Viscoelastic properties of the passive mechanical behavior of the porcine carotid artery: Influence of proximal and distal positions

The viscoelastic properties of porcine carotid tissue are investigated in this work. Experimental uniaxial stress relaxation tests along the longitudinal and circumferential directions of the vessel were performed for carotid strips extracted from 10 vessels. Directional and local differences - distal versus proximal position - in the tissue behavior were investigated. The experimental tests reveal a highly anisotropic, non-linear viscoelastic response and local dependence of the samples. The carotid artery shows anisotropic relaxation behavior for both proximal and distal samples. The highest stress relaxation was found in the circumferential tensile test for the highest applied strain at the distal position. For the circumferential direction, the relaxation stress was higher than in the longitudinal being at its highest in the distal position. These facts show that the stress relaxation is higher in the distal than in the proximal position. However, there are no differences between both positions in the longitudinal direction. In addition, a constitutive law that takes into account the fundamental features, including non-linear viscoelasticity, of the arterial tissue is proposed. The present results are correlated with the purely elastic response and the microstructural analysis of the tissue by means of histological quantification presented in a previous study.     

Passive viscoelastic properties of the rabbit aorta in vitro

Tensile tests on longitudinal and circumferential strips of the rabbit aorta have been performed. Stress-strain and relaxation parameters have been estimated with respect to four stress levels and three positions on the aorta. Stress-strain data indicate that in longitudinal direction the aorta becomes more compliant with distance from the aortic arch. The opposite tendency is found for the circumferential direction. Stress-relaxation is found to be strongly dependent on the stress level. The results are discussed with regard to arterial dynamics.     

Multiphoton microscopy observations of 3D elastin and collagen fiber microstructure changes during pressurization in aortic media

Elastin and collagen fibers play important roles in the mechanical properties of aortic media. Because knowledge of local fiber structures is required for detailed analysis of blood vessel wall mechanics, we investigated 3D microstructures of elastin and collagen fibers in thoracic aortas and monitored changes during pressurization. Using multiphoton microscopy, autofluorescence images from elastin and second harmonic generation signals from collagen were acquired in media from rabbit thoracic aortas that were stretched biaxially to restore physiological dimensions. Both elastin and collagen fibers were observed in all longitudinal–circumferential plane images, whereas alternate bright and dark layers were observed along the radial direction and were recognized as elastic laminas (ELs) and smooth muscle-rich layers (SMLs), respectively. Elastin and collagen fibers are mainly oriented in the circumferential direction, and waviness of collagen fibers was significantly higher than that of elastin fibers. Collagen fibers were more undulated in longitudinal than in radial direction, whereas undulation of elastin fibers was equibiaxial. Changes in waviness of collagen fibers during pressurization were then evaluated using 2-dimensional fast Fourier transform in mouse aortas, and indices of waviness of collagen fibers decreased with increases in intraluminal pressure. These indices also showed that collagen fibers in SMLs became straight at lower intraluminal pressures than those in EL, indicating that SMLs stretched more than ELs. These results indicate that deformation of the aorta due to pressurization is complicated because of the heterogeneity of tissue layers and differences in elastic properties of ELs, SMLs, and surrounding collagen and elastin.     

Mechanical properties of the tracheal mucosal membrane in the rabbit. II. Morphometric analysis.

Folding of the airway mucosal membrane provides a mechanical load that impedes airway smooth muscle contraction. Mechanical testing of rabbit tracheal mucosal membrane showed that the membrane is stiffer in the longitudinal than in the circumferential direction of the airway. To explain this difference in the mechanical properties, we studied the morphological structure of the rabbit tracheal mucosal membrane in both longitudinal and circumferential directions. The collagen fibers were found to form a random meshwork, which would not account for differences in stiffness in the longitudinal and circumferential directions. The volume fraction of the elastic fibers was measured using a point-counting technique. The orientation of the elastic fibers in the tissue samples was measured using a new method based on simple geometry and probability. The results showed that the volume fraction of the elastic fibers in the rabbit tracheal mucosal membrane was approximately 5% and that the elastic fibers were mainly oriented in the longitudinal direction. Age had no statistically significant effect on either the volume fraction or the orientation of the elastic fibers. Linear correlations were found between the steady-state stiffness and the quantity of the elastic fibers oriented in the direction of testing.     

Anisotropic and inhomogeneous mechanical characteristics of the retina

We studied contribution of blood vessels to the mechanical properties of retina. Previous studies revealed anisotropic and inhomogeneous retinal mechanical characteristics. To examine different vessel types and sizes, 3 strips of retinal samples were dissected in each of 5 pig eyes. One strip contained the superior-temporal vein in the axial direction, one strip contained the superior-temporal artery in the axial direction, and one strip did not contain any visible vessel. To examine different vessel orientations, 2 strips of retinal samples were dissected in each of 5 other pig eyes. One strip contained the superior-temporal vein in the axial direction, and one strip contained the superior-temporal vein in the circumferential direction. Tensile testing was performed on the samples. The venous and arterial samples were found to be significantly stiffer than the no-visible-vessel samples. No significant difference was observed neither between the venous and arterial samples, nor between the venous samples in the axial versus circumferential directions. We therefore conclude that the blood vessels contribute significantly to the stiffness of the retina, and the vessel size is the only determining factor that governs the anisotropic and inhomogeneous characteristics of the retina.     

Rheological properties of the thoracic aorta in normal and WHHL rabbits

The tension-strain, stress-strain and stress relaxation curves of longitudinal and circumferential strips of proximal thoracic aortas in normal and WHHL rabbits of different ages were determined using a tensile testing instrument. Wall distensibility of longitudinal and circumferential strips was the greatest in the normal aorta and decreased with advancing age in the atherosclerotic aorta. The wall thickness of the atherosclerotic aorta was positively related to age with a correlation coefficient of 0.66(p less than 0.01). The incremental elastic moduli calculated from the stress-strain curves increased with advancing age in the atherosclerotic aorta. Accordingly, the decreased distensibility of the atherosclerotic wall may be due to the increased wall thickness caused by the intimal thickening as well as to the increase in wall stiffness caused by the increased elastic modulus. The viscoelasticity of the atherosclerotic aorta was larger than that of the normal aorta. This reflects the mechanical effect of atherosclerotic changes that occurred in the thickened intima.     

Tensile characteristics of collenchyma cell walls at different calcium contents

Collenchyma fibres from celery ( Apium graveolens L.) were extracted with detergent and phenol-acetic acid-water to leave the intact cell walls, free from active enzymes. Under a small, constant stress the cell wall fibres showed elastic and plastic extension and viscoelastic deformation, but viscous flow was observed only at high stresses close to the breaking stress. After complete removal of calcium ions with cyclohex-anediamine tetraacetic acid (CDTA) and incubation for 18 h, comparable levels of these extensibility components were observed at much lower stresses. However, partial removal of calcium ions with citrate did not increase the plastic, elastic or viscoelastic components even when the residual calcium was reduced to 3.5% of the exchange capacity. The breaking stress of the fibres was rather more sensitive to calcium removal, being reduced by 50% at 7% calcium saturation. CDTA-extracted fibres broke by cell separation at very low stress. These characteristics did not appear compatible with removal of calcium ions, or their displacement by protons, as a mechanism for auxin-induced growth in this material: however such mechanisms are not excluded in other tissues or under other conditions. Strong chelating agents which remove enough calcium to weaken cell walls should be avoided in experiments on other mechanisms of auxin-induced growth.     

The role of viscoelasticity of collagen fibers in articular cartilage: theory and numerical formulation

The relative importance of fluid-dependent and fluid-independent transient mechanical behavior in articular cartilage was examined for tensile and unconfined compression testing using a fibril reinforced model. The collagen matrix of articular cartilage was modeled as viscoelastic using a quasi-linear viscoelastic formulation with strain-dependent elastic modulus, while the proteoglycan matrix was considered as linearly elastic. The collagen viscoelastic properties were obtained by fitting experimental data from a tensile test. These properties were used to investigate unconfined compression testing, and the sensitivity of the properties was also explored. It was predicted that the stress relaxation observed in tensile tests was not caused by fluid pressurization at the macroscopic level. A multi-step tensile stress relaxation test could be approximated using a hereditary integral in which the elastic fibrillar modulus was taken to be a linear function of the fibrillar strain. Applying the same formulation to the radial fibers in unconfined compression, stress relaxation could not be simulated if fluid pressurization were absent. Collagen viscoelasticity was found to slightly weaken fluid pressurization in unconfined compression, and this effect was relatively more significant at moderate strain rates. Therefore, collagen viscoelasticity appears to play an import role in articular cartilage in tensile testing, while fluid pressurization dominates the transient mechanical behavior in compression. Collagen viscoelasticity plays a minor role in the mechanical response of cartilage in unconfined compression if significant fluid flow is present.     

A nonlinear model of passive muscle viscosity

The material properties of passive skeletal muscle are critical to proper function and are frequently a target for therapeutic and interventional strategies. Investigations into the passive viscoelasticity of muscle have primarily focused on characterizing the elastic behavior, largely neglecting the viscous component. However, viscosity is a sizeable contributor to muscle stress and extensibility during passive stretch and thus there is a need for characterization of the viscous as well as the elastic components of muscle viscoelasticity. Single mouse muscle fibers were subjected to incremental stress relaxation tests to characterize the dependence of passive muscle stress on time, strain and strain rate. A model was then developed to describe fiber viscoelasticity incorporating the observed nonlinearities. The results of this model were compared with two commonly used linear viscoelastic models in their ability to represent fiber stress relaxation and strain rate sensitivity. The viscous component of mouse muscle fiber stress was not linear as is typically assumed, but rather a more complex function of time, strain and strain rate. The model developed here, which incorporates these nonlinearities, was better able to represent the stress relaxation behavior of fibers under the conditions tested than commonly used models with linear viscosity. It presents a new tool to investigate the changes in muscle viscous stresses with age, injury and disuse.     

Effects of stress shielding on the transverse mechanical properties of rabbit patellar tendons

With the aim of studying mechanisms of the remodeling of tendons and ligaments, the effects of stress shielding on the rabbit patellar tendon were studied by performing tensile and stress relaxation tests in the transverse direction. The tangent modulus, tensile strength, and strain at failure of non-treated, control patellar tendons in the transverse direction were 1272 kPa, 370 kPa, and 40.5 percent, respectively, whereas those of the tendons stress-shielded for 1 week were 299 kPa, 108 kPa, and 40.4 percent, respectively. Stress shielding markedly decreased tangent modulus and tensile strength in the transverse direction, and the decreases were larger than those in the longitudinal direction, which were determined in our previous study. For example, tensile strength in the transverse and longitudinal direction decreased to 29 and 50 percent of each control value, respectively, after 1 week stress shielding. In addition, the stress relaxation in the transverse direction of stress-shielded patellar tendons was much larger than that of nontreated, control ones. In contrast to longitudinal tensile tests for the behavior of collagen, transverse tests reflect the contributions of ground substances such as proteoglycans and mechanical interactions between collagen fibers. Ground substances provide lubrication and spacing between fibers, and also confer viscoelastic properties. Therefore, the results obtained from the present study suggest that ground substance matrix, and interfiber and fiber-matrix interactions have important roles in the remodeling response of tendons to stress.     

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.     

Effect of cyclic axial stretch of rat arteries on endothelial cytoskeletal morphology and vascular reactivity

Pulsatile fluid shear stress and circumferential stretch are responsible for the axial alignment of vascular endothelial cells and their actin stress fibers in vivo. We studied the effect of cyclic alterations in axial stretch independent of flow on endothelial cytoskeletal organization in intact arteries and determined if functional alterations accompanied morphologic alterations. Rat renal arteries were axially stretched (20%, 0.5 Hz) around their in vivo lengths, for up to 4h. Actin stress fibers were examined by immunofluorescent staining. We found that cyclic axial stretching of intact vessels under normal transmural pressure in the absence of shear stress induces within a few hours realignment of endothelial actin stress fibers toward the circumferential direction. Concomitant with this morphologic alteration, the sensitivity (log(EC(50))) to the endothelium-dependent vasodilator (acetylcholine) was significantly decreased in the stretched vessels (after stretching -5.15+/-0.79 and before stretching -6.71+/-0.78, resp.), while there was no difference in sodium nitroprusside (SNP) sensitivity. There was no difference in sensitivity to both acetylcholine and SNP in time control vessels. Similar to cultured cells, endothelial cells in intact vessels subjected to cyclic stretching reorganize their actin filaments almost perpendicular to the stretching direction. Accompanying this morphological alteration is a loss of endothelium-dependent vasodilation but not of smooth muscle responsiveness.     

A constituent-based model for the nonlinear viscoelastic behavior of ligaments

This paper presents a constitutive model for predicting the nonlinear viscoelastic behavior of soft biological tissues and in particular of ligaments. The constitutive law is a generalization of the well-known quasi-linear viscoelastic theory (QLV) in which the elastic response of the tissue and the time-dependent properties are independently modeled and combined into a convolution time integral. The elastic behavior, based on the definition of anisotropic strain energy function, is extended to the time-dependent regime by means of a suitably developed time discretization scheme. The time-dependent constitutive law is based on the postulate that a constituent-based relaxation behavior may be defined through two different stress relaxation functions: one for the isotropic matrix and one for the reinforcing (collagen) fibers. The constitutive parameters of the viscoelastic model have been estimated by curve fitting the stress relaxation experiments conducted on medial collateral ligaments (MCLs) taken from the literature, whereas the predictive capability of the model was assessed by simulating experimental tests different from those used for the parameter estimation. In particular, creep tests at different maximum stresses have been successfully simulated. The proposed nonlinear viscoelastic model is able to predict the time-dependent response of ligaments described in experimental works (Bonifasi-Lista et al., 2005, J. Orthopaed. Res., 23, pp. 67-76; Hingorani et al., 2004, Ann. Biomed. Eng., 32, pp. 306-312; Provenzano et al., 2001, Ann. Biomed. Eng., 29, pp. 908-214; Weiss et al., 2002, J. Biomech., 35, pp. 943-950). In particular, the nonlinear viscoelastic response which implies different relaxation rates for different applied strains, as well as different creep rates for different applied stresses and direction-dependent relaxation behavior, can be described.     

Structural response of relaxed and constricted arterioles

Analysis of viscoelastic properties of blood vessel walls presents difficult analytical problems in view of their non-homogeneity, anisotropy, and non-linear viscoelastic characteristics. The analytical technique used in this study is the numerical method of direct stiffness which has been successfully applied in preliminary studies. The direct stiffness method constitutes a finite element analysis, where the wall structure in an electron micrograph is represented by an assembly of triangular elements. Different material properties can be assigned to each triangular element; the method is thus ideally suited for computer analysis of non-homogeneous biological structures. Anatomical components considered in the analysis were two types of collagen fibers and smooth muscle cells. Typical circumferential stress-time and radius-time histories for selected points have been obtained in relaxed and constricted arterioles under static pressure loads. Plots of the stress distribution in cross sections of the vessel wall have been obtained in relaxed and constricted arterioles.     

Bladder tissue passive response to monotonic and cyclic loading

The fundamental passive mechanical properties of the bladder need to be known in order to design the most appropriate long-term surgical repair procedures and develop materials for bladder reconstruction. This study has focused on the bladder tissue viscoelastic behavior, providing a comprehensive analysis of the effects of fibers orientation, strain rate and loading history. Whole bladders harvested from one year old fat pigs (160 kg approximate weight) were dissected along the apex-to-base direction and samples were isolated from the lateral region of the wall, as well as along apex-to-base and transverse directions. Uniaxial monotonic (stress relaxation) and cyclic tests at different frequencies have been performed with the Bose Electroforce(?) 3200. Normalized stress relaxation functions have been interpolated using a second-order exponential series and loading and unloading stress-strain curves have been interpolated with a non-linear elastic model. The passive mechanical behavior of bladder tissue was shown to be heavily influenced by frequency and loading history, both in monotonic and cyclic tests. The anisotropy of the tissue was evident in monotonic and in cyclic tests as well, especially in tests performed on an exercised tissue and at high frequencies. In contrast, transverse and apex-to-base samples demonstrated an analogous relaxation behavior.     

Tissue softening of guinea pig oesophagus tested by the tri-axial test machine

Tissue softening is commonly reported during mechanical testing of biological tissues in vitro. The loss of stiffness may be due to viscoelasticity-induced softening (the time-history of load-caused softening) and strain-induced stress softening (the maximum previous load-caused softening). However, the knowledge about tissue softening behaviour is presently poor. The aims of this study were to distinguish whether the loss of the stiffness during preconditioning was due to strain softening or viscoelasticity and to test the tissue softening in circumferential and longitudinal direction in the guinea pig oesophagus. Eight repeated pressure controlled ramp distensions and eight uniaxial tensile-release ramp stretches in three series were done on eight guinea pig oesophagi. The stress–strain curves were used to display the time-dependency (viscoelasticity) and the maximum previous load-caused softening (strain softening) in circumferential and longitudinal directions. For both the longitudinal and the circumferential softening, the peak stress and stiffness produced during the first loading were bigger than those produced in the remaining loadings. The stress loss due to strain softening was about three times more than that due to viscoelasticity in the longitudinal direction. The strain increased more than two times between the strain softening and viscoelastic softening in the circumferential direction. With a stress level of 20 kPa, the stiffness in the circumferential direction lost more than that in the longitudinal direction (P<0.05), indicating the anisotropic softening properties in the oesophagus. In conclusion, the stiffness loss during preconditioning is mainly attributed to strain softening, appears irreversible and is anisotropic.     

Thermal stresses from large volumetric expansion during freezing of biomaterials.

Thermal stresses were studied in freezing of biomaterials containing significant amounts of water. An apparent specific heat formulation of the energy equation and a viscoelastic model for the mechanics problem were used to analyze the transient axi-symmetric freezing of a long cylinder. Viscoelastic properties were measured in an Instron machine. Results show that, before phase change occurs at any location, both radial and circumferential stresses are tensile and keep increasing until phase change begins. The maximum principal tensile stress during phase change increases with a decrease in boundary temperature (faster cooling). This is consistent with experimentally observed fractures at a lower boundary temperature. Large volumetric expansion during water to ice transformation was shown to be the primary contributor to large stress development. For very rapid freezing, relaxation may not be significant, and an elastic model may be sufficient.     

Viscous elements have little impact on measured passive length-tension properties of human gastrocnemius muscle-tendon units in vivo

Several studies have measured the elastic properties of a single human muscle-tendon unit in vivo. However the viscoelastic behavior of single human muscles has not been characterized. In this study, we adapted QLV theory to model the viscoelastic behavior of human gastrocnemius muscle-tendon units in vivo. We also determined the influence of viscoelasticity on passive length-tension properties of human gastrocnemius muscle-tendon units. Eight subjects participated in the experiment, which consisted of two parts. First, the stress relaxation response of human gastrocnemius muscle-tendon units was determined at a range of knee and ankle angles. Subsequently, passive ankle torque and ankle angle were collected during cyclic dorsiflexion and plantarflexion at a range of knee angles. Viscous parameters were determined by fitting the stress relaxation experiment data with a two-term exponential function, and elastic parameters were estimated by fitting the QLV model and viscous parameters to the cyclic experiment data. The model fitted the experimental data well at slow speeds (RMSE: 1.7 ± 0.5N) and at fast speeds (RMSE: 1.9 ± 0.2N). Muscle-tendon units demonstrated a large amount of stress relaxation. Nonetheless, viscoelastic passive length-tension curves estimated with the QLV model were similar to elastic passive length-tension curves obtained using a model that ignored viscosity. There was little difference in the elastic passive length-tension curves at different loading rates. We conclude that (a) the QLV model can be used to quantify viscoelastic behaviors of relaxed human gastrocnemius muscle-tendon units in vivo, and (b) over the range of velocities we examined, the velocity of loading has little effect on the passive length-tension properties of human gastrocnemius muscle-tendon units.     

The effect of auxin on stress relaxation in isolated Avena coleoptiles

In order to characterize further the mechanical properties of coleoptile cell walls, stress relaxation measurements were made on methanol-boiled sections of Avena coleoptiles. Relaxation was measured both in mechanically conditioned specimens and in specimens which had not been previously extended. In both cases the relaxation was proportional to log time. Mechanical conditioning increased the relaxation modules and decreased the relative rate of relaxation. In contrast, pretreatment of the live coleoptiles with indoleacetic acid reduced the relaxation modulus and the absolute rate of relaxation but did not affect the relative rate of relaxation. Essentially similar pictures of the mechanical properties of coleoptile walls are obtained from stress relaxation and creep tests; the wall behaves as a nonlinear viscoelastic material.     

Sarcomere Mechanics in Capillary Endothelial Cells

Tension generation in endothelial cells of the aorta, spleen, and eye occurs in actin stress fibers, and is necessary for normal cell function. Sarcomeres are the tension-generating units of actin stress fibers in endothelial cells. How sarcomeres generate and maintain tension in stress fibers is not well understood. Using femtosecond laser ablation, we severed living stress fibers and measured sarcomere contraction under zero tension. The length of the sarcomere decreased in two phases: an instantaneous initial response, followed by a slower change in length attributed to myosin activity. The latter phase ceased abruptly after a minimum sarcomere length was reached, suggesting a rigid resistance that prevents further contraction. Furthermore, severed, contracted stress fibers did not relax when treated with myosin inhibitors, indicating that contracted stress fibers do not store elastic potential energy. These novel measurements combined with modeling suggest that myosin-generated forces in adjacent sarcomeres are directly in balance, and argue against sarcomere models with springlike elements in parallel with myosin contractile elements. We propose a new model for tension generation in the sarcomere, which provides a mechanistic interpretation for our observations and previous observations of inhomogeneous sarcomere contraction and apparent stress fiber viscoelastic behavior.