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.     

Regional distribution of layer-specific circumferential residual deformations and opening angles in the porcine aorta

Information on the layer-specific residual deformations of aortic tissue and how these vary throughout the vessel is important for understanding the regionally-varying aortic functions and pathophysiology, but not so much can be found in the literature. Toward this end, porcine aortas were sectioned into eighteen rings, with one ring from each anatomical position radially cut to obtain the zero-stress state for the intact wall and the other ring dissected into intimal-medial and adventitial layers; these rings were then radially cut to reach the zero-stress state for the intima-media and adventitia. Peripheral variations in internal/external circumferences, thickness, and opening angle of the intact wall and its layers were measured through image analysis at the no-load and zero-stress states. Intact wall and layer circumferences at both states significantly declined along the aorta, as did intact wall and intimal-medial but not adventitial thickness. Adventitia exhibited the greatest opening angles, approaching 180 deg all over the aorta. The opening angles of the intima-media and intact wall were quite similar, with the highest values in the ascending aorta, the lowest at the diaphragm, and increasing subsequently. Bending-related residual stretches were released by radial cutting that were compressive internally and tensile externally, displaying distinct axial variation for the intima-media and intact wall, and non-significant variation for the adventitia. Evidence is provided for the release upon layer separation of compressive stretches in the intima-media and of tensile stretches in the adventitia, whose values were smallest in the descending thoracic aorta and highest near the iliac artery bifurcation.     

Regional distribution of axial strain and circumferential residual strain in the layered rabbit oesophagus

The oesophagus is subjected to large axial strains in vivo and the zero-stress state is not a closed cylinder but an open circular cylindrical sector. The closed cylinder with no external loads applied is called the no-load state and residual strain is the difference in strain between the no-load state and zero-stress state. To understand oesophageal physiology and pathophysiology, it is necessary to know the distribution of axial strain, the zero-stress state, the stress-strain relations of oesophageal tissue, and the changes of these states and relationships due to biological remodeling of the tissue under stress. This study is addressed to such biomechanical properties in normal rabbits. The oesophagi were marked on the surface in vivo, photographed, excised (in vitro state), photographed again, and sectioned into rings (no-load state) in an organ bath containing calcium-free Kreb's solution with dextran and EGTA added. The rings were cut radially to obtain the zero-stress state for the non-separated wall and further dissected to separate the muscle and submucosa layers. Equilibrium was awaited for 30min in each state and the specimens were photographed in no-load and the zero-stress states. The oesophageal length, circumferences, layer thicknesses and areas, and openings angle were measured from the digitised images. The oesophagus shortened axially by 35% after excision. The in vivo axial strain showed a significant variation with the highest values in the mid-oesophagus (p<0.001). Luminal area, circumferences, and wall and layer thicknesses and areas varied in axial direction (in all tests p<0.05). The residual strain was compressive at the mucosal surface and tensile at the serosal surface. The dissection studies demonstrated shear forces between the two layers in the non-separated wall in the no-load and zero-stress states. In conclusion, our data show significant axial variation in passive morphometric and biomechanical properties of the oesophagus. The oesophagus is a layered composite structure with nonlinear and anisotropic mechanical behaviour.     

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…     

Determination of homeostatic elastic moduli in two layers of the esophagus

The function of the esophagus is mechanical. To understand the function, it is necessary to know how the stress and strain in the esophagus can be computed, and how to determine the stress-strain relationship of the wall materials. The present article is devoted to the issue of determining the incremental elastic moduli in the layers of the esophagus under homeostatic conditions. The esophagus is treated as a two-layered structure consisting of an inner collagen-rich submucosa layer and an outer muscle layer. We adopt a theory based on small perturbation experiments at homeostatic conditions for determination of incremental moduli in circumferential, axial, and cross directions in the two layers. The experiments are inflation, axial stretching, circumferential bending, and axial bending. The analysis takes advantage of knowing the esophageal zero-stress state (an open sector with an opening angle of 59.4 +/- 13.2 deg). The neutral axis was located 27% +/- 1.9%away from the mucosal surface. It is demonstrated that under homeostatic conditions, the incremental moduli are layer and direction dependent. The incremental modulus is the highest in the axial direction. Furthermore, the axial moduli for the two layers are similar, whereas in the circumferential direction, the incremental modulus is a factor of 6 higher in the mucosa-submucosa layer compared to the muscle layer. Hence, the esophagus has to be treated as a composite, anisotropic body. With this additional information, we can then look forward to a vision of truly understanding the mechanical events of the esophagus.     

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.     

The zero-stress state of rat veins and vena cava

The zero-stress state of a vein is, like that of an artery, not a closed cylindrical tube, but is a series of segments whose cross-sections are open sectors. An opening angle of each sector is defined as the angle subtended between two radii joining the midpoint of the inner wall to the tips of the inner wall. Data on the opening angles (mean +/- standard deviation) of the veins and vena cava of the rat are presented. For the superior vena cava and subclavian, jugular, facial, renal, common iliac, saphenous, and plantar veins, the opening angle varies in the range of 25 to 75 deg. The inferior vena cava (below the heart), however, has noncircular, nonaxisymmetric cross-sections, a curved axis, and a rapid longitudinal variation of its "diameter"; its zero-stress state is not circular sectors; but the opening angle is still a useful characterization. The mean opening angle of the interior vena cava varies in the range of 40 to 150 deg in the thoracic portion, and 75 to 130 deg in the abdominal portion, with the larger values occurring about the middle of each portion. There are considerable length, diameter reductions, and wall thickening of the vena cava from the homeostatic state to the no-load state in vitro. Physically, the zero-stress state is the basis of the stress analysis of blood vessels. The change of opening angle is a convenient parameter to characterize any nonuniform remodeling of the vessel wall due to changes in physical stress or chemical environment.(ABSTRACT TRUNCATED AT 250 WORDS)     

Strain-energy function and three-dimensional stress distribution in esophageal biomechanics

Knowledge of the transmural stress and stretch fields in esophageal wall is necessary to quantify growth and remodeling, and the response to mechanically based clinical interventions or traumatic injury, but there are currently conflicting reports on this issue and the mechanical properties of intact esophagus have not been rigorously addressed. This paper offers multiaxial data on rabbit esophagus, warranted for proper identification of the 3D mechanical properties. The Fung-type strain-energy function was adopted to model our data for esophagus, taken as a thick-walled (1 or 2-layer) tubular structure subjected to inflation and longitudinal extension. Accurate predictions of the pressure–radius–force data were obtained using the 1-layer model, covering a broad range of extensions; the calculated material parameters indicated that intact wall was equally stiff as mucosa–submucosa, but stiffer than muscle in both principal axes, and tissue was stiffer longitudinally, concurring our histological findings (Stavropoulou et al., Journal of Biomechanics. 42 (2009) 2654–2663). Employing the material parameters of individual layers, with reference to their zero-stress state, a reasonable fit was obtained to the data for intact wall, modeled as a 2-layer tissue. Different from the stress distributions presented hitherto in the esophagus literature, consideration of residual stresses led to less dramatic homogenization of stresses under loading. Comparison of the 1- and 2-layer models of esophagus demonstrated that heterogeneity induced a more uniform distribution of residual stresses in each layer, a discontinuity in circumferential and longitudinal stresses at the interface among layers, and a considerable rise of stresses in mucosa, with a reduction in muscle.     

Zero-stress state of intra- and extraparenchymal airways from human, pig, rabbit, and sheep lung.

Alterations in airway wall anatomic properties and the consequential effects on airway narrowing have been assessed by use of computational models. In these models, it is generally assumed that at zero transmural pressure the airway wall exists in a zero-stress state. Many studies have shown that this is often not the case, as evidenced by a nonzero opening angle. In this study, we measured the opening angle of airway rings at zero transmural pressure to test this assumption. The airway tree was dissected from human, pig, sheep, and rabbit lungs. Airways were excised from the tree, and the opening angle was measured. There were obvious species and regional differences in opening angle. Rabbit airways from both extraparenchymal and intraparenchymal sites exhibited marked opening angles (7-82 degrees). Extraparenchymal airways from sheep had large opening angles (up to 50 degrees), but ovine intraparenchymal airways had small opening angles. Measurable opening angles were rarely observed in human and porcine airways of any size. The assumption of a stable zero-stress state at zero transmural pressure is therefore valid for human and porcine, but not rabbit and sheep, airways.     

Effect of Danshen on the Zero-Stress State of Rat's Abdominal Aorta

The objective of our study was to study the effect of danshen, a Chinese herbal medicine known to prevent hypertension, on the zero-stress state of rat's abdominal aorta. The zero-stress state of a blood vessel represents the release of residual stress on the vessel wall, and is the basic configuration of blood vessel affected solely by intrinsic parameters. At the in vivo state, the rat's abdominal aorta was subjected to blood pressure and flow and longitudinal stress. After dissecting from the abdominal aorta, the aortic specimens were cut into small rings at no-load state, in which the internal pressure, external pressure, and longitudinal stress in a short ring-shaped segment were all zero; by cutting radially to release the residual stress in the wall, the vessel ring opened up into a sector quickly, and the sector's configuration would not change at 20 min after cutting and was defined as the zero-stress state of a blood vessel, which was characterized by its residual strain and opening angle. Then aqueous extract of danshen prepared with methanol was added in the Krebs solution, and the changes of the aorta's zero-stress state were monitored by taking photos routinely for analysis to determine the opening angle and residual strain. Additionally, other sets of samples were tested in a Norepinephrine-Krebs solution as positive control or a Krebs solution as negative control, respectively. It was demonstrated that the zero-stress state of rat's abdominal aorta was affected by danshen extract and norepinephrine in two different patterns, while the Krebs solution did not have similar effects. The present work provides a new approach to study the anti-hypertension effect and mechanism of danshen.     

Duration of no-load state affects opening angle of porcine coronary arteries

The zero-stress state of a blood vessel has been extensively studied because it is the reference state for which all calculations of intramural stress and strain must be based. It has also been found to reflect nonuniformity in growth and remodeling in response to chemical or physical changes. The zero-stress state can be characterized by an opening angle, defined as the angle subtended by two radii connecting the midpoint of the inner wall. All prior studies documented the zero-stress state or opening angle with no regard to duration of the no-load state. Our hypotheses were that, given the viscoelastic properties of blood vessels, the zero-stress state may have "memory" of prior circumferential and axial loading, i.e., duration of the no-load state influences opening angle. To test these hypotheses, we considered ring pairs of porcine coronary arteries to examine the effect of duration in the no-load state after circumferential distension. Our results show a significant reduction in opening angle as duration of the no-load state increases, i.e., vessels that are reduced to the zero-stress state directly from the loaded state attain much larger opening angles at 30 min after the radial cut than rings that are in the no-load state for various durations. To examine the effect of axial loading, we found similar reductions in opening angle with duration in the no-load from the in situ state, albeit the effect was significantly smaller than that of circumferential loading. Hence, we found that the zero-stress state has memory of both circumferential and axial loading. These results are important for understanding viscoelastic properties of coronary arteries, interpretation of the enormous data on the opening angle and strain in the literature, and standardization of future measurements on the zero-stress state.     

Biomechanical and morphological properties in rat large intestine

Intestinal stress-strain distributions are important determinants of intestinal function and are determined by the mechanical properties of the intestinal wall, the physiological loading conditions and the zero-stress state of the intestine. In this study the distribution of morphometric measures, residual circumferential strains and stress-strain relationships along the rat large intestine were determined in vitro. Segments from four parts of the large intestine were excised, closed at both ends, and inflated with pressures up to 2kPa. The outer diameter and length were measured. The zero-stress state was obtained by cutting rings of large intestine radially. The geometric configuration at the zero-stress state is of fundamental importance because it is the basic state with respect to which the physical stresses and strains are defined. The outer and inner circumferences, wall thickness and opening angle were measured from digitised images. Subsequently, residual strain and stress-strain distributions were calculated. The wall thickness and wall thickness-to-circumference ratio increased in the distal direction. The opening angle varied between approximately 40 and approximately 125 degrees with the highest values in the beginning of proximal colon (F=1.739, P<0.05). The residual strain at the inner surface was negative indicating that the mucosa-submucosal layers of the large intestine in no-load state are in compression. The four segments showed stress-strain distributions that were exponential. All segments were stiffer in longitudinal direction than in the circumferential direction (P<0.05). The transverse colon seemed stiffest both in the circumferential and longitudinal directions. In conclusion, significant variations were found in morphometric and biomechanical properties along the large intestine. The circumferential residual strains and passive elastic properties must be taken into account in studies of physiological problems in which the stress and strain are important, e.g. large intestinal bolus transport function.     

A two-layered mechanical model of the rat esophagus. Experiment and theory

Background  

The function of esophagus is to move food by peristaltic motion which is the result of the interaction of the tissue forces in the esophageal wall and the hydrodynamic forces in the food bolus. The structure of the esophagus is layered. In this paper, the esophagus is treated as a two-layered structure consisting of an inner collagen-rich submucosa layer and an outer muscle layer. We developed a model and experimental setup for determination of elastic moduli in the two layers in circumferential direction and related the measured elastic modulus of the intact esophagus to the elastic modulus computed from the elastic moduli of the two layers.     

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     

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.     

Morphology and stress-strain properties along the small intestine in the rat

The stress-strain relationship is determined by the inherent mechanical properties of the intestinal wall, the geometric configurations, the loading conditions and the zero-stress state of the segment. The purpose of this project was to provide morphometric and biomechanical data for rat duodenum, jejunum and ileum. The circumferential strains were referenced to the zero-stress state. Large morphometric variations were found along the small intestine with an increase in the outer circumferential length and luminal area and a decrease in wall thickness in distal direction. The serosal residual strain was tensile and decreased in distal direction (P < 0.05). The mucosal residual strain was compressive and the absolute value decreased in distal direction (P < 0.001). The stress-strain experiments showed that the duodenum was stiffest. All segments were stiffest in longitudinal direction (P < 0.05). In conclusion, axial variation in morphometric and biomechanical properties was found in the small intestine. The zero-stress state must be considered in future biomechanical studies in the gastrointestinal tract.     

Zero-stress states of arteries

The no-load configuration of a living organ is, in general, not the zero-stress state. The difference can be revealed by cutting up an unloaded organ to such an extent that the stress becomes zero in the tissue everywhere. For the aorta, it is shown that the configuration of the zero-stress state differs considerably from being a cylindrical tube. It is, in fact, an open sector with opening angles varying along the arterial tree. This article presents data on the zero-stress state in the arteries of the rat in normal condition.     

Effect of osmolarity on the zero-stress state and mechanical properties of aorta

Some pathological conditions may affect osmolarity, which can impact cell, tissue, and organ volume. The hypothesis of this study is that changes in osmolarity affect the zero-stress state and mechanical properties of the aorta. To test this hypothesis, a segment of mouse abdominal aorta was cannulated in vivo and mechanically distended by perfusion of physiological salt (NaCl) solutions with graded osmolarities from 145 to 562 mosM. The mechanical (circumferential stress, strain, and elastic modulus) and morphological (wall thickness and wall area) parameters in the loaded state were determined. To determine the osmolarity-induced changes of zero-stress state, the opening angle was observed by immersion of the sectors of mouse, rat, and pig thoracic aorta in NaCl solution with different osmolarities. Wall volume and tissue water content of the rings were also recorded at different osmolarities. Our results show that acute aortic swelling due to low osmolarity leads to an increase in wall thickness and area, a change in the stress-strain relationship, and an increase in the elastic modulus (stiffness) in mouse aorta. The opening angle, wall volume, and water content decreased significantly with increase in osmolarity. These findings suggest that acute aortic swelling and shrinking result in immediate mechanical changes in the aorta. Osmotic pressure-induced changes in the zero-stress state may serve to regulate mechanical homeostasis.     

Biomechanical remodelling of obstructed guinea pig jejunum

Data on morphological and biomechanical remodelling are needed to understand the mechanisms behind intestinal obstruction. The effect of partial obstruction on mechanical properties with reference to the zero-stress state and on the histomorphological properties of the guinea pig small intestine was determined in this study. Partial obstruction and sham operation were surgically created in mid-jejunum of guinea pigs. The animals survived 2, 4, 7, and 14 days. The age-matched guinea pigs that were not operated served as normal controls. The segment proximal to the obstruction site was used for histological analysis, no-load state and zero-stress state data, and distension test. The segment for distension was immersed in an organ bath and inflated to 10 cm H₂O. The outer diameter change during the inflation was monitored using a microscope with CCD camera. Circumferential stresses and strains were computed from the diameter, pressure and the zero-stress state data. The opening angle and absolute value of residual strain decreased (P<0.01 and P<0.001) whereas the wall thickness, wall cross-sectional area, and the wall stiffness increased after 7 days obstruction (P<0.05, P<0.01). Histologically, the muscle and submucosa layers, especially the circumferential muscle layer increased in thickness after obstruction. The opening angle and residual strain mainly depended on the thickness of the muscle layer whereas the wall stiffness mainly depended on the thickness of the submucosa layer. In conclusion, the histomorphological and biomechanical properties of small intestine (referenced for the first time to the zero-stress state) remodel proximal to the obstruction site in a time-dependent manner.