Biomechanical behavior of the arterial wall and its numerical characterization.

This paper presents the biomechanical behavior of two types of rat arteries under the passive state of the vascular smooth muscles. The hyperelastic, highly nonlinear, incompressible and orthotropic stress-strain behavior of arteries is described by adopting a classical material model of the exponential type and a 'biphasic' one, recently proposed by the authors. In order to obtain the fundamental relations for a computer simulation the theoretical continuum-mechanical background is briefly reviewed in a compact manner as well. By using the new material model the study shows a significant improvement in approaching the experimental stress-strain data within the entire pressure domain (from 0 up to 200 mm Hg) and for various levels of prestretch encompassing physiological pressures and the individual in vivo axial prestretches.

Unstable radii in muscular blood vessels.

A model of a muscular blood vessel in equilibrium that predicts stable and unstable control of radius is presented. The equilibrium wall tension is modeled as the sum of a passive exponential function of radius and an active parabolic function of radius. The magnitude of the active tension is varied to simulate the variable level of smooth muscle activation. This tension-radius relationship is then converted to an equilibrium pressure-radius relationship via Laplace's law. This model predicts the traditional ability to control the radius below a critical level of activation. However, when the active tension is raised above this critical level, the pressure-radius relationship (with pressure plotted on the ordinate and radius on the abscissa) becomes N shaped with a relative maximal pressure (Pmax) and a relative minimal pressure (Pmin). For this N-shaped curve, there are three equilibrium radii for any pressure between Pmin and Pmax. Analysis shows that the middle radius is unstable and thus cannot be maintained at equilibrium. Previously unexplained experimental data reveal evidence of this instability.

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.

Passive elastic properties of the rat abdominal vena cava.

The quasistatic passive venous elastic properties were studied in-vitro on 6 cylindrical segments of abdominal vena cava from Wistar rats. Using noncontact methods of deformation measurement, diameter and axial force of the segments were analyzed as a function of simultaneous axial stretch and internal pressure in the physiological range of 0-2.7 kPa. The elasticity of the wall tissue was investigated in terms of moduli of elasticity in the circumferential, axial and radial directions. Results show that the pressure-diameter relationship is highly nonlinear, indicating that veins are extremely compliant at lowest pressures and rather stiff beginning from some 0.7 kPa of pressure. The axial force decreases with pressure at small prestretches, increases at large, but remains constant for the in-vivo prestretch. The venous wall tissue is markedly anisotropic in the entire physiological range of deformations.

Isotropy and anisotropy of the arterial wall.

The passive biomechanical response of intact cylindrical rat carotid arteries is studied in vitro and compared with the mechanical response of rubber tubes. Using true stress and natural strain in the definition of the incremental modulus of elasticity, the tissue wall properties are analyzed over wide ranges of simultaneous circumferential and longitudinal deformations. The type of loading chosen is 'physiological' i.e. symmetric: the cylindrical segments are subjected to internal pressure and axial prestretch without torsion or shear. Several aspects pertaining to the choice of parameters characterizing the material are discussed and the analysis pertaining to the deformational behavior of a hypothetical compliant tube with Hookean wall material is presented. The experimental results show that while rubber response can be adequately represented as linearly elastic and isotropic, the overall response of vascular tissue is highly non-linear and anisotropic. However, for states of deformation that occur in vivo, the elasticity of arteries is quite similar to that of rubber tubes and as such the arterial wall may be viewed as incrementally isotropic for the range of deformations that occur in vivo.

A stress-strain relation for a rat abdominal aorta.

Assuming the arterial wall is homogeneous, incompressible, isotropic and elastic, a stress-strain relation has been presented for a rat's abdominal aorta. As an illustrating example, the problem of simultaneous inflation and the axial stretch of a cylindrical artery under physiological loading has been solved and then the material coefficients are determined by comparing theoretical results with the existing experiments. The result indicates that the maximum deviation between the theory and experiment for various pressure levels is 3.7% which seems to be a good approximation of theory to the experiments. The variation of circumferential stress and the incremental pressure modulus with inner pressure are also depicted in the work.

In situ study of active and passive mechanical properties of rat tail artery.

The present study constitutes an effort to close the gap still existing between in vivo and in vitro experiments in vascular mechanics. Active and passive mechanical properties of the rat tail artery were studied in situ under conditions as close as possible to the natural in vivo state of the vessel. At constant pressure levels, changes in diameter as caused after dilatation by papaverine and constriction by norepinephrine, were automatically and continuously registered using a contact free measurement technique (Video Dimension Analyzer). In order to check consistency of the in situ findings with analogous results from in vitro studies, pressure-diameter and stress-strain data were compared with results from our own and other authors. The results show that, quite similarly to in vitro, the maximum isobaric response of the tail artery in situ occurs in the physiological pressure range of some 100 mm Hg. Although the results focus the attention on the importance of both the experimental protocols and the definitions used, they confirm the validity of in vitro investigations as a powerful tool in arterial rheology.

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.

[Length-force relation of rat-carotid artery at different transmural pression. Experiments and models (author's transl)]. / Das Kraft-Dehnungs-Verhalten von Rattenkarotiden in Längsrichtung bei verschiedenem Innendruck und seine modellmässige Deutung.

The relation between force and extension in longitudinal direction as a function of the internal pressure was examined in isolated carotid arteries of rats using a device for testing the stress-strain relation of fibers (Vibrodyn). In the region of small longitudinal extensions the longitudinal force is decreased by raising the internal pressure. The longitudinal force is increased as a function of the internal pressure in the region of high longitudinal extensions. This behaviour can be explained by a model of the arterial wall which takes into account the spiral structure of the fibers. We could find a good qualitative agreement between this model and our experiments. It can be concluded that the histological structure plays an important role in determining the elastic behaviour of the arterial wall.