A viscoplastic model for the active component in cardiac muscle

Abstract

The HMK model (Hunter et al. in Prog Biophys Mol Biol 69:289–331, 1998) proposes mechanobiological equations for the influence of intracellular calcium concentration $\hbox {Ca}_\mathrm{i}$ on the evolution of bound calcium concentration $\hbox {Ca}_\mathrm{b}$ and the tropomyosin kinetics parameter z, which model processes in the active component of the tension in cardiac muscle. The inelastic response due to actin-myosin crossbridge kinetics is modeled in the HMK model with a function Q that depends on the history of the rate of total stretch of the muscle fiber. Here, an alternative model is proposed which models the active component of the muscle fiber as a viscoplastic material. In particular, an evolution equation is proposed for the elastic stretch $\lambda _\mathrm{a}$ in the active component. Specific forms of the constitutive equations are proposed and used to match experimental data. The proposed viscoplastic formulation allows for separate modeling of three processes: the high rate deactivation of crossbridges causing rapid reduction in active tension; the high but lower rate reactivation of crossbridges causing recovery of active tension; and the low rate relaxation effects characterizing the Hill model of muscles.

Appendix: Details of the analysis of the dynamic stiffness

Substituting (88) into (6), (7) and (34) yields (89) where the functions $f_{i}$ and $C_{i}$ are given by

$$\begin{aligned} f_{1}= & {} \frac{r_{0}}{b} \left( \frac{\hbox {Ca}_\mathrm{i}}{\hbox {Ca}_\mathrm{bh}}\right) \left( \frac{\hbox {Ca}_\mathrm{bh}}{\hbox {Ca}_\mathrm{bh0}}-1-y_{1}\right) ,\nonumber \\ f_{2}= & {} \frac{\alpha _{0}}{bz_\mathrm{h0}} \left[ \frac{z_\mathrm{h}-z_\mathrm{h0}(1+y_{2})}{1-z_\mathrm{h0}(1+y_{2})}\right] ,\nonumber \\ f_{3}= & {} -(1+y_{3})\left( \frac{\varGamma }{b}\right) \ln \left[ \frac{\lambda _\mathrm{h0}(1+y_{3})}{\lambda _\mathrm{h}}\right] , \end{aligned}$$

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with R being set equal to unity in all of the equations in this appendix. Linearizing these functions about the homeostatic state (85) yields the following values for $A_{ij}, B_{j}$ and $C_{j}$ in (91)

$$\begin{aligned}&A_{11}=\frac{r_{0}\hbox {Ca}_\mathrm{i}}{b\hbox {Ca}_\mathrm{bh0}},\nonumber \\&A_{12}=\frac{r_{0}b_{0}}{b\gamma },\nonumber \\&A_{13}=\frac{r_{0}b_{0}T_{\mathrm{ref}}\beta _\mathrm{f}}{b\gamma T_\mathrm{h0}},\nonumber \\&A_{21}=\frac{\alpha _{0}z_\mathrm{h0}n_{0}{C_{500}}^{n_{0}}}{b(1-z_\mathrm{ho}){\hbox {Ca}_\mathrm{bh0}}^{n_{0}}},\nonumber \\&A_{22}=-\frac{\alpha _{0}}{b(1-z_\mathrm{ho})}, A_{23}=0, A_{31}=0,\nonumber \\&A_{32}=\frac{\varGamma _{0}T_\mathrm{h0}}{bT_{\mathrm{ref}}\beta _\mathrm{f}},\nonumber \\&A_{33}=-\frac{\varGamma _{0}}{b},\nonumber \\&B_{2}=\left[ \frac{\alpha _{0}z_\mathrm{h0} \lambda _{0}{C_{500}}^{n_{0}}}{b(1-z_\mathrm{h0}){\hbox {Ca}_\mathrm{bh0}}^{n_{0}}}\right] \left[ n_{\mathrm{ref}}\beta _{1}\ln \left( \frac{\hbox {Ca}_\mathrm{bh0}}{C_{500}}\right) \right. \nonumber \\&\quad \qquad \left. +\left\{ \frac{n_{0}\beta _{2}pC_{\mathrm{50ref}}\ln (10)}{C_{500}}\right\} 10^{(6-pC_{\mathrm{50ref}})}\right] ,\nonumber \\&B_{1}=-\frac{r_{0}b_{0}z_\mathrm{h0}T_{\mathrm{ref}}\beta _{0}\lambda _{0}}{{b\gamma }T_\mathrm{h0}},\nonumber \\&B_{3}=\frac{\varGamma _{0}\beta _{0}\lambda _{0}z_\mathrm{h0}}{b\beta _\mathrm{f}}, C_{1}=C_{2}=0, C_{3}=1. \end{aligned}$$

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