On the effects of leaflet microstructure and constitutive model on the closing behavior of the mitral valve

Abstract

Recent long-term studies showed an unsatisfactory recurrence rate of severe mitral regurgitation 3–5 years after surgical repair, suggesting that excessive tissue stresses and the resulting strain-induced tissue failure are potential etiological factors controlling the success of surgical repair for treating mitral valve (MV) diseases. We hypothesized that restoring normal MV tissue stresses in MV repair techniques would ultimately lead to improved repair durability through the restoration of MV normal homeostatic state. Therefore, we developed a micro- and macro- anatomically accurate MV finite element model by incorporating actual fiber microstructural architecture and a realistic structure-based constitutive model. We investigated MV closing behaviors, with extensive in vitro data used for validating the proposed model. Comparative and parametric studies were conducted to identify essential model fidelity and information for achieving desirable accuracy. More importantly, for the first time, the interrelationship between the local fiber ensemble behavior and the organ-level MV closing behavior was investigated using a computational simulation. These novel results indicated not only the appropriate parameter ranges, but also the importance of the microstructural tuning (i.e., straightening and re-orientation) of the collagen/elastin fiber networks at the macroscopic tissue level for facilitating the proper coaptation and natural functioning of the MV apparatus under physiological loading at the organ level. The proposed computational model would serve as a logical first step toward our long-term modeling goal—facilitating simulation-guided design of optimal surgical repair strategies for treating diseased MVs with significantly enhanced durability.

Appendices

Appendix 1: Computation of displacement errors at fiducial marker positions

Each of the shell elements for the MV median leaflet surfaces at the reference configuration (Fig. 5a) developed in Sect. 2.3 was first extruded toward the left atrial direction with a distance of half the elemental thickness in order to obtain the FE mesh for the atrial leaflet surfaces. Then, each of all the 266 fiducial marker was associated with one shell element whose element region entirely covers this fiducial marker, and the corresponding four shape functions $(N_{1},N_{2},N_{3},N_{4})$ were computed based on the positions of the four nodes of this identified element as well as the fiducial marker location on the expanded 2D parametric domain $(s,t)$ (Fig. 12). Similar extrusion procedure was performed for the FE meshes under deformations at 30 and 70 mmHg transvalvular pressure, respectively, and the FE-predicted position of each fiducial marker was obtained by interpolation using the above four shape functions and the current locations of the four nodes

$$\begin{aligned} \mathbf{x}_I^{{t\mathrm{(FE)}}}= & {} \sum _{J=1}^4 {\mathbf{N}_J \left( {{s}_I ,{t}_I } \right) \bar{{\mathbf{x}}}_J^t } , \mathbf{y}_I^{t\mathrm{(FE)}}\nonumber \\= & {} \sum _{J=1}^4 {\mathbf{N}_J \left( {{s}_I ,{t}_I } \right) \bar{{\mathbf{y}}}_J^t } , \mathbf{z}_I^{t\mathrm{(FE)}}\nonumber \\= & {} \sum _{J=1}^4 {\mathbf{N}_J \left( {{s}_I ,{t}_I } \right) \bar{{\mathbf{z}}}_J^t } , \end{aligned}$$

(9)

where $(\bar{{\mathbf{x}}}_{J}^{t} ,\bar{{\mathbf{y}}}_{J}^{t} ,\bar{{\mathbf{z}}}_{J}^{t} )$ is the current position (after extrusion) of node $J$ in the deformed configuration $\varOmega _{t}$, and superscript t denotes the state (30 or 70 mmHg). Finally, the displacement errors at the fiducial markers were evaluated by comparing the micro-CT segmented locations and the FE predictions as follows

$$\begin{aligned} {error}_{I}^t =\sqrt{\left( {\mathbf{x}_I^{t \mathrm{(FE)}} -\mathbf{x}_I^{t \mathrm{(in}{\text {-}}\mathrm{vitro})} } \right) ^{2}+\left( {\mathbf{y}_I^{t \mathrm{(FE)}} -\mathbf{y}_I^{t \mathrm{(in}{\text {-}}\mathrm{vitro})} } \right) ^{2}+\left( {\mathbf{z}_I^{t \mathrm{(FE)}} -\mathbf{z}_I^{t \mathrm{(in}{\text {-}}\mathrm{vitro})} } \right) ^{2}}.\nonumber \\ \end{aligned}$$

(10)

Appendix 2: Kinematics and strain computations in the convective curvilinear coordinates

We modified previously developed approaches (Sacks et al. 1993, 2002; Smith et al. 2000) for computation of the strain field within the fiducial marker array by adopting an in-surface convective curvilinear coordinate system ($u,v,n$) based on the local tangent plane (Fig. 12). This coordinate system was defined with unit vectors $\mathbf{e}_{v}$ parallel to the cylindrically circumferential direction, $\mathbf{e}_{n}$ being the local surface normal, and $\mathbf{e}_{v} =\mathbf{e}_{n} \times \mathbf{e}_{v}$. The 3D position vector of each marker ($x,y,z$) at different configurations was then translated and rotated into this ($u,v,n$) coordinate system. In this work, we utilized a point-based meshless surface interpolation to calculate the in-surface strain field within the MV leaflet delimited by the markers. The position vectors of each point at the reference state and the deformed state were given by

$$\begin{aligned} \mathbf{R}^0 =\mathbf{R}_{u}^0 \mathbf{e}_{u} +\mathbf{R}_{v}^0 \mathbf{e}_{v} +\mathbf{R}_{n}^0 \mathbf{e}_{n}\hbox { and }\mathbf{r}^t =\mathbf{r}_{u}^t \mathbf{e}_{u} +\mathbf{r}_{v}^t \mathbf{e}_{v} +\mathbf{r}_{n}^t \mathbf{e}_{n},\nonumber \\ \end{aligned}$$

(11)

where $\mathbf{R}_{u}^0 , \mathbf{R}_{v}^0$, and $\mathbf{R}_{n}^0$ are the $u, v, n$ components of the position vector associated with the reference state, and $\mathbf{r}_{u}^t , \mathbf{r}_{v}^t$, and $\mathbf{r}_{n}^t$ are the $u,v,n$ components of the position vector corresponding to the deformed state. In this study, these components were computed by using the spline interpolation

$$\begin{aligned} \mathbf{R}_J^0 \left( {s,t} \right)= & {} \sum _{I=1}^{NP} {\phi _{I} (s,t;s_I ,t_I )\left( {\mathbf{R}_J^0 } \right) _I }\hbox { and }\nonumber \\ \mathbf{r}_J^t \left( {s,t} \right)= & {} \sum _{I=1}^{NP} {\phi _{I} (s,t;s_I ,t_I )\left( {\mathbf{r}_J^t } \right) _I } . \end{aligned}$$

(12)

Here, subscript ${ J}\in \{u,v,n\}, (\mathbf{R}_J^0 )_I$ and $(\mathbf{r}_J^t )_I$ are the $J$-components of the position vectors of marker $I$ corresponding to the reference and deformed states, respectively, and $\phi _\mathrm{I} (s,t;s_I ,t_I )$ is the $C^{2}$-continuous cubic spline function of marker I with a compact support d covering NP markers which has the form

$$\begin{aligned} {\phi _{{I}}} \left( {z_{I} } \right) \!=\!\left\{ {{\begin{array}{ll} \frac{2}{3}-4z_I^2 +4z_I^3\;&{}\mathrm{for}\;0\le \left| {{z}_I } \right| \le 0.5 \\ \frac{4}{3}\!-\!4z_I \!+\!4z_I^2 \!-\!\frac{4}{3}z_I^3\;&{}\mathrm{for}\;0.5\!\le \! \left| {z_I } \right| \!\le \! 1 \\ 0\;&{}\mathrm{for}\;\left| {z_I } \right| >1 \\ \end{array} }} \right. .\qquad \end{aligned}$$

(13)

where $z_I (s,t;s_I ,t_I )\equiv {\sqrt{(s_I -s)^{2}+(t_I -t)^{2}}}/d, \mathrm{d}$ was chosen as twice of the average marker distance, and the covering fiducial makers were determined on the expanded 2D parametric domain ($s,t$) (Fig. 12). Hence, the covariant base vectors on the MV leaflet surfaces were determined by

$$\begin{aligned} \mathbf{G}_\alpha= & {} \mathbf{R}_{,\alpha }^0 =\left( {\frac{\partial R_\alpha ^0 }{\partial \alpha }} \right) \mathbf{e}_\alpha +\left( {\frac{\partial R_{n}^0 }{\partial \alpha }} \right) \mathrm{e}_n\hbox { and }\nonumber \\ \mathbf{g}_\alpha= & {} \mathbf{r}_{,\alpha }^t =\left( {\frac{\partial r_\alpha ^t }{\partial \alpha }} \right) \mathbf{e}_\alpha +\left( {\frac{\partial r_{n}^t }{\partial \alpha }} \right) \mathrm{e}_n, \end{aligned}$$

(14)

where subscript $\alpha \in \{u,v\}, \mathbf{G}_\alpha$ and $\mathbf{g}_\alpha$ are the covariant base vectors in the reference and deformed configurations, respectively, and the third covariant base vectors were computed by $\mathbf{G}_3 ={(\mathbf{G}_1 \times \mathbf{G}_2 )}/{||\mathbf{G}_1 \times \mathbf{G}_2 ||}$ and $\mathbf{g}_3 ={(\mathbf{g}_1 \times \mathbf{g}_2 )}/{||\mathbf{g}_1 \times \mathbf{g}_2 ||}$. Then, the contra-variant base vectors were computed by their definition

$$\begin{aligned} \mathbf{G}^{1}= & {} \frac{\mathbf{G}_2 \times \mathbf{G}_3 }{\mathbf{G}_1 \cdot \left( {\mathbf{G}_2 \times \mathbf{G}_3 } \right) },\quad \mathbf{G}^{2}=\frac{\mathbf{G}_3 \times \mathbf{G}_1 }{\mathbf{G}_1 \cdot \left( {\mathbf{G}_2 \times \mathbf{G}_3 } \right) },\nonumber \\ \mathbf{G}^{3}= & {} \frac{\mathbf{G}_1 \times \mathbf{G}_2 }{\mathbf{G}_1 \cdot \left( {\mathbf{G}_2 \times \mathbf{G}_3 } \right) },\quad \end{aligned}$$

(15)

$$\begin{aligned} \mathbf{g}^{1}= & {} \frac{\mathbf{g}_2 \times \mathbf{g}_3 }{\mathbf{g}_1 \cdot \left( {\mathbf{g}_2 \times \mathbf{g}_3 } \right) },\quad \mathbf{g}^{2}=\frac{\mathbf{g}_3 \times \mathbf{g}_1 }{\mathbf{g}_1 \cdot \left( {\mathbf{g}_2 \times \mathbf{g}_3 } \right) },\nonumber \\ \mathbf{g}^{3}= & {} \frac{\mathbf{g}_1 \times \mathbf{g}_2 }{\mathbf{g}_1 \cdot \left( {\mathbf{g}_2 \times \mathbf{g}_3 } \right) }. \end{aligned}$$

(16)

After carrying out the components of each base vector, the in-plane deformation gradient tensor $\mathbf{F}_{2\mathrm{D}}$ (used for the mapping algorithm presented in Sect. 2.3), right Cauchy–Green deformation tensor $\mathbf{C}_{2\mathrm{D}}$, and Euler–Almansi strain tensor $\mathbf{e}_{2\mathrm{D}}$ (served as validation data in Sects. 2.6, 2.7) were computed (Flugge 1972; Fung 1993) as the following:

$$\begin{aligned} \mathbf{F}_{2\mathrm{D}}= & {} \mathbf{g}_I \otimes \mathbf{G}^{I}, \mathbf{C}_{2\mathrm{D}} =\left( {\mathbf{F}_{2\mathrm{D}} } \right) ^{T}\left( {\mathbf{F}_{2\mathrm{D}} } \right) ,\nonumber \\ \left( {\mathbf{e}_{2\mathrm{D}} } \right) _{\alpha \beta }= & {} \frac{1}{2}\left( {\mathbf{g}_\alpha \cdot \mathbf{g}_\beta -\mathbf{G}_\alpha \cdot \mathbf{G}_\beta } \right) \alpha ,\beta =1,2. \end{aligned}$$

(17)

Finally, the corresponding principal stretches $(\lambda _\mathrm{R},\lambda _\mathrm{C})$ and the principal directions can then be obtained by solving the eigenvalue problem of the above Euler-Almansi strain tensor for the eigenvalues and eigenvectors.

Appendix 3: Summary of validations of the in-surface principal stretches and sensitivity study

The FE-predicted maximum and minimum stretches in the central regions of the MV two leaflets at 70 mmHg transvalvular pressure were reported as follows: $\lambda _\mathrm{R}=1.39 \pm 0.03$ and $\lambda _\mathrm{C}=1.12 \pm 0.04$ for the MV anterior leaflet, and $\lambda _\mathrm{R}=1.48 \pm 0.05$ and $\lambda _\mathrm{C}=1.11 \pm 0.03$ for the MV posterior leaflet, whereas the in vitro experimental measurements were $\lambda _\mathrm{R}=1.39 \pm 0.03$ and $\lambda _\mathrm{C}=1.12 \pm 0.04$ for the MVAL, and $\lambda _\mathrm{R}=1.48 \pm 0.04$ and $\lambda _\mathrm{C}=1.11 \pm 0.02$ for the MVPL. Furthermore, the principal directions associated with the principal stretches from both numerical predictions and experimental data were very smooth, especially in the central regions of the MVAL and MVPL, and were in good alignment with the radial and circumferential directions, respectively (Figs. 13, 14). For the sensitivity study on the leaflet microstructural information, the FE prediction accuracy with various levels of model fidelity was evaluated based on the displacement errors at the fiducial marker positions at 70 mmHg transvalvular pressure (Fig. 15).

Fig. 13

Comparisons of the maximum principal stretches $\lambda _\mathrm{R}$ and directions at 70 mmHg transvalvular pressure: a in vitro measurements and b FE predictions. (Left MVAL; right MVPL)

Fig. 14

Comparisons of the minimum principal stretches $\lambda _\mathrm{C}$ and directions at 70 mmHg transvalvular pressure: a in vitro measurements and b FE predictions. (Left MVAL; right MVPL)

Fig. 15

Comparison of the FE-predicted displacement errors at 70 mmHg transvalvular pressure with an increasing level of model fidelity: a level I—isotropic material, b level II—transversely isotropic material with uniformly curvilinear fiber directions and identical fiber dispersions, c level III—transversely isotropic material with uniformly curvilinear fiber directions and mapped fiber dispersions, and d level IV—transversely isotropic material with mapped fiber directions (via the proposed mapping technique) and mapped fiber dispersions