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

Classical fluid dynamics predicts that the pressure difference Deltap between any two points along a fully developed, viscous flow stream is linearly proportional to the flow rate Q (the Poiseuille relation). However, the passive urethral resistance relationship (PURR) widely used in modern urodynamics describes the pressure difference Deltap between two points along the urethra as linearly proportional to the flow rate squared (Q(2)). It is our hypothesis that this functional dependence may have its origins in the developing flow field within the urethra. That is, rather than being fully developed hydrodynamically, urethral flow is more likely representative of flow within the entry length of a rigid conduit. In our study, we used a canine model of the lower urinary tract to investigate the possibility of entrance effects. Although the most rigorous model of urethral fluid mechanics would include the elastic properties of the urethra into its configuration, the solutions from such a model would be unnecessarily complex and not readily 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 characterized its instantaneous flow using viscous flow theory for a rigid tube. All urodynamic analyses were performed on a surgically exposed urinary tract. Solid state pressure transducers were used to measure the intravesical and distal urethral pressures, whereas an ultrasonic flowmeter was used to obtain a simultaneous measure of the urinary flow rate. Detrusor contractions were induced using bilateral electrical stimulation of the pelvic nerves. Varying degrees of outlet obstruction were created using an inflatable sphincter cuff secured around the bladder outlet. The experimental data were evaluated using the well-known laminar entry length model of Atkinson and Goldstein. The peak Reynolds numbers under nonobstructed R(p)(e non-obs) and obstructed R(p)(e obs) outlet conditions ranged between 500 < R(p)(e non-obs) < 1,500 and 300 < R(p)(e obs) < 1,700, respectively. Under non-obstructed outlet conditions, 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 entrance lengths L(p)(e non-obs) ranged between 55 mm < L(p)(e non-obs) < 215 mm. These data suggest that flow in the canine urethra under both non-obstructed and obstructed outlet conditions is typically laminar. The data further support the hypothesis that non-obstructed flows are predominantly entry length in nature. Entry length flows are fluid dynamically described by a quadratic pressure-flow relationship, thus suggesting a physiological basis for Schäfer's quadratic pressure-flow relationship, and therefore, for the PURR. Neurourol. Urodynam. 18:521-541, 1999.