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

Biological materials: Structure and mechanical properties

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

http://biomechanics.stanford.edu/me338/me338_project04.pdf

Stress-dependent finite growth in soft elastic tissues

Forces that are associated with blood flow are major determinants of vascular morphogenesis and physiology. Blood flow is crucial for blood vessel development during embryogenesis and for regulation of vessel diameter in adult life. It is also a key factor in atherosclerosis, which, despite the systemic nature of major risk factors, occurs mainly in regions of arteries that experience disturbances in fluid flow. Recent data have highlighted the potential endothelial mechanotransducers that might mediate responses to blood flow, the effects of atheroprotective rather than atherogenic flow, the mechanisms that contribute to the progression of the disease and how systemic factors interact with flow patterns to cause atherosclerosis.

/pdf/mechanotransduction-in-vascular-physiology-and-atherogenesis-bh5vcvtol5.pdf

Mechanotransduction in vascular physiology and atherogenesis

http://maeresearch.ucsd.edu/~vlubarda/research/pdfpapers/ijss-02.pdf

On the mechanics of solids with a growing mass

The equations of compatibility which are pertinant for growth strain fields are collected and examples are given in simply-connected and multiply-connected regions. Compatibility conditions for infinitesimal strains are well known and the possibilities of Volterra dislocations in multiply-connected regions are enumerated. For finite growth strains in a multiply-connected regions, each case must be examined individually and no generalizations in terms of Volterra dislocations are available. Any incompatible growth strains give rise to residual stresses which are known to occur in many tissues such as the heart, arterial wall, and solid tumors.

Compatibility and the genesis of residual stress by volumetric growth.

The stress conjugate to logarithmic strain

Endothelial cells (ECs) line the mammalian vascular system and respond to the hemodynamic stimulus of fluid shear stress, the frictional force produced by blood flow. When ECs are exposed to shear stress, one of the fastest responses is an increase of K+ conductance, which suggests that ion channels are involved in the early shear stress response. Here we show that an applied shear stress induces a K+ ion current in cells expressing the endothelial Kir2.1 channel. This ion current shares the properties of the shear-induced current found in ECs. In addition, the shear current induction can be specifically prevented by tyrosine kinase inhibition. Our findings identify the Kir2.1 channel as an early component of the endothelial shear response mechanism.

https://www.pnas.org/content/pnas/99/11/7780.full.pdf

Anne Hoger

Papers

Stress-dependent finite growth in soft elastic tissues

On the mechanics of solids with a growing mass

Compatibility and the genesis of residual stress by volumetric growth.

The stress conjugate to logarithmic strain

Shear stress regulates the endothelial Kir2.1 ion channel.