Wave Propagation through a Viscous Fluid Contained in a Tethered, Initially Stressed, Orthotropic Elastic Tube

To give a realistic representation of the pulse propagation in arteries a theoretical analysis of the wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube is considered. The tube is assumed to be orthotropic and its longitudinal motion is constrained by a uniformly distributed additional mass, a dashpot and a spring. The fluid is assumed to be Newtonian. The analysis is restricted to propagation of small amplitude harmonic waves whose wavelength is large compared to the radius of the vessel. Elimination of arbitrary constants from the general solutions of the equations of motion of the fluid and the wall gives a frequency equation to determine the velocity of propagation. Two roots of this equation give the velocity of propagation of two distinct outgoing waves. One of the waves propagates slower than the other. The propagation properties of s lower waves are very slightly affected by the degree of anisotropy of the wall. The velocity of propagation of faster waves decreases as the ratio of the longitudinal modulus of elasticity to the circumferential modulus decreases; transmission of these waves is very little affected. The influence of the tethering on the propagation velocity of slower waves is negligibly small; transmission of these waves is seriously affected. In tethered tubes faster waves are completely attenuated.     

Wave Propagation through a Newtonian Fluid Contained within a Thick-Walled,Viscoelastic Tube

The propagation of harmonic pressure waves through a Newtonian fluid contained within a thick-walled, viscoelastic tube is considered as a model of arterial blood flow. The fluid is assumed to be homogeneous and Newtonian, and its motion to be laminar and axisymmetric. The wall is assumed to be isotropic, incompressible, linear, and viscoelastic. It is also assumed that the motion is such that the convective acceleration is negligible. The motion of the fluid is described by the linearized form of the Navier-Stokes equations and the motion of the wall by classical elasticity theory. The frequency dependence of the wall mechanical properties are represented by a three parameter, relaxation-type model. Using boundary conditions describing the continuity of stress and velocity components in the fluid and the wall, explicit solutions for the system of equations of the model have been obtained. The longitudinal fluid impedance has been expressed in terms of frequency and the system parameters. The frequency equation has been solved and the propagation constant also expressed in terms of frequency and system parameters. The results indicate that the fluid impedance is smaller than predicted by the rigid tube model or by Womersley''s constrained elastic tube model. Also, the velocity of propagation is generally slower and the transmission per wavelength less than predicted by Womersley''s elastic tube model. The propagation constant is very sensitive to changes in the degree of wall viscoelasticity.     

Wave Propagation through a Viscous Incompressible Fluid Contained in an Initially Stressed Elastic Tube

To have a better understanding of the flow of blood in arteries a theoretical analysis of the pressure wave propagation through a viscous incompressible fluid contained in an initially stressed tube is considered. The fluid is assumed to be Newtonian. The tube is taken to be elastic and isotropic. The analysis is restricted to tubes with thin walls and to waves whose wavelengths are very large compared with the radius of the tube. It is further assumed that the amplitude of the pressure disturbance is sufficiently small so that nonlinear terms of the inertia of the fluid are negligible compared with linear ones. Both circumferential and longitudinal initial stresses are considered; however, their origins are not specified. Initial stresses enter equations as independent parameters. A frequency equation, which is quadratic in the square of the propagation velocity is obtained. Two out of four roots of this equation give the velocity of propagation of two distinct outgoing waves. The remaining two roots represent incoming waves corresponding to the first two waves. One of the waves propagates more slowly than the other. As the circumferential and/or longitudinal stress of the wall increases, the velocity of propagation and transmission per wavelength of the slower wave decreases. The response of the fast wave to a change in the initial stress is on the opposite direction.     

A model of a radially expanding and contracting lymphangion

The lymphatic system is an extensive vascular network featuring valves and contractile walls that pump interstitial fluid and plasma proteins back to the main circulation. Immune function also relies on the lymphatic system's ability to transport white blood cells. Failure to drain and pump this excess fluid results in edema characterized by fluid retention and swelling of limbs. It is, therefore, important to understand the mechanisms of fluid transport and pumping of lymphatic vessels. Unfortunately, there are very few studies in this area, most of which assume Poiseuille flow conditions. In vivo observations reveal that these vessels contract strongly, with diameter changes of the order of magnitude of the diameter itself over a cycle that lasts typically 2-3s. The radial velocity of the contracting vessel is on the order of the axial fluid velocity, suggesting that modeling flow in these vessels with a Poiseuille model is inappropriate. In this paper, we describe a model of a radially expanding and contracting lymphatic vessel and investigate the validity of assuming Poiseuille flow to estimate wall shear stress, which is presumably important for lymphatic endothelial cell mechanotransduction. Three different wall motions, periodic sinusoidal, skewed sinusoidal and physiologic wall motions, were investigated with steady and unsteady parabolic inlet velocities. Despite high radial velocities resulting from the wall motion, wall shear stress values were within 4% of quasi-static Poiseuille values. Therefore, Poiseuille flow is valid for the estimation of wall shear stress for the majority of the lymphangion contractile cycle.     

Transmission characteristics of axial waves in blood vessels

The elastic behavior of blood vessels can be quantitatively examined by measuring the propagation characteristics of waves transmitted by them. In addition, specific information regarding the viscoelastic properties of the vessel wall can be deduced by comparing the observed wave transmission data with theoretical predictions. The relevance of these deductions is directly dependent on the validity of the mathematical model for the mechanical behavior of blood vessels used in the theoretical analysis. Previous experimental investigations of waves in blood vessels have been restricted to pressure waves even though theoretical studies predict three types of waves with distinctly different transmission characteristics. These waves can be distinguished by the dominant displacement component of the vessel wall and are accordingly referred to as radial, axial and circumferential waves. The radial waves are also referred to as pressure waves since they exhibit pronounced pressure fluctuations. For a thorough evaluation of the mathematical models used in the analysis it is necessary to measure also the dispersion and attenuation of the axial and circumferential (torsion) waves.

To this end a method has been developed to determine the phase velocities and damping of sinusoidal axial waves in the carotid artery of anesthetized dogs with the aid of an electro-optical tracking system. For frequencies between 25 and 150 Hz the speed of the axial waves was between 20 and 40 m/sec and generally increased with frequency, while the natural pressure wave travelled at a speed of about 10 m/sec. On the basis of an isotropic wall model the axial wave speed should however be approximately 5 times higher than the pressure wave speed. This discrepancy can be interpreted as an indication for an anisotropic behavior of the carotid wall. The carotid artery appears to be more elastic in the axial than in the circumferential direction.     

Microstructures in the Organ of Corti Help Outer Hair Cells Form Traveling Waves along the Cochlear Coil

According to the generally accepted theory of mammalian cochlear mechanics, the fluid in the cochlear scalae interacts with the elastic cochlear partition to generate transversely oscillating displacement waves that propagate along the cochlear coil. Using a computational model of cochlear segments, a different type of propagating wave is reported, an elastic propagating wave that is independent of the fluid-structure interaction. The characteristics of the propagating wave observed in the model, such as the wavelength, speed, and phase lag, are similar to those observed in the living cochlea. Three conditions are required for the existence of the elastic propagating wave in the cochlear partition without fluid-interaction: 1), the stiffness gradient of the cochlear partition; 2), the elastic longitudinal coupling; and 3), the Y-shaped structure in the organ of Corti formed by the outer hair cell, the Deiters cell, and the Deiters cell phalangeal process. The elastic propagating waves in the cochlear partition disappeared without the push-pull action provided by the outer hair cell and Deiters cell phalangeal process. The results suggest that the mechanical feedback of outer hair cells, facilitated by the organ of Corti microstructure, can control the tuning and amplification by modulating the cochlear traveling wave.     

An alternative ultrasonic method for measuring the elastic properties of cortical bone

We studied the elastic properties of bone to analyze its mechanical behavior. The basic principles of ultrasonic methods are now well established for varying isotropic media, particularly in the field of biomedical engineering. However, little progress has been made in its application to anisotropic materials. This is largely due to the complex nature of wave propagation in these media. In the present study, the theory of elastic waves is essential because it relates the elastic moduli of a material to the velocity of propagation of these waves along arbitrary directions in a solid. Transducers are generally placed in contact with the samples which are often cubes with parallel faces that are difficult to prepare. The ultrasonic method used here is original, a rough preparation of the bone is sufficient and the sample is rotated. Moreover, to analyze heterogeneity of the structure we measure velocities in different points on the sample. The aim of the present study was to determine in vitro the anisotropic elastic properties of cortical bones. For this purpose, our method allowed measurement of longitudinal and transverse velocities (C(L) and C(T)) in longitudinal (fiber direction) and the radial directions (orthogonal to the fiber direction) of compact bones. Young's modulus E and Poisson's ratio nu, were then deduced from the velocities measured considering the compact bone as transversely isotropic or orthotropic. The results are in line with those of other methods.     

Microstructures in the Organ of Corti Help Outer Hair Cells Form Traveling Waves along the Cochlear Coil

According to the generally accepted theory of mammalian cochlear mechanics, the fluid in the cochlear scalae interacts with the elastic cochlear partition to generate transversely oscillating displacement waves that propagate along the cochlear coil. Using a computational model of cochlear segments, a different type of propagating wave is reported, an elastic propagating wave that is independent of the fluid-structure interaction. The characteristics of the propagating wave observed in the model, such as the wavelength, speed, and phase lag, are similar to those observed in the living cochlea. Three conditions are required for the existence of the elastic propagating wave in the cochlear partition without fluid-interaction: 1), the stiffness gradient of the cochlear partition; 2), the elastic longitudinal coupling; and 3), the Y-shaped structure in the organ of Corti formed by the outer hair cell, the Deiters cell, and the Deiters cell phalangeal process. The elastic propagating waves in the cochlear partition disappeared without the push-pull action provided by the outer hair cell and Deiters cell phalangeal process. The results suggest that the mechanical feedback of outer hair cells, facilitated by the organ of Corti microstructure, can control the tuning and amplification by modulating the cochlear traveling wave.     

Numerical simulation for the propagation of nonlinear pulsatile waves in arteries.

The behavior of nonlinear pulsatile flow of incompressible blood contained in an elastic tube is examined. The theory takes into account the nonlinear convective terms of the Navier-Stokes equations. The motion of the arterial wall is characterized by a set of linearized differential equations. The region bounded by the flexible arterial wall is mapped into a fixed area in which numerical discretization takes place. The finite element method (Galerkin weighted residual approach) is used for the solution of this nonlinear system. The results obtained are pressure distribution, velocity profile, flow rate and wall displacements along the elastic tube (20 cm long).     

Theory of longitudinal plasma waves with allowance for ion motion

A study is made of the propagation of steady-state large-amplitude longitudinal plasma waves in a cold collisionless plasma with allowance for both electron and ion motion. Conditions for the existence of periodic potential waves are determined. The electric field, potential, frequency, and wavelength are obtained as functions of the wave phase velocity and ion-to-electron mass ratio. Taking into account the ion motion results in the nonmonotonic dependence of the frequency of the waves with the maximum possible amplitudes on the wave phase velocity. Specifically, at low phase velocities, the frequency is equal to the electron plasma frequency for linear waves. As the phase velocity increases, the frequency first decreases insignificantly, reaches its minimum value, and then increases. As the phase velocity increases further, the frequency continues to increase and, at relativistic phase velocities, again becomes equal to the plasma frequency. Finally, as the phase velocity approaches the speed of light, the frequency increases without bound.     

The Dissipation and Dispersion of Small Waves in Arteries and Veins with Viscoelastic Wall Properties

Theoretical and experimental evidence suggests that the dissipation of high frequency pressure waves in blood vessels is caused primarily by the viscoelastic behavior of the vessel wall. In this theoretical analysis the vessels are considered as fluid-filled circular cylindrical shells whose walls have isotropic and homogeneous viscoelastic properties and are subjected to an initial axial stretch and a transmural pressure. If the wall material is incompressible and behaves as a Voigt solid in shear, the results predict a decrease in wave amplitude per wavelength which is essentially independent of frequency over a wide range. This finding is in qualitative agreement with recent experiments on anesthetized dogs. A parametric study also shows a great sensitivity of the dissipation to changes in transmural pressure and axial stretch. Axisymmetric waves are only mildly dispersive, while all nonaxisymmetric waves are highly dispersive and exhibit much stronger damping per wavelength at low frequencies than do axisymmetric waves.     

An introduction to biofluid mechanics--basic models and applications

Cardiovascular disease is the primary cause of morbidity and mortality in the western world. Complex hemodynamics play a critical role in the development of atherosclerosis and the processes of aging, as well as many other disease processes. Biofluid mechanics play a major role in the cardiovascular system and it is important to understand the forces and movement of blood cells and whole blood as well as the interaction between blood cells and the vessel wall. Fundamental fluid mechanical, which are important for the understanding of the blood flow in the cardiovascular circulatory system of the human body aspects are presented. Measurement techniques for model studies such as LDA, ultrasound, and MRI studies will be discussed. Viscosity and flow behavior changes specifically the creation of vortices and flow disturbances can be used to show how medication can influence flow behavior. Experiments have shown that hemodynamics may have a strong influence on the creation of aneurysms and varicose veins. Other factors such as vessel wall structure are also important. In preliminary studies, it has been demonstrated that geometry and elasticity of vessel walls help determine flow behavior. High velocity fluctuations indicate flow disturbances that should be avoided. Health care practitioners must understand fluid dynamic factors such as flow rate ratio, pressure and velocity gradients, and flow behavior, velocity distribution, shear stress on the wall and on blood cells. These mechanical factors are largely responsible for the deposit of blood cells and lipids, a leading cause of atherosclerosis. The interaction between blood cells and of the cells with the vessel, leads to the formation of plaques and agglomerations. These deposits are found predominantly at arterial bends and bifurcations where blood flow is disturbed, where a secondary flow is created, and where flow separation regions are found. Experiments on hemodynamic effects in elastic silicon rubber models of the cardiovascular system with flow wire, stents, or patches for vessel surgery will be discussed. These studies can be important in improving diagnostics and therapeutic applications.     

Effects of Viscosity and Constraints on the Dispersion and Dissipation of Waves in Large Blood Vessels: I. Theoretical Analysis

The propagation of sounds and pulse waves within the cardiovascular system is subject to strong dissipative mechanisms. To investigate the effects of blood viscosity on dissipation as well as dispersion of small waves in arteries and veins, a parametric study has been carried out. A linearized analysis of axisymmetric waves in a cylindrical membrane that contains a viscous fluid indicates that there are two families of waves: a family of slow waves and one of fast waves. The faster waves are shown to be more sensitive to variations in the elastic properties of the medium surrounding the blood vessels and at high values of the frequency parameter α defined by α = √ρωR²₀/μ the blood viscosity attenuates them more strongly over a length than the slow waves. At low values of α, the effects of viscosity on attenuation are reversed; that is, the family of slow waves is much more attenuated than the family of fast waves. For the slow waves the radial displacement component generally exceeds the axial component except at very low frequencies. Conversely the axial displacements are much larger than the radial displacement for the faster waves. The presence of external constraints, however, can modify these results. In the case of the slow waves the phase angle between pressure and radial wall displacement is virtually negligible in the presence of mild external constraints, while the phase angles between pressure and fluid mass flow are at most 45°. The corresponding phase angles for the fast waves exhibit much larger variations with changes in the elastic properties of the surrounding medium.     

A multiscale approach for modelling wave propagation in an arterial segment

A mathematical model of blood flow through an arterial vessel is presented and the wave propagation in it is studied numerically. Based on the assumption of long wavelength and small amplitude of the pressure waves, a quasi-one-dimensional (1D) differential model is adopted. It describes the non-linear fluid-wall interaction and includes wall deformation in both radial and axial directions. The 1D model is coupled with a six compartment lumped parameter model, which accounts for the global circulatory features and provides boundary conditions. The differential equations are first linearized to investigate the nature of the propagation phenomena. The full non-linear equations are then approximated with a numerical finite difference method on a staggered grid. Some numerical simulations show the characteristics of the wave propagation. The dependence of the flow, of the wall deformation and of the wave velocity on the elasticity parameter has been highlighted. The importance of the axial deformation is evidenced by its variation in correspondence of the pressure peaks. The wave disturbances consequent to a local stiffening of the vessel and to a compliance jump due to prosthetic implantations are finally studied.     

Swimming of spermatozoa in a linear viscoelastic fluid.

A modified resistive force theory is developed for a spermatozoon swimming in a general linear viscoelastic fluid. The theory is based on a Fourier decomposition of the flagellar velocity, which leads to solving the Stokes flow equations with a complex viscosity. We use a model spermatozoon with a spherical head which propagates small amplitude sinusoidal waves along its flagellum. Results are obtained for the velocity of propulsion and the rate of working for a free swimming spermatozoon and the thrust on a fixed spermatozoon. There is no change in propulsive velocity for a viscoelastic fluid compared to a Newtonian fluid. The rate of working does change however, decreasing with increasing elasticity of the fluid, for a Maxwell fluid. Thus the theory predicts that a spermatozoon can swim faster in a Maxwell fluid with the same expenditure of energy for a Newtonian fluid.     

Stress wave velocities in bovine patellar tendon.

The velocity of longitudinal stress waves in an elastic body is given by the square root of the ratio of its elastic modulus to its density. In tendinous and ligamentous tissue, the elastic modulus increases with strain and with strain rate. Therefore, it was postulated that stress wave velocity would also increase with increasing strain and strain rate. The purpose of this study was to determine the velocity of stress waves in tendinous tissue as a function of strain and to compare these values to those predicted using the elastic modulus derived from quasi-static testing. Five bovine patellar tendons were harvested and potted as bone-tendon-bone specimens. Quasi-static mechanical properties were determined in tension at a deformation rate of 100 mm/s. Impact loading was employed to determine wave velocity at various strain levels, achieved by preloading the tendon. Following impact, there was a measurable delay in force transmission across the specimen and this delay decreased with increasing tendon strain. The wave velocities at tendon strains of 0.0075, 0.015, and 0.0225 were determined to be 260 +/- 52 m/s, 360 +/- 71 m/s, and 461 +/- 94 m/s, respectively. These velocities were significantly (p < 0.01) faster than those predicted using elastic moduli derived from the quasi-static tests by 52, 45, and 41 percent, respectively. This study has documented that stress wave velocity in patellar tendon increases with increasing strain and is underestimated with a modulus estimated from quasi-static testing.     

Wave Propagation in a Viscous Fluid Contained in an Orthotropic Elastic Tube

The problem of pressure wave propagation through a viscous fluid contained in an orthotropic elastic tube is considered in connection with arterial blood flow. Solutions to the fluid flow and elasticity equations are obtained for the presence of a reflected wave. Numerical results are presented for both isotropic and orthotropic elastic tubes. In particular, the pressure pulse, flow rate, axial fluid velocity, and wall displacements are plotted vs. time at various stations along the ascending aorta of man. The results indicate an increase in the peak value of the pressure pulse and a decrease in the flow rate as the pulse propagates away from the heart. Finally, the velocity of wave propagation depends mainly on the tangential modulus of elasticity of the arterial wall, and anisotropy of the wall accounts in part for the reduction of longitudinal movements and an increase in the hydraulic resistance.     

Pressure wave propagation in fluid-filled co-axial elastic tubes. Part 1: Basic theory

Our work is motivated by ideas about the pathogenesis of syringomyelia. This is a serious disease characterized by the appearance of longitudinal cavities within the spinal cord. Its causes are unknown, but pressure propagation is probably implicated. We have developed an inviscid theory for the propagation of pressure waves in co-axial, fluid-filled, elastic tubes. This is intended as a simple model of the intraspinal cerebrospinal-fluid system. Our approach is based on the classic theory for the propagation of longitudinal waves in single, fluid-filled, elastic tubes. We show that for small-amplitude waves the governing equations reduce to the classic wave equation. The wave speed is found to be a strong function of the ratio of the tubes' cross-sectional areas. It is found that the leading edge of a transmural pressure pulse tends to generate compressive waves with converging wave fronts. Consequently, the leading edge of the pressure pulse steepens to form a shock-like elastic jump. A weakly nonlinear theory is developed for such an elastic jump.     

Effect of initial stresses on the wave propagation in arteries

A theoretical analysis for the problem of wave propagation in arteries is presented. Blood is treated as a Newtonian, viscous incompressible fluid. On the basis of information derived from experimental investigations on the mechanical properties of wall tissues, the arterial wall is considered to be nonlinearly viscoelastic and orthotropic. The analysis is carried out for a cylindrical artery, under the purview of membrane theory, by taking account the effect of initial stresses. The motion of the wall and that of the fluid are assumed to be axisymmetric. Particular emphasis has been paid to the propagation of small amplitude harmonic waves whose wavelength is large compared to the radius of the vessel. By employing the equations of motion of the fluid and those for the wall, together with the equations of continuity, a frequency equation is derived by exploiting the conditions of continuity of the velocity of the arterial wall and that of blood on the endosteal surface of the wall. In order to illustrate the validity of the derived analytical expressions a quantitative analysis is made for the variations of the phase velocities as well as the transmission coefficient with frequency corresponding to different transmural pressures which relate to different initial stresses. Computational results indicate that phase velocities increase with the increase of transmural pressures.     

The effect of shear stress on solitary waves in arteries

In the present work, we study the propagation of solitary waves in a prestressed thick walled elastic tube filled with an incompressible inviscid fluid. In order to include the geometric dispersion in the analysis the wall inertia and shear deformation effects are taken into account for the inner pressure-cross-sectional area relation. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the long-wave approximation is examined. It is shown that, contrary to thin tube theories, the present approach makes it possible to have solitary waves even for a Mooney-Rivlin (M-R) material. Due to dependence of the coefficients of the governing Korteweg-deVries equation on initial deformation, the solution profile changes with inner pressure and the axial stretch. The variation of wave profiles for a class of elastic materals are depicted in graphical forms. As might be seen from these illustrations, with increasing thickness ratio, the profile of solitary wave is steepened for a M-R material but it is broadened for biological tissues.