Biaxial incremental homeostatic elastic moduli of coronary artery: two-layer model

The detailed mechanical properties of various layers of the coronary artery are important for understanding the function of the vessel. The present article is focused on the determination of the incremental modulus in different layers and directions in the neighborhood of the in vivo state. The incremental modulus can be defined for any material subjected to a large deformation if small perturbations in strain lead to small perturbations of stresses in a linear fashion. This analysis was applied to the porcine coronary artery, which was treated as a two-layered structure consisting of an inner intima-media layer and an outer adventitia layer. We adopted 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 were based on inflation and axial stretch. We demonstrate that under homeostatic conditions the incremental moduli are layer- and direction dependent. The incremental modulus is highest in the circumferential direction. Furthermore, in the circumferential direction, the media is stiffer than the whole wall, which is stiffer than the adventitia. In the axial direction, the adventitia is stiffer than the intact wall, which is stiffer than the media. Hence, the coronary artery must be treated as a composite, nonisotropic body. The data acquire physiological relevance in relation to coronary artery health and disease.     

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.     

Analysis of the passive mechanical properties of rat carotid arteries

The passive mechanical properties of rat carotid arteries were studied in vitro. Using a tensile testing machine and a piston pump, intact segments of carotid arteries were subjected to large deformations both in the longitudinal and circumferential directions. Internal pressure, external diameter, length and longitudinal force were measured during the experiment and compared with the in vivo dimensions of the segments prior to excision. The anisotropic mechanical properties of the vessel wall material were analyzed using incremental elastic moduli and incremental Poisson's ratios. The results suggest that there is a characteristic deformation pattern common to all vessels investigated which is highly correlated with the conditions of loading that occur in vivo. That is, under average physiological deformation of the vessel, the longitudinal force is nearly independent of internal pressure. In this range of loading the circumferential incremental elastic modulus is nearly independent of longitudinal strain. However, the longitudinal and radial incremental elastic moduli vary significantly with deformation in this direction. The values of the moduli in all three directions increase with raising internal pressure. The weak coupling between circumferential and longitudinal direction in the wall material of carotid arteries is shown by the small value of the corresponding incremental Poisson's ratios.     

Shear modulus of porcine coronary artery: contributions of media and adventitia

The epicardial coronary arteries experience significant torsion in the axial direction due to changes in the shape of the heart during the cardiac cycle. The objective of this study was to determine the torsional mechanical properties of the coronary arteries under various circumferential and longitudinal loadings. The coronary artery was treated as a two-layer composite vessel consisting of intima-medial and adventitial layers, and the shear modulus of each layer was determined. Eight porcine hearts were obtained at a local abattoir, and their right coronary and left anterior descending arteries were isolated and tested in vitro with a triaxial torsion machine (inflation, longitudinal stretch, and circumferential twist). After the intact vessel was tested, the adventitia was dissected away, leaving an intact media that was then tested under identical triaxial loading conditions. We proposed a biomechanical analysis to compute the shear modulus of the adventitia from the measured shear moduli of the intact vessel and the media. To validate our predictions, we used four additional hearts in which the shear modulus of the adventitia was measured after dissection of media. Our results show that the shear modulus does not depend on the shear stress or strain but varies linearly with circumferential and longitudinal stresses and in a nonlinear way with the corresponding strains. Furthermore, we found that the shear modulus of the adventitia is larger than that of the intact vessel, which is larger than the vessel media. These results may have important implications for baroreceptor sensitivity, circulation of the vasa vasorum, and coronary dissection.     

Biomechanical properties of the layered oesophagus and its remodelling in experimental type-1 diabetes

Passive biomechanical properties in term of the stress-strain relationship and the shear modulus were studied in separated muscle layer and mucosa-submucosa layer in the oesophagus of normal and STZ (streptozotocin)-induced diabetic rats. The mucosa-submucosa and muscle layers were separated using microsurgery and studied in vitro using a self-developed test machine. Stepwise elongation and inflation plus continuous twist were applied to the samples. A constitutive equation based on a strain energy function was used for the stress-strain analysis. Five material constants were obtained for both layers. The mucosa-submucosa layer was significantly stiffer than the muscle layer in longitudinal, circumferential and circumferential-longitudinal shear direction. The mechanical constants of the oesophagus show that the oesophageal wall was anisotropic, the stiffness in the longitudinal direction was higher than in the circumferential direction in the intact oesophagus (P < 0.001) and in the muscle layer (P < 0.05). Diabetes-induced pronounced increase in the outer perimeter, inner perimeter and lumen area in both the muscle and mucosa-submucosa layer. The growth of the mucosa-submucosa layer (P < 0.001) was more pronounced than the muscle layer (P < 0.05). Furthermore, the circumferential stiffness of the mucosa-submucosa layer increased 28 days after STZ treatment. In conclusion, the oesophagus is a non-homogeneous anisotropic tube. Thus, the mechanical properties differed between layers as well as in different directions. Morphological and biomechanical remodelling is prominent in the diabetic oesophagus.     

Passive elastic properties of the rat aorta

The passive anisotropic elastic properties of rat's aorta were studied in vitro by subjecting cylindrical segments of thoracic and abdominal aorta to a wide range of deformations. Using data on pressure, axial stretch, outer diameter, axial force and wall thickness, incremental moduli of elasticity in the circumferential, axial and radial directions were computed. Results indicate that while the elastic behavior of the aortic wall is globally anisotropic, there exists a state of deformation at which the vessel displays incremental isotropy. This state of deformation corresponds approximately to the loading conditions to which the aorta is exposed in situ. Values of the moduli, analyzed as a function of transmural pressure, show that the stiffness of the aortic wall is fairly constant at low pressures but raises steeply for pressures higher than physiological. For axial stretches as occurring in situ, the magnitudes of the circumferential and radial moduli do not differ significantly for the thoracic aorta; hence this vessel can be regarded as transversely isotropic over a wide range of pressures. The same observation is valid also for the abdominal aorta when pressures equal or smaller than physiological are considered. For both the thoracic and abdominal segments of the aorta, the circumferential and radial moduli are smaller than the axial modulus at low pressures, while the reverse is true for large pressures.     

Molecular modeling of the axial and circumferential elastic moduli of tubulin

Microtubules play a number of important mechanical roles in almost all cell types in nearly all major phylogenetic trees. We have used a molecular mechanics approach to perform tensile tests on individual tubulin monomers and determined values for the axial and circumferential moduli for all currently known complete sequences. The axial elastic moduli, in vacuo, were found to be 1.25 GPa and 1.34 GPa for α- and β-bovine tubulin monomers. In the circumferential direction, these moduli were 378 MPa for α- and 460 MPa for β-structures. Using bovine tubulin as a template, 269 homologous tubulin structures were also subjected to simulated tensile loads yielding an average axial elastic modulus of 1.10 ± 0.14 GPa for α-tubulin structures and 1.39 ± 0.68 GPa for β-tubulin. Circumferentially the α- and β-moduli were 936 ± 216 MPa and 658 ± 134 MPa, respectively. Our primary finding is that that the axial elastic modulus of tubulin diminishes as the length of the monomer increases. However, in the circumferential direction, no correlation exists. These predicted anisotropies and scale dependencies may assist in interpreting the macroscale behavior of microtubules during mitosis or cell growth. Additionally, an intergenomic approach to investigating the mechanical properties of proteins may provide a way to elucidate the evolutionary mechanical constraints imposed by nature upon individual subcellular components.     

Pipette aspiration technique for the measurement of nonlinear and anisotropic mechanical properties of blood vessel walls under biaxial stretch

A pipette aspiration technique was proposed for the measurement of nonlinear mechanical properties of arteries under biaxial stretching. A cross-shaped specimen of porcine thoracic aorta whose principal axes corresponded with the axial and circumferential directions of the aortic walls was excised. The intraluminal surface of the specimen was aspirated with a circular cross-sectioned glass pipette while the specimen was stretching in the axial and circumferential directions in 10% increments. The elastic modulus agreed with the incremental elastic modulus obtained through a conventional pressure-diameter test of the same specimen to within an error of 30% at a circumferential stretch ratio below 1.3 and an axial stretch ratio of 1.0, 1.1 or 1.2, which represent lower range of physiological stretch ratios for the porcine aorta. A rectangular cross-sectioned pipette was utilized to measure anisotropic properties of the specimen under biaxial stretching. When aspirated with such a pipette, the specimens' elastic properties along the length of the rectangular pipette cross section can be neglected. The elastic modulus was found to increase rapidly when the specimen was stretched in the direction of the pipette's width. Thus, pipette aspiration should have many advantages such as well measurement of the local nonlinear and anisotropic mechanical properties of blood vessel walls.     

Determination of layer-specific mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling

At autopsy, 13 nonstenotic human left anterior descending coronary arteries [71.5 +/- 7.3 (mean +/- SD) yr old] were harvested, and related anamnesis was documented. Preconditioned prepared strips (n = 78) of segments from the midregion of the left anterior descending coronary artery from the individual layers in axial and circumferential directions were subjected to cyclic quasi-static uniaxial tension tests, and ultimate tensile stresses and stretches were documented. The ratio of outer diameter to total wall thickness was 0.189 +/- 0.014; ratios of adventitia, media, and intima thickness to total wall thickness were 0.4 +/- 0.03, 0.36 +/- 0.03, and 0.27 +/- 0.02, respectively; axial in situ stretch of 1.044 +/- 0.06 decreased with age. Stress-stretch responses for the individual tissues showed pronounced mechanical heterogeneity. The intima is the stiffest layer over the whole deformation domain, whereas the media in the longitudinal direction is the softest. All specimens exhibited small hysteresis and anisotropic and strong nonlinear behavior in both loading directions. The media and intima showed similar ultimate tensile stresses, which are on average three times smaller than ultimate tensile stresses in the adventitia (1,430 +/- 604 kPa circumferential and 1,300 +/- 692 kPa longitudinal). The ultimate tensile stretches are similar for all tissue layers. A recently proposed constitutive model was extended and used to represent the deformation behavior for each tissue type over the entire loading range. The study showed the need to model nonstenotic human coronary arteries with nonatherosclerotic intimal thickening as a composite structure composed of three solid mechanically relevant layers with different mechanical properties. The intima showed significant thickness, load-bearing capacity, and mechanical strength compared with the media and adventitia.     

Anisotropic and inhomogeneous tensile behavior of the human anulus fibrosus: experimental measurement and material model predictions

The anulus fibrosus (AF) of the intervertebral disc exhibits spatial variations in structure and composition that give rise to both anisotropy and inhomogeneity in its material behaviors in tension. In this study, the tensile moduli and Poisson's ratios were measured in samples of human AF along circumferential, axial, and radial directions at inner and outer sites. There was evidence of significant inhomogeneity in the linear-region circumferential tensile modulus (17.4+/-14.3 MPa versus 5.6+/-4.7 MPa, outer versus inner sites) and the Poisson's ratio v21 (0.67+/-0.22 versus 1.6+/-0.7, outer versus inner), but not in the axial modulus (0.8+/-0.9 MPa) or the Poisson's ratios V12 (1.8+/-1.4) or v13 (0.6+/-0.7). These properties were implemented in a linear an isotropic material model of the AF to determine a complete set of model properties and to predict material behaviors for the AF under idealized kinematic states. These predictions demonstrate that interactions between fiber populations in the multilamellae AF significantly contribute to the material behavior, suggesting that a model for th     

Mechanical properties of the lateral cortex of mammalian auditory outer hair cells.

Mammalian auditory outer hair cells generate high-frequency mechanical forces that enhance sound-induced displacements of the basilar membrane within the inner ear. It has been proposed that the resulting cell deformation is directed along the longitudinal axis of the cell by the cortical cytoskeleton. We have tested this proposal by making direct mechanical measurements on outer hair cells. The resultant stiffness modulus along the axis of whole dissociated cells was 3 x 10(-3) N/m, consistent with previously published values. The resultant axial and circumferential stiffness moduli for the cortical lattice were 5 x 10(-4) N/m and 3 x 10(-3) N/m, respectively. Thus the cortical lattice is a highly orthotropic structure. Its axial stiffness is small compared with that of the intact cell, but its circumferential stiffness is within the same order of magnitude. These measurements support the theory that the cortical cytoskeleton directs electrically driven length changes along the longitudinal axis of the cell. The Young's modulus of the circumferential filamentous components of the lattice were calculated to be 1 x 10(7) N/m2. The axial cross-links, believed to be a form of spectrin, were calculated to have a Young's modulus of 3 x 10(6) N/m2. Based on the measured values for the lattice and intact cell cortex, an estimate for the resultant stiffness modulus of the plasma membrane was estimated to be on the order of 10(-3) N/m. Thus, the plasma membrane appears to be relatively stiff and may be the dominant contributor to the axial stiffness of the intact cell.     

A comparison of uniaxial and biaxial mechanical properties of the annulus fibrosus: a porcine model

The annulus fibrosus of the intervertebral disk experiences multidirectional tension in vivo, yet the majority of mechanical property testing has been uniaxial. Therefore, our understanding of how this complex multilayered tissue responds to loading may be deficient. This study aimed to determine the mechanical properties of porcine annular samples under uniaxial and biaxial tensile loading. Two-layer annulus samples were isolated from porcine disks from four locations: anterior superficial, anterior deep, posterior superficial, and posterior deep. These tissues were then subjected to three deformation conditions each to a maximal stretch ratio of 1.23: uniaxial, constrained uniaxial, and biaxial. Uniaxial deformation was applied in the circumferential direction, while biaxial deformation was applied simultaneously in the circumferential and compressive directions. Constrained uniaxial consisted of a stretch ratio of 1.23 in the circumferential direction while holding the tissue stationary in the axial direction. The maximal stress and stress-stretch ratio (S-S) moduli determined from the biaxial tests were significantly higher than those observed during both the uniaxial tests (maximal stress, 97.1% higher during biaxial; p=0.002; S-S moduli, 117.9% higher during biaxial; p=0.0004) and the constrained uniaxial tests (maximal stress, 46.8% higher during biaxial; S-S moduli, 82.9% higher during biaxial). These findings suggest that the annulus is subjected to higher stresses in vivo when under multidirectional tension.     

Effect of fiber orientation and strain rate on the nonlinear uniaxial tensile material properties of tendon

Tendons are exposed to complex loading scenarios that can only be quantified by mathematical models, requiring a full knowledge of tendon mechanical properties. This study measured the anisotropic, nonlinear, elastic material properties of tendon. Previous studies have primarily used constant strain-rate tensile tests to determine elastic modulus in the fiber direction. Data for Poisson's ratio aligned with the fiber direction and all material properties transverse to the fiber direction are sparse. Additionally, it is not known whether quasi-static constant strain-rate tests represent equilibrium elastic tissue behavior. Incremental stress-relaxation and constant strain-rate tensile tests were performed on sheep flexor tendon samples aligned with the tendon fiber direction or transverse to the fiber direction to determine the anisotropic properties of toe-region modulus (E0), linear-region modulus (E), and Poisson's ratio (v). Among the modulus values calculated, only fiber-aligned linear-region modulus (E1) was found to be strain-rate dependent. The E1 calculated from the constant strain-rate tests were significantly greater than the value calculated from incremental stress-relaxation testing. Fiber-aligned toe-region modulus (E(1)0 = 10.5 +/- 4.7 MPa) and linear-region modulus (E1 = 34.0 +/- 15.5 MPa) were consistently 2 orders of magnitude greater than transverse moduli (E(2)0 = 0.055 +/- 0.044 MPa, E2 = 0.157 +/- 0.154 MPa). Poisson's ratio values were not found to be rate-dependent in either the fiber-aligned (v12 = 2.98 +/- 2.59, n = 24) or transverse (v21 = 0.488 +/- 0.653, n = 22) directions, and average Poisson's ratio values in the fiber-aligned direction were six times greater than in the transverse direction. The lack of strain-rate dependence of transverse properties demonstrates that slow constant strain-rate tests represent elastic properties in the transverse direction. However, the strain-rate dependence demonstrated by the fiber-aligned linear-region modulus suggests that incremental stress-relaxation tests are necessary to determine the equilibrium elastic properties of tendon, and may be more appropriate for determining the properties to be used in elastic mathematical models.     

Anisotropic material behaviours of soft tissues in human trachea: an experimental study

BackgroundHuman trachea is a multi-component structure composed of cartilage, trachealis muscle, mucosa and submucosa membrane and adventitial membrane. Its mechanical properties are essential for an accurate prediction of tracheal deformation, which has a significant clinic relevance. Efforts have been made in quantifying the material behaviour of tracheal cartilage and trachealis muscle. However, the material behaviours of other components have been least investigated.MethodsThree human cadaveric trachea specimens were used in this study. Trachealis muscle, mucosa and submucosa membrane and adventitia membrane were excised to perform the uniaxial test in axial and circumferential directions. In total, 72 tissue strips were prepared and tested. Tangent modulus was used to quantified the stiffness of each tissue strip at various stretch levels.ResultsThe obtained results indicated that all types of tracheal soft tissues were highly non-linear and anisotropic. Trachealis muscle in the circumferential direction had the most excellent extensibility; and the adventitial collagen membrane in the circumferential direction was the stiffest.ConclusionThis study is helpful in understanding the material behaviour of trachea. Obtained results can be used for computational and analytic modelling to quantify the tracheal deformation.     

Determination of the axial and circumferential mechanical properties of the skin tissue using experimental testing and constitutive modeling

The skin, being a multi-layered material, is responsible for protecting the human body from the mechanical, bacterial, and viral insults. The skin tissue may display different mechanical properties according to the anatomical locations of a body. However, these mechanical properties in different anatomical regions and at different loading directions (axial and circumferential) of the mice body to date have not been determined. In this study, the axial and circumferential loads were imposed on the mice skin samples. The elastic modulus and maximum stress of the skin tissues were measured before the failure occurred. The nonlinear mechanical behavior of the skin tissues was also computationally investigated through a suitable constitutive equation. Hyperelastic material model was calibrated using the experimental data. Regardless of the anatomic locations of the mice body, the results revealed significantly different mechanical properties in the axial and circumferential directions and, consequently, the mice skin tissue behaves like a pure anisotropic material. The highest elastic modulus was observed in the back skin under the circumferential direction (6.67 MPa), while the lowest one was seen in the abdomen skin under circumferential loading (0.80 MPa). The Ogden material model was narrowly captured the nonlinear mechanical response of the skin at different loading directions. The results help to understand the isotropic/anisotropic mechanical behavior of the skin tissue at different anatomical locations. They also have implications for a diversity of disciplines, i.e., dermatology, cosmetics industry, clinical decision making, and clinical intervention.     

Variability in the muscle composition of rat esophagus and neural pathway of lower esophageal sphincter relaxation

Several studies from our laboratory show that axial stretch of the lower esophageal sphincter (LES) in an oral direction causes neurally mediated LES relaxation. Under physiological conditions, axial stretch of the LES is caused by longitudinal muscle contraction (LMC) of the esophagus. Because longitudinal muscle is composed of skeletal muscle in mice, vagal-induced LMC and LES relaxation are both blocked by pancuronium. We conducted studies in rats (thought to have skeletal muscle esophagus) to determine if vagus nerve-mediated LES relaxation is also blocked by pancuronium. LMC-mediated axial stretch on the LES was monitored using piezoelectric crystals. LES and esophageal pressures were monitored with a 2.5-Fr solid-state pressure transducer catheter. Following bilateral cervical vagotomy, the vagus nerve was stimulated electrically. LES, along with the esophagus, was harvested after in vivo experiments and immunostained for smooth muscle (smooth muscle α-actin) and skeletal muscle (fast myosin heavy chain). Vagus nerve-stimulated LES relaxation and esophageal LMC were reduced in a dose-dependent fashion and completely abolished by pancuronium (96 μg/kg) in six rats (group 1). On the other hand, in seven rats, LES relaxation and LMC were only blocked completely by a combination of pancuronium (group 2) and hexamethonium. Immunostaining revealed that the longitudinal muscle layer was composed of predominantly skeletal muscle in the group 1 rats. On the other hand, the longitudinal muscle layer of group 2 rats contained a significant amount of smooth muscle (P < 0.05). Our study shows tight coupling between axial stretch on the LES and relaxation of the LES, which suggests a cause and effect relationship between the two. We propose that the vagus nerve fibers that cause LMC induce LES relaxation through the stretch-sensitive activation of inhibitory motor neurons.     

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.     

Connecting incremental shear modulus and Poisson's ratio of lung tissue with morphology and rheology of microstructure

The width and curvature of the collagen and elastin fiber bundles in the human pulmonary interalveolar septa and alveolar mouths are measured. The data, together with the known mechanical properties of collagen and elastin fibers, are used to derive the incremental elastic moduli of the lung tissue. The constitutive equation for small incremental stress and strain superposed on a homeostatic inflated lung is linear and isotropic, and characterized by two material constants.     

Three-dimensional stress and strain distribution in a two-layer model of a coronary artery

For a right coronary artery, three-dimensional stress and strain distributions at a physiological intraluminal pressure and an axial extension ratio were computed on the basis of a two-layer elastic model. To validate the model, curves of external radius versus pressure and of axial force versus pressure were computed for three axial extension ratios. To analyze mechanical properties, stress-free configurations of media and adventitia, and the constitutive law of each layer in literature, were used. The present study showed that the peak circumferential stress and the peak axial stress appear in the media at the boundary between the media and adventitia. This result is due to the opening angle of the media being larger than π (rad) and the larger value of a material constant of the strain energy function for the media than for the adventitia. The circumferential stress and strain were discontinuous at the boundary. On the other hand, the radial stress was continuous at the boundary because of the boundary condition for stress. The circumferential stress and axial stress in the adventitia were almost uniformly distributed, and smaller than in the media. The residual stress and strain were also computed. The circumferential residual stress and strain were almost linearly distributed in each layer, although discontinuity appeared at the boundary between the two layers.