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.
References
1.
D.
Assimos
, A.
Krambeck
, N. L.
Miller
, M.
Monga
, M. H.
Murad
, C. P.
Nelson
, K. T.
Pace
, V. M.
Pais
, M. S.
Pearle
, G. M.
Preminger
, and H.
Razvi
, “Surgical management of stones: American Urological Association/Endourological Society Guideline
,” J. Urol.
196
(4
), 1161
–1169
(2016
).2.
G. D.
Innes
, F. X.
Scheuermeyer
, A. D.
McRae
, M. R.
Law
, J. M.
Teichman
, E.
Grafstein
, and J. E.
Andruchow
, “Which patients should have early surgical intervention for acute ureteral colic?
,” J. Urol.
205
(1
), 152
–158
(2021
).3.
A. D.
Rule
, J. C.
Lieske
, and V. M.
Pais
, “Management of kidney stones in 2020
,” JAMA
323
(19
), 1961
–1962
(2020
).4.
A.
Skolarikos
, G.
Alivizatos
, and J.
de la Rosette
, “Extracorporeal shock wave lithotripsy 25 years later: Complications and their prevention
,” Eur. Urol.
50
(5
), 981
–990
(2006
).5.
C.
Senocak
, C.
Ozcan
, T.
Sahin
, G.
Yilmaz
, E.
Ozyuvali
, S.
Sarikaya
, B.
Resorlu
, U.
Oguz
, O. F.
Bozkurt
, A.
Unsal
, and O.
Adsan
, “Risk factors of infectious complications after flexible uretero-renoscopy with laser lithotripsy
,” Urol. J.
15
(4
), 158
–163
(2018
).6.
J. A.
McAteer
, J. C.
Williams
, Jr., M. R.
Bailey
, R. O.
Cleveland
, and A. P.
Evan
, “Strategies for improved shock wave lithotripsy
,” Minerva Urologica E Nefrologica
57
(4
), 271
–287
(2005
).7.
J. J.
Rassweiler
, T.
Knoll
, K.-U.
Köhrmann
, J. A.
McAteer
, J. E.
Lingeman
, R. O.
Cleveland
, M. R.
Bailey
, and C.
Chaussy
, “Shock wave technology and application: An update
,” Eur. Urol.
59
(5
), 784
–796
(2011
).8.
O. A.
Sapozhnikov
, V. A.
Khokhlova
, M. R.
Bailey
, J. C.
Williams
, Jr., J. A.
McAteer
, R. O.
Cleveland
, and L. A.
Crum
, “Effect of overpressure and pulse repetition frequency on cavitation in shock wave lithotripsy
,” J. Acoust. Soc. Am.
112
(3
), 1183
–1195
(2002
).9.
O. A.
Sapozhnikov
, A. D.
Maxwell
, B.
MacConaghy
, and M. R.
Bailey
, “A mechanistic analysis of stone fracture in lithotripsy
,” J. Acoust. Soc. Am.
121
(2
), 1190
–1202
(2007
).10.
J.
Qin
, W. N.
Simmons
, G.
Sankin
, and P.
Zhong
, “Effect of lithotripter focal width on stone comminution in shock wave lithotripsy
,” J. Acoust. Soc. Am.
127
(4
), 2635
–2645
(2010
).11.
J.
Jendeberg
, H.
Geijer
, M.
Alshamari
, B.
Cierzniak
, and M.
Lidén
, “Size matters: The width and location of a ureteral stone accurately predict the chance of spontaneous passage
,” Eur. Radiol.
27
(11
), 4775
–4785
(2017
).12.
A. D.
Maxwell
, B. W.
Cunitz
, W.
Kreider
, O. A.
Sapozhnikov
, R. S.
Hsi
, J. D.
Harper
, M. R.
Bailey
, and M. D.
Sorensen
, “Fragmentation of renal calculi in vitro by focused ultrasound bursts
,” J. Urol.
193
(1
), 338
–344
(2015
).13.
A. D.
Maxwell
, B.
MacConaghy
, M. R.
Bailey
, and O. A.
Sapozhnikov
, “An investigation of elastic waves producing stone fracture in burst wave lithotripsy
,” J. Acoust. Soc. Am.
147
(3
), 1607
–1622
(2020
).14.
A. D.
Maxwell
, Y.-N.
Wang
, W.
Kreider
, B. W.
Cunitz
, F.
Starr
, D.
Lee
, Y.
Nazari
, J. C.
Williams
, Jr., M. R.
Bailey
, and M. D.
Sorensen
, “Evaluation of renal stone comminution and injury by burst wave lithotripsy in a pig model
,” J. Endourol.
33
(10
), 787
–792
(2019
).15.
O. A.
Sapozhnikov
, A. D.
Maxwell
, and M. R.
Bailey
, “Modeling of photoelastic imaging of mechanical stresses in transparent solids mimicking kidney stones
,” J. Acoust. Soc. Am.
147
(6
), 3819
–3829
(2020
).16.
R. O.
Cleveland
and O. A.
Sapozhnikov
, “Modeling elastic wave propagation in kidney stones with application to shock wave lithotripsy
,” J. Acoust. Soc. Am.
118
(4
), 2667
–2676
(2005
).17.
S.
Cao
, Y.
Zhang
, D.
Liao
, P.
Zhong
, and K. G.
Wang
, “Shock-induced damage and dynamic fracture in cylindrical bodies submerged in liquid
,” Int. J. Solids Struct.
169
, 55
–71
(2019
).18.
T. A.
Kishore
, R. N.
Pedro
, B.
Hinck
, and M.
Monga
, “Estimation of size of distal ureteral stones: Noncontrast CT scan versus actual size
,” Urology
72
(4
), 761
–764
(2008
).19.
V. R.
Singh
and J. B.
Dhawan
, “Ultrasonic velocity and attenuation measurement in kidney stones: Correlation to constituents and hardness
,” Biomed. Mater. Eng.
2
(2
), 79
–82
(1992
).20.
D.
Heimbach
, R.
Munver
, P.
Zhong
, J.
Jacobs
, A.
Hesse
, S. C.
Müller
, and G. M.
Preminger
, “Acoustic and mechanical properties of artificial stones in comparison to natural kidney stones
,” J. Urol.
164
(2
), 537
–544
(2000
).21.
J. J.
Faran
, Jr., “Sound scattering by solid cylinders and spheres
,” J. Acoust. Soc. Am..
23
(4
), 405
–418
(1951
).22.
23.
O. A.
Sapozhnikov
, “An exact solution to the Helmholtz equation for a quasi-Gaussian beam in the form of a superposition of two sources and sinks with complex coordinates
,” Acoust.. Phys.
58
(1
), 41
–47
(2012
).24.
R.
Hickling
, “Analysis of echoes from a solid elastic sphere in water
,” J. Acoust. Soc. Am.
34
(10
), 1582
–1592
(1962
).25.
O. A.
Sapozhnikov
and M. R.
Bailey
, “Radiation force of an arbitrary acoustic beam on an elastic sphere in a fluid
,” J. Acoust. Soc. Am.
133
(2
), 661
–676
(2013
).26.
L. D.
Landau
and E. M.
Lifshitz
, Theory of Elasticity
, 3rd ed. (Pergamon
, New York
, 1986
).27.
S. P.
Timoshenko
and J. N.
Goodier
, Theory of Elasticity
(McGraw-Hill
, New York
, 1982
).28.
P.
Zhong
, C. J.
Chuong
, and G. M.
Preminger
, “Propagation of shock waves in elastic solids caused by cavitation microjet impact. II: Application in extracorporeal shock wave lithotripsy
,” J. Acoust. Soc. Am.
94
(1
), 29
–36
(1993
).29.
Y.
Liu
and P.
Zhong
, “BegoStone – A new stone phantom for shock wave lithotripsy research
,” J. Acoust. Soc. Am.
112
(4
), 1265
–1268
(2002
).30.
J. A.
McAteer
, J. C.
Williams
, Jr., R. O.
Cleveland
, J.
Van Cauwelaert
, M. R.
Bailey
, D. A.
Lifshitz
, and A. P.
Evan
, “Ultracal-30 gypsum artificial stones for research on the mechanisms of stone breakage in shock wave lithotripsy
,” Urol. Res.
33
(6
), 429
–434
(2005
).31.
D.
Zhang
, S.
Li
, Z.
Zhang
, N.
Li
, X.
Yuan
, Z.
Jia
, and J.
Yang
, “Urinary stone composition analysis and clinical characterization of 1520 patients in central China
,” Sci. Rep.
11
, 6467
(2021
).32.
J. D.
Harper
, I.
Metzler
, M. K.
Hall
, T. T.
Chen
, A. D.
Maxwell
, B. W.
Cunitz
, B.
Dunmire
, J.
Thiel
, J. C.
Williams
, Jr., M. R.
Bailey
, and M. D.
Sorensen
, “First-in-human burst-wave lithotripsy (BWL) for kidney stone comminution
,” J. Endourol.
35
, 506
–511
(2021
).33.
Handbook of Mathematical Functions
, Applied Mathematics Series 55, edited by M.
Abramowitz
and I. A.
Stegun
(National Bureau of Standards
, Washington DC
, 1962
).© 2021 Acoustical Society of America.
2021
Acoustical Society of America
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