Comparison of approaches to quantify arterial damping capacity from pressurization tests on mouse conduit arteries.

Large conduit arteries are not purely elastic, but viscoelastic, which affects not only the mechanical behavior but also the ventricular afterload. Different hysteresis loops such as pressure-diameter, pressure-luminal cross-sectional area (LCSA), and stress-strain have been used to estimate damping capacity, which is associated with the ratio of the dissipated energy to the stored energy. Typically, linearized methods are used to calculate the damping capacity of arteries despite the fact that arteries are nonlinearly viscoelastic. The differences in the calculated damping capacity between these hysteresis loops and the most common linear and correct nonlinear methods have not been fully examined. The purpose of this study was thus to examine these differences and to determine a preferred approach for arterial damping capacity estimation. Pressurization tests were performed on mouse extralobar pulmonary and carotid arteries in their physiological pressure ranges with pressure (P) and outer diameter (OD) measured. The P-inner diameter (ID), P-stretch, P-Almansi strain, P-Green strain, P-LCSA, and stress-strain loops (including the Cauchy and Piola-Kirchhoff stresses and Almansi and Green strains) were calculated using the P-OD data and arterial geometry. Then, the damping capacity was calculated from these loops with both linear and nonlinear methods. Our results demonstrate that the linear approach provides a reasonable approximation of damping capacity for all of the loops except the Cauchy stress-Almansi strain, for which the estimate of damping capacity was significantly smaller (22 ± 8% with the nonlinear method and 31 ± 10% with the linear method). Between healthy and diseased extralobar pulmonary arteries, both methods detected significant differences. However, the estimate of damping capacity provided by the linear method was significantly smaller (27 ± 11%) than that of the nonlinear method. We conclude that all loops except the Cauchy stress-Almansi strain loop can be used to estimate artery wall damping capacity in the physiological pressure range and the nonlinear method is recommended over the linear method.

Chiral behavior in rat tail tendon fascicles.

Ex vivo tendon mechanical behavior has been well described under rotationally constrained uniaxial tensile testing. During standard loading of rat tail tendon (RTT) fascicles, apparent axial twist has been observed. To quantify this behavior, we designed a custom testing setup, utilizing magnetic suspension, to allow unconstrained axial rotation during tensile loading. We characterized the rotational behavior of single and paired RTT fascicles under cyclic loading. We also measured stress relaxation across loading cycles as well as "rotational relaxation". Single fascicle nonlinear stretch-twist coupling is well described by the asymptotic function Δθ=A(1-e-Bε) in which fascicles rotated a mean ±51.1° within about 1% applied axial strain. On average, paired fascicles rotated just over 10° less. Specimen cross-sectional diameter had a noticeable effect on the measured mechanical properties, particularly effective elastic modulus. Such stretch-twist coupling and size dependence cannot be understood via classical elasticity but is predicted by Cosserat (micropolar) elasticity. The current study demonstrates RTT fascicles are chiral based on observed axial load-induced twist. Additionally, our findings support existing research that suggests a helical fascicle structure. Potential consequences of helical substructures, mechanical and biological, merit further investigation.

Changes in large pulmonary arterial viscoelasticity in chronic pulmonary hypertension.

Conduit pulmonary artery (PA) stiffening is characteristic of pulmonary arterial hypertension (PAH) and is an excellent predictor of mortality due to right ventricular (RV) overload. To better understand the impact of conduit PA stiffening on RV afterload, it is critical to examine the arterial viscoelastic properties, which require measurements of elasticity (energy storage behavior) and viscosity (energy dissipation behavior). Here we hypothesize that PAH leads to frequency-dependent changes in arterial stiffness (related to elasticity) and damping ratio (related to viscosity) in large PAs. To test our hypothesis, PAH was induced by the combination of chronic hypoxia and an antiangiogenic compound (SU5416) treatment in mice. Static and sinusoidal pressure-inflation tests were performed on isolated conduit PAs at various frequencies (0.01-20 Hz) to obtain the mechanical properties in the absence of smooth muscle contraction. Static mechanical tests showed significant stiffening of large PAs with PAH, as expected. In dynamic mechanical tests, structural stiffness (κ) increased and damping ratio (D) decreased at a physiologically relevant frequency (10 Hz) in hypertensive PAs. The dynamic elastic modulus (E), a material stiffness, did not increase significantly with PAH. All dynamic mechanical properties were strong functions of frequency. In particular, κ, E and D increased with increasing frequency in control PAs. While this behavior remained for D in hypertensive PAs, it reversed for κ and E. Since these novel dynamic mechanical property changes were found in the absence of changes in smooth muscle cell content or contraction, changes in collagen and proteoglycans and their interactions are likely critical to arterial viscoelasticity in a way that has not been previously described. The impact of these changes in PA viscoelasticity on RV afterload in PAH awaits further investigation.

Relationship between tendon stiffness and failure: a metaanalysis.

Tendon is a highly specialized, hierarchical tissue designed to transfer forces from muscle to bone; complex viscoelastic and anisotropic behaviors have been extensively characterized for specific subsets of tendons. Reported mechanical data consistently show a pseudoelastic, stress-vs.-strain behavior with a linear slope after an initial toe region. Many studies report a linear, elastic modulus, or Young's modulus (hereafter called elastic modulus) and ultimate stress for their tendon specimens. Individually, these studies are unable to provide a broader, interstudy understanding of tendon mechanical behavior. Herein we present a metaanalysis of pooled mechanical data from a representative sample of tendons from different species. These data include healthy tendons and those altered by injury and healing, genetic modification, allograft preparation, mechanical environment, and age. Fifty studies were selected and analyzed. Despite a wide range of mechanical properties between and within species, elastic modulus and ultimate stress are highly correlated (R(2) = 0.785), suggesting that tendon failure is highly strain-dependent. Furthermore, this relationship was observed to be predictable over controlled ranges of elastic moduli, as would be typical of any individual species. With the knowledge gained through this metaanalysis, noninvasive tools could measure elastic modulus in vivo and reasonably predict ultimate stress (or structural compromise) for diseased or injured tendon.

Effects of collagen deposition on passive and active mechanical properties of large pulmonary arteries in hypoxic pulmonary hypertension.

Proximal pulmonary artery (PA) stiffening is a strong predictor of mortality in pulmonary hypertension. Collagen accumulation is mainly responsible for PA stiffening in hypoxia-induced pulmonary hypertension (HPH) in mouse models. We hypothesized that collagen cross-linking and the type I isoform are the main determinants of large PA mechanical changes during HPH, which we tested by exposing mice that resist type I collagen degradation (Col1a1[Formula: see text] and littermate controls (Col1a1[Formula: see text] to hypoxia for 10 days with or without [Formula: see text]-aminopropionitrile (BAPN) treatment to prevent cross-link formation. Static and dynamic mechanical tests were performed on isolated PAs with smooth muscle cells (SMC) in passive and active states. Percentages of type I and III collagen were quantified by histology; total collagen content and cross-linking were measured biochemically. In the SMC passive state, for both genotypes, hypoxia tended to increase PA stiffness and damping capacity, and BAPN treatment limited these increases. These changes were correlated with collagen cross-linking ([Formula: see text]). In the SMC active state, hypoxia increased PA dynamic stiffness and BAPN had no effect in Col1a1[Formula: see text] mice ([Formula: see text]). PA stiffness did not change in Col1a1[Formula: see text] mice. Similarly, damping capacity did not change for either genotype. Type I collagen accumulated more in Col1a1[Formula: see text] mice, whereas type III collagen increased more in Col1a1[Formula: see text] mice during HPH. In summary, PA passive mechanical properties (both static and dynamic) are related to collagen cross-linking. Type I collagen turnover is critical to large PA remodeling during HPH when collagen metabolism is not mutated and type III collagen may serve as a reserve.

Effect of age and exercise on the viscoelastic properties of rat tail tendon.

Tendon mechanical properties are thought to degrade during aging but improve with exercise. A remaining question is whether exercise in aged animals provides sufficient regenerative, systemic stimulus to restore younger mechanical behaviors. Herein we address that question with tail tendons from aged and exercised rats, which would be subject to systemic effects but not direct loading from the exercise regimen. Twenty-four month old rats underwent one of three treadmill exercise training protocols for 12 months: sedentary (walking at 0° incline for 5 min/day), moderate (running at 0° incline for 30 min/day), or high (running at 4° incline for 30 min/day). A group of 9 month old rats were used to provide an adult control, while a group of 3 month old rats provided a young control. Tendons were harvested at sacrifice and mechanically tested. Results show significant age-dependent differences in modulus, ultimate stress, relaxation rate, and percent relaxation. Relaxation rate was strain-dependent, consistent with nonlinear superposition or Schapery models but not with quasilinear viscoelasticity (QLV). Trends in exercise data suggest that with exercise, tendons assume the elastic character of younger rats (lower elastic modulus and ultimate stress).

Resonant ultrasound spectroscopy of cubes over the full range of Poisson's ratio.

Methods are developed for study of isotropic cubes via resonant ultrasound spectroscopy. To that end, mode structure maps are determined for freely vibrating isotropic cubes via finite element method over the full range of Poisson's ratio ν (-1 to +0.5). The fundamental torsional mode has the lowest frequency provided ν is between about -0.31 and +0.5. Experimental measurements for the mode structures of materials with Poisson's ratio +0.33, +0.3, +0.15, -0.15, and -0.72 are performed using resonant ultrasound spectroscopy and interpreted. Methods are developed to identify pertinent modes. The experimental results match well with the analysis with the exception of some splitting of some modes because of slight material anisotropy. The effects of slight imperfection of specimen shape on the first 10 modes are analyzed for various Poisson's ratios.

Damage mechanics of porcine flexor tendon: mechanical evaluation and modeling.

Porcine flexor tendons underwent cyclic and stress relaxation testing before and after strain exceeding elastic limit ("overstretch") to examine which mechanical parameters undergo changes following subfailure damage. From these data, we developed an "effective strain" damage model (in which the tendon is modeled as if being pulled to a lower strain). Damage was induced at three strain levels to determine the extent to which post-damage parameter changes were affected by overstretch strain level. We found that diffuse damage induced by overstretch decreased elastic and viscoelastic parameters obtained during testing. The stress response of tendon to strain is therefore altered following damage. We next compared the strain-dependent parameter behavior to damage-dependent behavior to determine the effective strain for each parameter. Effects of damage became more pronounced as strain during overstretch increased; following overstretch to 6.5, 9, or 13% strain, effective strain was 2.43 ± 0.33, 1.98 ± 0.3, or 0.88 ± 0.43% strain, respectively. By determining the effective strain and using it to calculate predicted values of post-damage mechanical parameters, it was possible to predict the stress relaxation behavior of tendons with Schapery's nonlinear viscoelastic model. Using this approach, a single parameter predicts both elastic and viscoelastic compromise from known, strain-dependent behaviors.

Quantification of collagen organization using fractal dimensions and Fourier transforms.

Collagen fibers and fibrils that comprise tendons and ligaments are disrupted or damaged during injury. Fibrillogenesis during healing produces a matrix that is initially quite disorganized, but remodels over time to resemble, but not replicate, the original roughly parallel microstructure. Quantification of these changes is traditionally a laborious and subjective task. In this work we applied two automated techniques, fast Fourier transformation (FFT) and fractal dimension analysis (FA) to quantify the organization of collagen fibers or fibrils. Using multi-photon images of collagen fibers obtained from rat ligament we showed that for healing ligaments, FA differentiates more clearly between the different time-points during healing. Using scanning electron microscopy images of overstretched porcine flexor tendon, we showed that combining FFT and FA measures distinguishes the damaged and undamaged groups more clearly than either method separately.

Resonant ultrasound spectroscopy of cylinders over the full range of Poisson's ratio.

Mode structure maps for freely vibrating cylinders over a range of Poisson's ratio, ν, are desirable for the design and interpretation of experiments using resonant ultrasound spectroscopy (RUS). The full range of isotropic ν (-1 to +0.5) is analyzed here using a finite element method to accommodate materials with a negative Poisson's ratio. The fundamental torsional mode has the lowest frequency provided ν is between about -0.24 and +0.5. For any ν, the torsional mode can be identified utilizing the polarization sensitivity of the shear transducers. RUS experimental results for materials with Poisson's ratio +0.3, +0.16, and -0.3 and a previous numerical study for ν = 0.33 are compared with the present analysis. Interpretation of results is easiest if the length∕diameter ratio of the cylinder is close to 1. Slight material anisotropy leads to splitting of the higher modes but not of the fundamental torsion mode.

Stress relaxation and recovery in tendon and ligament: experiment and modeling.

Accurate joint models require the ability to predict soft tissue behavior. This study evaluates the ability of constitutive equations to predict the nonlinear and viscoelastic behavior of tendon and ligament during stress relaxation testing in a porcine model. Three constitutive equations are compared in their ability to model relaxation, recovery and reloading of tissues. Quasi-linear viscoelasticity (QLV) can fit a single stress relaxation curve, but fails to account for the strain-dependence in relaxation. Nonlinear superposition can fit the single relaxation curve and will account for the strain-dependent relaxation behavior, but fails to accurately predict recovery behavior. Schapery's nonlinear viscoelastic model successfully fits a single relaxation curve, accounts for strain-dependent relaxation behavior, and accurately predicts recovery and reloading behavior. Comparing Schapery's model to QLV and nonlinear superposition, Schapery's method was uniquely capable of fitting the different nonlinearities that arise in stress relaxation curves from different tissues, e.g. the porcine digital flexor tendon and the porcine medial collateral ligament (MCL), as well as predicting subsequent recovery and relaxation curves after initial loads.

Viscoelastic relaxation and recovery of tendon.

Tendons exhibit complex viscoelastic behaviors during relaxation and recovery. Recovery is critical to predicting behavior in subsequent loading, yet is not well studied. Our goal is to explore time-dependent recovery of these tendons after loading. As a prerequisite, their strain-dependent viscoelastic behaviors during relaxation were also characterized. The porcine digital flexor tendon was used as a model of tendon behavior. Strain-dependent relaxation was observed in tests at 1, 2, 3, 4, 5, and 6% strain. Recovery behavior of the tendon was examined by performing relaxation tests at 6%, then dropping to a low but nonzero strain level. Results show that the rate of relaxation in tendon is indeed a function of strain. Unlike previously reported tests on the medial collateral ligament (MCL), the relaxation rate of tendons increased with increased levels of strain. This strain-dependent relaxation contrasts with quasilinear viscoelasticity (QLV), which predicts equal time dependence across various strains. Also, the tendons did not recover to predicted levels by nonlinear superposition models or QLV, though they did recover partially. This recovery behavior and behavior during subsequent loadings will then become problematic for both quasilinear and nonlinear models to correctly predict.

Post-yield relaxation behavior of bovine cancellous bone.

Relaxation studies were conducted on specimens of bovine cancellous bone at post-yield strains. Stress and strain were measured for 1000s and the relaxation modulus was determined. Fifteen cylindrical, cancellous bone specimens were removed from one bovine femur in the anterior-posterior direction. The relaxation modulus was found to be a function of strain. Therefore cancellous bone is non-linearly viscoelastic/viscoplastic in the plastic region. A power law regression was fit to the relaxation modulus data. The multiplicative constant was found to be statistically related through a power law relationship to both strain (p<0.0005) and apparent density (p<0.0005) while the power coefficient was found to be related through a power law relationship, E(t, epsilon)=A(epsilon)t(-n(epsilon)), to strain (p<0.0005), but not apparent density.

Mechanical instabilities of individual multiwalled carbon nanotubes under cyclic axial compression.

Individual multiwalled carbon nanotubes with a range of aspect ratios are subjected to cyclic axial compression to large strains using atomic force microscopy. Distinct elastic buckling and post-buckling phenomena are observed reproducibly and are ascribed to Euler, asymmetric shell buckling (i.e., kinking), and symmetric shell buckling. These show agreement with continuum theories that range from approximate to remarkable. Shell buckling yields reproducible incremental negative stiffness in the initial post-buckled regime.

Nonlinear viscoelasticity in rabbit medial collateral ligament.

The goal of this study was to characterize the viscoelastic behavior of the rabbit medial collateral ligament (MCL) at multiple levels of strain (between 0% and approximately 5%) and their corresponding stresses (between 0 and approximately 55 MPa) for stress relaxation and creep, respectively. We hypothesized that in the rabbit MCL the rate of stress relaxation would be strain dependent and the rate of creep would be stress dependent. Thirty MCLs from 15 rabbits were tested ex vivo for this study. Results show that within the physiologically relevant region of ligament behavior, the rate of stress relaxation is strain dependent in the rabbit MCL, with the rate of relaxation decreasing with increasing tissue strain. The rate of creep is stress dependent in the rabbit MCL, with the rate of creep decreasing with increasing stress. These results support our hypothesis, with the greatest nonlinearities in a physiologically relevant region of loading. As such, these nonlinearities should be considered when quantifying ligament viscoelastic behavior with a rabbit model.