The flow regimes and the pressure-flow relationship in the canine urethra

Classical fluid dynamics predicts that the pressure difference Delta p betw een any two points along a fully developed, viscous flow stream is Linearly proportional to the flow rate Q (the Poiseuille relation). However, the pa ssive urethral resistance relationship (PURR) widely used in modern urodyna mics describes the pressure difference Delta p between two points along the urethra as linearly proportional to the flow rate squared (Q(2)). It is ou r hypothesis that this functional dependence may have its origins in the de veloping now field within the urethra. That is, rather than being fully dev eloped hydrodynamically, urethral flow is more likely representative of flo w within the entry length of a rigid conduit. In our study, we used a canin e model of the lower urinary tract to investigate the possibility of entran ce effects. Although the most rigorous model of urethral fluid mechanics wo uld include the elastic properties of the urethra into its configuration, t he solutions from such a model would be unnecessarily complex and not readi ly lend themselves to the analysis of clinical data. Therefore, we chose to model the canine urethra at each instant in time as a rigid tube, and char acterized its instantaneous flow using viscous flow theory for a rigid tube . All urodynamic analyses were performed on a surgically exposed urinary trac t. Solid state pressure transducers were used to measure the intravesical a nd distal urethral pressures, whereas an ultrasonic flowmeter was used to o btain a simultaneous measure of the urinary flow rate. Detrusor contraction s were induced using bilateral electrical stimulation of the pelvic nerves. Varying degrees of outlet obstruction were created using an inflatable sph incter cuff secured around the bladder outlet. The experimental data were e valuated using the well-known laminar entry length model of Atkinson and Go ldstein. The peak Reynolds numbers under nonobstructed R-e non-obs(P) and obstructed R-e obs(P) outlet conditions ranged between 500 < R-e non-obs(P) < 1,500 a nd 300 < R-e obs(P) < 1,700, respectively. Under non-obstructed outlet cond itions, the urethral diameters D and total lengths l(T) ranged between 1.5 mm < D < 2.5 mm and 75 mm < l(T) < 95 mm, respectively, whereas the peak en trance lengths L-e non-obs(P) ranged between 55 mm < L-e non-obs(P) < 215 m m. These data suggest that now in the canine urethra under both non-obstructed and obstructed outlet conditions is typically laminar. The data further su pport the hypothesis that non-obstructed flows are predominantly entry leng th in nature. Entry length flows are fluid dynamically described by a quadr atic pressure-flow relationship, thus suggesting a physiological basis for Schafer's quadratic pressure-flow relationship, and therefore, for the PURR . (C) 1999 Wiley-Liss, Inc.