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

At autopsy, 13 nonstenotic human left anterior descending coronary arteries [71.5 ± 7.3 (mean ± SD) yr old] were harvested, and related anamnesis was documented. Preconditioned prepared strips (n =...

Determination of layer-specific mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling

Coronary heart disease

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

The mechanical behavior of the entire anulus fibrosus is determined essentially by the tensile properties of its lamellae, their fiber orientations, and the regional variation of these quantities. Corresponding data are rare in the literature. The paper deals with an in vitro study of single lamellar anulus lamellae and aims to determine (i) their tensile response and regional variation, and (ii) the orientation of lamellar collagen fibers and their regional variation. Fresh human body-disc-body units (L1–L2, n=11) from cadavers were cut midsagittally producing two hemidisc units. One hemidisc was used for the preparation of single lamellar anulus specimens for tensile testing, while the other one was used for the investigation of the lamellar fiber orientation. Single lamellar anulus specimens with adjacent bone fragments were isolated from four anatomical regions: superficial and deep lamellae (3.9±0.21 mm, mean ± SD, apart from the outer boundary surface of the anulus fibrosus) at ventro-lateral and dorsal positions. The specimens underwent cyclic uniaxial tensile tests at three different strain rates in 0.15 mol/l NaCl solution at 37°C, whereby the lamellar fiber direction was aligned with the load axis. For the characterization of the tensile behavior three moduli were calculated: Elow (0–0.1 MPa), Emedium (0.1–0.5 MPa) and Ehigh (0.5–1 MPa). Additionally, specimens were tested with the load axis transverse to the fiber direction. From the second hemidisc fiber angles with respect to the horizontal plane were determined photogrammetrically from images taken at six circumferential positions from ventral to dorsal and at three depth levels. Tensile moduli along the fiber direction were in the range of 28–78 MPa (regional mean values). Superficial lamellae have larger Emedium (p=0.017) and Ehigh (p=0.012) than internal lamellae, and the mean value of superficial lamellae is about three times higher than that of deep lamellae. Tensile moduli of ventro-lateral lamellae do not differ significantly from the tensile moduli of dorsal lamellae, and Elow is generally indifferent with respect to the anatomical region. Tensile moduli transverse to the fiber direction were about two orders of magnitude smaller (0.22±0.2 MPa, mean ± SD, n=5). Tensile properties are not correlated significantly with donor age. Only small viscoelastic effects were observed. The regional variation of lamellar fiber angle ϕ is described appropriately by a regression line |ϕ|=23.2+0.130×α (r2=0.55, p<0.001), where α is the polar angle associated with the circumferential position. The single anulus lamella may be seen as the elementary structural unit of the anulus fibrosus, and exhibits marked anisotropy and distinct regional variation of tensile properties and fiber angles. These features must be considered for appropriate physical and numerical modeling of the anulus fibrosus.

Single lamellar mechanics of the human lumbar anulus fibrosus

A detailed understanding of the mechanical procedure of balloon angioplasty requires three-dimensional (3D) modeling and efficient numerical simulations. We have developed a 3D model for eight distinct arterial components associated with specific mechanical responses. The 3D geometrical model is based on in vitro magnetic resonance imaging of a human stenotic postmortem artery and is represented by nonuniform rational B-spline surfaces. Mechanical tests of the corresponding vascular tissues provide a fundamental basis for the formulation of large strain constitutive laws, which model the typical anisotropic, highly nonlinear, and inelastic mechanical characteristics under supraphysiological loadings. The 3D finite-element realization considers the balloon–artery interaction and accounts for vessel-specific axial in situ prestretches. 3D stress states of the investigated artery during balloon expansion and stent deployment were analyzed. Furthermore, we studied the changes of the 3D stress state due to model simplifications, which are characterized by neglecting axial in situ prestretch, assuming plane strain states, and isotropic material responses, as commonly utilized in previous works. Since these simplifications lead to maximum stress deviations of up to 600%—where even the stress character may interchange—the associated models are, in general, inappropriate. The proposed approach provides a tool that has the potential (i) to improve procedural protocols and the design of interventional instruments on a lesion-specific basis, and (ii) to determine postangioplasty mechanical environments, which may be correlated with restenosis responses. © 2002 Biomedical Engineering Society.
 PAC2002: 8719Rr, 8761Lh, 8710+e

A layer-specific three-dimensional model for the simulation of balloon angioplasty using magnetic resonance imaging and mechanical testing.

Inflation and extension tests of arteries are essential for the understanding of arterial wall mechanics. Data for such tests of human arteries are rare. At autopsy we harvested 10 non-diseased external iliac arteries of aged subjects (52-87 yrs). Structural homogeneity was ensured by means of ultrasound imaging, and anamneses of patients were recorded. We measured the axial in situ stretches, load-free geometries and opening angles. Passive biaxial mechanical responses of preconditioned cylindrical specimens were studied in 37 degrees C calcium-free Tyrode solution under quasistatic loading conditions. Specimens were subjected to pressure cycles varying from 0 to 33.3 kPa (250 mmHg) at nine fixed axial loads, varying from 0 to 9.90N. For the description of the load-deformation behavior we employed five "two-dimensional" orthotropic strain-energy functions frequently used in arterial wall mechanics. The associated constitutive models were compared in regard to their ability of representing the experimental data. Histology showed that the arteries were of the muscular type. In contrast to animal arteries they exhibited intimal layers of considerable thickness. The average ratio of wall thickness to outer diameter was 7.7, which is much less than observed for common animal arteries. We found a clear correlation between age and the axial in situ stretch lambda is (r = -0.72, P = 0.03), and between age and distensibility of specimens, i.e. aged specimens are less distensible. Axial in situ stretches were clearly smaller (1.07 +/- 0.09, mean +/- SD) than in animal arteries. For one specimen lambda is was even smaller than 1.0, i.e. the vessel elongated axially upon excision. The nonlinear and anisotropic load-deformation behavior showed small hystereses. For the majority of specimens we observed axial stretches smaller than 1.3 and circumferential stretches smaller than 1.1 for the investigated loading range. Data from in situ inflation tests showed a significant increase of the axial stretch with intraluminal pressure. Thus, for this type of artery the axial in situ stretch of a non-pressurized vessel is not representative of the axial in vivo stretch. None of the constitutive models were able to represent the deformation behavior of the entire loading range. For the physiological loading range, however, some of the models achieved good agreement with the experimental data.

Passive Biaxial Mechanical Response of Aged Human Iliac Arteries

The current investigation addresses advanced FE analyses of intact lumbar intervertebral discs. Human disc body units basically consist of a fluid filled cavity (nucleus pulposus) approximately at the disc center and surrounding annulus tissue reinforced by collagen fibers (annulus fibrosus). For the latter component a constitutive model is presented which accounts for nonlinear anisotropic stress response. In contrast to state-of-the-art material descriptions mostly used in the literature (linear elasticity), the current approach provides numerical accuracy in the finite strain regime. Thus, local strain and stress distributions in the annulus fibrosus can properly be determined. Because of the highly nonlinear stress response of the constitutive equations, nonlinear material parameters are required. Although tensile properties of the human annulus fibrosus can already be found in the literature, these material data are not suitable for the current concept as will be shown. The entire constitutive theory...

An Anisotropic Model for Annulus Tissue and Enhanced Finite Element Analyses of Intact Lumbar Disc Bodies

Adventitial mechanics were studied on the basis of adventitial tube tests and associated stress analyses utilizing a thin-walled model. Inflation tests of 11 nonstenotic human femoral arteries (79....

Christian Schulze-Bauer

Papers

Single lamellar mechanics of the human lumbar anulus fibrosus

A layer-specific three-dimensional model for the simulation of balloon angioplasty using magnetic resonance imaging and mechanical testing.

Passive Biaxial Mechanical Response of Aged Human Iliac Arteries

An Anisotropic Model for Annulus Tissue and Enhanced Finite Element Analyses of Intact Lumbar Disc Bodies

Mechanics of the human femoral adventitia including the high-pressure response