Active Tissue Response

Therefore, the total stress in the fiber direction, $σ_{f}$ , is equal to the active stress, $σ_{a f}$ , plus the passive stress, $σ_{p f}$ :

σ_{f} = σ_{p f} + σ_{a f} .

The following time-varying elastance model (Walker et al.) defines the active stress in the cardiac muscle fiber direction:

σ_{a f} (t, E_{f f}) = \frac{T_{\max}}{2} \frac{{C a}_{0}^{2}}{{C a}_{0}^{2} + E {C a}_{50}^{2} (E_{f f})} (1 - \cos (ω (t, E_{f f}))),

where

E C a_{50} (E_{f f}) = \frac{C a_{0_{\max}}}{\sqrt{e^{B (l (E_{f f}) - l_{0})} - 1}}

\begin{matrix} ω (t, E_{f f}) = \end{matrix} {\begin{matrix} \begin{matrix} π \frac{t}{t_{0}} w h e n 0 \leq t \leq t_{0} \\ π \frac{t - t_{0} + t_{r} (l (E_{f f}))}{t_{r}} w h e n t_{0} \leq t \leq t_{0} + t_{r} (l (E_{f f})) \end{matrix} \\ 0 w h e n t \geq t_{0} + t_{r} (l (E_{f f})) \end{matrix}

t_{r} (l) = m l + b

l (E_{f f}) = l_{r} \sqrt{2 E_{f f} + 1}

and the following table defines its parameters.

Table 1. Constitutive Parameters for the Active Tissue Response
Parameters	Description
$T_{\max}$	Constitutive law contractility scaling factor (value directly scales ejection fraction)
${C a}_{0}$	The peak intercellular calcium concentration
${C a}_{0_{\max}}$	The maximum intercellular calcium concentration
$B$	Governs the shape of peak isometric tension-sarcomere length relation
$l_{0}$	The sarcomere length below which no active force develops
$t_{0}$	Time to reach the peak tension
$m, b$	Coefficients that govern the shape of the linear relaxation duration and sarcomere length relaxation
$E_{f f}$	Lagrangian strain tensor component aligned with the local muscle fiber direction
$l_{r}$	The initial sarcomere length

Active stress in the sheet direction, $σ_{s}$ , is the sum of the passive stress, $σ_{p s}$ , and a fraction of the stress in the fiber direction, $n * σ_{a f}$ (where $n$ is a scalar value less than 1.0 and represents the interaction between the adjacent muscle fibers):

σ_{s} = σ_{p s} + n * σ_{a f} .

The value of $n$ affects not only the total contractility of the chambers, but also the degree of twist developed in the chamber during the cardiac cycle. The magnitude of contractility for each chamber is tuned to provide the appropriate ejection fraction for that chamber. This involved the tuning of $T_{\max}, n$ (to limit the twist of the LV and RV), and $l_{0} .$