Attenuation of flow disturbances in tapered arterial grafts

Flow disturbances in tapered arterial grafts of angles of taper between 0.5 and 1.0 deg were measured in vitro using a pulsed ultrasound Doppler velocimeter. The increase in transition Reynolds numbers with angle of taper and axial distance was determined for steady flow. The instantaneous centerline velocities were measured distal to a 50 percent area stenosis (as a model of a proximal anastomosis), in steady and pulsatile flow, from which the disturbance intensities were calculated. A significant reduction in post-stenotic disturbance intensity was recorded in the tapered grafts, relative to a conventional cylindrical graft. In pulsatile flow with a large backflow component, however, there was an increase in disturbance intensity due to diverging flow during flow reversal. This was observed only in the 1.0 deg tapered graft. These findings indicate that taper is an important consideration in the design of vascular prostheses.     

Pressure losses in non-Newtonian flow through rigid wall tapered tubes

Pressure drop and pressure gradient were measured in steady Newtonian and non-Newtonian flow through tapered tubes having angles of taper, alpha, between 0.5 degree and 1.25 degrees. Aqueous solutions of polyacrylamide, characterized as power law fluids, were used for the non-Newtonian flow measurements. These solutions had power law parameters similar in magnitude to those of blood. The pressure drop-flow rate data compared well with the predictions of a semi-empirical flow model over a large range of flow rates (Re alpha up to 10 for Newtonian flow and 5.7 for non-Newtonian flow). The pressure gradient increased with distance, z, into the taper as the radius decreased. The linear relationship between pressure gradient and z, derived by Oka (Biorheology, 10, 207-212, 1973) was found to be valid only when alpha z was small. For the tapered tubes examined here, agreement was confined to the region near the inlet. If higher order terms in alpha z were taken into account the agreement was extended further into the taper. However, under higher flow conditions, when the inertial losses are not negligible, the semi-empirical model provides much better estimates of pressure gradient.     

Flow measurements in an atherosclerotic curved, tapered femoral artery model of man

Flow visualization and pressure measurements were made for physiological conditions in a model derived from a femoral angiogram of a patient with lesion localization on the inner curvature wall and with vessel taper. Effects of curvature and taper were evaluated separately in other curved, tapered, smooth and straight, tapered, smooth models. Double helical secondary flow patterns were modified by plaque on the inner wall, and flow separations were observed between plaques at higher flow rates and Reynolds numbers. Pressure drop data for the plaque simulation model were similar in trend with Reynolds number as for the smooth model, but flow resistances were 25 to 40 percent higher. Significant pressure drops were measured due to the mild taper which could be estimated from momentum considerations, and smaller increased pressure drops were found due to curvature effects at the higher Dean numbers. Flow resistances for in vivo pulsatile flow simulation were about 10 percent higher than for steady flow for the plaque model, whereas no differences were observed for the smooth model.     

Blood flow in tapered tubes with biorheological applications

A steady laminar flow of blood in a uniform tapered tube has been examined. Blood rheology is assumed to be described by a polar fluid. The analytical expressions for velocities (both axial and radial), total angular velocity, wall shear and pressure drop have been obtained. In literature, the parameters N (coupling number) and L (length ratio) have been chosen independently. But, in the present analysis, it is found that they are interrelated. Variation of the flow variables with suspension concentration and tapered angle have been investigated. Some of the theoretical models for the flow through tapered tubes have been critically examined. The pressure-flow relationship has been studied numerically over the flow rate range 0.01-0.1 cc/sec and compared with experimental results. It has been shown that the existing experimental results are for the tapered tubes of larger diameter which correspond to the flow under Newtonian conditions. Finally, some biological implications and future developments of this theory have been indicated.     

Flow-pressure drop measurement and calculation in a tapered femoral artery of a dog

This study describes the in vivo measurement of pressure drop and flow during the cardiac cycle in the femoral artery of a dog, and the computer simulation of the experiment based on the use of the measured flow, vessel dimensions and blood viscosity. In view of the experimental uncertainty in obtaining the accurate velocity profile at the wall region, the velocity pulse at the center was measured and numerical calculations were performed for the center Une instantaneous velocity and within the two limits of spatial distribution of inlet flow conditions: uniform and parabolic. Temporal and spatial variations of flow parameters, i.e., velocity profile, shear rate, non-Newtonian viscosity, wall shear stress, and pressure drop were calculated. There existed both positive and negative shear rates during a pulse cycle, i.e., the arterial wall experiences zero shear three times during a cardiac cycle. For the parabolic inlet condition, the taper of the artery not only increased the magnitude of the positive and negative shear rates, but caused a steep gradient in shear rate, a phenomenon which in turn affects wall shear stress and pressure. In contrast, for the uniform inlet condition, the flow through the tapered artery was predominantly the developing type, which resulted in reduction in magnitude of wall shear rate along the axial direction.     

Convection-diffusion interaction for oscillatory flow in a tapered tube

Transport of soluble material is analyzed for volume-cycle oscillatory flow in a tapered tube. The equations of motion are solved using a regular perturbation method for small taper angle and order unity amplitude over a range of the Womersley parameter. The transport equation is also solved by a regular perturbation method where uniform end concentrations and no wall flux are assumed. The time-averaged axial transport of solute is calculated for several tapered tubes. There is substantial modification of transport compared to the straight tube case and the results are interpreted with respect to pulmonary gas exchange.     

One-dimensional model for propagation of a pressure wave in a model of the human arterial network: comparison of theoretical and experimental results

Pulse wave evaluation is an effective method for arteriosclerosis screening. In a previous study, we verified that pulse waveforms change markedly due to arterial stiffness. However, a pulse wave consists of two components, the incident wave and multireflected waves. Clarification of the complicated propagation of these waves is necessary to gain an understanding of the nature of pulse waves in vivo. In this study, we built a one-dimensional theoretical model of a pressure wave propagating in a flexible tube. To evaluate the applicability of the model, we compared theoretical estimations with measured data obtained from basic tube models and a simple arterial model. We constructed different viscoelastic tube set-ups: two straight tubes; one tube connected to two tubes of different elasticity; a single bifurcation tube; and a simple arterial network with four bifurcations. Soft polyurethane tubes were used and the configuration was based on a realistic human arterial network. The tensile modulus of the material was similar to the elasticity of arteries. A pulsatile flow with ejection time 0.3 s was applied using a controlled pump. Inner pressure waves and flow velocity were then measured using a pressure sensor and an ultrasonic diagnostic system. We formulated a 1D model derived from the Navier-Stokes equations and a continuity equation to characterize pressure propagation in flexible tubes. The theoretical model includes nonlinearity and attenuation terms due to the tube wall, and flow viscosity derived from a steady Hagen-Poiseuille profile. Under the same configuration as for experiments, the governing equations were computed using the MacCormack scheme. The theoretical pressure waves for each case showed a good fit to the experimental waves. The square sum of residuals (difference between theoretical and experimental wave-forms) for each case was <10.0%. A possible explanation for the increase in the square sum of residuals is the approximation error for flow viscosity. However, the comparatively small values prove the validity of the approach and indicate the usefulness of the model for understanding pressure propagation in the human arterial network.     

Steady flow through collapsible tubes: measurements of flow and geometry.

Compliant tubes attain a complex three-dimensional geometry when the external pressure exceeds the internal pressure and the tube is partially collapsed. A new technique for remote measurement of dynamic surfaces was applied to classical experiments with collapsible tubes. This work presents measurements of the three-dimensional structure of the tube as well as pressure and flow measurements during static loading and during steady-state fluid flow. Results are shown for two tubes of the same material and internal diameter but with different wall thicknesses. The measured tube laws compare well with previously published data and suggest the possible existence of a similarity tube law. The steady flow measurements did not compare well with the one-dimensional theoretical predictions.     

Theoretical analysis of pressure pulse propagation in arterial vessels.

An original mathematical model of viscous fluid motion in a tapered and distensible tube is presented. The model equations are deduced by assuming a two-dimensional flow and taking into account the nonlinear terms in the fluid motion equations, as well as the nonlinear deformation of the tube wall. One distinctive feature of the model is the formal integration with respect to the radial coordinate of the Navier-Stokes equations by power series expansion. The consequent computational frame allows an easy, accurate evaluation of the effects produced by changing the values of all physical and geometrical tube parameters. The model is employed to study the propagation along an arterial vessel of a pressure pulse produced by a single flow pulse applied at the proximal vessel extremity. In particular, the effects of the natural taper angle of the arterial wall on pulse propagation are investigated. The simulation results show that tapering considerably influences wave attenuation but not wave velocity. The substantially different behavior of pulse propagation, depending upon whether it travels towards the distal extremity or in the opposite direction, is observed: natural tapering causes a continuous increase in the pulse amplitude as it moves towards the distal extremity; on the contrary, the reflected pulse, running in the opposite direction, is greatly damped. For a vessel with physical and geometrical properties similar to those of a canine femoral artery and 0.1 degree taper angle, the forward amplification is about 0.9 m-1 and the backward attenuation is 1.4 m-1, so that the overall tapering effect gives a remarkably damped pressure response. For a natural taper angle of 0.14 degrees the perturbation is almost extinct when the pulse wave returns to the proximal extremity.     

The impedance of curved artery models

The impedance (pressure drop/flow rate) of four curved artery models has been determined experimentally for steady and periodic flows simulating conditions in the aortic arch. Steady flow results indicate that very short entry lengths are required for flow development in curved artery models, and impedance is elevated above straight tube values by a factor of 3-4 for mean flow conditions in the aortic arch. Results for periodic flow with a nonzero mean show a significant elevation of mean flow impedance relative to values for steady flow at the mean flow rate--a factor of 2-3 for aortic arch flow conditions. The impedance of the first harmonic of periodic flows follows straight tube theory at high values of the unsteadiness parameter in agreement with available theory for curved tubes. The implications of the impedance measurements for wall shear stress in the aortic arch are discussed.     

Experimental analysis of the influence of stenotic geometry on steady flow.

We studied the flow behavior under steady flow conditions in four models of cylindrical stenoses at Reynolds numbers from 150 to 920. The flow upstream of the constrictions was always fully developed. The constriction ratios of the rigid tubes (D) to the stenoses (d) were d/D = 0.273; 0.505; 0.548; 0.786. The pressure drop at various locations in the stenotic models was measured with water manometers. The flow was visualized with a photoelasticity apparatus using an aqueous birefringent solution. We also studied the flow behavior at pulsatile flow in a dog aorta with a constriction of 71%. The flow through stenotic geometries depends on the Reynolds number of the flow generated in the tube and the constriction ratio d/D. At low d/D ratios, (with the increased constriction), the flow separation zones (recirculation zones, so-called reattachment length) and flow disturbances increased with larger Reynolds numbers. At lower values, eddies were generated. At high Re, eddies were observed in the pre-stenotic regions. The pressure drop is a function of the length and internal diameter of the stenosis, respective ratio of stenosis to the main vessel and the Reynolds numbers. At low Re-numbers and low d/D, distinct recirculation zones were found close to the stenosis. The flow is laminar in the distal areas. Further experiments under steady and unsteady flow conditions in a dog aorta model with a constriction of 71% showed similar effects. High velocity fluctuations downstream of the stenosis were found in the dog aorta. A videotape demonstrates these results.     

Interaction of peristaltic flow with pulsatile flow in a circular cylindrical tube

The effect of pulsatile flow on peristaltic transport in a circular cylindrical tube is analysed. The flow of a Newtonian viscous incompressible fluid in a flexible circular cylindrical tube on which an axisymmetric travelling sinusoidal wave is imposed, is considered. The initial flow in the tube is induced by an arbitrary periodic pressure gradient. A perturbation solution with amplitude ratio (wave amplitude/tube radius) as a parameter is obtained when the frequency of the travelling wave and that of the imposed pressure gradient are equal. The interaction effects of periodic wall induced flow and periodic pressure imposed flow are visualized through the presence of substantially different components of steady and higher harmonic oscillating flow in the first order flow solution. Numerical results show a strong variation of steady state velocity profiles with boundary wave number and Reynolds number and a strong phase shift behaviour of the flow in the radial direction.     

Modeling expiratory flow from excised tracheal tube laws.

Flow limitation during forced exhalation and gas trapping during high-frequency ventilation are affected by upstream viscous losses and by the relationship between transmural pressure (Ptm) and cross-sectional area (A(tr)) of the airways, i.e., tube law (TL). Our objective was to test the validity of a simple lumped-parameter model of expiratory flow limitation, including the measured TL, static pressure recovery, and upstream viscous losses. To accomplish this objective, we assessed the TLs of various excised animal tracheae in controlled conditions of quasi-static (no flow) and steady forced expiratory flow. A(tr) was measured from digitized images of inner tracheal walls delineated by transillumination at an axial location defining the minimal area during forced expiratory flow. Tracheal TLs followed closely the exponential form proposed by Shapiro (A. H. Shapiro. J. Biomech. Eng. 99: 126-147, 1977) for elastic tubes: Ptm = K(p) [(A(tr)/A(tr0))(-n) - 1], where A(tr0) is A(tr) at Ptm = 0 and K(p) is a parametric factor related to the stiffness of the tube wall. Using these TLs, we found that the simple model of expiratory flow limitation described well the experimental data. Independent of upstream resistance, all tracheae with an exponent n < 2 experienced flow limitation, whereas a trachea with n > 2 did not. Upstream viscous losses, as expected, reduced maximal expiratory flow. The TL measured under steady-flow conditions was stiffer than that measured under expiratory no-flow conditions, only if a significant static pressure recovery from the choke point to atmosphere was assumed in the measurement.     

Measurement of steady-flow instability and turbulence levels in Dacron vascular grafts.

Fluid dynamic properties of Dacron vascular grafts were studied under controlled steady-flow conditions over a Reynolds number range of 800 to 4500. Knitted and woven grafts having nominal diameters of 6 mm and 10 mm were studied. Thermal anemometry was used to measure centerline velocity at the downstream end of the graft; pressure drop across the graft was also measured. Transition from laminar flow to turbulent flow was observed, and turbulence intensity and turbulent stresses (Reynolds normal stresses) were measured in the turbulent regime. Knitted grafts were found to have greater pressure drop than the woven grafts, and one sample was found to have a critical Reynolds number (Rc) of less than one-half the value of Rc for a smooth-walled tube.     

The flow of a viscous liquid in a converging tube

The problem of the viscous flow of an incompressible Newtonian liquid in a converging tapered tube has been solved in spherical polar coordinates. The method of the solution involves the Stokes' stream function and a technique introduced by Stokes in the study of a sphere oscillating in a fluid. The theory for the flow in a rigid tube includes: (1) the pulsatile flow with both radial and angular velocity components; (2) the steady state flow with both radial and angular velocity components and (3) the very slow steady state flow with only a radial velocity component present. For a tapered elastic tube, the velocity of the propagated pulse wave is determined. The solution given is in terms of the elastic constants of the system and the coordinates for this type of geometry. The pulse velocity is then related to the velocity in an elastic cylindrical tube with the necessary correction terms to account for the tapered tube. Supported in part by the American Heart Association (No. 62F4EG). This work was done during the tenure of an Established Investigatorship of the American Heart Association.     

Measurement of oscillatory flow pressure gradient in an elastic artery model

In vitro experiments were conducted to measure the oscillatory flow pressure gradient along an elastic tube in order to assess the recent nonlinear theory of Wang and Tarbell. According to this theory, in an elastic tube with oscillatory flow, the mean (time-averaged) pressure gradient cannot be calculated using Poiseuille's law. The effect of wall motion creates a nonlinear convective acceleration, and an induced mean pressure gradient is required to balance the convective acceleration. The induced mean pressure gradient depends on the diameter variation over a cycle, the pulsatility and unsteadiness of the flow, and the phase difference between the pressure wave form and the flow wave form. The amplitude of the pressure gradient also depends on these parameters and may deviate significantly from Womersley's rigid tube theory. A flow loop was constructed to produce oscillatory flow in an elastic tube. Flow wave forms were measured with an ultrasonic flow probe, and ultrasonic diameter crystals were used to measure wall movement. A special device for pressure drop measurement was constructed using Millar catheter tip transducers to obtain both forward and backward pressure drops that were then averaged. This averaging method eliminated the static error of the pressure transducers. The pressure-flow phase angle was varied by clamping a distal elastic section at various locations. Pressure gradients were obtained for a range of phase angles between −55 ° and +49 °. The mean and amplitude of the measured pressure gradient were compared to theoretical values. Both positive and negative induced mean pressure gradients were measured over the range of phase angles. The measured pressure gradient amplitudes were always lower than predicted by Womersley's rigid tube theory. The experimental means and amplitudes are in good agreement with the elastic tube theoretical values. Thus, the experiments verify the theory of Wang and Tarbell.     

Numerical modeling of simulated blood flow in idealized composite arterial coronary grafts: steady state simulations

This paper presents a comparative study of simulated blood flow in different configurations of simplified composite arterial coronary grafts (CACGs). Even though the composite arterial grafting is increasingly used in cardiac surgery, it is still questionable whether or not the blood flow in such grafts can adequately meet the demands of the native myocardial circulation. A computational fluid dynamics (CFD) model was developed to conduct computer-based studies of simulated blood flow in four different geometric configurations of CACGs, corresponding to routinely used networks in cardiac surgery coronary grafts (T, Y, Pi and sequential). The flow was assumed three-dimensional, laminar and steady and the fluid as Newtonian, while the vessel walls were considered as inelastic and impermeable. It was concluded that local haemodynamics, practically described by velocity, pressure drop, wall shear stress (WSS) and flow rates, may be strongly influenced by the local geometry, especially at the anastomotic sites. The computations were made at mean flow rates of 37.5, 75 and 150ml/min. The side-branch outflow rates, computed for each bypass graft, showed noticeable differences. The results, which were found both qualitatively and quantitatively consistent with other studies, indicate that the Pi-graft exhibits significantly less uniform distribution of outflow rates than the other geometric configurations. Moreover, prominent variations in WSS and velocity distribution among the assessed CACGs were predicted, showing remarkable flow interactions among the arterial branches. The lowest shear stress regions were found on the lateral walls of bifurcations, which are predominantly susceptible to the occurrence of coronary artery disease (CAD). In contrast, the highest WSS were observed at the turn of the arterial branches.     

A mathematical model of flow through a collapsible tube--I. Model and steady flow results

A one-dimensional model is presented to describe the flow through a collapsible tube whose ends are tethered to rigid tubes and which is enclosed in a pressurized chamber. Results are presented for the special case of steady flow. Predicted pressure drop versus flow rate (delta P-Q) characteristics agree qualitatively with available experimental data. The significance of the model and of various physical parameters, in regard to the shape of these characteristics, is discussed.     

Pressure drop and flow rate measurements in a human aortic bifurcation cast for steady and pulsatile flow

Pressure drop and flow rate measurements in a rigid cast of a human aortic bifurcation under both steady and physiological pulsatile flow conditions are reported. Integral momentum and mechanical energy balances are used to calculate impedance, spatially averaged wall shear stress and viscous dissipation rate from the data. In the daughter branches, steady flow impedance is within 30% of the Poiseuille flow prediction, while pulsatile flow impedance is within a factor of 2 of fully developed, oscillatory, straight tube flow theory (Womersley theory). Estimates of wall shear stress are in accord with measurements obtained from velocity profiles. Mean pressure drop and viscous dissipation rate are elevated in pulsatile flow relative to steady flow at the mean flow rate, and the exponents of their Reynolds number dependence are in accord with available theory.     

Errors in the Measurement of Root Pressure and Exudation Volume Flow Rate Caused by Damage during the Transfer of Unsupported Roots between Solutions

Plants of Zea mays were grown with their roots confined to growing tubes, consisting of cylindrical or spherical glass tubes fitted at the bottom with a stopcock. Nutrient solution was circulated past the roots, and when a plant was 21 to 25 days old, the stopcock was closed, the root excised from the plant and connected to an apparatus which measured root pressure and exudation volume flow rate. The stopcock was opened and solution was again circulated through the growing tube without dropping the level of the liquid bathing the root in the process. Measurements of pressure and flow rate were made continuously during a period in which the solution was replaced, first by draining and refilling the tube in situ, and second by replacing the growing tube with a beaker of solution. Both these manipulations caused at least temporary and frequently permanent drops in root pressure and flow rate. Plants were also grown in cylindrical tubes with a support medium of either glass beads or Raschig rings which filled the growing tubes. It is shown that the solution bathing these roots could be repeatedly replaced by draining and refilling with no visible effect on the measurements. It is recommended, therefore, that in future, support be provided for the roots of all experimental plants grown by solution culture.