In Vivo Dynamic Deformation of the Mitral Valve Annulus

Abstract

Though mitral valve (MV) repair surgical procedures have increased in the United States [Gammie, J. S., et al. Ann. Thorac. Surg. 87(5):1431–1437, 2009; Nowicki, E. R., et al. Am. Heart J. 145(6):1058–1062, 2003], studies suggest that altering MV stress states may have an effect on tissue homeostasis, which could impact the long-term outcome [Accola, K. D., et al. Ann. Thorac. Surg. 79(4):1276–1283, 2005; Fasol, R., et al. Ann. Thorac. Surg. 77(6):1985–1988, 2004; Flameng, W., P. Herijgers, and K. Bogaerts. Circulation 107(12):1609–1613, 2003; Gillinov, A. M., et al. Ann. Thorac. Surg. 69(3):717–721, 2000]. Improved computational modeling that incorporates structural and geometrical data as well as cellular components has the potential to predict such changes; however, the absence of important boundary condition information limits current efforts. In this study, novel high definition in vivo annular kinematic data collected from surgically implanted sonocrystals in sheep was fit to a contiguous 3D spline based on quintic-order hermite shape functions with C² continuity. From the interpolated displacements, the annular axial strain and strain rate, bending, and twist along the entire annulus were calculated over the cardiac cycle. Axial strain was shown to be regionally and temporally variant with minimum and maximum values of −10 and 4%, respectively, observed. Similarly, regionally and temporally variant strain rate values, up to 100%/s contraction and 120%/s elongation, were observed. Both annular bend and twist data showed little deviation from unity with limited regional variations, indicating that most of the energy for deformation was associated with annular axial strain. The regionally and temporally variant strain/strain rate behavior of the annulus are related to the varied fibrous-muscle structure and contractile behavior of the annulus and surrounding ventricular structures, although specific details are still unavailable. With the high resolution shape and displacement information described in this work, high fidelity boundary conditions can be prescribed in future MV finite element models, leading to new insights into MV function and strategies for repair.

Appendix

Appendix A—Quintic Hermite Shape Functions

The six shape functions $N_{\alpha }^{i}$ of the 1D quintic finite element used in this study are defined as follows24:

$$ \begin{aligned} N_{1}^{0} &= - {\frac{1}{16}}\left( {\xi - 1} \right)^{3} \left( {3\xi^{2} + 9\xi + 8} \right) \\ N_{1}^{1} &= - {\frac{1}{16}}\left( {\xi - 1} \right)^{3} \left( {3\xi + 5} \right)\left( {\xi + 1} \right) \\ N_{1}^{2} &= - {\frac{1}{16}}\left( {\xi - 1} \right)^{3} \left( {\xi + 1} \right)^{2} \\ N_{2}^{0} &= {\frac{1}{16}}\left( {\xi + 1} \right)^{3} \left( {3\xi^{2} - 9\xi + 8} \right) \\ N_{2}^{1} &= - {\frac{1}{16}}\left( {\xi + 1} \right)^{3} \left( {3\xi - 5} \right)\left( {\xi - 1} \right) \\ N_{2}^{2} &= {\frac{1}{16}}\left( {\xi + 1} \right)^{3} \left( {\xi - 1} \right)^{2} \\ \end{aligned} $$

(A.1)

where the subscript indicates the node number, the superscript the order of the derivative with respect to the local coordinate ξ, and the coordinate values at nodes 1 and 2 are −1 and 1, respectively.

Appendix B—Generalized Least Squares Regression

With the data properly oriented, the parameter estimation for the interpolation is done using the general matrix formulation for linear least squares. The function describing the component being fit is defined by:

$$ f = a_{0} \cdot \psi_{0}^{0} + a_{1} \cdot \psi_{1}^{0} + a_{2} \cdot \psi_{2}^{0} + a_{3} \cdot \psi_{0}^{1} + a \cdot \psi_{1}^{1} + a_{5} \cdot \psi_{2}^{1} + e $$

(B.1)

where e is error and [a] is a vector containing the nodal fit parameters for the function f. Expressed in matrix form the relationship can be written concisely as:

$$ \{f\} = [ \psi ]\{ A \} + \{ e\}. $$

(B.2)

In this case, f is the component of the data being fit (r(θ) or z(θ)) and e is the error associated with the fit at a given point. The rows of matrix [ψ] are the shape functions evaluated at there isoparametric coordinate corresponding to the data point observed. It can be shown that the minimization of the sum square of errors associated with the partial derivatives of f with respect to each of the parameters can be reduced to the following expression:

$$ [ \psi]^{\text{T}} [ \psi]\{A\} = \{ {[ \psi ]^{\text{T}}\{ f \}}\} $$

(B.3)

So the parameters are computed by simply inverting [ψ]^T [ψ] and left multiplying both sides of the equation. After estimating the parameters for each element, the curve can be interpolated by evaluating the polynomials for each element at any point within the domain. In the case of smoothing via the Sobolev algorithm, the Sobolev penalty function is added to the matrix [ψ]^T [ψ]. The nodal parameters for the smoothed fit are computed using:

$$ A = [ {[ \psi]^{\text{T}} [ \psi ] + S} ] ^{-1} \cdot [ {[ \psi ]^{\text{T}} f} ] $$

(B.4)

The described general matrix formulation for linear least squares is then used in the same manner to determine the parameters for each element.