Ramesh N. Vaishnav

Residual stress and strain in aortic segments

In the study of stresses and strains in vascular segments, it is generally assumed that the traction-free configuration assumed by a segment when there is no axial force and there are no intravascular and extravascular pressures is stress-free. To investigate the degree of validity of this assumption, 286 oval shaped rings were excised from three bovine and six porcine aortas and photographed. Radial cuts were made in these rings which opened up into horseshoe shapes and were also photographed. Smoothed boundary lengths at intimal and adventitial levels in the rings and their cut open configurations were measured from the photographs and the residual strains in the annular configuration relative to the open configuration were computed. It was found that: the average maximum residual intimal engineering strain in the uncut configuration was -0.082 for all nine aortas and -0.096 and -0.077 for the bovine and porcine aortas alone, respectively; the average maximum residual adventitial strain was 0.085 for all aortas, and 0.102 and 0.078 for the bovine and porcine aortas alone, respectively; an estimated average beneficial compressive stress of -0.188 X 10(5) Pa (corresponding to a strain level of -0.082) is available at the intimal level to counteract the in vivo tensile stress due to the intravascular pressure; an estimated average initial tensile stress of 0.195 X 10(5) Pa (corresponding to a strain level of 0.085) exists at the adventitial level which adds to the in vivo tensile stress due to the intravascular pressure. Although these stress levels are not large in comparison with the in vivo stress in the arterial wall, a detailed stress analysis must take into account these initial stresses.

ESTIMATION OF RESIDUAL STRAINS IN AORTIC SEGMENTS

Publisher Summary In characterizing the mechanical properties of the arterial tissue, it is customary to assume that the configuration that a cylindrical arterial segment assumes when the intravascular and extravascular pressures and the longitudinal force acting on it are zero is stress-free. Similarly, also in uniaxial stress tests on arterial strips it is generally assumed that the initial straight configuration of the strip prior to the application of axial load is stress-free. This crucial assumption has never been experimentally substantiated. This chapter discusses this assumption and show that it is not warranted. It presents the results that indicate that the aorta is in a state of residual strain and stress when it is in a configuration which is circumferentially intact but free from external forces. This is contrary to what is generally assumed to be the case. It is of interest to compare the residual stresses with the normal, in vivo circumferential stress. The maximum residual stress is approximately 14 to 17% of the mean in vivo circumferential engineering stress. A similar error is also implicit when an initially curved strip is straightened out prior to subjecting it to a uniaxial tensile stress test. These findings indicate that there is advantageous residual stress in the artery by virtue of which the actual maximum in vivo circumferential stress is lower than what would be computed on the basis of assuming the natural configuration to be stress-free.

Distribution of Stresses and of Strain-Energy Density through the Wall Thickness in a Canine Aortic Segment

In this paper we presented a method for determining the distribution of circumferential (SΘ), longitudinal (Sz), and radial (Sr) stress and of strain-energy density (W) in canine aortic segments under physiological loading. Aortic segments from 13 dogs were studied in vitro. Intraluminal pressure and longitudinal force were varied over ranges of 25 to 200 cm H2O and 0 to 100 g, respectively, and the external radius and the segment length were measured at each step. A nonlinear theory of large deformation of the aortic tissue based on the assumptions of incompressibility and curvilinear orthotropy was applied to the data. The 3, 7, or 12 constitutive constants of the theory were calculated without using the usual thin-wall assumption. These constants were then used to calculate the distribution of stresses and of W. The results indicated that (I) SΘ, Sz, Sr, and W were largest in magnitude at the endothelial surface and decreased toward the adventitial surface, (2) the decrease was most marked in the inner third of the vessel wall, (3) in general SΘ > Sz > W > Sr, and (4) at a given longitudinal stretch an increase in intraluminal pressure increased SΘ, Sz, |Sr|, and W, with the effect being most marked on SΘ.

Submerged laminar jet impingement on a plane

Submerged laminar jet impingement on a plane is studied using computation. Steady-state Navier-Stokes equations for the axisymmetric case are solved numerically. The extent of the infinite flow is approximated by applying the boundary conditions at a finite but sufficiently large distance. The tube-exit velocity profile is assumed to be either a fully developed parabolic profile or a flat profile. For the former case, two different nozzle heights from the target plane are considered. The presence of a toroid-shaped eddy at low values of Reynolds number, Re , leads to some interesting observations such as the manner in which the wall shear stress depends on Re . An increase in the height of the nozzle exit from the target plane decreases the wall shear stress, more so at lower values of Re . A change from the parabolic exit velocity profile to the flat profile leads to a decrease in wall shear stress due to decreased momentum flux. The study was motivated by experiments designed to measure the yield shear strength of the vascular endothelium wherein a small saline jet was used to erode the tissue by normal impingement.

Dynamic Anisotropic Viscoelastic Properties of the Aorta in Living Dogs

The purpose of this paper is to describe the incremental dynamic anisotropic viscoelastic properties of the middle descending thoracic aorta in ten living dogs. A segment of the middle descending thoracic aorta was isolated in situ and connected to a reservoir filled with oxygenated blood. The height of the reservoir could be adjusted to impress a known pressure on the segment. In addition, a sinusoidal fluid displacement pump was connected to this assembly to superimpose small sinusoidal pressures at various frequencies (0–5 Hz); in another phase of the experiment small sinusoidal changes in length were also imposed on the segment. The pressure, the longitudinal force, and the length of the segment were monitored continuously with specially designed transducers. From these data, the incremental elastic (E′) and the viscous (E″) moduli in the circumferential (&thetas;), longitudinal (z), and radial (γ) directions were calculated around a given state of strain in the physiological range. In general, the viscoelastic moduli were functions of the initial stretches, indicating the nonlinear nature of the arterial wall. Within the physiological range of pressure, Ez′ > E&thetas;′ > Eγ′ (P < 0.01) and Ez″ > Eγ″ > E&thetas;″ (P < 0.01). These results indicated that the vessel wall was anisotropic, having its largest values of elastic and viscous moduli in the longitudinal direction. The ratio of E′ /E′ < 0.123 at 2 Hz, indicating a small viscous component relative to its elastic counterpart. The elastic moduli increased markedly from 0–2 Hz and then settled down to a relatively constant value; this finding was consistent with previously published data. The values of Ez′ and Ez″ from the in vivo experiments were higher than those from the in vitro experiments, and the difference was attributed to the vascular tethering present in the in vivo state.

A general procedure to evaluate robot positioning errors

A new, explicit, and complete formulation that describes the geometric errors due to both origin translation and misalign ments of axes in the positioning of an open-loop robot ma nipulator has been presented. The formulation does not use the usual Denavit-Hartenberg approach. The results clearly display the role of each quantity involved and allow easy physical interpretation of each error term. First, a general kinematic formulation for an ideal robot with an arbitrary number of links is developed. The geometric errors in axes locations and orientations are then shown to be skew coordi nate transformations with origin translations and are incor porated into the analysis using general tensor algebra. The final forms of forward and backward transformations contain up to 9 (N + 2) error parameters for a robot with N physical links. This theoretical formulation is applied to a specific six-degrees-of-freedom robot. It is observed that of the possi ble 72 errors in this robot, only 53 can independently contrib ute to the error at the end effector and are the only ones that can be determined by measurements made at the end effec tor. It is shown how a correlation matrix analysis can be used to isolate these 53 parameters. In addition. it is shown how, using a successive orthogonalization technique, one can eliminate from consideration error parameters (eight in the present case) that are too small to be significant but contrib ute to the ill-conditioning of the coefficient matrix. Applica tion of the procedure to computer-simulated data shows that the formulation and the numerical techniques developed here combine to form a powerful method to calibrate a robot.

Collapse and viscoelasticity of diseased human arteries

A laboratory study of the hydrostatic collapse of diseased tibial arteries demonstrated hysteresis in the pressure-flow behaviour which resembled that seen in the stress-strain relations of the arterial tissue. The pressures at which the vessels collapsed were found to be considerably lower than expected on the basis of theoretical elastic models. Also, the pressures at which the vessels reopened were consistently lower than the pressures at which they collapsed. These findings were explained on the basis of viscoelasticity. The difference between collapse and opening pressure may provide insight into the mechanical properties of vessels, and a clue to errors in non-invasive measurements of blood pressure which depend upon collapse of arteries.

ANISOTROPY OF THE AORTIC TISSUE IN THE AXIAL AND TRANSVERSE DIRECTIONS

ABSTRACT Mechanical properties of the porcine aortic tissue were studied by subjecting circumferential and longitudinal strips of the tissue to uniaxial states of stress. Stress-strain curves and transverse strain vs. axial strain curves were obtained for strips in the circumferential (6) and longitudinal (z) directions. The fact that these curves were distinct suggested anisotropic behavior of the tissue. To quantify the degree of separation of these curves directly, indexes of anisotropy were computed as ratios of stresses and strains in the θ and z directions at equal values of axial strains or stresses. It is shown that various stress and strain ratios, which would have a uniform value of unity if the tissue were isotropic, are far from having a constant value of one.

INTERCONVERSIONS OF VARIOUS TYPES OF SECANT AND INCREMENTAL MODULI FOR SOFT BIOLOGICAL TISSUE

ABSTRACT Elastic behavior of most biological tissue is nonlinear in terms of the usual definitions of stress and strain. Because of the large geometric changes involved, two different types of stress (engineering stress and true stress) and several different types of strain (for example, engineering strain, Green-St. Venant strain and natural strain) can be used to describe uniaxial test data for such materials, with each choice of a stress-strain pair yielding a different curve. Linearizations using secant or incremental moduli of elasticity can be carried out in terms of many alternate definitions. For a choice of two stress and three strain descriptions we may define six sets of secant moduli and nine sets of incremental moduli. Here presented are explicit relationships in tabular form among the different types of secant and incremental moduli at arbitrary levels of deformation. It is hoped that the tables will serve as ready tools to compare and interconvert uniaxial test data presented by different investigators.

VARIATION OF POISSON'S RATIOS OF THE AORTIC TISSUE

ABSTRACT Uniaxial tests were carried out on 185 circumferential and longitudinal strips from seven porcine aortic segments. Axial strain as well as one of the two transverse strains were measured at several levels of deformation. The second transverse strain was computed from the requirement of tissue incompressibility. Four Poissons ratios were computed at several values of axial strains as the negatives of the ratios of the averaged transverse strains and the axial strains. The results indicate that (1) the four Poissons ratios were distinct and in general unequal at the corresponding levels of the axial strains, (2) two Poissons ratios decreased and two increased with increasing axial strains, and (3) the values of the four Poissons ratios ranged from 0.12 to 0.75 approximately. These findings are consistent with the accepted picture of the aortic tissue as a nonlinear orthotropic material capable of undergoing large deformations.