A parametric study on mathematical formulation and geometrical construction of a stentless aortic heart valve

Abstract

This study presents a novel methodology for constructing an accurate geometrical model of a stentless aortic heart valve replacement (AVR). The main objective is to propose an optimized AVR model that can be used as an ideal scaffold for tissue engineering applications or a biocompatible prosthesis. Current techniques available for creating heart valve geometry, including leaflets, are very complicated and are not precise, due to the extensive use of various complicated parameters. This paper introduces an alternative design procedure that uses limited and effective numbers of controlling parameters to construct the whole valve including the sinus of valsalva. In doing so the hyperbolic curves for multithickness leaflets are used and a 3D elliptical formulation is incorporated for the surface geometry of the sinus of valsalva. Still, the feasibility and the precision of the mathematical method are established by performing standard deviation analysis on the constructed surfaces. The surface fitting residuals are found as low as error 0.2351 mm with standard deviation of 8.83e−5 over the commissural lines. Preliminary validation to the proposed AVR model performance is achieved by testing the generated AVR model under quasi static condition while obtaining the mesh independent setup. The numerical model showed a rapid response of the leaflets to the transvalvular pressure where adequate values of stress are measured over the commissural lines.

Appendices

Appendix 1

$$ \begin{gathered} A = \left[ {\begin{array}{*{20}c} {\sum\limits_{i = 1}^{m} {x_{i}^{ 4} } \, } & {\sum\limits_{i = 1}^{m} {x_{i}^{ 2} y_{i}^{ 2} } } & {\sum\limits_{i = 1}^{m} {x_{i}^{ 2} z_{i}^{ 2} } } & {\sum\limits_{i = 1}^{m} {x_{i}^{ 2} y_{i} z_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i}^{ 3} z_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i}^{ 3} y_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i}^{ 3} } } & {\sum\limits_{i = 1}^{m} {x_{i}^{ 2} y_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i}^{ 2} z_{i} } } \\ {} & {\sum\limits_{i = 1}^{m} {y_{i}^{ 4} } } & {\sum\limits_{i = 1}^{m} {y_{i}^{ 2} z_{i}^{ 2} } } & {\sum\limits_{i = 1}^{m} {y_{i}^{ 3} z_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i} y_{i}^{ 2} z_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i} y_{\text{i}}^{ 3} } } & {\sum\limits_{i = 1}^{m} {x_{i} y_{i}^{ 2} } } & {\sum\limits_{i = 1}^{m} {y_{i}^{ 3} } } & {\sum\limits_{i = 1}^{m} {y_{i}^{ 2} z_{i} } } \\ {} & {} & {\sum\limits_{{{\text{i}} = 1}}^{m} {z_{\text{i}}^{ 4} } } & {\sum\limits_{i = 1}^{m} {y_{i} z_{i}^{ 3} } } & {\sum\limits_{i = 1}^{m} {x_{i} z_{i}^{ 3} } } & {\sum\limits_{i = 1}^{m} {x_{i} y_{i} z_{i}^{ 2} } } & {\sum\limits_{i = 1}^{m} {x_{i} z_{\text{i}}^{ 2} } } & {\sum\limits_{i = 1}^{m} {y_{i} z_{\text{i}}^{ 2} } } & {\sum\limits_{i = 1}^{m} {z_{\text{i}}^{ 3} } } \\ {} & {} & {} & {\sum\limits_{i = 1}^{m} {y_{i}^{ 2} z_{i}^{ 2} } } & {\sum\limits_{i = 1}^{m} {x_{i} y_{i} z_{\text{i}}^{ 2} } } & {\sum\limits_{i = 1}^{m} {x_{i} y_{i}^{ 2} z_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i} y_{i} z_{i} } } & {\sum\limits_{i = 1}^{m} {y_{i}^{2} z_{i} } } & {\sum\limits_{i = 1}^{m} {y_{i} z_{\text{i}}^{ 2} } } \\ {} & {} & {} & {} & {\sum\limits_{i = 1}^{m} {x_{i}^{ 2} z_{i}^{ 2} } } & {\sum\limits_{i = 1}^{m} {x_{i}^{ 2} y_{i} z_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i}^{ 2} z_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i} y_{i} z_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i} z_{\text{i}}^{ 2} } } \\ {} & {} & {} & {} & {} & {\sum\limits_{i = 1}^{m} {x_{i}^{ 2} y_{i}^{ 2} } } & {\sum\limits_{i = 1}^{m} {x_{i}^{ 2} y_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i} y_{i}^{ 2} } } & {\sum\limits_{i = 1}^{m} {x_{i} y_{i} z_{i} } } \\ {} & {} & {} & {} & {} & {} & {\sum\limits_{i = 1}^{m} {x_{i}^{ 2} } } & {\sum\limits_{i = 1}^{m} {x_{i} y_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i} z_{i} } } \\ {} & {} & {} & {} & {} & {} & {} & {\sum\limits_{i = 1}^{m} {y_{i}^{ 2} } } & {\sum\limits_{i = 1}^{m} {y_{i} z_{i} } } \\ {} & {} & {} & {} & {} & {} & {} & {} & {\sum\limits_{{{\text{i}} = 1}}^{m} {z_{i}^{ 2} } } \\ \end{array} } \right] \hfill \\ B = \left[ {\begin{array}{*{20}c} {\sum\limits_{i = 1}^{m} {x_{i}^{2} } } & {\sum\limits_{i = 1}^{m} {y_{i}^{2} } } & {\sum\limits_{i = 1}^{m} {z_{i}^{2} } } & {\sum\limits_{i = 1}^{m} {y_{i} z_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i} z_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i} y_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i} } } & {\sum\limits_{i = 1}^{m} {y_{i} } } & {\sum\limits_{i = 1}^{m} {y_{i} } } \\ \end{array} } \right]^{\tau } \hfill \\ X = [a\;b\;c\;2f\;2g\;2h\;2p\;2q\;2r]^{\tau } \hfill \\ \end{gathered} $$

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$$ \begin{gathered} \Updelta = \left[ {\begin{array}{*{20}c} a & h & g & p \\ h & b & f & q \\ g & f & c & r \\ p & q & r & l \\ \end{array} } \right] \quad D = \left[ {\begin{array}{*{20}c} a & h & g \\ h & b & f \\ g & f & c \\ \end{array} } \right] \hfill \\ I = a = b + c \qquad \qquad J = ab + bc + ca - f^{2} - g^{2} - h^{2} \hfill \\ \end{gathered} $$

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Appendix 2

• Fluid domain (See Fig. 9)

  Fig. 9

  

  Meshing statistic in fluid domain a quality, b aspect ratio, c skewness

• Solid domain (See Fig. 10)

  Fig. 10

  

  Meshing statistic in solid domain a quality, b aspect ratio, c skewness

Appendix 3

See Tables 4, 5, 6.