Improving Burst Wave Lithotripsy Effectiveness for Small Stones and Fragments by Increasing Frequency: Theoretical Modeling and Ex Vivo Study

Abstract

Introduction and Objective: In clinical trial NCT03873259, a 2.6-mm lower pole stone was treated transcutaneously and ex vivo with 390-kHz burst wave lithotripsy (BWL) for 40 minutes and failed to break. The stone was subsequently fragmented with 650-kHz BWL after a 4-minute exposure. This study investigated how to fragment small stones and why varying the BWL frequency may more effectively fragment stones to dust. Methods: A linear elastic theoretical model was used to calculate the stress created inside stones from shock wave lithotripsy (SWL) and different BWL frequencies mimicking the stone's size, shape, lamellar structure, and composition. To test model predictions about the impact of BWL frequency, matched pairs of stones (1-5 mm) were treated at (1) 390 kHz, (2) 830 kHz, and (3) 390 kHz followed by 830 kHz. The mass of fragments >1 and 2 mm was measured over 10 minutes of exposure. Results: The linear elastic model predicts that the maximum principal stress inside a stone increases to more than 5.5 times the pressure applied by the ultrasound wave as frequency is increased, regardless of the composition tested. The threshold frequency for stress amplification is proportionate to the wave speed divided by the stone diameter. Thus, smaller stones may be likely to fragment at a higher frequency, but not at a lower frequency below a limit. Unlike with SWL, this amplification in BWL occurs consistently with spherical and irregularly shaped stones. In water tank experiments, stones smaller than the threshold size broke fastest at high frequency (p = 0.0003), whereas larger stones broke equally well to submillimeter dust at high, low, or mixed frequencies. Conclusions: For small stones and fragments, increasing frequency of BWL may produce amplified stress in the stone causing the stone to break. Using the strategies outlined here, stones of all sizes may be turned to dust efficiently with BWL.

Keywords: lithotripsy; urinary stones; urolithiasis.

FIG. 1.

Experimental setup. Color images are available online.

FIG. 2.

(a) Photograph of the extracted stone on 1-mm graph paper, and (b) slice of a volume image of the stone made by microcomputed tomography after the in vivo exposure to BWL. The inner material is COM and the outer is COD. The fissure at the top of the stone appears to be naturally present and not caused by BWL. BWL = burst wave lithotripsy; COD = calcium oxalate dihydrate; COM = calcium oxalate monohydrate. Color images are available online.

FIG. 3.

Simulated image of the peak maximum principal stress along the centerline of the stone. The upper column shows stress induced by 390-kHz lithotripsy burst and the lower shows stress induced by a 650-kHz lithotripsy burst. The scale (−6 to 6) is maximum stress divided by the incident pressure and represents the amplification of the applied pressure. A single cycle, similar to the shape of an SWL pulse, produces little stress and negligible amplification in the 2.6-mm-diameter spherical stone regardless of the frequency. The second column shows the results from a 10-cycle BWL pulse, where the lower frequency produces little stress in the stone, but the higher frequency yields 5.5 times the applied pressure within the stone. Adding a COD shell (column 3) increases the stress within the stone slightly, because of COD's slower sound speed; however, significant amplification of the incident pressure within the stone is still only achieved at 650 kHz. The trend is similar for the irregularly shaped stone (scale in this column is stress, not amplification). SWL = shock wave lithotripsy. Color images are available online.

FIG. 4.

Peak maximum principal stress within a COM stone normalized to the peak pressure of the applied BWL pulse vs BWL frequency for two stone sizes. For the 4.5-mm stone (blue line), the pressure amplification is more than 5.5 at both 350 and 650 kHz. However, for the smaller 2.6-mm stone (orange line), the amplification is 2.8 at 390 kHz, but 5.5 at 650 kHz. A dashed line is shown to the right of the peak to reflect that the amplification remains consistent, but there is a shift off-center and variations with specific resonances are not shown but are discussed further in the article by Sapozhnikov and colleagues. The plots show a threshold in minimum frequency that must be used to achieve the maximum pressure amplification for a certain stone diameter. Color images are available online.

FIG. 5.

Time history of the total strain energy (a) and damage potential with BWL at 390 kHz (b) and 650 kHz (c) inside the 2.6-mm asymmetric stone. Multiple cycles are required to reach the maximum strain energy for a given a frequency. The total strain energy is nearly double for the higher frequency in this small stone. In addition, 650-kHz BWL produces a potential damage throughout the full width of the stone, whereas 390 kHz only yields a damage potential at the stress concentrating surface feature. Color images are available online.

FIG. 6.

The average normalized mass fraction of initially small stone (1–3 mm) fragments >1 mm remaining at each time point (a) and the probability of the stone being completely broken to <1 mm fragments (b) (upper). COM stones in the 1–3-mm-size range exposed to the higher frequency (830-kHz) BWL burst broke faster and more completely than similar stones exposed to the lower frequency (390-kHz) BWL burst (p = 0.0003). The calculated probability curve for the 390-kHz frequency flattened after 2.5 minutes, indicating the stones were not breaking further with additional exposure. The lower row shows the average normalized mass fraction of initially large stone (3–5 mm) fragments >2 mm (c) and 1 mm (d) remaining at each time point (lower). There are no statistical differences in rate of fragmentation between large stones exposed to low-frequency (390 kHz) BWL, high-frequency (830 kHz) BWL, or low frequency (2.5 minutes) followed by high frequency (mixed) (p = 0.2055). The results suggest that although it may not be possible to break a small stone at low frequency, it may be possible to use a higher frequency over a broad range of stone sizes (1–5 mm), without a loss of fragmentation effectiveness. Color images are available online.