Int Urogynecol J (1998) 9:103-107 9 1998 Springer-Verlag London Ltd International Urogynecology Journal Original Article A Mathematical Model for Micturition Gives New Insights into Pressure Measurement and Function P. E. Papa Petros 1 and M. B. Bush 2 XRoyal Perth Hospital, Perth, Western Australia; 2Department of Mechanical and Materials Engineering, University of Western Australia, Perth, Western Australia Abstract: Our objective was to analyze the factors contributing to the development of detrusor pressure during micturition in the female with reference to a mathematical model. One hundred patients with pre- dominantly stress incontinence were investigated with micturition pressure studies. Frictional and dynamic losses were estimated at various flow rates using a mathematical model. Almost 25% of patients recorded a micturition pressure below 11 cmH20 at peak flow (mean 23 cmH20, range 0-91). Large inter- and intrapatient variations in micturition pressures were recorded on retesting. The low pressures were explained by a recently described external opening mechanism, backward stretching of the vagina during micturition by the muscles of the pelvic floor. This opened out the outflow tract and created the potential for a falsely high Pabd. The large variability in micturition pressures on retesting was attributed to changes in urethral radius being magnified to the fourth power. It was concluded that, micturition itself, and the components for pressure generation, are complex non-linear entities which appear to be greatly modified by the external striated pelvic floor opening mechanism. Addressing anatomical defects in this mechanism may be a fruitful route of future enquiry in females with emptying problems. Keywords: Micturition; Muscle force; Pelvic floor; Urethral resistance; Urine flow Correspondence and offprint requests to: Dr P. E. Papa Petros, Suite 14A, Surgicentre 38 Ranelagh Cres., South Perth, Western Australia 6151, Australia. Introduction Voluntary micturition is said to occur by activation of the micturition reflex and relaxation of the striated muscle of the urethra and pelvic floor [1]. Quantification of urine flow, detrusor pressure, bladder work force and urethral resistance are all based on this concept. The flow rate through an open tube is governed by the pressure difference across the tube, the tube's geometry, the surface roughness and the fluid properties. The pressure difference, AP = Pves - Po is the difference between the pressure of the fluid contained within the bladder, the intravesical pressure, Pves, and the pressure acting at the tube exit, Po. The intravesical pressure is the sum of the abdominal pressure and the pressure caused by bladder muscle contraction, the 'detrusor pressure'. The intravesical pressure at a given flow rate varies according to the urethral resistance, and, in turn, with the fourth power of the radius [2]. Existing mathematical models are based on frictional resistance to flow through the urethral tube [3] or urodynamic parameters through an elastic tube [4-6]. The urethra is generally assumed to be a straight, smooth-walled circular tube carrying fully developed turbulent flow. Measured detrusor pressure and flow rate are used to determine an 'effective' urethral diameter and therefore, urethral restriction. Recent dynamic video X-ray studies have demonstrated that the urethra is not a straight tube, either at rest (Fig. 1) or during micturition (Fig. 2) [8,9]. Furthermore, contraction of specific muscles of the pelvic floor, the levator plate and longitudinal muscle of the anus, may play an important role in acting as an external mechanism for opening out the urethra during micturi- tion [8,9]. The importance of abrupt changes in cross- section (e.g. 'funneling') of the tube shape (Fig. 2) and