Biomechanics and Modeling in Mechanobiology

, Volume 15, Issue 4, pp 909–932 | Cite as

An inverse modeling approach for semilunar heart valve leaflet mechanics: exploitation of tissue structure

  • Ankush Aggarwal
  • Michael S. SacksEmail author
Original Paper


Determining the biomechanical behavior of heart valve leaflet tissues in a noninvasive manner remains an important clinical goal. While advances in 3D imaging modalities have made in vivo valve geometric data available, optimal methods to exploit such information in order to obtain functional information remain to be established. Herein we present and evaluate a novel leaflet shape-based framework to estimate the biomechanical behavior of heart valves from surface deformations by exploiting tissue structure. We determined accuracy levels using an “ideal” in vitro dataset, in which the leaflet geometry, strains, mechanical behavior, and fibrous structure were known to a high level of precision. By utilizing a simplified structural model for the leaflet mechanical behavior, we were able to limit the number of parameters to be determined per leaflet to only two. This approach allowed us to dramatically reduce the computational time and easily visualize the cost function to guide the minimization process. We determined that the image resolution and the number of available imaging frames were important components in the accuracy of our framework. Furthermore, our results suggest that it is possible to detect differences in fiber structure using our framework, thus allowing an opportunity to diagnose asymptomatic valve diseases and begin treatment at their early stages. Lastly, we observed good agreement of the final resulting stress–strain response when an averaged fiber architecture was used. This suggests that population-averaged fiber structural data may be sufficient for the application of the present framework to in vivo studies, although clearly much work remains to extend the present approach to in vivo problems.


Heart valves Inverse model Semilunar leaflets  Tissue microstructure 

Nomenclature and symbol definitions

\(\Gamma \)

Fiber orientation distribution function

\(\theta , \theta _0 \)

Fiber direction and the preferred fiber direction respectively


Fiber splay in terms of orientation index

\(\sigma \)

Fiber splay in terms of standard deviation of orientation distribution function

\(\Psi , \Psi _f , \Psi _m , \Psi ^{\mathrm{ens}}\)

Elastic strain energy (total, fiber contribution, matrix contribution and ensemble respectively)

\(\overline{\lambda }, \mu \)

Lamé constants in compressible neo-Hookean model

\({\overline{E}} , \nu \)

Young’s modulus and Poisson’s ratio


Fiber direction vector in reference configuration

\(\mathbf{E}, \mathbf{C}\)

Strain (Green–Lagrange and right Cauchy–Green tensors respectively)

\(I_1 , J\)

First and third invariants of right Cauchy–Green strain tensor

\(d_e \)

Anisotropy parameter

\(E_{\mathrm{ens}} \)

Ensemble strain

\(c_0 ,c_1 \)

Material parameters in the fiber part

\(w_{\theta i} \)

Weights in the Gauss quadrature rule

\({\varvec{\chi }}\)

Lagrange multiplier in contact constraint

\(\mu _f \)

Coefficient of friction between leaflets

\(\Delta t_{\min } ,\Delta t_{\max } \)

Time step (minimum and maximum)


Displacement vector


Total energy of the system

\(\mathbf{x}_i^\alpha ,\widetilde{\mathbf{x}}_i^\alpha \)

Coordinates of point cloud and its projection on the deformed mesh

\({\mathcal {F}}, {\mathcal {F}}_{\mathrm{min}} , \mathbf{J}\)

Objective function, its value at the absolute minima and corresponding gradient vector

\(\lambda \)

Damping-like parameter in Levenberg-Marquardt algorithm

\(\Theta _j (\mathbf{v}_{1j} ,\mathbf{v}_{2j} ,\mathbf{v}_{3j} ,\mathbf{v}_{4j} )\)

Linear quadrilateral element defined by its four vertices

\(\overline{c}_0 ,\overline{c}_1\)

Known material parameters in the fiber part



We gratefully acknowledge the help from Will Zhang with parameter estimation and analysis of biaxial data. This work was supported by the American Heart Association Southwest Affiliate Postdoctoral Fellowship 14POST18720037 to A.A. and National Institutes of Health research grant R01 HL108330 to M.S.S. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper. URL:

Funding This study was funded by National Institutes of Health (Grant Number R01 HL108330) and American Heart Association (Grant Number 14POST18720037).

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Center for Cardiovascular Simulation, Institute for Computational Engineering and Sciences, Department of Biomedical EngineeringThe University of Texas at AustinAustinUSA

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