University of Pavia) and Stefano Zampini (King Abdullah University of Science and Technology)