Active axial stress in mouse aorta.

The study verifies the development of active axial stress in the wall of mouse aorta over a range of physiological loads when the smooth muscle cells are stimulated to contract. The results obtained show that the active axial stress is virtually independent of the magnitude of pressure, but depends predominately on the longitudinal stretch ratio. The dependence is non-monotonic and is similar to the active stress-stretch dependence in the circumferential direction reported in the literature. The expression for the active axial stress fitted to the experimental data shows that the maximum active stress is developed at longitudinal stretch ratio 1.81, and 1.56 is the longitudinal stretch ratio below which the stimulation does not generate active stress. The study shows that the magnitude of active axial stress is smaller than the active circumferential stress. There is need for more experimental investigations on the active response of different types of arteries from different species and pathological conditions. The results of these studies can promote building of refined constrictive models in vascular rheology.

Residual strains in conduit arteries.

Residual strains and stresses are those that exist in a body when all external loads are removed. Residual strains in arteries can be characterized by the opening angle of the sector-like cross-section which arises when an unloaded ring segment is radially cut. A review of experimental methods for measuring residual strains and the main results about the variation of the opening angle with arterial localization, age, smooth muscle activity, mechanical environment and certain vascular pathologies are presented and discussed. It is shown that, in addition to their well-established ability to homogenize the stress field in the arterial wall, residual strains make arteries more compliant and thereby improve their performance as elastic reservoirs and ensure more effective local control of the arterial lumen by smooth muscle cells. Finally, evidence that, in some cases, residual strains remain in arteries even after they have been cut radially is discussed.

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.

Model of geometrical and smooth muscle tone adaptation of carotid artery subject to step change in pressure.

Recent experimental studies have shown significant alterations of the vascular smooth muscle (VSM) tone when an artery is subjected to an elevation in pressure. Therefore, the VSM participates in the adaptation process not only by means of its synthetic activity (fibronectins and collagen) or proliferative activity (hypertrophy and hyperplasia) but also by adjusting its contractile properties and its tone level. In previous theoretical models describing the time evolution of the arterial wall adaptation in response to induced hypertension, the contribution of VSM tone has been neglected. In this study, we propose a new biomechanical model for the wall adaptation to induced hypertension, including changes in VSM tone. On the basis of Hill's model, total circumferential stress is separated into its passive and active components, the active part being the stress developed by the VSM. Adaptation rate equations describe the geometrical adaptation (wall thickening) and the adaptation of active stress (VSM tone). The evolution curves that are derived from the theoretical model fit well the experimental data describing the adaptation of the rat common carotid subjected to a step increase in pressure. This leads to the identification of the model parameters and time constants by characterizing the rapidity of the adaptation processes. The agreement between the results of this simple theoretical model and the experimental data suggests that the theoretical approach used here may appropriately account for the biomechanics underlying the arterial wall adaptation.

A model of stress-induced geometrical remodeling of vessel segments adjacent to stents and artery/graft anastomoses.

The mismatch between the elastic properties and initial geometry of a host artery and an implanted stent or graft cause significant stress concentration at the zones close to junctions. This may contribute to the often observed intimal hyperplasia, resulting in late lumen loss and eventual restenosis. This study proposes a mathematical model for stress-induced thickening of the arterial wall at the zones close to an implanted stent or graft. The host artery was considered initially as a cylindrical shell with constant thickness that was clamped to the stent or graft, which was assumed to be non-deformable in the circumferential direction. It was assumed that the abnormal circumferential and axial stresses due to the bending of the arterial wall cause wall thickening that tends to restore the stress state close to that existing far from the junction. The linear equations of a cylindrical shell with variable thickness were coupled to an evolution equation for the wall thickness. These equations were solved numerically and a parametric study was performed using finite difference method and explicit time step. The results show that the remodeling process is self-limiting and leads to local thickening that gradually decreases with distance from the edge of the stent/graft. Model predictions were tested against morphological findings existing in the literature. Recommendations on stent designs that reduce stress concentrations are discussed.

Theoretical study of the effects of vascular smooth muscle contraction on strain and stress distributions in arteries.

To 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.

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.

Theoretical study of the effect of stress-dependent remodeling on arterial geometry under hypertensive conditions.

Remodeling of arterial geometry was studied on the basis of a theoretical model. Sustained hypertension was simulated by a step increase in blood pressure. The artery was considered to be a thick-walled two-layer tube made of nonlinear elastic incompressible material. The basic hypothesis is that the artery remodels its zero-stress configuration in such a way that the strain and stress distributions in the arterial wall under hypertensive conditions are the same as under normotensive conditions. Using this hypothesis, a method for determining the geometrical dimensions of the zero-stress configuration of the hypertensive artery was proposed. To ensure uniqueness of the solution, two side conditions on the remodeling process are imposed: (a) the inner radius of the artery in the unloaded state remains unchanged; and (b) the ratio between the thickness of the inner and outer layer of the hypertensive artery in the zero-stress configuration is known. The model predicts that the arterial wall remodeling causes: (i) an increase of the wall thickness both in the unloaded and physiological states; (ii) an increase of the inner diameter of the hypertensive vessel under high pressure compared to the diameter of the normotensive artery under normal pressure; (iii) the opened-up configuration which arises when the unloaded arterial segment is cut radially still contains residual strains and stresses. These results are consistent with published experimental findings. It is speculated that the origin of residual stresses that exist in the unloaded and opened-up configurations is the stress-modulated growth.

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.

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.

Analysis of the strain and stress distribution in the wall of the developing and mature rat aorta.

The variation of wall stress distribution with age in the thoracic and abdominal aortas of normotensive rats was studied. Dimensions of the zero-stress configurations were measured at the ages of 4, 8, 12, 20, and 52 weeks. Using data from previously published inflation tests, the circumferential stress-strain relationship was obtained in each age group. The calculated stress distribution showed that the average circumferential stress remained practically constant after the age of 20 weeks. The circumferential stress at the innermost part of the arterial wall was greater than the stress at the outermost part, but the difference was maintained at a moderate level with adjustments in the zero-stress configuration. It is speculated that, after the age of 20 weeks, changes in arterial geometry and rheological properties tend to maintain a constant stress distribution under varying conditions of loading. This distribution was achieved by enhanced growth at the inner part of the media in comparison with the growth at its outer margins and suggests that during development and maturity, the growth of the aorta is modulated by circumferential stress.

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

Spontaneous low frequency diameter oscillations have been observed in 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.

Effects of transmural pressure and muscular activity on pulse waves in arteries.

Propagation of small amplitude harmonic waves through a viscous incompressible fluid contained in an initially stressed elastic cylindrical tube is considered as a model of the pulse wave propagation in arteries. The nonlinearity and orthotropy of the vascular material is taken into account. Muscular activity is introduced by means of an "active" tension in circumferential direction of the vessel. The frequency equation is obtained and it is solved numerically for the parameters of a human abdominal aorta. Conclusions concerning pressure-dependence, age-dependence, and muscular activation-dependence of the wave characteristics are drawn which are in accord with available experimental data.