Mathematical modelling of unsteady flows during ureteroscopy
Keywords: fluid dynamics
ureteroscopy
mathematical modelling

Ureteroscopy is a procedure used to treat kidney stones, which consists of passing a flexible tool known as a ureteroscope through the urinary system to access the kidney. The scope is hollow along its length, creating a working channel through which working tools can be passed. Once a stone is located, a laser fibre is inserted down the working channel to deliver a high-power laser which reduces the kidney stone to dust. Throughout the procedure constant fluid irrigation is delivered through the scope working channel, which fills and dilates the kidney cavity, and then exits via the space in between the ureteroscope and ureter. Often an access sheath is inserted within the ureter to provide better dilation of the channel. The fluid washes the dust created during lasering out of the kidney, providing the clinician with a clear field of view of the kidney interior via a camera at the tip of the scope. Traditionally this fluid irrigation is provided by a hanging saline bag at a level above the patient, producing a constant, gravity driven flow. Instead, we consider the case where the fluid is driven via a peristaltic pump, providing the clinician with better control over the irrigation delivery. Due to the nature of how these pumps operate, previously unseen time dependence is introduced to the flow, causing it to oscillate in time. Understanding the prevalence of these oscillations throughout the system is key to determining how this new method differs from the traditional set-up, and what benefits the improved delivery method can bring.
This thesis is concerned with the mathematical modelling of oscillatory fluid irrigation throughout the ureteroscopy system, with an aim to determine the prevalence of the fluid oscillations throughout the system and how the time dependence can be harnessed to improve the procedure by decreasing dust wash-out times.
We begin by modelling the flow throughout the cylindrical geometries of the system, namely the tubing connecting the pump and scope, the scope working channel with working tool inserted, and the access sheath with scope inserted. The flow is driven by an upstream oscillatory pressure, with atmospheric pressure at the outlet of the access sheath, where at the junctions between each section we assume continuity of flux and pressure. The model is solved analytically, and it is found that the flow and pressure oscillations are significantly dampened by the time they reach the kidney cavity due to the small radii of the ureteroscope working channel and working tool. The assumptions made at the junctions between each section are then addressed by numerically simulating the flow within these more complicated geometries. Alongside this, we present a model of the peristaltic pump used to drive the fluid, and show how the characteristics of the flow mean, frequency, and amplitude are related to pump rotations per minute. Both models are validated by comparing the theoretical predictions with experimental data sets and good agreement is shown.
Next we turn to the modelling of fluid flow within the kidney cavity, and the transport of kidney stone dust within said fluid. To represent the cavity, we consider an idealised, two-dimensional, rectangular domain, with one inlet and two outlets on the same side of the domain. A parabolic, oscillatory inlet flow introduces the time-dependence to the system. The problem is solved numerically via finite element formulation. Dust wash-out time is used as a metric, and the parameters of the inflow are varied with an objective to reduce the wash-out time. It is found that an inlet flow which oscillates with a low frequency and high amplitude aids in disturbing vortical structures within the flow, leading to a significantly reduced dust wash-out time due to the resulting mixing effect. We go on to compare the wash-out times under this oscillatory regime to a simulation of a typical industry method termed flushing, and find that this new method performs favourably. The robustness of these results is then examined by demonstrating the impact of changes to the cavity geometry on dust wash-out times, and qualitatively similar wash-out results are seen for all geometries.
Finally, the previous modelling efforts are combined together to create a model of the full ureteroscopy system. Fluid irrigation is modelled through tubing, to scope working channel, and into the kidney, before exiting via the access sheath, all driven by a flux which is determined from the pump model. In addition to this, we introduce the idea of an air dampener upstream of the ureteroscope to the model and consider the impact it has on the oscillatory flows downstream of the scope. The model shows that the air dampener effectively reduces the amplitude of the oscillations as intended. The predictions from the modelling are again validated by experimental data sets and good agreement is found by accounting for a scaling factor on the working tool resistance.