A Microstructurally Based Multi-Scale Constitutive Model of Active Myocardial Mechanics

Abstract

Contraction of cardiac muscle cells provides the work for ventricular pumping. The primary component of this contractile stress development in myocardium acts along the axis of the myofilaments; however, there may be a component directed transversely as well. Biaxial testing of tonically activated cardiac tissue has shown that myocardium can generate active stresses in the transverse direction that are as high as 50% of those developed along the fiber axis. The microstructural basis for this is not clear. We hypothesized that transverse active stresses are generated at the crossbridge and myofilament lattice scales and transmitted via the myocardial laminar sheets as plane stress objects. To test this hypothesis, we developed a multi-scale constitutive model accounting for crossbridge and myofilament lattice structures as well as multicellular myofiber and sheet angle dispersions. Integrating these properties in a finite element model of an actively contracting myocardial tissue slice suggested that these mechanisms may be sufficient to explain the results of biaxial tests in contracted myocardium.

Invited Book Chapter in “Structure-Based Mechanics of Tissues and Organs: A Tribute to Yoram Lanir”

Appendices

A.1 Appendix 1: Fiber-Sheet Dispersion Effects on Active Stress

Here we give details of the derivation of the fiber-sheet dispersion effects on active stress from (22.5). We used a Von-Mises distribution for the three angles. The probability density of a Von-Mises distribution is given by the following equation:

$$ f\left(\gamma \right)=\frac{{\mathrm{e}}^{\kappa \cos \gamma }}{2\pi {I}_0\left(\kappa \right)} $$

(22.8)

where I ₀ is the modified Bessel function of order 0 and κ is called the concentration parameter that controls the standard deviation of the distribution. The components of the stress tensor can be computed to be

$$ \begin{aligned}[c]{\boldsymbol{T}}_{11}&={S}_{\mathrm{f}}\left[\nu \left({I}_{\sin^2\varphi }{I}_{\cos^2\theta } + {I}_{\cos^2\beta }{I}_{\cos^2\varphi }{I}_{\sin^2\theta}\right)\ \right.\\[4pt] {}&\left.+ \left({I}_{\cos^2\varphi }{I}_{\cos^2\theta } + {I}_{\cos^2\beta }{I}_{\sin^2\varphi }{I}_{\sin^2\theta}\right)\right]\\[4pt] {}{\boldsymbol{T}}_{22}&={S}_{\mathrm{f}}\left[\nu \left({I}_{\sin^2\beta }{I}_{\cos^2\varphi}\right) + \left({I}_{\sin^2\beta }{I}_{\sin^2\varphi}\right)\right]\\[4pt] {}{\boldsymbol{T}}_{33}&={S}_{\mathrm{f}}\left[\nu \left({I}_{\sin^2\varphi }{I}_{\sin^2\theta } + {I}_{\cos^2\beta }{I}_{\cos^2\varphi }{I}_{\cos^2\theta}\right)\right.\\[4pt]{}&\left.+ \left({I}_{\cos^2\varphi }{I}_{\sin^2\theta } + {I}_{\cos^2\beta }{I}_{\sin^2\varphi }{I}_{\cos^2\theta}\right)\right]\end{aligned} $$

(22.9)

where the integrals I can be computed numerically from the distribution. For a standard dispersion of 12° for φ and θ, and a 30° for b, we get the active stress components to be given by

$$ \begin{aligned}[c]{\boldsymbol{T}}_{11}&={S}_{\mathrm{f}}\left[0.067\ \nu +0.924\right]\\[4pt] {}{\boldsymbol{T}}_{22}&={S}_{\mathrm{f}}\left[0.201\ \nu +0.008\right]\\[4pt] {}{\boldsymbol{T}}_{33}&={S}_{\mathrm{f}}\left[0.724\ \nu +0.067\right]\end{aligned} $$

(22.10)

These equations were then used in the finite element model and the k computed from the lattice model is used as the input to these models.

A.2 Appendix 2: Strain Dependence of Angle Distributions

In continuum mechanics, deformations of bodies create changes in angles. For example, consider the two-dimensional example in Fig. A.1. Suppose the fibers in this tissue are originally oriented at an angle γ ₀. After undergoing deformation, this angle is represented by γ. The relationship between γ and γ ₀ can be derived from continuum mechanics principles (Fung 1993), and is given by:

$$ \begin{aligned}[c] \cos \gamma &=\frac{{\mathbf{u}}_1\mathbf{C}{\mathbf{u}}_2}{\sqrt{{\mathbf{u}}_1\mathbf{C}{\mathbf{u}}_1}\sqrt{{\mathbf{u}}_2\mathbf{C}{\mathbf{u}}_2}}\\ {}&=\frac{\left[\begin{array}{c}\hfill 1\hfill \\ {}\hfill 0\hfill \end{array}\right]\left[\begin{array}{cc}\hfill {C}_{11}\hfill & \hfill {C}_{12}\hfill \\ {}\hfill {C}_{12}\hfill & \hfill {C}_{22}\hfill \end{array}\right]\left[\begin{array}{c}\hfill \cos {\gamma}_0\hfill \\ {}\hfill \sin {\gamma}_0\hfill \end{array}\right]}{\sqrt{C_{11}}\sqrt{\left[\begin{array}{c}\hfill \cos {\gamma}_0\hfill \\ {}\hfill \sin {\gamma}_0\hfill \end{array}\right]\left[\begin{array}{cc}\hfill {C}_{11}\hfill & \hfill {C}_{12}\hfill \\ {}\hfill {C}_{12}\hfill & \hfill {C}_{22}\hfill \end{array}\right]\left[\begin{array}{c}\hfill \cos {\gamma}_0\hfill \\ {}\hfill \sin {\gamma}_0\hfill \end{array}\right]}}\\ {}&=\frac{C_{11} \cos {\gamma}_0+{C}_{12} \sin {\gamma}_0}{\sqrt{C_{11}}\sqrt{C_{11}{ \cos}^2{\gamma}_0+2{C}_{12} \sin {\gamma}_0 \cos {\gamma}_0+{C}_{22}{ \sin}^2{\gamma}_0}}\end{aligned} $$

(22.11)

Fig. A.1

Schematic diagram representing the change in angle γ as a body deforms. In this example, due to horizontal shortening and vertical lengthening, γ > γ ₀. The angle would also be affected by shearing deformation (not shown)

In terms of the strain components E, the cos(γ) can be computed from the equation,

$$ \cos \gamma =\frac{\left(2{E}_{11}+1\right) \cos {\gamma}_0+2{E}_{12} \sin {\gamma}_0}{\sqrt{\left(2{E}_{11}+1\right)}\sqrt{\Big(2{E}_{11}{ \cos}^2{\gamma}_0+4{E}_{12} \sin {\gamma}_0 \cos {\gamma}_0+2{E}_{22}{ \sin}^2{\gamma}_0+1}} $$

(22.12)

In order to understand the strain dependence of the fiber dispersion functions, several numerical experiments were performed. Samples of 5000 angles were drawn from a Von-Mises distribution of known κ, the concentration parameter, which gives a standard deviation of 12°. The change in the angle γ is computed for different values of biaxial strains, and the new standard deviation and the κ parameter were computed for the resulting distribution (Fig. A.2). This was then compared with directly computing the change in the standard deviation angle using (22.12). It can be seen from Figs. A.3 and A.4 that the predicted standard deviations are within few degrees of the predicted values. Under shear strain, the mean is not zero, but this deviation in the mean is <2° for reasonable shear strains.

Fig. A.2

Effect of strain on fiber distribution. A positive transverse strain increases the standard deviation of the angle distribution while a positive fiber strain decreases the standard deviation

Fig. A.3

Comparison of actual standard deviation with predicted values for different strains

Next, the strain dependence of the active stress components was computed. The concentration parameter was varied from 10 to 40 for φ and θ, and from 2 to 10 for the sheet angle β. These correspond to a standard deviation of 18°–9° for φ and θ, and 48°–18° for β, respectively. It can be seen from Fig. A.5 that the strain dependence is very small for practical values of standard deviation of fiber dispersion and strains. Consequently, the strain dependence can be ignored for typical strains in a myocardium. In addition, if the strain values are extreme, the strain dependence can be incorporated by computing the new standard deviation of the distribution and using the concentration parameter that corresponds to this standard deviation value in the simulations.

Fig. A.4

Effect of combined biaxial strains on standard deviation and its prediction

Fig. A.5

Effect of the concentration parameter κ on the diagonal components of the active stress tensor. It can be seen that the strain dependence is very small and we can ignore it for practical simulations