A new model is presented to describe flow in segments of collapsible tube mounted between two rigid tubes and surrounded by a pressurized container. The new features of the model are the inclusion of (a) longitudinal wall tension and (b) energy loss in the separated flow downstream of the time-dependent constriction in a collapsing tube, in a manner which is consistent with the one-dimensional equations of motion. As well as accurately simulating steady-state collapse, the model predicts self-excited oscillations whose amplitude is large enough to be observable only if the flow in the collapsible tube becomes supercritical somewhere (fluid speed exceeding long-wave propagation speed). The dynamics of the oscillations is dominated by longitudinal movement of the point of flow separation, in response to the adverse pressure gradient associated with waves propagating backwards and forwards between the (moving) narrowest point of the constriction and the tube outlet.

A separated-flow model for collapsible-tube oscillations

The treatment of benign prostatic hyperplasia and the definition of bladder-outlet obstruction has preoccupied urologists and researchers in recent years. Bladder-outlet obstruction can be defined only by pressure-flow measurement. Various methods of analysis of pressure-flow data have been proposed. The Abrams-Griffiths nomogram is an easy method of classifying these data to distinguish between the presence or absence of obstruction. Using the values for the maximal flow and the corresponding voiding detrusor pressure a point can be plotted on the nomogram that determines whether the bladder outlet is obstructed, unobstructed, or equivocally obstructed. For those that fall in the equivocal zone, further criteria for the mean slope of the pressure-flow plot and the minimal voiding detrusor pressure are used to determine whether there is obstruction or not. The nomogram's prognostic value in predicting the outcome of prostatectomy has been studied and found to be excellent. The Abrams-Griffiths nomogram can be modified by assigning an Abrams-Griffiths number to each set of pressure-flow data. This number is easy to calculate and use and gives a continuous variable that can be used to evaluate the effects of therapy. Although the Abrams-Griffiths nomogram and number are somewhat simplistic, none of the more complex methods of pressure-flow analysis have been shown to be better predictors of treatment outcome to date.

The Abrams-Griffiths nomogram

Keywords:

urethral resistance;
pressure/flow relation;
external stream velocity;
flow rate controlling zone;
flow rate/volume relation;
bladder outlet obstruction;
voiding efficiency

Urethral resistance? Urodynamic concepts of physiological and pathological bladder outlet function during voiding

The mechanics of the urethra and of micturition.

Flows in thin-walled, collapsible tubes are of fundamental importance to various physiologic phenomena and clinical devices.A one-dimensional, unsteady theory is developed for flows generated either by externally applied pressures or by body forces. Part 1 deals with small-amplitude, linearized flows, part 2 with large amplitude, nonlinear flows. Experimental results for a tube collapsing under external pressure are given in part 3, together with theoretical interpretations and comparative results of numerical simulations.Several new and unanticipated phenomena are revealed. These are in part associated with the highly nonlinear ‘equation of state’ (transmural pressure versus area) for a partially collapsed tube, and in part with whether the flow speed is sub- or supercritical relative to the speed of area waves. For instance, a flow produced by a spatially uniform external pressure applied to a limited region becomes choked at a flow-limiting throat at which point the fluid speed reaches the local wave speed. This throat forms at the edge of the pressurized region. The critical velocity can be exceeded with the application of certain types of spatially graded external pressures.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112079001348

Unsteady flow in a collapsible tube subjected to external pressure or body forces

A one-dimensional theory of steady flow through a tube with non-uniform elastic properties is developed. The importance of the speed of pressure waves on the tube (the sonic speed) is pointed out, and the characteristics and stability of subsonic and supersonic flow are examined. Subsonic flow is always stable. Stable supersonic flow is sometimes possible, but latent instability may lead to a localized transition (a hydraulic jump) to subsonic flow, associated with energy dissipation through turbulence.

Hydrodynamics of male micturition. I. Theory of steady flow through elastic-walled tubes.

The theory of the flow of an incompressible, inviscid fluid through an elastic tube is developed and applied to a simple model of the female urethra during micturition. The importance of the fact that the flow velocity is comparable with the velocity of longitudinal elastic waves on the urethra is stressed. Flow rates and urethral profiles at different bladder pressures are calculated and found to agree qualitatively with experimental observations. In particular, it is shown that an elastic constriction near the mid-point of the urethra leads directly to an asymmetrical profile, funnelling down from the bladder neck to the constriction, and remaining narrow between the constriction and the external meatus, even in a urethra elastically symmetrical about the constriction. It is shown that the flow rate is governed by the elastic properties of the constriction, and that the effect of viscosity on the flow is relatively small, justifying the idealization that the fluid is inviscid.

Urethral elasticity and micturition hydrodynamics in females.

Measurements on a normal male of the radius and speed of the urine stream, and the relation between them, are described. The tendency of the radius to increase with increasing stream speed is evidence that the urethral walls are elastic. The theoretical connection between the stream characteristics and urethral elasticity is developed and used to deduce the elastic properties of the urethra at the point which controls the stream (the distal constriction). Direct measurements of the pressure in the resting urethra of the same subject, using catheters of different radii, give further information on the elastic properties of the urethra between the external meatus and the external sphincter. The flow and pressure measurements together show that in this subject the stream is controlled about 3 cm from the external meatus. At this point the urethra behaves like an elastic walled tube of unstretched radius (0·16±0·02) cm, subject to an external pressure of (18±3) cm H2O, and the urethral walls have an average elastic constant per unit length of order 6×103 dyn cm−2 (6 cm H2O).

Hydrodynamics of male micturition. II. Measurements of stream parameters and urethral elasticity.

The theory of flow through elastic-walled tubes, developed previously (Griffiths, 1971a) is applied to steady flow through collapsible tubes, in particular to one model system (Conrad, 1969). The experimental observations are well accounted for. The collapse to small cross-section which sometimes occurs in this and similar systems is a sign that the mean flow velocity locally exceeds the sonic velocity (the velocity of pressure waves on the tube). The consequences for the study of venous blood flow are briefly examined.

Steady fluid flow through veins and collapsible tubes.

Observations of flow through a collapsible tube with two elastic constrictions in series show that it exhibits a type of negative-resistance behaviour, where increasing the resistance of the downstream constriction decreases the overall resistance to flow. This behaviour occurs because the resistance to flow of an elastic constriction is greater in the transonic configuration than in the sonic, and it cannot be accounted for by the 1-dimensional theory of flow through such tubes. The possibility of conventional negative resistance is demonstrated, and experimental observations of relaxation oscillations are presented and interpreted in terms of a negative-resistance region.

D. J. Griffiths

Papers

Hydrodynamics of male micturition. I. Theory of steady flow through elastic-walled tubes.

Urethral elasticity and micturition hydrodynamics in females.

Hydrodynamics of male micturition. II. Measurements of stream parameters and urethral elasticity.

Steady fluid flow through veins and collapsible tubes.

Negative-resistance effects in flow through collapsible tubes: 1 relaxation oscillations.