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

Blood flow in arteries is dominated by unsteady flow phenomena. The cardiovascular system is an internal flow loop with multiple branches in which a complex liquid circulates. A nondimensional frequency parameter, the Womersley number, governs the relationship between the unsteady and viscous forces. Normal arterial flow is laminar with secondary flows generated at curves and branches. The arteries are living organs that can adapt to and change with the varying hemodynamic conditions. In certain circumstances, unusual hemodynamic conditions create an abnormal biological response. Velocity profile skewing can create pockets in which the direction of the wall shear stress oscillates. Atherosclerotic disease tends to be localized in these sites and results in a narrowing of the artery lumen—a stenosis. The stenosis can cause turbulence and reduce flow by means of viscous head losses and flow choking. Very high shear stresses near the throat of the stenosis can activate platelets and thereby induce thrombosis, which can totally block blood flow to the heart or brain. Detection and quantification of stenosis serve as the basis for surgical intervention. In the future, the study of arterial blood flow will lead to the prediction of individual hemodynamic flows in any patient, the development of diagnostic tools to quantify disease, and the design of devices that mimic or alter blood flow. This field is rich with challenging problems in fluid mechanics involving three-dimensional, pulsatile flows at the edge of turbulence.

Blood flow in arteries

Atherosclerotic lesion localization to regions of disturbed flow within certain arterial geometries, in humans and experimental animals, suggests an important role for local hemodynamic forces in atherogenesis. To explore how endothelial cells (EC) acquire functional/dysfunctional phenotypes in response to vascular region-specific flow patterns, we have used an in vitro dynamic flow system to accurately reproduce arterial shear stress waveforms on cultured human EC and have examined the effects on EC gene expression by using a high-throughput transcriptional profiling approach. The flow patterns in the carotid artery bifurcations of several normal human subjects were characterized by using 3D flow analysis based on actual vascular geometries and blood flow profiles. Two prototypic arterial waveforms, “athero-prone” and “athero-protective,” were defined as representative of the wall shear stresses in two distinct regions of the carotid artery (carotid sinus and distal internal carotid artery) that are typically “susceptible” or “resistant,” respectively, to atherosclerotic lesion development. These two waveforms were applied to cultured EC, and cDNA microarrays were used to analyze the differential patterns of EC gene expression. In addition, the differential effects of athero-prone vs. athero-protective waveforms were further characterized on several parameters of EC structure and function, including actin cytoskeletal organization, expression and localization of junctional proteins, activation of the NF-κB transcriptional pathway, and expression of proinflammatory cytokines and adhesion molecules. These global gene expression patterns and functional data reveal a distinct phenotypic modulation in response to the wall shear stresses present in atherosclerosis-susceptible vs. atherosclerosis-resistant human arterial geometries.

/pdf/distinct-endothelial-phenotypes-evoked-by-arterial-waveforms-9l8hurfr2c.pdf

Distinct endothelial phenotypes evoked by arterial waveforms derived from atherosclerosis-susceptible and -resistant regions of human vasculature

Toward a biomechanical tool to evaluate rupture potential of abdominal aortic aneurysm: identification of a finite strain constitutive model and evaluation of its applicability

Residual strain effects on the stress field in a thick wall finite element model of the human carotid bifurcation.

Blood flow patterns are believed to be involved in the formation and progression of arterial diseases. It is possible that the normal physiologic movement of blood vessels during the cardiac cycle affects blood flow patterns significantly. For example, the contraction of the heart in systole and subsequent relaxation in diastole create movements of the coronary arteries, as evidenced in real-time angiography. The effects of this movement on coronary artery flow patterns have never been previously analyzed. This work was undertaken to provide a preliminary estimate of the importance of the effects of such physiologic movements on blood flow patterns in the coronary arteries. A Womersley-type solution was used to determine the effect of axial movement on the wall shear rate in a simplified model of the coronary arteries. The pulsatile pressure gradient was derived from previously published coronary artery flow waveforms. The axial movement function was obtained from a three-dimensional reconstruction of a biplanar coronary angiogram. Significant changes in wall shear rate were noted when the movement was taken into account. The maximum and minimum wall shear rates were 10 percent smaller and 107 percent larger in magnitude respectively, and the Oscillatory Shear Index (OSI) was doubled. Most of the changes in wall shear rate were observed in systole, when the pressure gradient is minimal and the movement is strongest. The results indicate that blood vessel movement during the cardiac cycle has a significant effect on hemodynamic phenomena which have been associated with the development of atherosclerosis.

Preliminary Analysis of the Effects of Blood Vessel Movement on Blood Flow Patterns in the Coronary Arteries

The effects of time-varying curvature on velocity profiles in a model of the coronary arteries

Lateral deformation and movement effects on flow through distensible tube models of blood vessels.

A finite element model of the human carotid bifurcation was developed to investigate the possibility that pressure-induced arterial wall stresses lead to the formation of atherosclerosis. The effect of material non-homogeneity was investigated by measuring the distribution of collagen and elastin in cadaver specimens and applying different material constants where appropriate. Both the homogeneous and non-homogeneous models included the residual stresses. The histology results revealed that the outer wall of the apex contained proportionally more collagen. Elsewhere, relative distributions of collagen and elastin were uniform through the thickness. The finite element model results demonstrated that the area of highest stress concentration was the lateral wall of the carotid sinus. The outer wall stress at the apex was higher for the non-homogeneous model owing to the higher collagen content there, but the stresses elsewhere in the model were relatively unaffected. A statistical comparison of the stress distribution to previously published data on intimal thickness showed that areas with a high ratio of inner to outer wall stress coincided with greatest intimal thickening (r = 0.800, P < 0.00005). The inclusion of material non-homogeneity had little effect on this correlation.

A. Delfino

Papers

Residual strain effects on the stress field in a thick wall finite element model of the human carotid bifurcation.

Preliminary Analysis of the Effects of Blood Vessel Movement on Blood Flow Patterns in the Coronary Arteries

The effects of time-varying curvature on velocity profiles in a model of the coronary arteries

Lateral deformation and movement effects on flow through distensible tube models of blood vessels.

Wall stresses in the carotid bifurcation: Effects of wall non-homogeneity and correlation with intimal thickness