Abstract
In a multiscale simulation of a beating heart, the very large difference in the time scales between rapid stochastic conformational changes of contractile proteins and deterministic macroscopic outcomes, such as the ventricular pressure and volume, have hampered the implementation of an efficient coupling algorithm for the two scales. Furthermore, the consideration of dynamic changes of muscle stiffness caused by the cross-bridge activity of motor proteins have not been well established in continuum mechanics. To overcome these issues, we propose a multiple time step scheme called the multiple step active stiffness integration scheme (MusAsi) for the coupling of Monte Carlo (MC) multiple steps and an implicit finite element (FE) time integration step. The method focuses on the active tension stiffness matrix, where the active tension derivatives concerning the current displacements in the FE model are correctly integrated into the total stiffness matrix to avoid instability. A sensitivity analysis of the number of samples used in the MC model and the combination of time step sizes confirmed the accuracy and robustness of MusAsi, and we concluded that the combination of a 1.25 ms FE time step and 0.005 ms MC multiple steps using a few hundred motor proteins in each finite element was appropriate in the tradeoff between accuracy and computational time. Furthermore, for a biventricular FE model consisting of 45,000 tetrahedral elements, one heartbeat could be computed within 1.5 h using 320 cores of a conventional parallel computer system. These results support the practicality of MusAsi for uses in both the basic research of the relationship between molecular mechanisms and cardiac outputs, and clinical applications of perioperative prediction.
Highlights
Demands for the prediction of outcomes from various types of operations are emerging in clinical problems of heart disease
To update the variables in both the Monte Carlo (MC) and finite element (FE) models from the FE time step at T to the time step at T + T, first, the stretch λT and stretch rate λ T in the fiber orientation of the FE model are used as the initial half-sarcomere length (HSL) λT·SL0/2 at T and its shorting velocity −λ T·SL0/2 in the time interval [T, T T] of the half-sarcomere model (Figure 3A), where SL0 is the sarcomere length under the unloaded condition
The stretch λT+ T is implicitly integrated into the active tension, which results in the appropriate evaluation of active stiffness in the FE model and the stability of the Newton iterations
Summary
Demands for the prediction of outcomes from various types of operations are emerging in clinical problems of heart disease. 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) These approaches appear to have difficulties, in reproducing the realistic relaxation phase that is important to ease the influx of blood from the atria to the ventricles
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