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5.12 Active Contraction Model

A time varying “elastance” active contraction model 1 was added to the transversely isotropic materials. When active contraction is activated, the total Cauchy stress \(\boldsymbol{\sigma}\) is defined as the sum of the active stress tensor \(\boldsymbol{\sigma}^{a}=T^{a}\mathbf{a}\otimes\mathbf{a}\) and the passive stress tensor \(\boldsymbol{\sigma}^{p}\):

\[ \begin{equation} \boldsymbol{\sigma}=\boldsymbol{\sigma}^{p}+\boldsymbol{\sigma}^{a},\label{eq516} \end{equation} \]

where _a_ is the deformed fiber vector (unit length), defined as \(\lambda\mathbf{a}=\mathbf{F}\cdot\mathbf{a}\). The time varying elastance model is a modification of the standard Hill equation that scales the standard equation by an activation curve \(C\left(t\right)\). The active fiber stress \(T^{a}\) is defined as:

\[ \begin{equation} T^{a}=T_{\max}\frac{Ca_{0}^{2}}{Ca_{0}^{2}+ECa_{50}^{2}}C\left(t\right),\label{eq517} \end{equation} \]

where \(T_{max}=135.7\,\text{kPa}\) is the isometric tension under maximal activation at the peak intracellular calcium concentration of \(\text{Ca}_{0}=4.35\,\mu\text{M}\). The length dependent calcium sensitivity is governed by the following equation:

\[ \begin{equation} ECa_{50}=\frac{\left(\text{Ca}_{0}\right)_{\max}}{\sqrt{\exp\left[B(l-l_{0})\right]-1}}\,,\label{eq518} \end{equation} \]

where \(\left(\text{Ca}_{0}\right)_{\max}=4.35\,\mu\text{M}\) is the maximum peak intracellular calcium concentration, \(B=4.75\,\mu\text{m}^{-1}\) governs the shape of the peak isometric tension-sarcomere length relation, \(l_{0}=1.58\,\mu\text{m}\) is the sarcomere length at which no active tension develops, and \(l\) is the sarcomere length which is the product of the fiber stretch \(\lambda\) and the sarcomere unloaded length \(l_{r}=2.04\,\mu\text{m}\).

  1. Guccione, J.M.; McCulloch, A.D.. "Mechanics of active contraction in cardiac muscle: part I - constitutive relations for fiber stress that describe deactivation." J. Biomechanical Engineering, vol. vol. 115, pp. 72-83 (1993). ↩