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force-velocity-Estrada

Module: solid

Category: materialprop

Type string: "force-velocity-Estrada"

Parameters

Name Description Default Units
ascl ascl 0 []
Tmax Tmax 1 []
ca0 ca0 1 []
camax camax 0 []
beta beta 0 []
l0 l0 0 []
refl refl 0 []
alpha1 alpha1 0 []
alpha2 alpha2 0 []
alpha3 alpha3 0 []
A1 A1 0 []
A2 A2 0 []
A3 A3 0 []
a_t a_t 0 []
force_velocity force_velocity true []

Description

This material model was formulated by Estrada et al. 1 and is called force-velocity-Estrada. It is a modification of the active_contraction material based on the formulation of Guccione et al. 2 described in active-contraction, based on the fading-memory formulation poroposed by Hunter et al. 3. The active stress is evaluated as \(T^{a}\mathbf{a}\otimes\mathbf{a}, where \mathbf{a}\) is the unit vector along the fiber in the current configuration, and

\[ T^{a}=e\left(t\right)\underbrace{T_{\max}\frac{Ca_{0}^{2}}{Ca_{0}^{2}+ECa_{50}^{2}}}_{\begin{array}{c} \text{instantaneous length}\\ \text{dependence} \end{array}}\underbrace{\frac{1+aQ\left(t,\lambda\left(t\right)\right)}{1-Q\left(t,\lambda\left(t\right)\right)}}_{\text{force-velocity}} \]

where

\[ ECa_{50}=\frac{\left(\text{Ca}_{0}\right)_{\max}}{\sqrt{\exp\left[\beta\left(\lambda\left(t\right)l_{r}-l_{0}\right)\right]-1}} \]

and

\[ Q\left(t,\lambda\left(t\right)\right)=\sum_{k=1}^{3}A_{k}\int_{-\infty}^{t}e^{-\alpha_{k}\left(t-\tau\right)}\dot{\lambda}\left(\tau\right)\,d\tau \]

In these expressions, \(\lambda\left(t\right)\) is the fiber stretch ratio at the current time. Here, \(T^{a}\) "is the product of three distinct components: a time-varying normalized activation, \(e\left(t\right)\), that defines the time course of force generation throughout the cardiac cycle; an instantaneous length-dependent term that scales the peak possible isometric tension based on the current fiber stretch; and a force-velocity term that dampens the instantaneous force generation based on the rate of shortening of the fibers" 1. The activation curve \(e\left(t\right)\) is represented by the ascl property that takes an optional attribute, lc, which defines the time-dependent load controller.

Example:

This example defines a transversely isotropic material with a Mooney-Rivlin basis. It defines a homogeneous fiber direction and uses the active fiber contraction feature.

<material id="3" type="trans iso Mooney-Rivlin">
  <c1>13.85</c1>
  <c2>0.0</c2>
  <c3>2.07</c3>
  <c4>61.44</c4>
  <c5>640.7</c5>
  <k>100.0</k>
  <lam_max>1.03</lam_max>
  <fiber type="vector">1,0,0</fiber>
  <active_contraction type="force-velocity-Estrada">
    <ascl lc="1">1</ascl>
    <ca0>4.35</ca0>
    <beta>4.75</beta>
    <l0>1.58</l0>
    <refl>1.58</refl>
    <Tmax>135.7</Tmax>
    <alpha1>30.30303</alpha1>
    <A1>50</A1>
    <a_t>0.5</a_t>
    <force_velocity>1</force_velocity>
  </active_contraction>
</material>

  1. Estrada, Ana Cristina, Yoshida, Kyoko, Clarke, Samantha A, and Holmes, Jeffrey W, "Longitudinal Reinforcement of Acute Myocardial Infarcts Improves Function by Transmurally Redistributing Stretch and Stress", J Biomech Eng 142, 2 (2020). ↩↩

  2. Guccione, J.M. and McCulloch, A.D., "Mechanics of active contraction in cardiac muscle: part I - constitutive relations for fiber stress that describe deactivation", J. Biomechanical Engineering vol. 115, no. 1 (1993), pp. 72-83. ↩

  3. Hunter, P J, McCulloch, A D, and ter Keurs, H E, "Modelling the mechanical properties of cardiac muscle", Prog Biophys Mol Biol 69, 2-3 (1998), pp. 289-331. ↩