Biophysically detailed mathematical models of multiscale cardiac active mechanics

Abstract

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 lengthdependent 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.

Polygonal surface processing and mesh generation tools for the numerical simulation of the cardiac function

“…The proposed algorithms have been combined into ad‐hoc pipelines to test them in some complex cardiac applications, like the mesh generation of a fully‐detailed ventricular geometry including the papillary muscles and the trabeculae carneae (Section 3.2) or a complete fluid‐dynamics mesh of the left‐heart (Section 3.3). Moreover, as discussed in Section 3.4, they have already been successfully used in other papers to address a broad range of applications 28,91‐96 . These examples highlight the flexibility of these tools and their ability to be easily applicable to a large variety of cardiac mesh generations through the building of ad‐hoc application‐dependent pipelines.…”

“…In this work electrophysiology simulations on a four‐chambers cardiac geometry were performed, as shown in Figure 17A. This complex mesh has been generated by exploiting most of the proposed algorithms of this paper, especially the ones concerning tagging, connections, and mesh‐size function definition; in their multiscale study of the cardiac active mechanics, Regazzoni et al 92 used our algorithms to generate a left‐ventricular mesh for electro‐mechanical simulations, as shown in Figure 17B; Fumagalli et al 28 performed a computational hemodynamics study of the left heart to assess the pathological systolic anterior motion of the mitral valve, as shown in Figure 17C. They used both the harmonic‐connection and harmonic‐extension algorithms in order to connect the patient‐specific ventricle with a template geometry and to extend the image‐based displacement field on the whole domain.…”

“…Some examples of numerical studies of the cardiac function performed on meshes generated with the tools proposed in this paper: (A) the electrical activation time of an electrophysiology simulation of the whole four‐chambers heart, see Piersanti et al 91 ; (B) the resulting displacement field at the end of systole of a left‐ventricular electro‐mechanical simulation, see Regazzoni et al 92 ; (C) the velocity field at the end of systole of a patient‐specific hemodynamics simulation of the systolic anterior motion of the mitral valve, see Fumagalli et al 28 …”

Mathematical and numerical models for the cardiac electromechanical function

“…3.1). In [71,75] a more rigorous coupling between the regulatory units and the crossbridge dynamics is studied: the population of crossbridges is split into two familiesaccording to the permissivity state of the associated regulatory unit -and by introducing additional terms accounting for the probability fluxes between the two families. Another model of the whole force generation process is proposed in [51].…”

A Multiple Step Active Stiffness Integration Scheme to Couple a Stochastic Cross-Bridge Model and Continuum Mechanics for Uses in Both Basic Research and Clinical Applications of Heart Simulation

“…Although previous efforts to construct numerical models using a type of mean field approximation have successfully reproduced specific tissue-level phenomena (Hunter et al, 1998;Niederer et al, 2006;Negroni and Lascano, 2008;Rice et al, 2008;Guérin et al, 2011;Chapelle et al, 2012;Washio et al, 2012;Syomin and Tsaturyan, 2017;Regazzoni et al, 2018;Caruel et al, 2019), these models have not yet been fully exploited in real-life heart simulations. The uses of ordinary differential equation (ODE) models that adopt the phenomenological approximations of the force-pCa relationship and the force-velocity relationship have become mainstream instead (Smith et al, 2004;Kerckhoffs et al, 2007;Gurev et al, 2011;Shavik et al, 2017;Dabiri et al, 2019;Azzolin et al, 2020;Regazzoni et al, 2020). However, these approaches appear to have difficulties, particularly in reproducing the realistic relaxation phase that is important to ease the influx of blood from the atria to the ventricles.…”