Maximizing mechanical stress in small urinary stones during burst wave lithotripsy

Abstract

Unlike shock wave lithotripsy, burst wave lithotripsy (BWL) uses tone bursts, consisting of many periods of a sinusoidal wave. In this work, an analytical theoretical approach to modeling mechanical stresses in a spherical stone was developed to assess the dependence of frequency and stone size on stress generated in the stone. The analytical model for spherical stones is compared against a finite-difference model used to calculate stress in nonspherical stones. It is shown that at low frequencies, when the wavelength is much greater than the diameter of the stone, the maximum principal stress is approximately equal to the pressure amplitude of the incident wave. With increasing frequency, when the diameter of the stone begins to exceed about half the wavelength in the surrounding liquid (the exact condition depends on the material of the stone), the maximum stress increases and can be more than six times greater than the incident pressure. These results suggest that the BWL frequency should be elevated for small stones to improve the likelihood and rate of fragmentation.

FIG. 1.

(Color online) Maximum in space and time of the maximum principal stress, $\underset{\text{space}}{\text{max}}[\underset{\text{time}}{\text{max}}(T_{\text{max}})]$, normalized by the incident plane wave amplitude p₀, versus dimensionless frequency ka, where k is the wavenumber in the liquid, a is the stone radius. The modeled spherical stone is made from COM material. The blue line corresponds to the case when no losses are present in the stone, the red line represents simulations when an absorption that grows linearly with frequency is introduced to the elastic waves in the stone: $k_t\to k_t(1+0.01\hspace{0.17em}i)$, $k_l\to k_l(1+0.01\hspace{0.17em}i)$. Images below the curve represent spatial distribution of the maximum-in-time of the maximum principal stress, $\underset{\text{time}}{\text{max}}(T_{\text{max}})$, for the frequencies where the curve has local maxima (i.e., the resonance frequencies).

FIG. 2.

(Color online) The distribution of the maximum-in-time of the maximum principal stress for a COM stone (left), as well as the pattern of the stone deformation in the process of its oscillations (center and right) at the 1st and 2nd lowest resonance frequencies, $ka=3.590$ and $ka=5.368$. Here, $t_0$ is a moment in time and $T$ is the wave period.

FIG. 3.

(Color online) Normalized maximum stress $\underset{\text{space}}{\text{max}}[\underset{\text{time}}{\text{max}}(T_{\text{max}})]/p_0$ in a stone as a function of dimensionless frequency ka varying stone composition: COD, MAPH, U-30, and COM. Arrows indicate the lowest resonance frequency for quasi-longitudinal standing waves in a thin rod with length equal to the stone diameter.

FIG. 4.

(Color online) Maximum principal stress $\underset{\text{time}}{\text{max}}(T_{\text{max}})$ at the center (r = 0) of 5 and 10 mm diameter COM stones normalized by the incident plane wave amplitude p₀. The exact analytical solution is presented as solid lines, whereas the finite-difference modeling results are shown as circles with 10 kHz frequency steps. Note, we discuss only the peak at the stone center to consider the resonance behavior here, but higher values exist above the lowest resonance off axis.

FIG. 5.

(Color online) Normalized maximum stress $\underset{\text{space}}{\text{max}}[\underset{\text{time}}{\text{max}}(T_{\text{max}})]/p_0$ in an axisymmetric COM stone as a function of the dimensionless frequency ka for stones of different shapes (shown in different colors): spherical, cylindrical, and in the form of two connected cones. For nonspherical stones, the parameter a corresponds to the radius of a sphere of equal volume.

FIG. 6.

(Color online) Maximum principal stress $\underset{\text{time}}{\text{max}}(T_{\text{max}})$ at the center (r = 0) of a COM stone normalized by the incident plane wave amplitude p₀ as a function of the dimensionless frequency ka. Vertical axis is presented in logarithmic scale, the curves of different colors correspond to different degree of the elastic waves' absorptions characterized by loss tangent $\text{tan}\hspace{0.17em}\delta$.