Biophysically detailed mathematical models of multiscale cardiac active mechanics

Author summary Computer-based numerical simulations of the heart are increasingly assuming a recognized role in the context of computational medicine and cardiology. They are based on mathematical models describing the different physical phenomena occurring during an heartbeat. Among these models, a pivotal role is played by those describing how cardiomyocytes-the cardiac muscle cells-produce active force, driven by changes in calcium concentration. However, due to the intrinsic complexity of these subcellular mechanisms, the computational cost associated with the solution of cardiac active force models is often prohibitive. For this reason, phenomenological models are typically used in place of biophysically detailed ones in organ-scale simulations. In this paper, we propose some new biophysically detailed mathematical models of cardiac force generation. Our models are rigorously derived on the basis of physically motivated assumptions that allow to drastically reduce the computational cost associated to their resolution, making them suitable for organ-scale numerical simulations, without renouncing to their biophysical detail. We propose four novel mathematical models, describing the microscopic mechanisms of force generation in the cardiac muscle tissue, which are suitable for multiscale numerical simulations of cardiac electromechanics. Such models are based on a biophysically accurate representation of the regulatory and contractile proteins in the sarcomeres. Our models, unlike most of the sarcomere dynamics models that are available in the literature and that feature a comparable richness of detail, do not require the time-consuming Monte Carlo method for their numerical approximation. Conversely, the models that we propose only require the solution of a system of PDEs and/or ODEs (the most reduced of the four only involving 20 ODEs), thus entailing a significant computational efficiency. By focusing on the two models that feature the best trade-off between detail of description and identifiability of parameters, we propose a pipeline to calibrate such parameters starting from experimental measurements available in literature. Thanks to this pipeline, we calibrate these models for room-temperature rat and for body-temperature human cells. We show, by means of numerical simulations, that the proposed models correctly predict the main features of force generation, including the steady-state force-calcium and force-length relationships, the length-dependent prolongation of twitches and increase of peak force, the force-velocity relationship. Moreover, they correctly reproduce the Frank-Starling effect, when employed in multiscale 3D numerical simulation of cardiac electromechanics.