A strain energy function for arteries accounting for wall composition and structure

Zulliger, Martin A.; Fridez, Pierre; Hayashi, Kozaburo; Stergiopulos, Nikolaos

doi:10.1016/j.jbiomech.2003.11.026

Cited by 280 publications

(157 citation statements)

References 41 publications

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“…This model captures the actual unfolding of the collagen fibres but has the disadvantage that the energy function associated with the fibres involves an integral over the probability distribution, which makes the model difficult to implement efficiently into numerical codes and therefore limits its applicability. An engagement strain was also discussed by Speirs et al (2008), who used a finite element implementation to analyse the inflation and extension of a thick-walled tube by using the models (2.21) and (2.22) and compared the results with those from the engagement model of Zulliger et al (2004a). Baek et al (2007a), Hu et al (2007) and Zeinali-Davarani et al (2009) considered a model of the form of equation (2.29) but with four families of fibres instead of two.…”

Section: (A) Arterial Wall Modelling and Its Applicationsmentioning

confidence: 99%

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Constitutive modelling of arteries

2010

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This review article is concerned with the mathematical modelling of the mechanical properties of the soft biological tissues that constitute the walls of arteries. Many important aspects of the mechanical behaviour of arterial tissue can be treated on the basis of elasticity theory, and the focus of the article is therefore on the constitutive modelling of the anisotropic and highly nonlinear elastic properties of the artery wall. The discussion focuses primarily on developments over the last decade based on the theory of deformation invariants, in particular invariants that in part capture structural aspects of the tissue, specifically the orientation of collagen fibres, the dispersion in the orientation, and the associated anisotropy of the material properties. The main features of the relevant theory are summarized briefly and particular forms of the elastic strain-energy function are discussed and then applied to an artery considered as a thickwalled circular cylindrical tube in order to illustrate its extension-inflation behaviour. The wide range of applications of the constitutive modelling framework to artery walls in both health and disease and to the other fibrous soft tissues is discussed in detail. Since the main modelling effort in the literature has been on the passive response of arteries, this is also the concern of the major part of this article. A section is nevertheless devoted to reviewing the limited literature within the continuum mechanics framework on the active response of artery walls, i.e. the mechanical behaviour associated with the activation of smooth muscle, a very important but also very challenging topic that requires substantial further development. A final section provides a brief summary of the current state of arterial wall mechanical modelling and points to key areas that need further modelling effort in order to improve understanding of the biomechanics and mechanobiology of arteries and other soft tissues, from the molecular, to the cellular, tissue and organ levels.

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Section: (A) Arterial Wall Modelling and Its Applicationsmentioning

confidence: 99%

“…As with the model in Zulliger et al (2004a), they used an engagement stretch associated with the un-crimping of collagen fibres.…”

Section: (B) Modelling Other Soft Tissuesmentioning

confidence: 99%

Constitutive modelling of arteries

2010

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“…First, although elastin is often modelled as an isotropic material [25,[30][31][32], qualitative observations of elastin orientation suggested nearly circumferentially and nearly axially oriented fibres. Second, although we did not quantify their organization, there are nearly circumferentially oriented smooth muscle cells.…”

Section: 'Microstructurally Motivated' Passive Constitutive Modelmentioning

confidence: 99%

Quantification of the passive and active biaxial mechanical behaviour and microstructural organization of rat thoracic ducts

Caulk

Nepiyushchikh

Shaw

et al. 2015

J. R. Soc. Interface.

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Mechanical loading conditions are likely to play a key role in passive and active (contractile) behaviour of lymphatic vessels. The development of a microstructurally motivated model of lymphatic tissue is necessary for quantification of mechanically mediated maladaptive remodelling in the lymphatic vasculature. Towards this end, we performed cylindrical biaxial testing of Sprague -Dawley rat thoracic ducts (n ¼ 6) and constitutive modelling to characterize their mechanical behaviour. Spontaneous contraction was quantified at transmural pressures of 3, 6 and 9 cmH 2 O. Cyclic inflation in calcium-free saline was performed at fixed axial stretches between 1.30 and 1.60, while recording pressure, outer diameter and axial force. (doi:10.1023/A:1010835316564)) was used to quantify the passive mechanical response, and the model of Rachev and Hayashi was used to quantify the active (contractile) mechanical response. The average error between data and theory was 8.9 + 0.8% for passive data and 6.6 + 2.6% and 6.8 + 3.4% for the systolic and basal conditions, respectively, for active data. Multi-photon microscopy was performed to quantify vessel wall thickness (32.2 + 1.60 mm) and elastin and collagen organization for three loading conditions. Elastin exhibited structural 'fibre families' oriented nearly circumferentially and axially. Sample-to-sample variation was observed in collagen fibre distributions, which were often non-axisymmetric, suggesting material asymmetry. In closure, this paper presents a microstructurally motivated model that accurately captures the biaxial active and passive mechanical behaviour in lymphatics and offers potential for future research to identify parameters contributing to mechanically mediated disease development.

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“…17,18 Many works either chose a purely phenomenological approach like the Fung-type model, or consider histostructural information, such as the Holzapfel-Gasser-Ogden constitutive model 19 or fiber family model [20][21][22] to capture the nonlinear hyperelastic mechanical behavior of the soft biological tissues, especially the arterial wall. Since the fiber family constitutive based material model has the potential ability to address the mechanical properties of the skin tissue better than that of phenomenological model, [23][24][25] it would be useful and practically valuable to determine the material coefficients of the rat and mice skins in different anatomical locations, including the abdomen and back, from uniaxial data with the previously proposed model by Holzapfel. Therefore, the objective of this study is to experimentally and numerically determine the anisotropic mechanical properties of the rat and mice skin tissues using histostructural and uniaxial test data.…”

Section: Introductionmentioning

confidence: 99%

Mechanical characterization of the rat and mice skin tissues using histostructural and uniaxial data

2015

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The skin tissue has been shown to behave like a nonlinear anisotropic material. This study was aimed to employ a constitutive fiber family equation to characterize the nonlinear anisotropic mechanical behavior of the rat and mice skin tissues in different anatomical locations, including the abdomen and back, using histostructural and uniaxial data. The rat and mice skin tissues were excised from the animals' body and then the histological analyses were performed on each skin type to determine the mean fiber orientation angle. Afterward, the preconditioned skin tissues were subjected to a series of quasi-static axial and circumferential loads until the incidence of failure. The crucial role of fiber orientation was explicitly added into a proposed strain energy density function. The material coefficients were determined using the constrained nonlinear optimization method based on the axial and circumferential extension data of the rat and mice samples at different anatomical locations. The material coefficients of the skins were given with R 2 0.998. The results revealed a significant load-bearing capacity and stiffness of the rat abdomen compared to the rat back tissues. In addition, the mice abdomen showed a higher stiffness in the axial direction in comparison with circumferential one, while the mice back displayed its highest stiffness in the circumferential direction. The material coefficients of the rat and mice skin tissues were determined and well compared to the experimental data. The optimized fiber angles were also compared to the experimental histological data, and in all cases less than 11.85% differences were observed in both the skin tissues.

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A strain energy function for arteries accounting for wall composition and structure

Cited by 280 publications

References 41 publications

Constitutive modelling of arteries

Constitutive modelling of arteries

Quantification of the passive and active biaxial mechanical behaviour and microstructural organization of rat thoracic ducts

Mechanical characterization of the rat and mice skin tissues using histostructural and uniaxial data

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