Alternate method for the analysis of residual strain in the arterial wall

If an artery is cut transversely into rings, and the rings are then cut radially, they spring open into sectors. This phenomenon implies the existence of residual stresses and strains in the arterial wall in the non-loaded state. In the present paper, we propose a new method to calculate the residual strain from the measured wall dimensions and a polar angle of a specimen in the stress-free state, assuming that the wall is homogeneous and incompressible, and that a radially cut, stress-free specimen forms a circular sector. For this analysis, edge angles were measured at the edges of the opened-up specimen. Residual strains were obtained for the descending thoracic aorta, the common carotid artery, and the femoral artery in the rabbit. The results obtained indicated that the magnitude of residual strain was largest in the femoral artery and smallest in the aorta among the three arteries. The opening angle did not depend upon the length of a ring specimen if the ratio of the length to the diameter was ≤ 3.     

Changes of opening angle in hypertensive and hypotensive arteries in 3-day organ culture

To study the effect of pressure changes on the opening angle of arteries in organ culture, tubular segments of porcine common carotid arteries were cultured with pulsatile flow perfusion under hypertensive (150+/-20 mmHg), normotensive (100+/-20 mmHg), or hypotensive (30+/-10 mmHg) pressure while maintaining the arteris at a physiological wall shear stress of approximately 15 dyn/cm(2) for up to 3 days. Arteries were then cut into short ring segments by sections perpendicular to the axis and then cut open radially to observe the opening angle in aerated phosphate buffered saline solution (37 degrees C). Norepinephrine (NE, 10 microM), carbacol (CCh, 100 microM), and sodium nitroprusside (SNP, 10 microM) were added after the radial cut at 30, 20, and 30 min intervals, the opening angles were measured, respectively. Results show that hypertensive arteries developed a significantly larger opening angle than normotensive and hypotensive arteries, associated with a significant increase in cell proliferation. In addition, with smooth muscle contraction activated by NE, the opening angle decreases significantly in hypertensive arteries but has little change in hypotensive and normotensive arteries, indicating an enhancement of smooth muscle contraction on the lumen side of the hypertensive arterial wall. In comparison, hypotensive pressure has little effect on arterial opening angle and cell proliferation.     

Residual strains in porcine and canine trachea

Residual strains exist in canine and porcine tracheas. They are revealed by cutting the trachea first perpendicular to its axis into rings, then radially into sectors. Each sector is characterized by an opening angle which is defined as the angle subtended between two radii joining the middle point of the inner wall to the tips of the inner wall. The trachea being non-axisymmetric, the opening angle depends on the position of the radial cut. The trachea being also nonuniform in the axial direction, the opening angle varies along the length of the trachea. In the dog, the opening angle of the trachea cut at the anterior position (cartilaginous) is about 100 degrees at the larynx; it increases fairly linearly to 180 degrees midway down the trachea; then increases slowly to about 200 degrees at the lower end where the trachea bifurcates into the main bronchi. Dog trachea cut in the posterior (muscular) position have an opening angle of about 50 degrees at the larynx, which increases to about 70 degrees three-quarters of the way down the trachea, then drops to 60 degrees at the lower end. In the pig, the opening angle of the trachea is much smaller, the values at anterior and posterior cuts are similar (without significant difference), and their mean value decreases from about 15 degrees at the laryngeal end to about 5 degrees at the lower end. These species and regional differences are discussed in relation to tracheal geometry and structure.(ABSTRACT TRUNCATED AT 250 WORDS)     

Species dependence of the zero-stress state of aorta: pig versus rat.

The zero-stress state of an aorta can be characterized by the angle with which each segment of the vessel opens up when it is cut radially. The opening angle varies with the region of the aorta: significantly with respect to the axial location, less significantly with respect to polar angle of the radial cut. Both pig and rat aortas have large opening angles in the neighborhood of 130 deg in the aortic arch region. In the thoracic region, the species difference is evident. The opening angle of the pig aorta in the middle thoracic region is rather constant in the neighborhood of 60 deg. The opening angle of the rat aorta in the thoracic region varies considerably, decreasing to 10 deg at the lower end of the thoracic region. In the abdominal region the opening angle of the pig increases from 60 to about 80 deg, that of the rat increases from about 10 to 90 deg. The potassium ion has effect on vascular smooth muscle, but has little effect on the opening angle. This suggests that the opening angle is not sensitive to smooth muscle contraction, similar to a previously known result that the opening angle is not affected by papaverine. The vessel wall thickness and vessel diameter were measured. It is shown that the ratio of the wall thickness to diameter of the pig is considerably larger than that of the rat throughout the aorta.(ABSTRACT TRUNCATED AT 250 WORDS)     

Origin of axial prestretch and residual stress in arteries

The structural protein elastin endows large arteries with unique biological functionality and mechanical integrity, hence its disorganization, fragmentation, or degradation can have important consequences on the progression and treatment of vascular diseases. There is, therefore, a need in arterial mechanics to move from materially uniform, phenomenological, constitutive relations for the wall to those that account for separate contributions of the primary structural constituents: elastin, fibrillar collagens, smooth muscle, and amorphous matrix. In this paper, we employ a recently proposed constrained mixture model of the arterial wall and show that prestretched elastin contributes significantly to both the retraction of arteries that is observed upon transection and the opening angle that follows the introduction of a radial cut in an unloaded segment. We also show that the transmural distributions of elastin and collagen, compressive stiffness of collagen, and smooth muscle tone play complementary roles. Axial prestresses and residual stresses in arteries contribute to the homeostatic state of stress in vivo as well as adaptations to perturbed loads, disease, or injury. Understanding better the development of and changes in wall stress due to individual extracellular matrix constituents thus promises to provide considerable clinically important insight into arterial health and disease.     

Postsurgical changes of the opening angle of canine autogenous vein graft.

The opening angles of 30 canine autogenous vein grafts were measured to determine the postsurgical change of residual strain in the vein graft. Canine femoral veins were grafted to femoral arteries in the end-to-end anastomosis fashion. When harvested, the vein grafts were cut into short segments and the segments were cut open radially. The opened-up configurations were taken as the zero-stress states of the vessels. Opening angle, defined as the angle between the two lines from the middle point to the tips of the inner wall, was used to describe the zero-stress states. Results show that the opening angles (mean +/- SD) are 63.0 +/- 30.6 deg for normal femoral veins, and -0.4 +/- 4.6, 6.1 +/- 19.4, 25.4 +/- 20.1, and 47.8 +/- 11.4 deg for vein grafts at 1 day, 1 week, 4 and 12 weeks postsurgery, respectively. The postsurgical changes in opening angle reveal nonuniform transmural tissue remodeling in the vascular wall. The relations between the changes in opening angle and the changes in the morphology of the vein grafts are discussed. Intimal hyperplasia is correlated to the opening angle and is suggested to be the main factor for the postsurgical increase in opening angle. The longitudinal strain in the vein graft is found to decrease postsurgically.     

Opening angle and residual strain in a three-layered model of pig oesophagus

Studies of various biological tissues have shown that residual strains are important for tissue function. Since a force balance exists in whole wall thickness specimens cut radially, it is evident that layer separation is an important procedure in the understanding of the meaning of residual stresses and strains. The present study investigated the zero-stress state and residual strain distribution in a three-layer model of the pig oesophagus. The middle part of the oesophagus was obtained from six slaughterhouse pigs. Four 3-mm-wide rings were serially cut from each oesophagus. Two of them were used for separating the wall into mucosa-submucosa, inner and outer muscle layers. The remaining two rings were kept as intact rings. The inner and outer circumferences and wall thickness of different layers in intact and separated rings were measured from the digital images in the no-load state and zero-stress state. The opening angle was measured and the residual strain at the inner and outer surface of different layers and the intact wall were computed. Compared with intact sectors (62.8+/-9.8 degrees ), the opening angles were smaller in the inner muscle sectors (37.2+/-11.4 degrees , P<0.01), whereas the opening angles of mucosa-submucosa (63.9+/-6.8 degrees ) and outer muscle sectors (63.9+/-6.8 degrees ) did not differ (P>0.1). Referenced to the zero-stress state of the intact sectors, the inner and outer residual strains of the intact rings was -0.128+/-0.043 and outer residual strain was 0.308+/-0.032. Referenced to the "true" zero-stress state of separated three-layered sectors, the inner residual strain of intact rings were -0.223+/-0.021 (P<0.01) and 0.071+/-0.022 (P<0.01). Referenced to the "true" zero-stress state, the residual strain distribution of different layers in intact rings was shown that the inner surface residual strain was negative at mucosa-submucosa and inner muscle layers and was positive at outer muscle layer, whereas the outer surface residual strain was negative at the mucosa-submucosa layer and positive at the inner and outer muscle layers. For the separated different layered rings, the inner residual strain was negative and outer residual strain was positive; however, the absolute values did not differ (P>0.1). In conclusion, it is possible to microsurgically separate the oesophagus into three layers, i.e., mucosa-submucosa, inner muscle and outer muscle layers, the residual strain differ between the layers, and the residual strain distribution was more uniform after the layers were separated.     

Effect of smooth muscle tone on morphometry and residual strain in rat duodenum, jejunum and ileum

It is difficult to measure gastrointestinal smooth muscle (SM) tone except in sphincter regions. Since tone affects the biomechanical properties, the aim of the present study was to evaluate intestinal SM tone by studying the morphometry and biomechanical properties with and without muscle tone. Circumferential rings of 0.8-1mm in width were cut from the rat duodenum, jejunum and ileum. Sectors were obtained by cutting the rings opposite to the mesentery. The rings and the sectors were immersed in physiological Krebs solution in order to maintain the tone and into Krebs solution without Ca(++) and with EGTA to abolish the tone. The circumferences, area, the circularity and residual strain of the mucosal and serosal surfaces, opening angle, and opening angle tone/non-tone ratio were measured or computed. The tone affects the opening angle and residual strain in the intestinal sectors. The opening angle in the tissue sectors with tone was smaller (P<0.05) than those without tone in all three segments. The opening angle tone/non-tone ratio was 0.40+/-0.05, 0.43+/-0.06 and 0.36+/-0.11 for duodenum, jejunum and ileum, respectively, and did not differ among the three intestinal segments. The residual strain between sectors with and without SM tone differed in duodenal and jejunal mucosa and in the serosa of all three segments (P<0.05). The intestinal rings with tone showed axial variation for luminal area (P<0.001), for wall area (P<0.05), and for the mucosal and serosal residual strains (P<0.05). In conclusion, the intestinal mechanical properties are affected by intestinal SM tone. The tone can be evaluated by measuring the opening angle and residual strains of sectors in intestinal segments with and without SM tone.     

Propagation of dissection in a residually-stressed artery model

This paper studies dissection propagation subject to internal pressure in a residually-stressed two-layer arterial model. The artery is assumed to be infinitely long, and the resultant plane strain problem is solved using the extended finite element method. The arterial layers are modelled using the anisotropic hyperelastic Holzapfel–Gasser–Ogden model, and the tissue damage due to tear propagation is described using a linear cohesive traction–separation law. Residual stress in the arterial wall is determined by an opening angle \(\alpha \) in a stress-free configuration. An initial tear is introduced within the artery which is subject to internal pressure. Quasi-static solutions are computed to determine the critical value of the pressure, at which the dissection starts to propagate. Our model shows that the dissection tends to propagate radially outwards. Interestingly, the critical pressure is higher for both very short and very long tears. The simulations also reveal that the inner wall buckles for longer tears, which is supported by clinical CT scans. In all simulated cases, the critical pressure is found to increase with the opening angle. In other words, residual stress acts to protect the artery against tear propagation. The effect of residual stress is more prominent when a tear is of intermediate length (\(\simeq \)90\(^\circ \) arc length). There is an intricate balance between tear length, wall buckling, fibre orientation, and residual stress that determines the tear propagation.     

Opening angles and residual strains in normal rat trachea

The no-load state and zero-stress state of the normal rat trachea were analyzed. It was found that there exist compressive residual strains in the inner wall region of the rat trachea and tensile residual strains in the outer wall region. The fact that the opening angle of the rat trachea cut at the cartilaginous region is significantly larger than that cut at the muscular portion shows that residual strains exist mainly in the muscular region in the rat trachea. It was also indicated that the opening angles and residual strains expressed by cutting at the muscular portion are basically identical along longitudinal location and those expressed by cutting in the cartilaginous region tend to increase in the longitudinal direction in the normal rat, and that there exists quantitatively positive correlation between the opening angles and residual strains in rat trachea. The results will help to further understand the opening angles and residual strains in the trachea and study tracheal remodeling in response to mechanical environment.     

Opening angles and residual strains in normal rat trachea

The no-load state and zero-stress state o1 the normal rat trachea were analyzed. It was found that there exist compressive residual strains in the inner wall region of the rat trachea and tensile residual strains in the outer wall region. The fact that the opening angle of the rat trachea cut at the cartilaginous region is significantly larger than that cut at the muscular portion shows that residual strains exist mainly in the muscular region in the rat trachea. It was also indicated that the opening angles and residual strains expressed by cutting at the muscular portion are basically identical along longitudinal location and those expressed by cutting in the cartilaginous region tend to increase in the longitudinal direction in the normal rat, and that there exists quantitatively positive correlation between the opening angles and residual strains in rat trachea. The results will help to further understand the opening angles and residual strains in the trachea and study tracheal remodeling in response     

Zero-stress states of human pulmonary arteries and veins

The zero-stressstates of the pulmonary arteries and veins fromorder3 toorder9 were determined in six normal humanlungs within 15 h postmortem. The zero-stress state of each vessel was obtained by cutting the vessel transversely into a series of short rings, then cutting each ring radially, which caused the ring to springopen into a sector. Each sector was characterized by its opening angle.The mean opening angle varied between 92 and 163° in the arterialtree and between 89 and 128° in the venous tree. There was atendency for opening angles to increase as the sizes of the arteriesand veins increased. We computed the residual strains based on theexperimental measurements and estimated the residual stresses accordingto Hooke's law. We found that the inner wall of a vessel at the statein which the internal pressure, external pressure, and longitudinalstress are all zero was under compression and the outer wall was intension, and that the magnitude of compressive stress was greater thanthe magnitude of tensile stress.

     

Stress-modulated growth, residual stress, and vascular heterogeneity.

A simple phenomenological model is used to study interrelations between material properties, growth-induced residual stresses, and opening angles in arteries. The artery is assumed to be a thick-walled tube composed of an orthotropic pseudoelastic material. In addition, the normal mature vessel is assumed to have uniform circumferential wall stress, which is achieved here via a mechanical growth law. Residual stresses are computed for three configurations: the unloaded intact artery, the artery after a single transmural cut, and the inner and outer rings of the artery created by combined radial and circumferential cuts. The results show that the magnitudes of the opening angles depend strongly on the heterogeneity of the material properties of the vessel wall and that multiple radial and circumferential cuts may be needed to relieve all residual stress. In addition, comparing computed opening angles with published experimental data for the bovine carotid artery suggests that the material properties change continuously across the vessel wall and that stress, not strain, correlates well with growth in arteries.     

Morphometry of the amount of smooth muscle cells in the media of various rabbit arteries

Our objective in this study was to evaluate the relative amount of smooth muscle cells in the medial layer of various rabbit arteries. The fixation of smooth muscle cells in the arterial wall is difficult and the differential effect of glutaraldehyde (GA) and fixative vehicle on cell ultrastructure in different tissues is controversial. We compared the effect of various concentrations of the vehicle and glutaraldehyde (osmolarity ranges for total fixative, 350-1030 mOsm) on the arterial wall ultrastructure. We found that a 600 mOsm GA solution (isotonic vehicle; 2.5% GA) adequately preserves arterial wall structures. The relative amount of smooth muscle cells in the media differed in various segments along the arterial tree. It ranged from 35% (thoracic aorta) to 74% (tibial artery). The importance of weighting the contractile response of different arteries in vitro to their relative smooth muscle cell content is discussed.     

Study on cartilaginous and muscular strains of rat trachea

Han and Fung (1991)[1] studied the zero-stressstates of porcine and canine tracheas by cutting themidpoints of cartilage and muscle respectively. Themethod of Fung, termed Once Cutting method in thispaper, was also used by Liu, Wang and Teng (2002)[2]in studying residual strain of rat tracheas. They all re-ported that the no-load state of trachea is not itszero-stress state, but the residual stress (strain) existsin no-load tracheal ring. The tracheal ring would openup into a figure of “C…     

Use of tensorial description in tissue remodeling: examples of F-actin distributions in pulmonary arteries in hypoxic hypertension

A molecular configuration tensor Pij was introduced to analyze the distribution of fibrous proteins in vascular cells for studying cells and tissues biomechanics. We have used this technique to study the biomechanics of vascular remodeling in response to the changes of blood pressure and flow. In this paper, the remodeling of the geometrical arrangement of F-actin fibers in the smooth muscle cells in rat's pulmonary arteries in hypoxic hypertension was studied. The rats were exposed to a hypoxia condition of 10% for 0, 2, 12, and 24 hr at sea level. Remodeling of blood vessels were studied at the in vivo state under normal perfusion, no-load state when small rings from blood vessels were excised, and zero-stress state after the rings were cut open radially to release the residual stress. Tissue remodeling in response to changes in blood pressure is reflected in the zero-stress state. The tensor components were determined by analyzing the configuration of phalloidin stained F-actin fibers in the media layer of pulmonary arteries. The values of P31, P32, P33 in the in-vivo state, the no-load state, and the zero-stress state are obtained. This study demonstrated the distributions of fibrous molecules in tissue remodeling can be described quantitatively using the molecular configuration tensor.     

Complex distributions of residual stress and strain in the mouse left ventricle: experimental and theoretical models

 Most soft biological tissues, including ventricular myocardium, are not stress free when all external loads are removed. Residual stress has implications for mechanical performance of the heart, and may be an indicator of patterns of regional growth and remodeling. Cross-sectional rings of arrested ventricles opened up when a radial cut was made (initial mean opening angles were 64 ± 17°), but further circumferential cuts revealed the presence of additional residual stresses in the tissue with further opening of the rings. In normal mouse hearts, the inner half of a short-axis ring opened more than the outer half, and this change was dependent on apex–base location. At the apex the inner section vs. outer section opening angles were 226 ± 47° vs. 89 ± 28°, while at the base the same two angles were 160 ± 30° vs. 123 ± 35°. A simple theoretical cylindrical shell model with incompressible hyperelastic material properties was used to model the experimental deformations based on the cutting experiments. The model predicts different residual stress fields depending on the nature of the opening after the circumferential cut (which is done after the conventional radial cut). The observed opening angles were consistent with steep stress gradients near the endocardium compared with those predicted if the first cut was assumed to relieve all residual stresses. These results imply a more complex distribution of residual stress and strain in ventricular myocardium than previously thought. Received: 23 May 2002 / Accepted: 30 September 2002 We would like to acknowledge the surgical skills and data analysis of Zuangjie Li. This work was supported in part by National Heart, Lung, and Blood Institute Grants HL-43026 and HL-64321.     

Study on cartilaginous and muscular strains of rat trachea

This paper introduces a new method, termed Twice Cutting, for obtaining the zero-stress states of cartilage and muscle of trachea. The method applied cuts at the two junctions of tracheal cartilage and muscle perpendicular to the tangent lines of cartilage at its tips. The cartilaginous and muscular opening angles are defined for the first time in Twice Cutting methods. Based on the analysis of cartilaginous and muscular geometric information in no-load and zero-stress states, it is found that there are compressive and tensile residual strains in the inner and outer walls of the cartilage respectively. Residual strains at the muscular inner wall of tracheal rings near bifurcation are negative, whereas those of other rings are positive, and residual strains at outer wall of all rings are positive. This phenomenon of tracheal muscle residual strains is different from those of vessel etc. The results also show that the absolute values of cartilaginous strains are considerably smaller than that of muscular ones, with the ratio being around 0.05. The values of all the tracheal parameters, including residual strains and opening angles, are reducing with the increasing value of tracheal rings’ position. So the consequences obtained in this paper not only indicate that the trachea is a non-uniform tissue along the circumferential and axial directions, but also reveal the differences between the trachea and other living tissues, such as vessel, esophagus. This is a basic research for further work, such as determining stress in trachea, to which the cartilaginous and muscular zero-stress states should be referred.     

Linked opening angle and histological and mechanical aspects of the proximal pulmonary arteries of healthy and pulmonary hypertensive rats and calves

Understanding how arterial remodeling changes the mechanical behavior of pulmonary arteries (PAs) is important to the evaluation of pulmonary vascular function. Early and current efforts have focused on the arteries' histological changes, their mechanical properties under in vitro mechanical testing, and their zero-stress and no-load states. However, the linkage between the histology and mechanical behavior is still not well understood. To explore this linkage, we investigated the geometry, residual stretch, and histology of proximal PAs in both adult rat and neonatal calf hypoxic models of pulmonary hypertension (PH), compared their changes due to chronic hypoxia across species, and proposed a two-layer mechanical model of artery to relate the opening angle to the stiffness ratio of the PA outer to inner layer. We found that the proximal PA remodeling in calves was quite different from that in rats. In rats, the arterial wall thickness, inner diameter, and outer layer thickness fraction all increased dramatically in PH and the opening angle decreased significantly, whereas in calves, only the arterial wall thickness increased in PH. The proposed model predicted that the stiffness ratio of the calf proximal PAs changed very little from control to hypertensive group, while the decrease of opening angle in rat proximal PAs in response to chronic hypoxia was approximately linear to the increase of the stiffness ratio. We conclude that the arterial remodeling in rat and calf proximal PAs is different and the change of opening angle can be linked to the change of the arterial histological structure and mechanics.