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,
Anglesea Road,
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,
Anglesea Road,
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

This paper is concerned with proposing a suitable structurally motivated strain energy function, denoted by Weelastinnetwork, 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 Weelastinnetwork 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 Weelastinnetwork 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 Weelastinnetwork 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 Weelastinnetwork 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.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Holzapfel, G. A. , Gasser, T. C. , and Ogden, R. W. , 2000, “ A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models,” J. Elasticity, 61(1–3), pp. 1–48. [CrossRef]
Anssari-Benam, A. , Bucchi, A. , Screen, H. R. C. , and Evans, S. L. , 2017, “ A Transverse Isotropic Viscoelastic Constitutive Model for the Aortic Valve Tissue,” R. Soc. Open Sci., 4(1), p. 160585. [CrossRef] [PubMed]
Freed, A. D. , Einstein, D. R. , and Vesely, I. , 2005, “ Invariant Formulation for Dispersed Transverse Isotropy in Aortic Heart Valves: An Efficient Means for Modeling Fiber Splay,” Biomech. Model. Mechanobiol., 4(2–3), pp. 100–117. [CrossRef] [PubMed]
Gasser, T. C. , Ogden, R. W. , and Holzapfel, G. A. , 2006, “ Hyperelastic Modelling of Arterial Layers With Distributed Collagen Fibre Orientation,” J. R. Soc. Interface, 3(6), pp. 15–35. [CrossRef] [PubMed]
Holzapfel, G. A. , and Ogden, R. W. , 2010, “ Constitutive Modelling of Arteries,” Proc. R. Soc. A, 466(2118), pp. 1551–1597. [CrossRef]
Holzapfel, G. A. , Niestrawska, J. A. , Ogden, R. W. , Reinisch, A. J. , and Schriefl, A. J. , 2015, “ Modelling Non-Symmetric Collagen Fibre Dispersion in Arterial Walls,” J. R. Soc. Interface, 12(106), p. 20150188. [CrossRef] [PubMed]
Humphrey, J. D. , 2003, “ Review Paper: Continuum Biomechanics of Soft Biological Tissues,” Proc. R. Soc. A, 459(2029), pp. 3–46. [CrossRef]
Holzapfel, G. A. , 2006, “ Determination of Material Models for Arterial Walls From Uniaxial Extension Tests and Histological Structure,” J. Theor. Biol., 238(2), pp. 290–302. [CrossRef] [PubMed]
Anssari-Benam, A. , Bader, D. L. , and Screen, H. R. C. , 2011, “ A Combined Experimental and Modelling Approach to Aortic Valve Viscoelasticity in Tensile Deformation,” J. Mater. Sci. Mater. Med., 22(2), pp. 253–262. [CrossRef] [PubMed]
Anssari-Benam, A. , Barber, A. H. , and Bucchi, A. , 2016, “ Evaluation of Bioprosthetic Heart Valve Failure Using a Matrix-Fibril Shear Stress Transfer Approach,” J. Mater. Sci. Mater. Med., 27(2), p. 42. [CrossRef] [PubMed]
Vesely, I. , 1997, “ The Role of Elastin in Aortic Valve Mechanics,” J. Biomech., 31(2), pp. 115–123. [CrossRef]
Weinberg, E. J. , and Kaazempur-Mofrad, M. R. , 2005, “ On the Constitutive Models for Heart Valve Leaflet Mechanics,” Cardiovasc. Eng., 5(1), pp. 37–43. [CrossRef]
Weinberg, E. J. , and Kaazempur Mofrad, M. R. , 2007, “ Transient, Three-Dimensional, Multiscale Simulations of the Human Aortic Valve,” Cardiovasc. Eng., 7(4), pp. 140–155. [CrossRef] [PubMed]
Weinberg, E. J. , Shahmirzadi, D. , and Kaazempur Mofrad, M. R. , 2010, “ On the Multiscale Modeling of Heart Valve Biomechanics in Health and Disease,” Biomech. Model. Mechanobiol., 9(4), pp. 373–387. [CrossRef] [PubMed]
Bischoff, J. E. , Arruda, E. M. , and Grosh, K. , 2002, “ Orthotropic Hyperelasticity in Terms of an Arbitrary Molecular Chain Model,” J. Appl. Mech., 69(2), pp. 198–201. [CrossRef]
Kuhl, E. , Garikipati, K. , Arruda, E. M. , and Grosh, K. , 2005, “ Remodeling of Biological Tissue: Mechanically Induced Reorientation of a Transversely Isotropic Chain Network,” J. Mech. Phys. Solids, 53(7), pp. 1552–1573. [CrossRef]
Zhang, Y. , Dunn, M. L. , Drexler, E. S. , McCowan, C. N. , Slifka, A. J. , Ivy, D. D. , and Shandas, R. , 2005, “ A Microstructural Hyperelastic Model of Pulmonary Arteries Under Normo- and Hypertensive Conditions,” Ann. Biomed. Eng., 33(8), pp. 1042–1052. [CrossRef] [PubMed]
Arruda, E. M. , and Boyce, M. C. , 1993, “ A Three-Dimensional Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials,” J. Mech. Phys. Solids, 41(2), pp. 389–412. [CrossRef]
Lewinsohn, A. D. , Anssari-Benham, A. , Lee, D. A. , Taylor, P. M. , Chester, A. H. , Yacoub, M. H. , and Screen, H. R. C. , 2011, “ Anisotropic Strain Transfer Through the Aortic Valve and Its Relevance to the Cellular Mechanical Environment,” Proc. Inst. Mech. Eng. H, 225(8), pp. 821–830. [CrossRef] [PubMed]
Rock, C. A. , Han, L. , and Doehring, T. C. , 2014, “ Complex Collagen Fiber and Membrane Morphologies of the Whole Porcine Aortic Valve,” PLoS One, 9(1), p. e86087. [CrossRef] [PubMed]
Elias-Zuniga, A. , and Beatty, M. F. , 2002, “ Constitutive Equations for Amended Non-Gaussian Network Models of Rubber Elasticity,” Int. J. Eng. Sci., 40(20), pp. 2265–2294. [CrossRef]
Scott, M. , and Vesely, I. , 1995, “ Aortic Valve Cusps Microstructure: The Role of Elastin,” Ann. Thorac. Surg., 60(2), pp. S391–S394. [CrossRef] [PubMed]
Tseng, H. , and Grande-Allen, K. J. , 2011, “ Elastic Fibers in the Aortic Valve Spongiosa: A Fresh Perspective on its Structure and Role in Overall Tissue Function,” Acta Biomater., 7(5), pp. 2101–2108. [CrossRef] [PubMed]
Sacks, M. S. , 2003, “ Incorporation of Experimentally-Derived Fiber Orientation Into a Structural Constitutive Model for Planar Collagenous Tissues,” ASME J. Biomech. Eng., 125(2), pp. 280–287. [CrossRef]
James, H. M. , and Guth, E. , 1943, “ Theory of the Elastic Properties of Rubber,” J. Chem. Phys., 11(10), pp. 455–481. [CrossRef]
Treloar, L. R. G. , 1954, “ The Photoelastic Properties of Short-Chain Molecular Networks,” Trans. Faraday Soc., 50, pp. 881–896. [CrossRef]
Anssari-Benam, A. , Viola, G. , and Korakianitis, T. , 2010, “ Thermodynamic Effects of Linear Dissipative Small Deformations,” J. Therm. Anal. Calorim., 100(3), pp. 941–947. [CrossRef]
Beatty, M. F. , 2003, “ An Average-Stretch Full-Network Model for Rubber Elasticity,” J. Elasticity, 70(1–3), pp. 65–86. [CrossRef]
Miehe, C. , Göktepe, S. , and Lulei, F. , 2004, “ A Micro-Macro Approach to Rubber-Like Materials—Part I: The Non-Affine Micro-Sphere Model of Rubber Elasticity,” J. Mech. Phys. Solids, 52(11), pp. 2617–2660. [CrossRef]
Boyce, M. C. , and Arruda, E. M. , 2000, “ Constitutive Models of Rubber Elasticity: A Review,” Rubber Chem. Technol., 73(3), pp. 504–523. [CrossRef]
Anssari-Benam, A. , Gupta, H. S. , and Screen, H. R. C. , 2012, “ Strain Transfer Through the Aortic Valve,” ASME J. Biomech. Eng., 134(6), p. 061003. [CrossRef]
Lee, C.-H. , Zhang, W. , Liao, J. , Carruthers, C. A. , Sacks, J. I. , and Sacks, M. S. , 2015, “ On the Presence of Affine Fibril and Fiber Kinematics in the Mitral Valve Anterior Leaflet,” Biophys. J., 108(8), pp. 2074–2087. [CrossRef] [PubMed]
Jayyosi, C. , Affagard, J. S. , Ducourthial, G. , Bonod-Bidaud, C. , Lynch, B. , Bancelin, S. , Ruggiero, F. , Schanne-Klein, M. C. , Allain, J. M. , Bruyère-Garnier, K. , and Coret, M. , 2017, “ Affine Kinematics in Planar Fibrous Connective Tissues: An Experimental Investigation,” Biomech. Model Mechanobiol., 16(4), pp. 1459–1473. [CrossRef] [PubMed]
Cohen, A. , 1991, “ A Padé Approximant to the Inverse Langevin Function,” Rheol. Acta, 30(3), pp. 270–273. [CrossRef]
Billiar, K. L. , and Sacks, M. S. , 2000, “ Biaxial Mechanical Properties of the Natural and Glutaraldehyde Treated Aortic Valve Cusp—Part II: A Structural Constitutive Model,” ASME J. Biomech. Eng., 122(4), pp. 327–335. [CrossRef]
Zou, Y. , and Zhang, Y. , 2009, “ An Experimental and Theoretical Study on the Anisotropy of Elastin Network,” Ann. Biomed. Eng., 37(8), pp. 1572–1583. [CrossRef] [PubMed]


Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In