Effect of hypertension on elasticity and geometry of aortic tissue from dogs.

Inflation-extension experiments were carried out on segments of the descending thoracic aortas from 4 normotensive and 4 hypertensive dogs rendered hypertensive using either unilateral or bilateral renal artery constriction. Intravascular pressures up to 200 mm Hg and axial forces up to 200 g were used. The external diameter of the segment and the distance between two longitudinally spaced gage marks were recorded photographically at each pressure-force level combination. Dimensions in the underformed configuration were measured at the end of the inflation-extension experiment. Data were analyzed for changes in geometry and force-deformation response. Results indicate that: 1. Under sustained hypertension the wall thickness in the underformed configuration increases with a concurrent reduction in the in-situ longitudinal extension ratio. 2. This dual tissue response accomplishes substantial reductions in the circumferential and longitudinal stresses from the levels that would be reached at equivalent pressures in the absence of these geometric changes. 3. At comparable intravascular pressures the extensibility in the circumferential direction is slightly greater for the hypertensive aortas as compared to normals. However, the stress-extension ratio relationship in the circumferential direction is similar in the two groups. 4. The stress-extension ratio relationship in the longitudinal direction indicates that the hypertensive aorta is stiffer than its normotensive counterpart.

A theoretical method for estimating small vessel distensibility in humans.

A simple theoretical approach is presented for estimating vascular distensibility of small blood vessels from noninvasively obtained pressure-flow data in the hand and forearm of human subjects. To the extent that Poiseuille's law applies to blood flow in these vascular beds, conductance (the reciprocal of vascular resistance) can be calculated from these data as the ratio of blood flow to mean arterial pressure. The fourth root of the conductance is proportional to the radius of the vascular bed. The slope of the relation between the logarithm of the radius of the vascular bed and the transmural pressure is proportional to the vascular extensibility (E), which, in turn, for small deformations and constant vascular length, is proportional to the distensibility of small blood vessel. Data obtained from the hands of six hypertensive subjects were compared with that obtained from six normotensive subjects, all with their vascular beds in a maximally dilated state. Also compared were data obtained from four normal subjects with their vascular beds in the resting state and when the beds were maximally dilated. The results indicate that 1) in the hypertensive subjects, the small blood vessels of the maximally dilated vascular bed of the hand are significantly (p less than 0.02) less distensible (E = 0.126 +/- 0.034/mm Hg) than those in the normotensive subjects (E = 0.272 +/- 0.047/mm Hg) and 2) the small blood vessels of the normal forearm at resting levels of vasomotor tone are more distensible (E = 1.00 +/- 0.38/mm Hg) than in the maximally dilated state (E = 0.51 +/- 0.08/mm Hg).

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.

Incremental formulations in vascular mechanics.

The pulsatile deformations of the large arteries can be viewed as small time-varying deformations superposed on large deformations. This motivates the study of the incremental deformations of the vascular tissue. Unfortunately, because of the variety of possible choices of definitions of stresses and strains, the choice of the characterizing incremental moduli is not unique, which has led to much ambiguity and confusion in the literature. This communication systematically presents some of the options available for characterization of orthotropic incremental deformations of the vascular tissue, and provides explicit formulas for interconversions of incremental elastic moduli for uniaxial tests on strips of incompressible tissue. Relative merits of various choices are discussed.

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.

Mathematical characterization of the nonlinear thermorheological behavior of the vascular tissue.

A characterization of the passive nonlinear thermorheological response of incompressible, curvilinearly orthotropic arterial tissue is presented in the framework of modern continuum thermodynamics. The stress tensor, the specific entropy, the specific internal energy and the heat flux vector and expressed as functionals of the histories of local deformation, temperature and the temperature gradient. These functionals are systematically reduced by subjecting them to the requirements of Clausius-Duhem inequality and material frame indifference. The reduced functionals are then specialized to reflect the material frame indifference. The reduced functionals are then specialized to reflect the material symmetry characterizing the tissue by using the histories of the joint invariants of the Green-St. Venant strain tensor and temperature as the independent argument functions. The functionals are expressed in terms of series of multiple integrals and terms upto and including second order integrals are retained. An approach toward experimental determination of the 14 constitutive functions to describe two stress differences is outlined. It is believed that the characterization presented here will provide a rational basis for simpler thermorheological descriptions and experimental programs to include important thermorheologic considerations.

Static linear and nonlinear elastic properties of normal and arterialized venous tissue in dog and man.

Ten normal and four transplanted canine jugular vein segments and four human saphenous vein segments were studied to determine the in vitro static elastic properties of venous tissue and their modification by transplantation into the arterial system. Both the intraluminal pressure and the longitudinal force were varied, and the resulting dimensions were recorded photographically. Venous segments manifested a hysteresis response but showed minimum tendency to creep. The pressure-strain relationships were curvilinear with an initial, highly compliant phase over the physiological venous pressure range followed by a relatively noncompliant phase. This transition occurred at lower pressures for jugular segments than it did for saphenous segments. In contrast, comparable-sized canine carotide artery segments did not show this essentially noncompliant phase over the pressure range studied (0 to 200 cm H2O). At comparable pressures and strains, the jugular vein segments were stiffer than the saphenous vein segments in both the circumferential and the longitudinal directions. At comparable strains, the saphenous vein moduli were similar to those in the carotid artery segments. Jugular segments transplanted into arterial circuits became virtually noncompliant and markedly inhomogeneous, with wall thickening and a histologic picture of intimal proliferation. They showed no tendency to "arterialize," that is, they failed to assume either the elastic or the histologic characteristics of arterial tissue.

A microindentation technique to measure rheological properties of the vascular intima.

A microindentor was developed to measure the depth of indentation of the intimal surface of arterial tissue loaded by flat-ended, cylindrical probes. The depth of indentation depended on the initial stretch of the tissue which required a rigid support (plaster of Paris) beneath the adventitial surface. Probe tips used ranged from 550-mum down to 65-mum diameter while loads ranged from 800 to 15 mg. The depth of indentation was markedly time dependent; that obtained 30 s after loading (variation of 30) was reproducible and served as a useful parameter of viscoelasticity of the aortic intima and supporting tissues of dog and man. The mean variation of 30 (0.19-mm diam tip, 120-mg load), obtained from longitudinal series of indentations of nine dog aortas, ranged from 40.8 to 68.8 mum while coefficient of variation in these series ranged from 4.8 to 15.9%. Intimal pads were found to have greater resistance to identation than adjacent tissue; likewise the tissue on the dorsal, intimal surface of the aorta had lower variation of 30 values compared with the rest of the intima. Lipid-filled intimal regions were about twice as complaint as macroscopically spared areas. The technique should prove useful in understanding the microrheological response of the blood vascular interface to hydrodynamic stresses.

Nonlinear anisotropic elastic properties of the canine aorta.

A nonlinear theory of large elastic deformations of the aortic tissue has been developed. The wall tissue has been considered to be incompressible and curvilinearly orthotropic. The strain energy density function for the tissue is expressed as a polynomial in the circumferential and longitudinal Green-St. Venant strains. Limiting application to states of strains wherein the geometric axes are the principal axes and truncating the energy expression to include terms with highest degrees 2, 3, and 4, three expressions with 3, 7, and 12 constitutive constants are obtained. Results of application of these expressions to data from three series of in vitro and in vivo experiments involving 31 dogs have been presented. Whereas all the three expressions are found to be applicable to various degrees, the third-degree expression for the strain energy density function with seven constitutive constants is particularly recommended for general use.

Stress distribution in the canine left ventricle during diastole and systole.

A model is proposed for stress analysis of the left ventricular wall (LV wall) based on the realistic assumption that the myocardium is essentially composed of fiber elements which carry only axial tension and vary in orientation through the wall. Stress analysis based on such a model requires an extensive study of muscle fiber orientation and curvature through the myocardium. Accordingly, the principal curvatures were studied at a local site near the equator in ten dog hearts rapidly fixed in situ at end diastole and end systole; the fiber orientation for these hearts had already been established in a previous study. The principal radii of curvature were (a) measured by fitting templates to the endocardial and epicardial wall surfaces in the circumferential and longitudinal directions and (b) computed from measured lengths of semiaxes of ellipsoids of revolution representing the LV wall ("ellipsoid" data). The wall was regarded as a tethered set of nested shells, each having a unique fiber orientation. Results indicate the following. (a) Fiber curvature, k, is maximum at midwall at end systole; this peak shifts towards endocardium at end diastole. (b) The pressure or radial stress through the wall decreases more rapidly near the endocardium than near the epicardium at end diastole and at end systole when a constant tension is assumed for each fiber through the wall. (c) At end diastole the curve for the circumferential stress vs. wall thickness is convex with a maximum at midwall. In the longitudinal direction the stress distribution curve is concave with a minimum at midwall. Similar distributions are obtained at end systole when a constant tension is assumed for each fiber through the wall. (d) The curvature and stress distributions obtained by direct measurements at a selected local site agree well with those computed from "ellipsoid" data.