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.     

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.     

Experimental assessment of wall shear flow in models

The blood flow immediately adjacent to the wall of a blood vessel or an artificial surface is of great interest. This flow defines the shear stress at the wall and is known to have a great physiological importance. The use of models is a viable method to investigate this flow. However, even in models the shear stress at the wall is difficult to assess. A new optical method is based on transparent models and uses particles in the model fluid, which are only visible near the wall. This is achieved with a model fluid having a defined opacity. This fluid obscures particles in the center of the models, but permits the observation and recording of particles close to the wall. The method has been applied for Hagen-Poiseuille flow and for the likewise well researched flow in a tube with a sudden expansion.     

The velocity of the arterial pulse wave: a viscous-fluid shock wave in an elastic tube

Background

The arterial pulse is a viscous-fluid shock wave that is initiated by blood ejected from the heart. This wave travels away from the heart at a speed termed the pulse wave velocity (PWV). The PWV increases during the course of a number of diseases, and this increase is often attributed to arterial stiffness. As the pulse wave approaches a point in an artery, the pressure rises as does the pressure gradient. This pressure gradient increases the rate of blood flow ahead of the wave. The rate of blood flow ahead of the wave decreases with distance because the pressure gradient also decreases with distance ahead of the wave. Consequently, the amount of blood per unit length in a segment of an artery increases ahead of the wave, and this increase stretches the wall of the artery. As a result, the tension in the wall increases, and this results in an increase in the pressure of blood in the artery.

Methods

An expression for the PWV is derived from an equation describing the flow-pressure coupling (FPC) for a pulse wave in an incompressible, viscous fluid in an elastic tube. The initial increase in force of the fluid in the tube is described by an increasing exponential function of time. The relationship between force gradient and fluid flow is approximated by an expression known to hold for a rigid tube.

Results

For large arteries, the PWV derived by this method agrees with the Korteweg-Moens equation for the PWV in a non-viscous fluid. For small arteries, the PWV is approximately proportional to the Korteweg-Moens velocity divided by the radius of the artery. The PWV in small arteries is also predicted to increase when the specific rate of increase in pressure as a function of time decreases. This rate decreases with increasing myocardial ischemia, suggesting an explanation for the observation that an increase in the PWV is a predictor of future myocardial infarction. The derivation of the equation for the PWV that has been used for more than fifty years is analyzed and shown to yield predictions that do not appear to be correct.

Conclusion

Contrary to the theory used for more than fifty years to predict the PWV, it speeds up as arteries become smaller and smaller. Furthermore, an increase in the PWV in some cases may be due to decreasing force of myocardial contraction rather than arterial stiffness.     

Numerical solution of unsteady blood flow through an indented tube with atherosclerosis

The unsteady blood flow through an indented tube with atherosclerosis in the presence of mild stenosis has been studied numerically by finite difference method. The effects of hematocrit, frequency parameter, height of stenosis, parameter determining the shape of the constriction on velocity field, volumetric flow rate, pressure gradient of the fluid in stenotic region and wall shear stress at the surface of stenosis are obtained and shown graphically.     

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.     

In vitro evaluation of flow divertors in an elastase-induced saccular aneurysm model in rabbit

Endovascular coiling is an acceptable treatment of intracranial aneurysms, yet long term follow-ups suggest that endovascular coiling fails to achieve complete aneurysm occlusions particularly in wide-neck and giant aneurysms. Placing of a stentlike device across the aneurysm neck may be sufficient to occlude the aneurysm by promoting intra-aneurysmal thrombosis; however, conclusive evidence of its efficacy is still lacking. In this study, we investigate in vitro the efficacy of custom designed flow divertors that will be subsequently implanted in a large cohort of animals. The aim of this study is to provide a detailed database against which in vivo results can be analyzed. Six custom designed flow divertors were fabricated and tested in vitro. The design matrix included three different porosities (75%, 70%, and 65%). For each porosity, there were two divertors with one having a nominal pore density double than that of the other. To quantify efficacy, the divertors were implanted in a compliant elastomeric model of an elastase-induced aneurysm model in rabbit and intra-aneurysmal flow changes were evaluated by particle image velocimetry (PIV). PIV results indicate a marked reduction in intra-aneurysmal flow activity after divertor implantation in the innominate artery across the aneurysm neck. The mean hydrodynamic circulation after divertor implantation was reduced to 14% or less of the mean circulation in the control and the mean intra-aneurysmal kinetic energy was reduced to 29% or less of its value in the control. The intra-aneurysmal wall shear rate in this model is low and implantation of the flow divertor did not change the wall shear rate magnitude appreciably. This in vitro experiment evaluates the characteristics of local flow phenomena such as hydrodynamic circulation, kinetic energy, wall shear rate, perforator flow, and changes of these parameters as a result of implantation of stentlike flow divertors in an elastomeric replica of elastase-induced saccular aneurysm model in rabbit. These initial findings offer a database for evaluation of in vivo implantations of such devices in the animal model and help in further development of cerebral aneurysm bypass devices.     

Linear propagation of pulsatile waves in viscoelastic tubes

An experimental and theoretical analysis is made of pulsatile wave propagation in deformable latex tubes as a model of the propagation of pressure pulses in arteries. A quasi one-dimensional linear model is used in which, in particular, attention is paid to the viscous phenomena in fluid and tube wall. The agreement between experimental and theoretical results is satisfactory. It appeared that the viscoelastic behaviour of the tube wall dominates the damping of the pressure pulse. Several linear models are used to describe the wall behaviour. No significant differences between the results of these models were found.     

Inflation waves induced by axial acceleration of the aorta

The phenomenon of high-amplitude inflation waves resulting from a sharp axial acceleration of the aorta, as may occur in road accidents, is investigated theoretically. The aorta is modeled as an axisymmetric tapered membranic shell (tube) made of an incompressible, nonlinear viscoelastic material with cylindrical orthotropy. It is filled with an inviscid, incompressible fluid whose flow is considered as quasi-one dimensional along the tube axis. The equations of motion of the tube and of the fluid are solved numerically, by using a two-step explicit scheme, for several axial acceleration profiles. The solutions shows that an inflation wave is generated and it propagates in opposite direction to that of the acceleration. The wall stresses, deformations and their time derivatives as well as fluid velocity and pressure are determined along the tube at different time intervals. Peak axial and circumferential stresses are high, with the latter far exceeding the former. These stresses may cause rupture of the aorta.     

A numerical study of flow in curved tubes simulating coronary arteries

Numerical simulations of pulsatile flow in coronary arteries which take into account the curvature associated with the bending of arteries over the surface of the heart are presented for resting, excited and drug induced states. The study was motivated by reported observations of atherosclerotic plaque localization on the inner curvature of coronary arteries. The simulated flow field appears quasi-steady under resting conditions with wall shear stress always highest on the outside wall and only a single secondary flow vortex in the half tube. However, reversal of wall shear stress direction at the inside wall does occur under resting flow conditions and this is not a quasi-steady characteristic. The flow field is markedly unsteady under excited conditions with wall shear stress sometimes peaking on the inside wall and an increase in the magnitude of wall shear stress reversal on the inside wall. However, only a single secondary flow vortex in the half tube is observed. Implications of the simulations for the role of fluid mechanics in coronary artery atherosclerosis are also discussed.     

Elastic expulsion from a long tube

The paper presents a theoretical analysis of elastic expulsion from a long pressurized tube following instantaneous severance. The governing equations of motion and continuity are used, together with an equation representing radial equilibrium of the tube wall, to construct a differential equation for the inner radius of the tube as a function of axial position and time. Similarity solutions for this equation are presented for a Newtonian fluid. The effect of the relaxed/fully-extended radius ratio on wall dilatation and expulsion rate is discussed.     

Stenotic effects in a tube of elliptic cross-section at low reynolds numbers

With an objective to understanding arteriosclerosis, the blood flow in a cylindrical tube with local constriction is analysed. The cross-section of the tube is an ellipse, the axes of which are in an arbitrary position with respect to the axis of the tube. Blood is taken to be a Newtonian and homogeneous fluid. The cross-sectional area varies slowly with the longitudinal distance and the area change is so adjusted to take account of stenosis. The transverse velocity field and the effects of inertia on the primary velocity and pressure distribution are calculated to a first order in the relevant small parameter and effects of asymmetry on the wall shear stress and impedance are presented.     

Response of the quiescent heart tube to mechanical stretch in the intact chick embryo

Quiescent heart tubes in intact eight-somite chick embryos were mechanically stretched by injecting excess fluid into the heart lumen. The stretch stimuli sometimes caused precocious focal twitches in the ventricular portion of the primitive heart tube. The contractile response to mechanical stretch was confirmed by recording electrograms before, during, and after the pressure pulse injections. The results prompt us to suggest that the precise timing of the initial heartbeats in the intact embryo may involve an increase in the intraluminal fluid volume and pressure stretching the heart tube wall.     

Blood flow in small curved tubes

Blood flow in small curved tubes is modeled by the two-fluid model where a relatively cell-free fluid layer envelops a fluid core of higher viscosity. The parameters in the model are successfully curve fitted to experimental data for straight tubes. The curved tube equations are then solved by perturbation theory. It was found that curvature in general lowers the tube resistance, but increases the shear stress near the inside wall.     

Developing steady laminar flow through uniform straight tubes with varying wall cross curvature

Numerical calculations are used to determine not only the wall shear stress but also the entry length in a laminar steady flow of an incompressible Newtonian fluid. The fluid is conveyed through rigid straight tubes with axially uniform cross sections, which mimic collapsed vessels. For each tube configuration, the "Navier-Stokes" equations are solved using the finite element method. The numerical tests are performed with the same value of the volume flow-rate whatever the tube configuration for three "Reynolds numbers". The wall shear stress is computed and determined along the axis of the tube, then the entry length is estimated by introducing two indexes by using: (i) the axial fluid velocity, and (ii) the wall shear stress. The results are analysed in order to exhibit the mechanical environment of cultured endothelial cells in the flow chamber for which the test conditions will be well-defined. For example, in a tube configuration where the opposite walls are in contact for which the inner perimeter and the area of the cross section are respectively given by 45 mm and 37.02 mm(2), the computed entry lengths with the criteria defined by (i) and (ii) are equals to about 118 and 126 mm, respectively for R(e0) = 500.     

The securing of a polyethylene tube prior to intravascular dye injection in anatomic microcirculation research: the three-tie technique

A simple and reliable method for securing the cannulated polyethylene tube to the blood vessel wall prior to intravascular dye injection is described. The three-tie technique has been used in many vascular anatomic studies without any leakage of the injection fluid taking place around the tube.     

Flow in arteries in the presence of stenosis

Of concern in the paper is a study of blood flow in an arterial segment having a stenosis. The artery is modelled as an initially stressed orthotropic elastic tube filled with a viscous incompressible fluid. The analysis is based on the assumption of the presence of a mild stenosis in the artery. Blood is treated as a Newtonian fluid. The effect of the surrounding connective tissues on the motion of the wall has been incorporated. Propagation of small amplitude harmonic waves, generated due to the flow of blood where the wave length is large compared to the radius of the arterial segment, is is considered in detail.     

Low pressure condition of a lipid core in an eccentrically developed carotid atheromatous plaque: a static finite element analysis

Plaque ruptures in atherosclerotic carotid arteries cause cerebral strokes. Accumulation of lipoproteins in the deep intimal layer forms a lipid core (LC), whose progression may be enhanced by mechanical conditions on the arterial wall. In this study, we investigated the pressure conditions of a liquid LC through numerical simulations of a sliced segment finite element (FE) model and a three-dimensional (3D) symmetric FE model. A model of an LC filled with nearly incompressible fluid was compared with incompressible and soft neo-Hookean LC models in a static FE analysis. Material constants for a nonlinear hyperelastic model of the arterial wall were identified based on an inflation test using a tube specimen. The results from the FE analysis of a sliced segment model show an LC fluid pressure as low as 1.9 kPa at a blood pressure of 16 kPa. A neo-Hookean LC model with a Young’s modulus of 0.06 kPa produced an almost uniform pressure in the LC within an error of 1.3 %. The 3D model predicted a similar level of LC pressure. Such low fluid pressure in the LC region may enhance the infiltration of lipoproteins and other substances from the lumen and facilitate transport through microvessels from the adventitia to the LC.     

Temperature transitions of protein properties in human red blood cells.

Human red blood cells (RBC) undergo a sudden change from blocking to passing through 1.3 +/- 0.2-micrometer micropipettes at a transition temperature (Tc) of 36.4 degrees C. For resealed RBC ghosts this transition occurs at 28.3 degrees C (Tg). These findings are attributed to an elastomeric transition of hemoglobin from being gel-like to a fluid and to an elastomeric transition of membrane proteins such as spectrin. Spectrin shows a uniform distribution along the aspirated RBC tongue above Tg in contrast to the linear gradient below Tg.