Obstructive jaundice in the biliary tract can infect blood and result in mortality with a high rate. Percutaneous transhepatic biliary drainage (PTBD) with catheters is a useful solution discharging the obstructive jaundice. However, the elements of fluid mechanics showing clinical performance of a PTBD catheter have been documented little so far. In the article, empirical relationships between bile flow rate and pressure gradient in PTBD catheters were studied in terms of equivalent friction factor for the first time. Firstly, an equivalent friction factor in a catheter was raised and determined based on existing in vitro experimental data of bile flow through the catheters with different materials, various inner diameters and lengths under various pressure differences. Then, an empirical correlation of bile flow rate through a catheter was established based on pressure gradient, inner diameter and bile viscosity. The correlation was used to identify effects of catheter inner diameter and bile viscosity on the bile flow rate under the physiological bile pressure difference across obstructed common bile ducts. The feature of minor hydraulic losses in the catheters was clarified, too. The proposed equivalent friction factor was proportional to Reynolds number in a power of -0.654 in comparison with a power of -1 for the fully developed laminar flow in circular pipes. The bile flow rate through a catheter was proportional to inner diameter, kinematic viscosity, and pressure gradient in the powers of 3.2, -0.5 and 0.74, respectively. The minor hydraulic losses could be significant when Reynolds number was greater than 100.
The elements of fluid mechanics of bile flow through biliary drainage catheters
