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Research Papers

# Modeling the Deformation of the Elastin Network in the Aortic Valve

[+] Author and Article Information
Afshin Anssari-Benam

The BIONEER Centre,
Cardiovascular Engineering Research
Laboratory (CERL),
School of Engineering,
University of Portsmouth,
Portsmouth PO1 3DJ, UK
e-mail: afshin.anssari-benam@port.ac.uk

Andrea Bucchi

The BIONEER Centre,
Cardiovascular Engineering Research
Laboratory (CERL),
School of Engineering,
University of Portsmouth,
Portsmouth PO1 3DJ, UK

1Corresponding author.

Manuscript received March 4, 2017; final manuscript received September 6, 2017; published online October 19, 2017. Assoc. Editor: Jonathan Vande Geest.

J Biomech Eng 140(1), 011004 (Oct 19, 2017) (12 pages) Paper No: BIO-17-1094; doi: 10.1115/1.4037916 History: Received March 04, 2017; Revised September 06, 2017

## Abstract

This paper is concerned with proposing a suitable structurally motivated strain energy function, denoted by $Weelastin network$, for modeling the deformation of the elastin network within the aortic valve (AV) tissue. The AV elastin network is the main noncollagenous load-bearing component of the valve matrix, and therefore, in the context of continuum-based modeling of the AV, the $Weelastin network$ strain energy function would essentially serve to model the contribution of the “isotropic matrix.” To date, such a function has mainly been considered as either a generic neo-Hookean term or a general exponential function. In this paper, we take advantage of the established structural analogy between the network of elastin chains and the freely jointed molecular chain networks to customize a structurally motivated $Weelastin network$ function on this basis. The ensuing stress–strain (force-stretch) relationships are thus derived and fitted to the experimental data points reported by (Vesely, 1998, “The Role of Elastin in Aortic Valve Mechanics,” J. Biomech., 31, pp. 115–123) for intact AV elastin network specimens under uniaxial tension. The fitting results are then compared with those of the neo-Hookean and the general exponential models, as the frequently used models in the literature, as well as the “Arruda–Boyce” model as the gold standard of the network chain models. It is shown that our proposed $Weelastin network$ function, together with the general exponential and the Arruda–Boyce models provide excellent fits to the data, with R2 values in excess of 0.98, while the neo-Hookean function is entirely inadequate for modeling the AV elastin network. However, the general exponential function may not be amenable to rigorous interpretation, as there is no structural meaning attached to the model. It is also shown that the parameters estimated by the Arruda–Boyce model are not mathematically and structurally valid, despite providing very good fits. We thus conclude that our proposed strain energy function $Weelastin network$ is the preferred choice for modeling the behavior of the AV elastin network and thereby the isotropic matrix. This function may therefore be superimposed onto that of the anisotropic collagen fibers family in order to develop a structurally motivated continuum-based model for the AV.

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## Figures

Fig. 1

(a) AV cut open: three leaflets of the valve attached to the aortic root (adapted from Ref. [19]). (b) Schematic of a leaflet's cross section (adapted from Ref. [20]) and an idealized elastin network structure for a representative volume element (RVE) of an AV leaflet. (c) Geometry of a single elastin chain in a 3D cube RVE. (d) The two principal loading directions of the AV leaflet: circumferential and radial.

Fig. 2

Experimental and modeling tension-stretch data for the intact AV elastin network in (a) the circumferential and (b) theradial loading directions for six representative datasets. The experimental data points were collated from Ref. [11]. The markers represent the experimental data points, and the continuous lines represent the modeling outcomes.

Fig. 3

(a) The energy function Weelastin network proposed in Eq. (16) plotted versus λ1 and λ2; (b) the contours of Weelastin network in (λ1,λ2) plane; and (c) in (E11,E22) plane, for mean values of n and N (see Table 1). The graphs highlight the convexity of the proposed strain energy function.

Fig. 4

A representative example of how the neo-Hookean model in Eq. (28)3 provides a fit to the experimental data for: (a) circumferential sample; and (b) radial sample. The fitting results demonstrate the unsuitability of this function for characterizing the behavior of the AV elastin network, i.e., the AV isotropic matrix.

Fig. 5

Experimental data fitted using the Arruda–Boyce model in Eq. (28)1: (a) samples loaded in the circumferential direction; and (b) samples loaded in the radial direction. The markers represent the experimental data points, and the continuous lines show the model predications.

Fig. 6

Experimental data fitted using the general exponential model in Eq. (28)4: (a) samples loaded in the circumferential direction; and (b) samples loaded in the radial direction. The markers represent the experimental data points, and the continuous lines show the model predications.

Fig. 7

Comparison between the inverse Langevin function versus its series expansion (first five terms) and Padè approximations: (a) Padè approximation provides a more accurate description of the original function in the entire domain compared with the series expansion approximation. The typical deformation domain is shown in the inset. (b) Plots of the inverse Langevin, its series expansion, and Padè approximation functions versus I1 for the average values of N calculated in Table 1. Note that in all panels, the hollow circles represent the inverse Langevin function, while the dotted and continuous lines represent the series expansion and the Padè approximations, respectively.

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