Interlayer micromechanics of the aortic heart valve leaflet

Abstract

While the mechanical behaviors of the fibrosa and ventricularis layers of the aortic valve (AV) leaflet are understood, little information exists on their mechanical interactions mediated by the GAG-rich central spongiosa layer. Parametric simulations of the interlayer interactions of the AV leaflets in flexure utilized a tri-layered finite element (FE) model of circumferentially oriented tissue sections to investigate inter-layer sliding hypothesized to occur. Simulation results indicated that the leaflet tissue functions as a tightly bonded structure when the spongiosa effective modulus was at least 25 % that of the fibrosa and ventricularis layers. Novel studies that directly measured transmural strain in flexure of AV leaflet tissue specimens validated these findings. Interestingly, a smooth transmural strain distribution indicated that the layers of the leaflet indeed act as a bonded unit, consistent with our previous observations (Stella and Sacks in J Biomech Eng 129:757–766, 2007) of a large number of transverse collagen fibers interconnecting the fibrosa and ventricularis layers. Additionally, when the tri-layered FE model was refined to match the transmural deformations, a layer-specific bimodular material model (resulting in four total moduli) accurately matched the transmural strain and moment-curvature relations simultaneously. Collectively, these results provide evidence, contrary to previous assumptions, that the valve layers function as a bonded structure in the low-strain flexure deformation mode. Most likely, this results directly from the transverse collagen fibers that bind the layers together to disable physical sliding and maintain layer residual stresses. Further, the spongiosa may function as a general dampening layer while the AV leaflets deforms as a homogenous structure despite its heterogeneous architecture.

Appendices

Appendix 1

1.1 Transmural deformation analysis

The images captured using the micro-imaging system (Fig. 2d) were analyzed by locating the markers that had been airbrushed onto the edge of the tissue. A custom program was written in LabVIEW (National Instruments, Austin, TX) to post-process the images taken by the micro-imaging system so that displacement fields could be determined. The markers were identified, numbered, and their areas and centroid coordinates were determined. This procedure was performed simultaneously for the reference image and for the deformed image. The software then displayed both altered images concurrently so that the user of the program could match markers between the reference image and the deformed image.

From the resulting images, the coordinates of the reference markers were referred to as the ($X_{1}, X_{2})$ system, and the deformed coordinates were referred to as the ($x_{1}, x_{2})$ system. The displacements, $u$ and $v$, were calculated from the former using $u=x_1 -X_1$ and $v=x_2 -X_2$, respectively. These quantities were then fitted to the surface described by Eq. (5).

$$\begin{aligned} u&= a_0 +a_1 X_1 + a_2 X_2 + a_3 X_1 X_2 +a_4 X_1^2 + a_5 X_2^2\nonumber \\ v&= b_0 + b_1 X_1 +b_2 X_2 + b_3 X_1 X_2 + b_4 X_1^2 +b_5 X_2^2 \end{aligned}$$

(5)

The surface fit to the $u$ and $v$ coordinates achieved an $r^{2}$ value of approximately 0.9. A higher order fit could have been used resulting in a higher $r^{2}$, this would produce a rough surface due to variations in marker location from thresholding. The lower order fit maintains a smooth surface, true to the nature of the sample tested, by not overfitting the curve to all variations in marker location. By evaluating Eq. (6), the deformation gradient was obtained.

$$\begin{aligned} \mathbf{F}=\mathbf{H}+\mathbf{I}=\left[ {{\begin{array}{ll} {\frac{\partial u }{\partial X_1 }}&{} {\frac{\partial u }{\partial X_2 }} \\ {\frac{\partial v}{\partial X_1 }}&{} {\frac{\partial v}{\partial X_2 }} \\ \end{array} }} \right] +\left[ {{\begin{array}{ll} 1&{} 0 \\ 0&{} 1 \\ \end{array} }} \right] =\left[ {{\begin{array}{ll} {\lambda _1 }&{} {k _1 } \\ {k _2 }&{} {\lambda _2 } \\ \end{array} }} \right] \end{aligned}$$

(6)

The F was then decomposed into its stretch and rotation tensors, U and R, respectively, Eq. (7). The polar decomposition of the deformation tensor removes rigid body motion effects into the rotation tensor, leaving only the stretch deformation information in the stretch tensor. The rigid body rotation information in R was calculated to determine the degree of rotation experienced by the tissue during flexure. Higher levels of rigid body rotation were determined to be coincident with measurements taken away from the center of the tissue.

$$\begin{aligned}&\mathbf{F}=\mathbf{RU}\nonumber \\&\mathbf{U}^{2}=\mathbf{F}^\mathbf{T}\mathbf{F}\nonumber \\&\mathbf{R}=\mathbf{FU}^{-1} \end{aligned}$$

(7)

The stretch tensor components $U_{11}$ and $U_{22 }$ correspond to local tissue strains in the $X_{1}$ and $X_{2}$ directions, respectively. Thus, the location of the neutral axis was determined by plotting $U_{11}$ against the thickness of the tissue. The depth of the tissue that coincided with the $U_{11}$ value of unity was the corresponding location of the neutral axis. Rotation that occurred in the displacement field was characterized by determining the angle of rotation, $\alpha$, incurred in the deformed system from the reference state.

Appendix 2

1.1 Parametric interlayer bonding study supplement

The reported results for the parametric bonding study represented a curvature change of $0.2\,\hbox {mm}^{-1 }$ solely for sake of clarity, as the same trends were observed at curvature changes of 0.1, 0.2 and 0.3 mm$^{-1}$ (Fig. 9a–c). These choices of curvature change were taken from our in vitro measurements (Sugimoto and Sacks 2013). Not surprisingly, we noted that the presence of interlayer sliding (not magnitude) either simulated (Fig. 9) or experimentally (Fig. 10) was not a function of the level of bending (i.e., $\Delta \kappa )$, but only of the ratio of the ventricularis and fibrosa:spongiosa moduli for the simulations. Thus, the estimated $\mu _{S}$ threshold is independent of imposed curvature. Greater bending simply created greater sliding.

Fig. 9

Transmural deformation results of parametric bonding simulation, $\varLambda _{1}$ is plotted against the normalized leaflet thickness for a curvature change of 0.1, 0.2, and 0.3 mm$^{-1}$. A tri-layered rectangular strip represented the AV and shear moduli values ranging from 1.0 Pa to 45 kPa were assigned to the central spongiosa layer to emulate varying degrees of connectivity between the outside layers and identify the following relationship: $\mu _{F}=\mu _{V}:\mu _{S}$. Results indicate for all curvature levels that for measurable sliding to occur between the fibrosa and ventricularis, the spongiosa must possess a shear modulus less than 1 kPa

Fig. 10

Experimental results obtained from transmural bending tests performed on native aortic valve tissue. The tissue was bent to three different changes of curvature, 0.1, 0.2, and 0.3 mm$^{-1}$. As curvature increased, the deformation increased as expected, yet no sliding is observed

1.2 Parametric out-of-plane warping study supplement

To investigate the effects of leaflet geometry on the simulation findings of out-of-plane warping effects, a parametric simulation was performed varying the thickness of the leaflets as well as the curvature change. The specimen length and width remained constant for the simulations. Figure 7 demonstrates the significant change in net axial stretch between the center of the specimen and the edge. Therefore, this change in absolute axial stretch ($\varLambda _{1}$) was used as a metric of warping and plotted against the change in specimen thickness (Fig. 11). Results found that increasing thickness of the specimens exaggerated the degree of warping. Furthermore, as expected, this warping effect increases with increasing curvature change.

Fig. 11

The effect of leaflet thickness on the out-of-plane warping estimated by the simulation (Fig. 7). The degree of warping, measured by the absolute change in net axial stretch ($\varLambda _{1})$, increases with increasing leaflet thickness. Additionally, this relationship is maintained and exaggerated with increasing curvature (0.1, 0.2 and 0.3 mm$^{-1})$