A temperature model for laser lithotripsy

Abstract

Objective: To derive and validate a mathematical model to predict laser-induced temperature changes in a kidney during kidney stone treatment.

Methods: A simplified mathematical model to predict temperature change in the kidney for any given renal volume, irrigation flow rate, irrigation fluid temperature, and laser power was derived. We validated our model with matched in vitro experiments.

Results: Excellent agreement between the mathematical model predictions and laboratory data was obtained.

Conclusion: The model obviates the need for repeated experimental validation. The model predicts scenarios where risk of renal tissue damage is high. With real-time knowledge of flow rate, irrigating fluid temperature and laser usage, safety warning levels could be predicted. Meanwhile, clinicians should be aware of the potential risk from thermal injury and take measures to reduce the risk, such as using room temperature irrigation fluid and judicious laser use.

Keywords: In vitro experiments; Lithotripsy; Mathematical modelling; Thermal tissue damage.

Fig. 1

The laser fibre is 10 mm distal to the scope tip. Two thermocouples measured temperature over time positioned at the scope tip and 9 mm from the base of the container. These are indicated by a circle and triangle, respectively. Diagram not to scale

Fig. 2

The unknown parameter was obtained by fitting the analytical solution for $Q=0$ mL/min, $\mathcal{W}=40$ Watts: a data from Set A, b data from Set B. The best-fit values are a $\beta \approx 1.14$ Watts/${}^{\circ }\text{C}$ and b $\beta \approx 1.36$ Watts/${}^{\circ }\text{C}$

Fig. 3

A comparison of the model predictions (solid lines) with the experimental data from Set A (symbols)

Fig. 4

A dimensional comparison of the model predictions (solid lines) with the experimental data (symbols). Triangles are from the thermocouple 9 mm from the base of the container and circles from the thermocouple at the level of the scope tip (see Fig. 1)

Fig. 5

Predicted temperature change after 60 s of laser firing for $T_0=37{}^{\circ }\text{C}$. In the top row $T_{\text{in}}=37{}^{\circ }\text{C}$ and in the bottom row $T_{\text{in}}=23{}^{\circ }\text{C}$. The colours provide $\mathrm{\Delta }T$ and white regions indicate where $\mathrm{\Delta }T<0$. a, d $V=30$ mL and Q, W varied. b, e $Q=20$ mL min${}^{-1}$ and V, W varied. c, f $W=20$ Watts and Q, V varied

Fig. 6

a and b Predicted $t_{\text{f}}^{\mathrm{safe}}$ (in minutes), as a function of $T_{\text{in}}$ and Q, such that $t_{43}=120$ min for a conditions for experiment Set A and b conditions for experiment Set B. c and d Example temperature curves for labelled points (i) (gray) and (ii) (black) in (a) and (b), respectively. Red dots indicates $t_{\text{f}}^{\mathrm{safe}}$, the dashed red line shows $T=43{}^{\circ }\text{C}$, and the dashed blue line shows $T=37{}^{\circ }\text{C}$. In all figures, $T_0=37{}^{\circ }\text{C}$ and $W=40$ Watts. The two thicker black lines in (a ) and (b ) denote $T^{\star }=37{}^{\circ }\text{C}$ and $T^{\star }=43{}^{\circ }\text{C}$

Fig. 7

The laser fibre is 20 mm distal to the scope tip. Two thermocouples measured temperature over time positioned at the scope tip and 9 mm from the base of the container. These are indicated by a circle and triangle, respectively. Diagram not to scale

Fig. 8

A dimensional comparison of the model predictions (solid lines) with the experimental data (symbols). Triangles are from the thermocouple 9 mm from the base of the container and circles from the thermocouple at the level of the scope tip (see Fig. 7)

Fig. 9

The considered domain