Invariant formulation for dispersed transverse isotropy in aortic heart valves

Freed, Alan D.; Einstein, Daniel R.; Veselý, Ivan

doi:10.1007/s10237-005-0069-8

Cited by 115 publications

(102 citation statements)

References 25 publications

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“…However, because of the complexity of the model it has not proved to be a practical option for efficient numerical implementation. More recently models based on invariants have been adopted including those of Freed et al (2005), Holzapfel et al (2005a,b) and Gasser & Holzapfel (2006a). The model of Freed et al (2005), for example, uses a structure tensor based on the Gaussian distribution of fibre directions for both two and three dimensions, and the model is applied to bioprosthetic valves; see also Freed (2008).…”

Section: Modelling Fibre Dispersionmentioning

confidence: 99%

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: Modelling Fibre Dispersionmentioning

confidence: 99%

Constitutive modelling of arteries

2010

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“…where the abbreviation ψ λ i = ∂ψ fibers (λ i )/∂λ i has been introduced, with the individual strain energies ψ fibers summarized in (25 …”

Section: Stress and Elasticity Tensorsmentioning

confidence: 99%

“…Thereby, the collagen fiber waveforms have a pre-defined arrangement; no statistical distribution is used. In different works, such as [25][26][27], the distribution of the fiber orientations has been addressed; however, therein, the mechanical properties of the collagen fibers 2 A c c e p t e d m a n u s c r i p t within the tissue were considered to be independent of the degree of crimping.…”

Section: Introductionmentioning

confidence: 99%

A constitutive model for fibrous tissues considering collagen fiber crimp

Cacho¹,

Elbischger²,

Rodrı́guez³

et al. 2007

International Journal of Non-Linear Mechanics

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A micromechanically-based constitutive model for fibrous tissues is presented. The model considers the randomly crimped morphology of individual collagen fibers, a morphology typically seen in photomicrographs of tissue samples. It describes the relationship between the fiber endpoints and its arc-length in terms of a measurable quantity, which can be estimated from image data. The collective mechanical behavior of collagen fibers is presented in terms of an explicit expression for the strain-energy function, where a fiber-specific random variable is approximated by a Beta distribution. The model-related stress and elasticity tensors are provided. Two representative numerical examples are analyzed with the aim of demonstrating the peculiar mechanism of the constitutive model and quantifying the effect of parameter changes on the mechanical response. In particular, a fibrous tissue, assumed to be (nearly) incompressible, is subject to a uniaxial extension along the fiber direction, and, separately, to pure shear. It is shown that the fiber crimp model can reproduce several of the expected characteristics of fibrous tissues.

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“…Some of these models have been extended to further incorporate a measure of dispersion in the fibers orientations (i.e. they capture the distribution of the fiber orientations about a principal direction of reinforcement) [31,166,168,[180][181][182][183].…”

Section: Introductionmentioning

confidence: 99%

“…[31,166,[180][181][182], stability of the fiber-terms in compression is still problematic, but a physical motivation to switch off fiber terms is now less clear. With fibers dispersed from the principal direction, single fibers oriented far from this principal direction may be in tension while the principal fiber direction is in compression (see, e.g., [167] for a related discussion on a dispersed model in tension).…”

Section: Introductionmentioning

confidence: 99%

Cardiovascular Mechanics

SpringerReference

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Modeling in biomechanics plays an important role in simulating biological functions and has great potential to aid medical clinicians in determining the cause of a disease, the type of treatment or by aiding in the training of a surgical procedure. Cardiovascular diseases (CVDs) are the leading cause of mortality today. This work therefore aims at developing a framework for modeling CVDs, such as cerebral aneurysms or heart diseases with increased myofiber dispersion as seen in, e.g., hypertrophic cardiomyopathy.To this end, a three-dimensional growth model of a human saccular cerebral aneurysm is presented that includes the anisotropy of the medial layer. It is shown that including fibers in the media reduces the maximum principal stress, thickness increase and shear stress in the aneurysm wall. It is also shown that the axial pre-stretch has a large impact on the stress levels and thickness increase in the aneurysm wall.In addition, the constituents needed for the numerical implementation of a structurally based constitutive law describing the behavior of passive myocardium is shown. A comparison is made between this invariant based model and a commonly used Green-Lagrange strain component based model and it is shown that using material parameters retrieved when both models is fitted using a simple shear mode experiment, the invariant based model is better suited to predict the stress in the myocardium for other modes of deformation. The passive cardiac model is coupled together with an evolution equation responsible for generating the active stress. A model of the left ventricle (LV) is presented where pressure is calculated as a response to the change in the ventricular volume in order to ensure physiologically realistic pressure-volume loops. The influence of myocardial fiber and sheet distribution is investigated by using two different setups, a generic setup and one based on experiments. The results implies that spatial heterogeneity may play a critical role in mechanical contraction of the LV and that geometrical descriptions of deformation are needed when evaluating the accuracy of a ventricular model.Further, a novel approach to model the dispersion of both the fiber and sheet orientations evident in, especially diseased, myocardium is presented. Analytical and numerical simulations show that the dispersion parameter has great effect on myocardial deformation and stress development. The results also show that the dispersion has a significant impact on pressure-volume loops of an LV, and in future simulations the presented dispersion model for myocardium may advantageously be used together with models of, e.g., growth and remodeling of various cardiac diseases. In cases where fiber-reinforced models are extended to include the effect of distributed fiber orientations, neither the mathematical nor physical motivation for tension-compression fiber switching is clear, and in fact several choices exist for the material modeler. Therefore, methods to study such switching mechanisms is explored by ana...

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Invariant formulation for dispersed transverse isotropy in aortic heart valves

Cited by 115 publications

References 25 publications

Constitutive modelling of arteries

Constitutive modelling of arteries

A constitutive model for fibrous tissues considering collagen fiber crimp

Cardiovascular Mechanics

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