A novel constitutive model for passive right ventricular myocardium: evidence for myofiber–collagen fiber mechanical coupling

Abstract

The function of right ventricle (RV) is recognized to play a key role in the development of many cardiopulmonary disorders, such as pulmonary arterial hypertension (PAH). Given the strong link between tissue structure and mechanical behavior, there remains a need for a myocardial constitutive model that accurately accounts for right ventricular myocardium architecture. Moreover, most available myocardial constitutive models approach myocardium at the length scale of mean fiber orientation and do not explicitly account for different fibrous constituents and possible interactions among them. In the present work, we developed a fiber-level constitutive model for the passive mechanical behavior of the right ventricular free wall (RVFW). The model explicitly separates the mechanical contributions of myofiber and collagen fiber ensembles, and accounts for the mechanical interactions between them. To obtain model parameters for the healthy passive RVFW, the model was informed by transmural orientation distribution measurements of myo- and collagen fibers and was fit to the mechanical testing data, where both sets of data were obtained from recent experimental studies on non-contractile, but viable, murine RVFW specimens. Results supported the hypothesis that in the low-strain regime, the behavior of the RVFW is governed by myofiber response alone, which does not demonstrate any coupling between different myofiber ensembles. At higher strains, the collagen fibers and their interactions with myofibers begin to gradually contribute and dominate the behavior as recruitment proceeds. Due to the use of viable myocardial tissue, the contribution of myofibers was significant at all strains with the predicted tensile modulus of $\sim$32 kPa. This was in contrast to earlier reports (Horowitz et al. 1988) where the contribution of myofibers was found to be insignificant. Also, we found that the interaction between myo- and collagen fibers was greatest under equibiaxial strain, with its contribution to the total stress not exceeding 20 %. The present model can be applied to organ-level computational models of right ventricular dysfunction for efficient diagnosis and evaluation of pulmonary hypertension disorder.

Appendices

Appendix 1

In this appendix, we discuss the local convexity of the energy function $\Psi (\mathbf{C})$ with respect to the right Cauchy–Green tensor $\mathbf{C}$. Recalling from relation (6) that the ground matrix term $\Psi ^{{g}}$ is neo-Hookean and convex in $\mathbf{C}$, it suffices to only assess the convexity of the anisotropic part of the energy, given by

$$\begin{aligned} \Psi ^{\mathrm{Aniso}}(\mathbf{C})={\phi }^{{m}} \Psi ^{{m}}(\mathbf{C})+{\phi }^{{c}}\left[ { \Psi ^{{c}}(\mathbf{C})+\Psi ^{{m}-{c}}(\mathbf{C})} \right] . \end{aligned}$$

(33)

By definition, the local convexity of $\Psi ^{\mathrm{Aniso}}(\mathbf{C})$ requires that the fourth-order tensor $\mathbf{L}= \partial ^{2} \Psi ^{\mathrm{Aniso}}(\mathbf{C})/\partial \mathbf{C}\partial \mathbf{C}$ be positive semi-definite, i.e.,

$$\begin{aligned} \mathbf{A}\cdot \mathbf{L} \mathbf{A}\ge 0, \end{aligned}$$

(34)

for all second order tensors $\mathbf{A}$. We note that, although the tensor $\mathbf{C}$ is subjected to the incompressibility constraint $\hbox {det}(\mathbf{C})=1$, we investigate the convexity of $\Psi (\mathbf{C})$ in 2D matrix $\mathbf{C}$ with three independent components in the plane containing the unit vectors ${\mathbf{n}}^{{m}}$ and ${\mathbf{n}}^{{c}}$.

Substituting the energy expressions in relations (11) and (12) into (33), it follows that

$$\begin{aligned}&\Psi ^{\mathrm{Aniso}}(\mathbf{C})\nonumber \\&\quad =\frac{1}{{H}} \int \limits _0^{{H}} {\left[ {{\phi }^{{m}}\int \limits _{-\pi /2}^{\pi /2} {\Gamma ^{{m}}\left( {\theta ^{{m}},{z}} \right) \Psi _{\mathrm{ens}}^{{m}} \left( {{{I}}^{{m}}} \right) \hbox {d}\theta ^{{m}} {\hbox {d}z}} } \right. }\nonumber \\&\qquad +\,{\phi }^{{c}} \int \limits _{-\pi /2}^{\pi /2} {\Gamma ^{{c}}\left( {\theta ^{{c}},{z}} \right) \Psi _{\mathrm{ens}}^{{c}} \left( {{{I}}^{{c}}} \right) \hbox {d}\theta ^{{c}} {\hbox {d}z}}\nonumber \\&\qquad +\,{\phi }^{{c}}\int \limits _{-\pi /2}^{\pi /2} \int \limits _{-\pi /2}^{\pi /2} \Gamma ^{{m}}\left( {\theta ^{{m}},{z}} \right) \Gamma ^{{c}}\left( {\theta ^{{c}},{z}} \right) \nonumber \\&\qquad \times \,\Psi _{\mathrm{ens}}^{{m}-{c}}\left. \left( {{{I}}^{{m}},{{I}}^{{c}}} \right) \hbox {d}\theta ^{{c}} \hbox {d}\theta ^{{m}} \right] {\hbox {d}z}, \end{aligned}$$

(35)

in which use has been made of $\theta _0^{{c}} =\pi /2$. Recalling that the distribution functions $\Gamma ^{{m}}$ and $\Gamma ^{{c}}$ satisfy the condition (5), it is easy to see that the convexity of the above expression in $\mathbf{C}$ is equivalent to the convexity of the function

$$\begin{aligned}&\Psi ^{{*}}\left( {{{I}}^{{m}},{{I}}^{{c}};{{R}}^{{m}},{{R}}^{{c}}} \right) \nonumber \\&\quad ={\phi }^{{m}}\int \limits _{-\pi /2}^{\pi /2} {{{R}}^{{m}}\left( {\theta ^{{m}}} \right) \Psi _{\mathrm{ens}}^{{m}} \left( {{{I}}^{{m}}} \right) \hbox {d}\theta ^{{m}}}\nonumber \\&\qquad +\,{\phi }^{{c}}\int \limits _{-\pi /2}^{\pi /2} {{{R}}^{{c}}\left( {\theta ^{{c}}} \right) \Psi _{\mathrm{ens}}^{{c}} \left( {{{I}}^{{c}}} \right) \hbox {d}\theta ^{{c}}}+{\phi }^{{c}}\int \limits _{-\pi /2}^{\pi /2} \nonumber \\&\qquad \times \,{\int \limits _{-\pi /2}^{\pi /2} {{{R}}^{{m}}\left( {\theta ^{{m}}} \right) {{R}}^{{c}}\left( {\theta ^{{c}}} \right) \Psi _{\mathrm{ens}}^{{m}-{c}} \left( {{{I}}^{{m}},{{I}}^{{c}}} \right) \hbox {d}\theta ^{{c}} \hbox {d}\theta ^{{m}}}},\nonumber \\ \end{aligned}$$

(36)

where ${{R}}^{{m}}$ and ${{R}}^{{c}}$ are two distribution functions satisfying the condition (5). Making use of this condition, the above energy function can be rewritten as

$$\begin{aligned}&\Psi ^{{*}}\left( {{{I}}^{{m}},{{I}}^{{c}};{{R}}^{{m}},{{R}}^{{c}}} \right) \nonumber \\&\quad =\int \limits _{-\pi /2}^{\pi /2} {\int \limits _{-\pi /2}^{\pi /2} {{{R}}^{{m}}\left( {\theta ^{{m}}} \right) {{R}}^{{c}}\left( {\theta ^{{c}}} \right) \Psi _{\mathrm{ens}}^{*} \left( {{{I}}^{{m}},{{I}}^{{c}}} \right) \hbox {d}\theta ^{{m}} \hbox {d}\theta ^{{c}}} },\nonumber \\ \end{aligned}$$

(37)

with

$$\begin{aligned} \Psi _{\mathrm{ens}}^{*} ({{I}}^{{m}},{{I}}^{{c}})= & {} {\phi }^{{m}} \Psi _{\mathrm{ens}}^{{m}} ({{I}}^{{m}})+{\phi }^{{c}}[\Psi _{\mathrm{ens}}^{{c}} ({{I}}^{{c}})\nonumber \\&+\,\Psi _{\mathrm{ens}}^{{m}-{c}} ({{I}}^{{m}},{{I}}^{{c}})]. \end{aligned}$$

(38)

Therefore, it follows that the convexity of the function $\Psi _{\mathrm{ens}}^{*} \left( {{{I}}^{{m}},{{I}}^{{c}}} \right)$ in $\mathbf{C}$ implies the convexity of $\Psi ^{\mathrm{Aniso}}(\mathbf{C})$. In other words, $\Psi ^{\mathrm{Aniso}}(\mathbf{C})$ is convex if the tensor ${\mathbf{L}}^{{*}}= \partial ^{2} \Psi _{\mathrm{ens}}^{*} \left( {{{I}}^{{m}},{{I}}^{{c}}} \right) /\partial \mathbf{C}\partial \mathbf{C}$ is positive semi-definite. The tensor ${\mathbf{L}}^{{*}}$ is obtained as

$$\begin{aligned}&{\mathbf{L}}^{{*}}=\left( {\Psi _{\mathrm{ens}}^{*} } \right) _{,{{I}}^{{m}} {{I}}^{{m}}}{\mathbf{n}}^{{m}}\otimes {\mathbf{n}}^{{m}}\otimes {\mathbf{n}}^{{m}}\otimes {\mathbf{n}}^{{m}}\nonumber \\&\qquad +\,\left( {\Psi _{\mathrm{ens}}^{*}} \right) _{,{{I}}^{{c}} {{I}}^{{c}}} {\mathbf{n}}^{{c}}\otimes {\mathbf{n}}^{{c}}\otimes {\mathbf{n}}^{{c}}\otimes {\mathbf{n}}^{{c}} +\left( {\Psi _{\mathrm{ens}}^{*} } \right) _{,{{I}}^{{m}} {{I}}^{{c}}}\nonumber \\&\qquad ({\mathbf{n}}^{{m}}\otimes {\mathbf{n}}^{{m}}\otimes {\mathbf{n}}^{{c}}\otimes {\mathbf{n}}^{{c}}+{\mathbf{n}}^{{c}}\otimes {\mathbf{n}}^{{c}}\otimes {\mathbf{n}}^{{m}}\otimes {\mathbf{n}}^{{m}}), \end{aligned}$$

(39)

in which subscript commas followed by an index denote derivatives with respect to the corresponding variables. Making use of the above expression, the condition (34) can be written as

$$\begin{aligned} {{a}}_1^2 \left( {\Psi _{\mathrm{ens}}^{*}}\right) _{,{{I}}^{{m}} {{I}}^{{m}}} +{{a}}_{2}^{2} \left( {\Psi _{\mathrm{ens}}^{*}} \right) _{,{{I}}^{{c}} {{I}}^{{c}}}+2 {{a}}_1 {{a}}_2 \left( {\Psi _{\mathrm{ens}}^{*}} \right) _{,{{I}}^{{m}} {{I}}^{{c}}} \ge 0,\nonumber \\ \end{aligned}$$

(40)

where ${{a}}_1 ={\mathbf{n}}^{{m}}\cdot \mathbf{A}{\mathbf{n}}^{{m}}$ and ${{a}}_2 ={\mathbf{n}}^{{c}}\cdot \mathbf{A}{\mathbf{n}}^{{c}}$. The above condition can be further recast into

$$\begin{aligned} \left( {{\begin{array}{l} {{{a}}_1} \\ {{{a}}_2} \\ \end{array} }} \right) \cdot \left( {{\begin{array}{ll} {\left( {\Psi _{\mathrm{ens}}^{*}} \right) _{,{{I}}^{{m}} {{I}}^{{m}}}}&{} {\left( {\Psi _{\mathrm{ens}}^{*}} \right) _{,{{I}}^{{m}} {{I}}^{{c}}}} \\ {\left( {\Psi _{\mathrm{ens}}^{*}} \right) _{,{{I}}^{{m}} {{I}}^{{c}}}}&{} {\left( {\Psi _{\mathrm{ens}}^{*}} \right) _{,{{I}}^{{c}} {{I}}^{{c}}}}\\ \end{array} }} \right) \left( {{\begin{array}{l} {{{a}}_1} \\ {{{a}}_2} \\ \end{array} }} \right) \ge 0, \end{aligned}$$

(41)

which is always satisfied for arbitrary real values of ${{a}}_1$ and ${{a}}_2$ if $\Psi _{\mathrm{ens}}^{*} \left( {{{I}}^{{m}},{{I}}^{{c}}} \right)$ is locally convex in ${{I}}^{{m}}$ and ${{I}}^{{c}}$ jointly. Two conditions to guarantee the convexity of $\Psi _{\mathrm{ens}}^{*} \left( {{{I}}^{{m}},{{I}}^{{c}}} \right)$ are

$$\begin{aligned}&\left( {\Psi _{\mathrm{ens}}^{*} } \right) _{,{{I}}^{{m}} {{I}}^{{m}}} \ge 0,\quad \left( {\Psi _{\mathrm{ens}}^{*} } \right) _{,{{I}}^{{m}} {{I}}^{{m}}}\left( {\Psi _{\mathrm{ens}}^{*}} \right) _{,{{I}}^{{c}} {{I}}^{{c}}}\nonumber \\&\quad -\left[ {\left( {\Psi _{\mathrm{ens}}^{*} } \right) _{,{{I}}^{{m}} {{I}}^{{c}}}} \right] ^{2}\ge 0. \end{aligned}$$

(42)

It is further simple to show that $\left( {\Psi _{\mathrm{ens}}^{*} } \right) _{,{{I}}^{{m}} {{I}}^{{m}}}$ is positive for the ensemble energy functions proposed in this work for all deformation (under the aforementioned conditions of positive material parameters). The remaining condition (42)$_{2}$ will be satisfied if

$$\begin{aligned}&\left\{ \frac{{\phi }^{{m}} {{k}}_1^{{m}} }{2 {{I}}^{{m}}}\exp \left[ {{{k}}_2^{{m}} \left( {\sqrt{{{I}}^{{m}}}-1} \right) ^{2}} \right] \right. \nonumber \\&\quad \times \, \left[ {2 {{k}}_2^{{m}} (\sqrt{{{I}}^{{m}}}-1)^{2}+1/\sqrt{{{I}}^{{m}}}} \right] \nonumber \\&\quad \left. +\,{\phi }^{{c}} {{k}}_1^{{mc}} {{k}}_2^{{mc}} \int \limits _{\lambda _{{lb}} }^{\lambda _{{ub}} } {D(\lambda _{{s}} )\psi \hbox {d}\lambda _{{s}} } \right\} \nonumber \\&\quad \times \, {\phi }^{{c}}\left\{ \frac{{k}^{{c}}}{2 {{I}}^{{c}}\sqrt{{{I}}^{{c}}}}\int \limits _{\lambda _{{lb}} }^{\lambda _{{ub}} } \frac{D(\lambda _{{s}} )}{\lambda _{{s}} } \hbox {d}\lambda _{{s}} \right. \nonumber \\&\left. \quad +\,{{k}}_1^{{mc}} {{k}}_2^{{mc}} \int \limits _{\lambda _{{lb}} }^{\lambda _{{ub}} } {\frac{D(\lambda _{{s}} )}{\lambda _{{s}}^4 } \exp \left[ {{{k}}_2^{{mc}} \left( {{\tilde{I}}-2} \right) } \right] \hbox {d}\lambda _{{s}} } \right\} \nonumber \\&\quad -\,\left\{ {\phi }^{{c}} {{k}}_1^{{mc}} {{k}}_2^{{mc}} \int \limits _{\lambda _{{lb}} }^{\lambda _{{ub}} } \frac{D(\lambda _{{s}} )}{\lambda _{{s}}^2 } \left\{ {\exp \left[ {{{k}}_2^{{mc}} ({\tilde{I}}-2)} \right] }\right. \right. \nonumber \\&\quad \left. \left. -\,\exp \left[ {{{k}}_2^{{mc}} ({{I}}^{{m}}-1)} \right] \right\} \hbox {d}\lambda _{{s}} \right\} ^{2}\ge 0, \end{aligned}$$

(43)

where the function $\psi \left( {{{I}}^{{m}},{{I}}^{{c}}} \right)$ is given in (22).

Appendix 2

Here, we express the energy term $\Psi ^{{m}-{m}}$ used to detect possible interactions between myofiber ensembles in the low-strain regime (where $\Psi ^{{c}}$ and $\Psi ^{{m}-{c}}$ do not contribute) as follows

$$\begin{aligned}&\Psi ^{{m}-{m}}\nonumber \\&\quad =\frac{1}{{H}} \frac{{{k}}_1^{{m}-{m}} }{2 {{k}}_2^{{m}-{m}} } \int \limits _0^{{H}} \left\{ \int \limits _{-\pi /2}^{\pi /2} \int \limits _{-\pi /2}^{-\pi /2} \Gamma ^{{m}}\left( {\theta ^{{m}},Z} \right) \Gamma ^{{m}}\left( {\theta ^{{m}*},Z} \right) \right. \nonumber \\&\qquad \times \, \left\{ {\exp \left[ {{{k}}_2^{{m}-{m}} \left( {\sqrt{{{I}}^{{m}}+{{I}}^{{m}*}}-\sqrt{2}} \right) ^{2}} \right] -1} \right\} \hbox {d}\theta ^{{m}} \hbox {d}\theta ^{{m}*} \nonumber \\&\qquad -\, \int \limits _{-\pi /2}^{\pi /2} \left[ {\Gamma ^{{m}}\left( {\theta ^{{m}},{z}} \right) } \right] ^{2}\nonumber \\&\qquad \times \,\left. \left\{ {\exp \left[ {2 {{k}}_2^{{m}-{m}} \left( {\sqrt{{{I}}^{{m}}}-1} \right) ^{2}} \right] -1} \right\} \hbox {d}\theta ^{{m}} \right\} {\hbox {d}z}. \end{aligned}$$

(44)

In the above expression, ${{k}}_1^{{m}-{m}}$ and ${{k}}_2^{{m}-{m}}$ are unknown parameters, and the angular variables $\theta ^{{m}}$ and $\theta ^{{m}*}$ (associated with invariants ${{I}}^{{m}}$ and ${{I}}^{{m}*}$) have been used to differentiate two myofiber ensembles with different orientations. The second term in the above expression subtracts the energy for the case of $\theta ^{{m}}=\theta ^{{m}*}$ accounted for in the first term (enforcing that each myofiber ensemble does not interact with itself).