Hemodynamic Flow in Anisotropic, Viscoelastic Thick-Wall Vessels

Abstract

Hemodynamic flow in anisotropic, viscoelastic thick-wall vessels is analyzed. For small amplitude harmonic waves, the wavelengths of which are large compared to the vessel radius, the fluid motion is governed by the linearized form of the Navier-Stokes equations and the vessel motion is described by the classical Navier equations for solids. Compressible and incompressible transversely isotropic vessels with frequency dependent material moduli are considered. Longitudinal constraints are also incorporated. The quantities calculated are: the velocities of propagation, transmission per wavelength, fluid impedance, and the displacements and stresses at the fluid-vessel interface. The results indicate that the earlier anisotropic analyses, which incorporate membrane or thick shell theories, agree qualitatively with the present thick-wall analysis. However, quantitative differences among the analyses do occur, especially as the compressibility of the vessel material decreases. For the thick-wall analysis, the effects of anisotropy on the various quantities of interest are established. In particular, it is shown that the fluid impedance is relatively insensitive to the degree of anisotropy, but the resistance and inductance are very sensitive to anisotropy at the high and low values of frequency respectively. In general, anisotropy significantly affects the first mode transmission per wavelength and second mode phase velocity. The displacements and stresses at the fluid-vessel interface are also altered by anisotropy. In addition, the arterial system was found to be essentially independent of moderate variations in the axial shear modulus of the material. Furthermore, it is demonstrated that the degree of viscoelasticity associated with the vessel material substantially alters the transmission per wavelength but affects the phase velocity to a much lesser degree. For the suggested value of the spring parameter associated with the longitudinal constraint, it is found that the additional mass and the viscous parameters have little effect on the system behavior. The overall effect of the longitudinal constraint is to completely subdue the second mode of transmission, increase the first mode phase velocity, decrease the first mode transmission per wavelength and axial displacement, and to increase the modulus and decrease the phase of the first mode fluid impedance. In addition to confirming the importance of anisotropy, viscoelasticity, and the longitudinal constraint, the investigation demonstrates that an appropriate thick-wall analysis is required in order to adequately describe the fluid-vessel behavior.