Distensible Tube Research Articles

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° 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° the perturbation is almost extinct when the pulse wave returns to the proximal extremity.

Unsteady wall shear stress in a distensible tube

An asymptotic expression of the wall shear stress (WSS) in an elastic tube is deduced for small values of the Womersley parameter. In the case of a rigid tube this asymptotic expression is shown to compare better with the exact solution than Poiseuille's or Lambossy's approximations. Its integration in a one-dimensional model of the internal carotid artery blood flow predicts more marked systolic and less marked diastolic WSS than those predicted by the commonly used Poiseuille's approximation.

A transmission line modelling approach to the interpretation of uterine Doppler waveforms.

An electrical analogue model of an artery that terminates into a vascular bed is presented. The model consists of an uniform transmission line that represents the artery and a load impedance that represents the vascular bed. The transmission line parameters are based on a well-established first-order approximation of the Navier-Stokes equations for fluid flow in distensible tubes. The model can be used to predict the incident and reflected components of both the arterial pressure and flow waveforms. In addition, it can predict the vessel diameter change and the mean blood velocity waveforms. In this study, the model was applied to the uterine artery so that the characteristics of the utero-placental circulation can be related to Doppler ultrasound recordings. It was found that the presence of the dicrotic notch in the uterine artery time-velocity waveform is the result of wave reflection and that a persistent notch past 20 weeks' gestation may be indicative of an abnormally high placental bed resistance. It is shown that the simulation results are consistent with the known physiological data and the clinically recorded uterine Doppler waveforms.

Reflection in liquid filled tube systems: Modelling the effects of mismatched tube segments

Here we employ an experimentally tested theory of wave propagation in liquid filled distensible tubes to study the interaction of periodic pressure waves with a finite segment of tubing whose characteristic impedance differs from those of the adjoining tubes. Such a tube system occurs when surgical bypass techniques are employed in the arterial system. In this case the source of the pressure pulse (the heart) is periodic and reflections from the insert could reinforce subsequent pulses arriving from the heart causing abnormally high pressures in the adjoining artery. This paper sheds some light on this phenomenon.

Dynamics of the upper urinary tract: I. Peristaltic flow through a distensible tube of limited length

A theoretical analysis and numerical solutions are presented for viscosity-dominated flow through a uniform, highly distensible tube with a slit-shaped cross section undergoing peristalsis. At flow rates below an intrinsic upper limit to the peristaltic carrying capacity of the tube, provided that there is always at least one peristaltic contraction wave in the tube, two different types of steady peristaltic flow are possible, with well defined boluses separated by regions where the tube is nearly completely occluded (flow with isolated boluses and flow with boluses in contact). Above this limit two other non-steady types of flow are possible, in which the boluses are less well defined and the contraction waves are in general not completely occlusive (flow with leaky boluses and open-tube flow). In flow with isolated boluses, boluses in contact and leaky boluses, the upstream pressure is determined by the mechanical properties (active and passive) of the tube undergoing peristalsis. In open-tube flow it is primarily determined by the resistance of the outlet from the tube.

Model of a distensible tube admitting localized waves

Model of a distensible tube admitting localized waves

Pulse-wave model of brachial arterial pressure modulation in aging and hypertension.

We apply a pulse-wave theory to a model of the human arm arterial system that predicts the changes in the arterial pressure waveform as it traverses the vasculature (increased pulse pressure, sharper main wave, disappearance of the aortic incisura, and appearance of a diastolic dicrotic wave) and also predicts the observed modulation of the waveform during phenylephrine-induced vasoconstriction and nitroglycerin-induced vasodilation. The model considers the arm arterial system as a tapered, distensible tube that bifurcates and then ends in a loop, with sidebranch networks represented by distributed Windkessels. The model uses verifiable values for realistic parameters. We found that the vertical modulation of the dicrotic wave, in humans, decreased with advancing age and with high blood pressure. The model explained these findings in terms of increasing vascular rigidity and decreasing small vessel vasodilator responsiveness. We noted a significant negative correlation between the arterial level of plasma norepinephrine and the amount of modulation of the dicrotic wave after nitroglycerin among subjects 40 yr old or younger, suggesting a sympathetic neurogenic contribution to the vascular abnormalities observed in relatively young patients with essential hypertension.

Wave propagation and reflection in liquid filled distensible tube systems exhibiting dissipation and dispersion

In a previous paper the present authors developed a model describing wave propagation in liquid filled distensible tubes and tested it against impulse experiments involving water filled latex rubber tubes. This model incorporates both dissipative and dispersive mechanism which are absent from the commonly employed linear long wave-length (LLW) theory of haemodynamics. This higher order theory is here employed to study propagation of an impulse in a semi infinite tube, reflection of an impulse from the distal end of a finite length tube, and reflection and transmission of impulses impinging on a function connecting dissimilar liquid filled tubes. Both open and closed type reflections are treated and numerical results presented graphically. To the best of our knowledge this is the first time that such a higher order theory has been employed to treat reflection and transmission of waves in tube systems.

Impulse propagation in liquid filled distensible tubes: Theory and experiment for intermediate to long wavelengths

A theory of wave propagation in liquid filled distensible tubes, which extends the region of validity of the linear long wavelength theory commonly employed in haemodynamics, is presented and tested against experiments on water filled latex tubes. The theory which is linear and involves but a single free parameter provides excellent agreement with the impulse experiments.

Simulation of coronary artery revascularization

Simulation of the commonly constructed geometries of aorto-coronary bypass anastomoses was carried out using especially fabricated distensible tubes and a pulsatile pump. The system pressure was maintained between 80 and 120 mm Hg. The total mean flow was set at 250 ml min −1 (Reynolds number of 200) and the pulsatile frequency was varied from 0 to 2 Hz. A water-glycerine mixture having a density and viscosity similar to that of blood was used throughout. A 16 mm film of the front of black dye injected proximal to the anastomosis was made as the dye approached and passed through the anastomosis. Anastomotic geometries consisted of: end to side, parallel, 45° angle, and 90° angle. Stenoses, located in the tube representing the coronary artery, were simulated using a bevelled insert which represented an 80–85% area reduction. Flow visualization revealed that distensible tubes gave more realistic flow patterns than rigid tubes, a result particularly evident when a stenosis was present. Pulsatile flow demonstrated considerably more mixing than steady flow. The use of pulsatile flow in distensible tubing with a partial stenosis showed retrograde flow through the stenosis which was not evident for either steady flow or for flow in rigid tubing. The flow at the anastomatic site of the graft having an angle of 0° showed a jetting action with a zone of recirculating fluid being present whereas for a 90 degree graft a distinct helical flow was formed distal to the anastomosis.

Propagation and reflection of waves in finite length liquid-filled distensible shells

Employing a previously established theory for wave propagation in liquid-filled distensible tubes [1] we analyse the propagation and subsequent reflection of a transient pulse from the distal end of a finite length tube. Conditions pertaining to pulse generation at the proximal end of the tube are specified so as to approximate the conditions of experiments carried out on water-filled latex tubes [1]. Laplace transforms are employed and numerical results presented graphically.

Pressure pulses in fluid filled distensible tubes

By comparing experimental records with model solutions we are led to propose a dispersion relation governing the propagation of pressure pulses in fluid filled distensible tubes. This relation contains a single undetermined parameter having the dimension of time. We show how this parameter may be interpreted and obtain an estimate of its value. Some comments concerning the speed of propagation of pressure waves in the haemodynamics contest are made.

Inflation waves in non-linear distensible tubes

A mathematical model for large amplitude wave propagation in a thin walled distensible tube is developed. The tube wall is considered as a membranic shell made of an incompressible, non-linear viscoelastic material with cylindrical orthotropy. The fluid is regarded as incompressible and inviscid and the flow is quasi-one-dimensional. The case of a pressure step applied at one end of a uniform straight tube is solved as an example. The system of partial differential equations, describing the motions of the fluid and the wall, are integrated numerically by using a two-step explicit scheme. Flow and deformation variables as well as the wave velocity are determined in time and space.

Solitons in distensible tubes

The unsteady flow of an incompressible fluid in distensible tubes is investigated by means of solutions of a nonlinear wave equation. The formalism applied is related to the theory of flat water waves; the propagation of impulses is described as solitons. Relations connecting parameters describing the propagation and the physical properties, respectively, are given. Various orders of approximation are compared. Dispersion relations of the linearized version are considered; flow patterns are calculated for typical cases.

Effect of pulsatility on oxygen transport to the human arterial wall

Oxygen transfer in fully-developed, pulsating, laminar flow in rigid and distensible tubes was simulated as part of a study of vessel-wall hypoxia and atherogenesis. The model used for the computations is based on dimensions and flows in the human thoracic aorta. The pulsatile velocity fields employed are based on those of Womersley with the modification that the radial convection is written relative to the moving wall. Womersley-type pulsatility, superposed on an axial Poiseuille flow, was found to negligibly affect oxygen transport to the wall. Caution should be exercised, however, in extending this conclusion to the complex pulsatility in actual living vessels.

The effect of the skin friction on the solution of the one-dimensional equations of pulsatile flow in distensible tubes.

The one-dimensional equations are used in the calculation of blood flow in arteries. The majority of the treatments use the method of characteristics and because of the nature of the method it is necessary to use a simplified value of the skin friction. A commonly used simplification is to assume the zero frequency value of the skin friction. The effect of the use of this approximation is compared with results using the full linear theory value. It is shown that the phase difference between the skin friction and the flux has an appreciable effect on the velocity wave calculated from a given pressure wave.

Propagation characteristics in distensible tubes containing a visco-elastic fluid

The propagation characteristics of a visco-elastic fluid in a distensible tube tube are studied. The linear visco-elastic nature of the fluid is described by a complex coefficient of viscosity η ∗ . The equation of motion of the vessel wall takes into account the pulsatile nature of the wall. Results are presented for wave propagation velocity, the resistance and the reactance of the fluid and the wall impedance. It is seen that the visco-elastic influence is significant for high values of the frequency of oscillation in various arterial vessels.

Shock waves in fluid-filled distensible tubes.

Flow of a liquid through distensible tubes is of interest primarily in biological systems, and some properties of shock waves in such tubes are discussed. In shock-fixed coordinates, these flows are steady, and the shock is associated with an increase of pressure and cross-sectional area. Shock transition is analyzed for two flow models, namely, immediate flow separation, when the flow enters the shock zone, and no separation. Shock properties are expressed in terms of the speed index (ratio of the velocity of the shock to that of a small-amplitude wave) and dissipation (loss of total pressure). Examples are worked out for the thoracic aorta of an anesthetized dog, a perfectly elastic tube, and a partially collapsed tube. Appreciable differences in shock velocity and dissipation result if either flow separation or no separation is assumed.

Radiology now. Wide ureters, a dilemma in diagnosis.

The presence of an abnormally wide ureter, or ureters, on an intravenous urogram is a fairly common finding in any radiology department. All too often it is implied that such dilatation must be due to some form of obstruction, and this unfortunate misinterpretation may lead to a needless or incorrect operation. The ureter is a distensible muscular tube whose function is to propel urine from the kidneys to the bladder. It responds to an increased fluid load by an increase in its peristaltic activity and, if the load is great enough, by compensatory hypertrophy, and then dilatation. This type of dilatation is observed in diabetes insipidus, where the urine output may approach 20 litres per day. A much commoner example of dilatation due to an increased fluid load is seen with vesico-ureteric reflux, particularly in children, where it can produce marked widening of the ureters with or without associated calyceal abnormality.

Radiology now. Dynamic bladder studies.

“Persisting stress incontinence following surgery: cystogram please” is a request to strike despair into many radiologists. What useful information is to be gleaned here from the micturition cysto-urethrogram (MCU)? Or from adult MCUs in general? How good are we at diagnosing bladder outflow obstruction by this examination, or explaining common presenting symptoms like frequency, urgency or incontinence? Not very good, it must be said. It is instructive to inquire why this at times inconclusive, lumbering investigation in the adult turns so readily into a highly rewarding, delicate study in the child. The questions asked there are of course different. With the common presenting problem of childhood urinary infection we want to know about vesicoureteric reflux, about major bladder disarrays like diverticula, and in boys about outflow obstruction. These are apt questions for the MCU, so they are answered. Bladder outflow obstruction in boys is usually sited in the urethra well below the bladder neck (e.g. urethral valve). Watching opaque medium passing briskly through a distensible tube of predictable calibre like the urethra is a good way of detecting obstruction along its length. Contrast this with the common questions we have to ask about adult MCUs: is bladder behaviour abnormal, or is there outflow obstruction at or just below the bladder neck? Is it likely that we could learn much about the action of a balloon or its neck by watching its contents pouring out?