Alexander Rachev

Theoretical Study of the Effects of Vascular Smooth Muscle Contraction on Strain and Stress Distributions in Arteries

AbstractTo study the effects of smooth muscle contraction and relaxation on the strain and stress distribution in the vascular wall, a mathematical model was proposed. The artery was assumed to be a thick-walled orthotropic tube made of nonlinear, incompressible elastic material. Considering that the contraction of smooth muscle generates an active circumferential stress in the wall, a numerical study was performed using data available in the literature. The results obtained showed that smooth muscle contraction affects the residual strains which exist in a ring segment cut out from the artery and exposed to no external load. When the ring specimen is cut radially, it springs open with an opening angle. The predicted monotonic increase of the opening angle with increasing muscular tone was in agreement with recent experimental results reported in the literature. It was shown that basal muscular tone, which exists under physiological conditions, reduces the strain gradient in the arterial wall and yields a near uniform stress distribution. During temporary changes in blood pressure, the increase in muscular tone induced by elevated pressure tends to restore the distribution of circumferential strain in the arterial wall, and to maintain the flow-induced wall shear stress to normal level.

Experimental investigation of the distribution of residual strains in the artery wall

Arterial wall stresses are thought to be a major determinant of vascular remodeling both during normal growth and throughout the development of occlusive vascular disease. A completely physiologic mechanical model of the arterial wall should account not only for its residual strains but also for its structural nonhomogeneity. It is known that each layer of the artery wall possesses different mechanical properties, but the distribution of residual strain among the different mechanical components, and thus the true zero stress state, remain unknown. In this study, two different sets of experiments were carried out in order to determine the distribution of residual strains in artery walls, and thus the true zero stress state. In the first, collagen and elastin were selectively eliminated by chemical methods and smooth muscle cells were destroyed by freezing. Dissolving elastin provoked a decrease in the opening angle, while dissolving collagen and destroying smooth muscle cells had no effect. In the second, different wall layers of bovine carotid arteries were removed from the exterior or luminal surfaces by lathing or drilling frozen specimens, and then allowing the frozen material to thaw before measuring residual strain. Lathing material away from the outer surface caused the opening angle of the remaining inner layers to increase. Drilling material from the inside caused the opening angle of the remaining outer layers to decrease. Mechanical nonhomogeneity, including the distribution of residual strains, should thus be considered as an important factor in determining the distribution of stress in the artery wall and the configuration of the true zero stress state.

A model for geometric and mechanical adaptation of arteries to sustained hypertension

This study aimed to model phenomenologically the dynamics of arterial wall remodeling under hypertensive conditions. Sustained hypertension was simulated by a step increase in blood pressure. The arterial wall was considered to be a thick-walled tube made of nonlinear elastic incompressible material. Remodeling rate equations were postulated for the evolution of the geometric dimensions of the hypertensive artery at the zero-stress state, as well as for one of the material constants in the constitutive equations. The driving stimuli for the geometric adaptation are the normalized deviations of wall stresses from their values under normotensive conditions. The geometric dimensions are modulated by the evolution of the deformed inner radius, which serves to restore the level of the flow-induced shear stresses at the arterial endothelium. Mechanical adaptation is driven by the difference between the area compliance under hypertensive and normotensive conditions. The predicted time course of the geometry and mechanical properties of arterial wall are in good qualitative agreement with published experimental findings. The model predicts that the geometric adaptation maintains the stress distribution in arterial wall to its control level, while the mechanical adaptation restores the normal arterial function under induced hypertension.

Hemodynamics and Wall Mechanics of a Compliance Matching Stent: In Vitro and In Vivo Analysis

PURPOSEnEvidence is emerging that the abrupt compliance mismatch that exists at the junction between the stent ends and the host arterial wall disturbs both the vascular hemodynamics and the natural wall stress distribution. These stent-induced alterations are greatly reduced by smoothing the compliance mismatch between the stent and host vessel. A stent that provides this smooth transition in compliance, the compliance matching stent (CMS), has been developed. This study attempts to evaluate the hemodynamics and wall mechanical consequences of the CMS both in vitro and in vivo.nnnMATERIALS AND METHODSnFinite element analysis was used to assess the solid mechanical behavior (compliance and stress) of the CMS in a stent/artery hybrid structure. A similar analysis was performed with a Palmaz stent. In vivo hemodynamics and wall mechanical changes induced by the CMS were investigated in a swine model from direct measurements of flow, pressure, diameter, and histology in the stented segment of superficial femoral arteries after 7 days.nnnRESULTSnFinite element analysis showed that the abrupt compliance mismatch was substantially smoothed between the vessel portions with and without the stent with CMS segments. Circumferential stress was also markedly reduced with the CMS compared to other stent. The in vivo results showed that the CMS was efficient in compliance matching and did not dampen flow or pressure waves downstream the stent. Concurrent histology showed limited thrombus and inflammatory cell accumulation around the stent struts.nnnCONCLUSIONnThese results indicate that the stent/artery hybrid structure can be compliance matched with proper stent design and that this structure limits solid mechanical stress and hemodynamic disturbances. It remains to be seen whether compliance-matched vascular stents reduce in-stent restenosis.

Theoretical study of dynamics of arterial wall remodeling in response to changes in blood pressure

The dynamics of arterial wall remodeling was studied on the basis of a phenomenological mathematical model. Sustained hypertension was simulated by a step increase in blood pressure. Remodeling rate equations were postulated for the evolution of the geometrical dimensions that characterize the zero stress state of the artery. The driving stimuli are the deviations of the extreme values of the circumferential stretch ratios and the average stress from their values at the normotensive state. Arterial wall was considered to be a thick-walled tube made of nonlinear elastic incompressible material. Results showed that thickness increases montonically with time whereas the opening angle exhibits a biphasic pattern. Geometric characteristics reach asymptotically a new homeostatic steady state, in which the stress and strain distribution is practically identical with the distribution under normotensive conditions. The model predictions are in good agreement with published experimental findings.

Assessing the Homogeneity of the Elastic Properties and Composition of the Pig Aortic Media

Most previous studies of arterial wall elasticity and rheology have assumed that the properties of the wall are uniform across the thickness of the media and, therefore, that the relationship between stress and strain may be described by a constitutive equation based on a single strain energy function. The few studies where this assumption has been questioned, focussed on differences between the adventitia and the media rather than on differences within the media itself. Here, we report in vitro elasticity and residual strain measurements performed separately on the inner and outer half of the pig aortic media, together with a histomorphometric assessment of the radial distribution of elastin, collagen and smooth muscle cell numbers. Although we found that the pressure-diameter relationships of the two halves were dissimilar, when allowance was made for their different unloaded dimensions, their material properties agreed closely, a result in keeping with the observed uniform radial distribution of scleroprotein and vascular smooth muscle. We also found a difference in the opening angle (which is often taken as a measure of residual strain) between the inner and outer medial halves. However, strain analysis showed that the opening angle is an extremely sensitive measure of residual strain and that the difference in the actual magnitudes of residual strain between the two halves of the media was small. We conclude that the media of the porcine thoracic aorta has similar elastic properties throughout its thickness and that this uniformity is matched by a uniform distribution of matrix protein and vascular smooth muscle cells. Furthermore, the distribution of strain in the media can adequately be described by a single-layer model with uniform elastic properties throughout its thickness.

A Model of Arterial Adaptation to Alterations in Blood Flow

Mechanisms of arterial adaptation to changes in blood flow rates were tested by comparing the predictions of a proposed theoretical model with available experimental data. The artery was modeled as an elastic membrane made of a nonlinear, incompressible, elastic material. Stimulation of the vascular smooth muscle was modeled through the generation of an active component of circumferential stress. The muscular tone was modulated by flow-induced shear stress sensed by the arterial endothelium, and is responsible for the vasomotor adjustment of the deformed arterial diameter in response to changes in blood flow. This study addresses the hypothesis that the synthetic and proliferative activity of smooth muscle cells, leading to a change in arterial dimensions, is shear stress dependent and is associated with changes in the contractile state of the smooth muscle cells and changes in the circumferential wall stress. Remodeling to a step change in flow was formulated as an initial-value problem for a system of first order autonomous differential equations for the evolution of muscular tone and evolution of arterial geometry. The governing equations were solved numerically for model parameters identified from experimental data available in the literature. The model predictions for the time variation of the geometrical dimensions and their asymptotic values were found to be in qualitative agreement with available experimental data. Experiments for validating the introduced hypotheses and further generalizations of the model were discussed.

A theoretical investigation of low frequency diameter oscillations of muscular arteries.

Spontaneous low frequency diameter oscillations have been observedin vivo in some muscular arteries. The aim of this paper is to propose a possible mechanism for their appearance. A lumped parameter mathematical model for the mechanical response of an artery perfused with constant flow is proposed, which takes into account the active behavior of the vascular smooth muscle. The system of governing equations is reduced into two nonlinear autonomous differential equations for the arterial circumferential stretch ratio, and the concentration of calcium ions, Ca2+, within the smooth muscle cells. Factors controlling the muscular tone are taken into account by assuming that the rate of change of Ca2+ depends on arterial pressure and on shear stress acting on the endothelium. Using the theory of dynamical systems, it was found that the stationary solution of the set of governing equations may become unstable and a periodic solution arises, yielding self-sustained diameter oscillations. It is found that a necessary condition for the appearance of diameter oscillations is the existence of a negative slope of the steady state pressure-diameter relationship, a phenomenon known to exist in arterioles. A numerical parametric study was performed and bifurcation diagrams were obtained for a typical muscular artery. Results show that low frequency diameter oscillations develop when the magnitude of the perfused inflow, the distal resistance, as well as the length of the artery are within a range of critical values.

A theoretical study of mechanical stability of arteries.

This study proposes a mathematical model for studying stability of arteries subjected to a longitudinal extension and a periodic pressure. An artery was considered as a straight composite beam comprised of an external thick-walled tube and a fluid core. The dynamic criterion for stability was used, based on analyzing the small transverse vibrations superposed on the finite deformation of the vessel under static load. In contrast to the case of a static pressurization, in which buckling is only possible if the load produces a critical axial compressive force, a loss of stability of arteries under periodic pressure occurs under many combinations of load parameters. Instability occurs as a parametric resonance characterized by an exponential increase in the amplitude of transverse vibrations over several bands of pressure frequencies. The effects of load parameters were analyzed on the basis of the results for a dynamic and static stability of a rabbit thoracic aorta. Under normal physiological loads the artery is in a stable configuration. Static instability occurs under high distending pressures and low longitudinal stretch ratios. When the artery is subjected to periodic pressure, an independent increase in the mean pressure, amplitude of the periodic pressure, or frequency, most often, but not always, increases the risk of stability loss. In contrary, an increase in longitudinal stretch ratio most likely, but not certain, stabilizes the vessel. It was shown that adaptive geometrical remodeling due to an increase in mean pressure and flow does not affect artery stability.

Remodeling of Arteries in Response to Changes in their Mechanical Environment

Arteries are subjected to mechanical forces, which may vary in time. A long-lasting alteration in pressure and/or blood flow rate causes an adaptive response termed remodeling. At the macro-level remodeling is manifest as a change in arterial geometry and a change in mechanical properties of the arterial tissue. A review of the main experimental findings concerning pressure- and flow-induced remodeling of large arteries is presented. Theoretical models of volumetric and global growth based on a continuum mechanics approach are discussed. Some specific biomechanical problems of arterial remodeling associated with abnormal narrowing of the arterial lumen are considered.