Gradient flows and variational principles for cardiac electrophysiology: Toward efficient and robust numerical simulations of the electrical activity of the heart

The computer simulation of the electrical activity of the heart has experienced tremendous advances in the last decade. However, the acceptance of computational methods in the medical community will largely depend on their reliability, efficiency and robustness. In this work, we present a gradient-flow reformulation of the cardiac electrophysiology equations, and propose a minimax variational principle for the time-discretized electrophysiology problem. Based on results from variational analysis, we derive bounds on the time-step size that guarantee the existence and uniqueness of the saddle point, and in turn of the weak solution of the electrophysiology incremental problem. We also show conditions under which the minimax problem is equivalent to an effective minimization principle, which is amenable to a Rayleigh–Ritz finite-element analysis. The derived time-step bounds guarantee the strict convexity of the objective function resulting from spatial discretization, thus ensuring the convergence of gradient-descent methods. The proposed theory is applied to the widely employed FitzHugh–Nagumo model, which is shown to conform to the variational framework proposed in this work. The applicability of the method and its implications on the robustness of time integration are demonstrated by way of numerical simulations of the electrical behavior in a single-cell and 3D wedge and biventricular geometries. We envision that the proposed framework will open the door to the development of robust and efficient electrophysiology models and simulations.