## Abstract

We present a thermodynamically based model that captures the remodeling effects in cardiac muscle cells. This work begins with the formulation of the kinematics of a cardiomyocyte resulting from a prescribed macroscopic deformation and the reorganization of the internal structure. Specifically, relations between the macroscopic deformation and the number of sarcomeres, the sarcomere stretch, and the number of myofibrils in the cell are determined. The remodeling process is split into two separate phases—(1) elongation/shortening of the existing myofibrils by addition/detachment of sarcomeres and (2) formation of new myofibrils. The remodeling associated with each phase is modeled through a dissipation postulate. We show that remodeling is based on a competition between the internal energy, the entropy, the energy supplied to the system by ATP and other sources to drive the remodeling process, and dissipation mechanisms. While the variations in entropy associated with phase (1) are neglected, the substantial entropy loss associated with the formation of new myofibrils is determined. To illustrate the merit of the proposed framework, we compute the response of cardiomyocytes subjected to isometric axial stretch that are either free to deform or fixed in the transverse direction. We also examine the predictions of this model for cardiomyocytes subjected to various cyclic loadings. The proposed framework is capable of capturing a wide range of remodeling effects and agrees with experimental observations.

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## Acknowledgements

This research was supported, in part, through an Otis Williams Postdoctoral Fellowship granted by the Santa Barbara Foundation.

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## Appendix: Loss of entropy due to myofibril formation

### Appendix: Loss of entropy due to myofibril formation

Here, we derive an expression for the loss of entropy during myofibril formation. To this end, we make the following conjectures: First, the number of unbounded proteins that can form free sarcomeres is significantly larger than the number of sarcomeres forming new myofibrils. Second, all sarcomeres are indistinguishable.

Recall Boltzmann’s entropy formula

where \(k_{\rm b}\) is the Boltzmann constant and \(\Omega\) denotes the number of possible configurations available to the system. We begin by examining the entropy of a single myofibril comprising \(n\) sarcomeres. Since the sarcomeres are indistinguishable and aligned along the direction of the myofibril, there is only one possible configuration available to the system. Accordingly, the entropy of a myofibril is \(S=0\).

Next, we examine a reservoir of \(n_{\rm u}\) unbounded protein packets. We discretize the available space for unbounded proteins in the cell to \(n_{\rm L}\gg n_{\rm u}\) lattice sites, where each site can contain only one protein packet. The protein packets are assumed to be identical and therefore there are

configurations that are available to the system. By employing Stirling’s approximation, we find the referential entropy

As a result of the remodeling processes \(\chi\) proteins convert into sarcomeres to form new myofibrils. Following the assumption that there is a very large reservoir of protein packets, we can say \(\chi \ll n_{\rm u}\). By substituting \(n_{\rm u}\) with \(n_{\rm u}-\chi\) in Eq. (29), we compute the entropy in the deformed configuration

To first order around \(\chi /n_{\rm u}=0\), the loss of entropy is

where the constant \(\zeta =k_{\rm b}\ln \left( n_{\rm L}/n_{\rm u}-1\right) >0\) and the negative sign signifies a decrease in entropy. The rate of reduction in entropy is as follows:

Note that the rate of reorganization of free sarcomeres that form myofibrils with \(n\) sarcomeres is \(\dot{\chi }=n\,\dot{\eta }\), leading to expression (22).

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Cohen, N., Deshpande, V.S., Holmes, J.W. *et al.* A microscopically motivated model for the remodeling of cardiomyocytes.
*Biomech Model Mechanobiol* **18, **1233–1245 (2019). https://doi.org/10.1007/s10237-019-01141-5

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### Keywords

- Remodeling in cardiomyocytes
- Cardiomyocytes
- Multi-scale modeling
- Statistical mechanics
- Actin/myosin interaction