A lumped-parameter model for kidney pressure during stone removal

In this paper, we consider a lumped-parameter model to predict renal pressures and flow rate during a minimally invasive surgery for kidney stone removal, ureterorenoscopy. A ureteroscope is an endoscope designed to work within the ureter and the kidney and consists of a long shaft containing a narrow, cylindrical pipe, called the working channel. Fluid flows through the working channel into the kidney. A second pipe, the ‘access sheath’, surrounds the shaft of the scope, allowing fluid to flow back out of the urinary system. We modify and extend a previously developed model ( Oratis et al., 2018) through the use of an exponential, instead of linear, constitutive law for kidney compliance and by exploring the effects of variable flow resistance, dependent on the presence of auxiliary ‘working tools’ in the working channel and the cross-sectional shapes of the tools, working channel, scope shaft and access sheath. We motivate the chosen function for kidney compliance and validate the model predictions, with ex vivo experimental data. Although the predicted and measured flow rates agree, we find some disagreement between theory and experiment for kidney pressure. We hypothesize that this is caused by spatial pressure variation in the renal pelvis, i.e. unaccounted for in the lumped-parameter model. We support this hypothesis through numerical simulations of the steady Navier–Stokes equations in a simplified geometry. We also determine the optimal cross-sectional shapes for the scope and access sheath (for fixed areas) to minimize kidney pressure and maximize flow rate.