In this paper we develop a new constitutive law for the description of the (passive) mechanical response of arterial tissue. The artery is modeled as a thick-walled nonlinearly elastic circular cylindrical tube consisting of two layers corresponding to the media and adventitia (the solid mechanically relevant layers in healthy tissue). Each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component of the material and symmetrically disposed with respect to the cylinder axis. The resulting constitutive law is orthotropic in each layer. Fiber orientations obtained from a statistical analysis of histological sections from each arterial layer are used. A specific form of the law, which requires only three material parameters for each layer, is used to study the response of an artery under combined axial extension, inflation and torsion. The characteristic and very important residual stress in an artery in vitro is accounted for by assuming that the natural (unstressed and unstrained) configuration of the material corresponds to an open sector of a tube, which is then closed by an initial bending to form a load-free, but stressed, circular cylindrical configuration prior to application of the extension, inflation and torsion. The effect of residual stress on the stress distribution through the deformed arterial wall in the physiological state is examined. The model is fitted to available data on arteries and its predictions are assessed for the considered combined loadings. It is explained how the new model is designed to avoid certain mechanical, mathematical and computational deficiencies evident in currently available phenomenological models. A critical review of these models is provided by way of background to the development of the new model.

https://hal.archives-ouvertes.fr/hal-01297725/document

A new constitutive framework for arterial wall mechanics and a comparative study of material models

Constitutive relations are fundamental to the solution of problems in continuum mechanics, and are required in the study of, for example, mechanically dominated clinical interventions involving soft biological tissues. Structural continuum constitutive models of arterial layers integrate information about the tissue morphology and therefore allow investigation of the interrelation between structure and function in response to mechanical loading. Collagen fibres are key ingredients in the structure of arteries. In the media (the middle layer of the artery wall) they are arranged in two helically distributed families with a small pitch and very little dispersion in their orientation (i.e. they are aligned quite close to the circumferential direction). By contrast, in the adventitial and intimal layers, the orientation of the collagen fibres is dispersed, as shown by polarized light microscopy of stained arterial tissue. As a result, continuum models that do not account for the dispersion are not able to capture accurately the stress–strain response of these layers. The purpose of this paper, therefore, is to develop a structural continuum framework that is able to represent the dispersion of the collagen fibre orientation. This then allows the development of a new hyperelastic free-energy function that is particularly suited for representing the anisotropic elastic properties of adventitial and intimal layers of arterial walls, and is a generalization of the fibre-reinforced structural model introduced by Holzapfel & Gasser (Holzapfel & Gasser 2001 Comput. Meth. Appl. Mech. Eng. 190, 4379–4403) and Holzapfel et al. (Holzapfel et al. 2000 J. Elast. 61, 1–48). The model incorporates an additional scalar structure parameter that characterizes the dispersed collagen orientation. An efficient finite element implementation of the model is then presented and numerical examples show that the dispersion of the orientation of collagen fibres in the adventitia of human iliac arteries has a significant effect on their mechanical response.

https://royalsocietypublishing.org/doi/pdf/10.1098/rsif.2005.0073

Hyperelastic modelling of arterial layers with distributed collagen fibre orientations

An important factor in the transition from an open to a closed circulatory system was a change in vessel wall structure and composition that enabled the large arteries to store and release energy during the cardiac cycle. The component of the arterial wall in vertebrates that accounts for these properties is the elastic fiber network organized by medial smooth muscle. Beginning with the onset of pulsatile blood flow in the developing aorta, smooth muscle cells in the vessel wall produce a complex extracellular matrix (ECM) that will ultimately define the mechanical properties that are critical for proper function of the adult vascular system. This review discusses the structural ECM proteins in the vertebrate aortic wall and will explore how the choice of ECM components has changed through evolution as the cardiovascular system became more advanced and pulse pressure increased. By correlating vessel mechanics with physiological blood pressure across animal species and in mice with altered vessel compliance, we show that cardiac and vascular development are physiologically coupled, and we provide evidence for a universal elastic modulus that controls the parameters of ECM deposition in vessel wall development. We also discuss mechanical models that can be used to design better tissue-engineered vessels and to test the efficacy of clinical treatments.

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Vascular Extracellular Matrix and Arterial Mechanics

In the cardiovascular system, morphology and functionality are closely related. Altered flow conditions play an important role in the development of arterial disease. In turn, these flow conditions are modified by arterial wall changes. A detailed understanding of the local hemodynamic environment, the influence of wall modifications on flow patterns and the long-term adaptations of the vascular wall after surgical procedures can have useful clinical applications, especially in view of reconstruction and revascularization operations. Some of these alterations are not well-understood, making it quite difficult to foresee short- and long-term evolution of the atherosclerotic disease and to plan an aggressive approach. Computational fluid dynamics (CFD) allows the carrying out of simulations at low cost and in completely controlled conditions. Three different issues are relevant to this subject and are addressed in this paper: (1) the definition of suitable mathematical models; (2) pre-processing of clinical data; and (3) the development of appropriate numerical techniques

Computational vascular fluid dynamics: problems, models and methods

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 thick-walled 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.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2010.0058

Constitutive modelling of arteries

Biomechanical behavior of the arterial wall and its numerical characterization

Analysis of the passive mechanical properties of rat carotid arteries

Large strain analysis of soft biological membranes: Formulation and finite element analysis

Isotropy and anisotropy of the arterial wall.

To explore the mechanical non-linear behaviour of anisotropic arterial walls on a computational basis, the formulation of a continuum based elastic potential is a major task and challenge to the analyst. The present communication is concerned with the constitutive modelling and numerical analysis of vascular segments covering finite strains. Special attention is paid to a two term potential that constitutes an essential foundation for accurate simulation within the entire strain domain. Axisymmetrical membrane elements are assembled to match the geometry of blood vessels. Numerical results confirm the theoretical approach by referring to experimental data of different rat arteries.

Hans Werner Weizsäcker

Papers

Biomechanical behavior of the arterial wall and its numerical characterization

Analysis of the passive mechanical properties of rat carotid arteries

Large strain analysis of soft biological membranes: Formulation and finite element analysis

Isotropy and anisotropy of the arterial wall.

A new axisymmetrical membrane element for anisotropic, finite strain analysis of arteries