| Summary: | <p>Laser lithotripsy, a common treatment for kidney stones, involves passing a flexible scope containing a laser fibre into the kidney via the bladder and ureter. Within the kidney, laser pulses are fired at the stone to break it into pieces small enough to flush out. During the procedure, laser energy is also transferred to the surrounding liquid, resulting in localised boiling and the production of vapour bubbles. While bubbles can have undesirable effects – for example, causing retropulsion of the stone – they can also be utilised to make stone ablation more efficient. The aim of this work is to contribute to improving the speed and ease of laser lithotripsy kidney stone treatment via mathematical modelling, focusing on the vapour bubbles that form during the treatment.</p>
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<p>The size, shape and duration of a laser-induced bubble is dependent on the total energy of the laser pulse and the way in which this energy is temporally distributed. The time dependent power profile of the pulse is referred to as ‘pulse shape’ and can be manipulated to produce bubbles with desirable properties. Using two different modelling approaches, we predict properties of the bubble as a function of the pulse shape, as well as understanding the physics driving bubble expansion. Previous models of laser-induced bubbles assume the energy transfer from the laser to the fluid to be instantaneous, which is not a good approximation for the ∼ 100 μs pulses employed in lithotripsy. To reflect the effect that the laser parameters have on the bubble size, shape and duration, we model the laser as a time and space dependent source of energy in the liquid-vapour fluid system.</p>
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<p>Our first model considers the simplified case of a single, spherical vapour bubble in liquid, with phase change at the bubble wall. With this model, we build upon an existing method of coupling the Rayleigh-Plesset equation to an energy equation in the surrounding liquid and an energy flux balance at the bubble wall. Mathematical analysis of the governing equations reveals the physical effects occurring at three stages of bubble expansion and the time-scales at which these stages occur. We then examine the effect on the bubble of varying pulse power, pulse duration, and total pulse energy.</p>
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<p>A second modelling approach tackles the challenging problem of breaking spherical symmetry in bubble modelling by utilising the ‘volume of fluid’ method on an axisymmetric domain. With localised phase change due to laser heating, we are able to reflect the elongated shapes that occur during laser lithotripsy and study how the proportions of the bubble — length and width — as well as its size and duration are affected by the properties of the pulse. Axisymmetry gives the flexibility to introduce solid boundaries near the bubble and measure the effect this has on both the bubble and the boundary.</p>
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<p>Through numerical and analytical methods alongside experimental work we develop an understanding of the physical factors driving bubble expansion and provide a framework to guide the future development and clinical usage of lasers.</p>
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