A theoretical model for the propagation of shock wave from an axisymmetric reflector was developed by modifying the initial conditions for the conventional solution of a nonlinear parabolic wave equation (i.e., the Khokhlov–Zabolotskaya–Kuznestsov equation). The ellipsoidal reflector of an HM-3 lithotripter is modeled equivalently as a self-focusing spherically distributed pressure source. The pressure wave form generated by the spark discharge of the HM-3 electrode was measured by a fiber optic probe hydrophone and used as source conditions in the numerical calculation. The simulated pressure wave forms, accounting for the effects of diffraction, nonlinearity, and thermoviscous absorption in wave propagation and focusing, were compared with the measured results and a reasonably good agreement was found. Furthermore, the primary characteristics in the pressure wave forms produced by different reflector geometries, such as that produced by a reflector insert, can also be predicted by this model. It is interesting to note that when the interpulse delay time calculated by linear geometric model is less than about 1.5μs, two pulses from the reflector insert and the uncovered bottom of the original HM-3 reflector will merge together. Coupling the simulated pressure wave form with the Gilmore model was carried out to evaluate the effect of reflector geometry on resultant bubble dynamics in a lithotripter field. Altogether, the equivalent reflector model was found to provide a useful tool for the prediction of pressure wave form generated in a lithotripter field. This model may be used to guide the design optimization of reflector geometries for improving the performance and safety of clinical lithotripters.

1.
C.
Chaussy
and
G. J.
Fuchs
, “
Current state and future developments of noninvasive treatment of human urinary stones with extracorporeal shock wave lithotripsy
,”
J. Urol. (Baltimore)
141
,
782
792
(
1989
).
2.
K.
Kerbl
,
J.
Rehman
,
J.
Landman
,
D.
Lee
,
C.
Sundaram
, and
R. V.
Clayman
, “
Current management of urolithiasis: Progress or regress?
,”
J. Endourol
16
,
281
288
(
2002
).
3.
J. E.
Lingman
, “
Extracorporeal shock wave lithotripsy—Development, instrumentation, and current status
,”
Urol. Clin. North Am.
24
,
185
211
(
1997
).
4.
Y. F.
Zhou
,
F. H.
Cocks
,
G. M.
Preminger
, and
P.
Zhong
, “
Innovations in shock wave lithotripsy technology: Updates in experimental studies
,”
J. Urol. (Baltimore)
172
,
1892
1898
(
2004
).
5.
M.
Delius
, “
Medical applications and bioeffects of extracorporeal shock waves
,”
Shock Waves
4
,
55
72
(
1994
).
6.
A. P.
Evan
and
J. A.
McAteer
, “
Q-effects of shock wave lithotripsy
,” in
Kidney Stones, Medical and Surgerical Management
, edited by
F. L.
Coe
 et al (
Lippincott-Raven
,
Philadephia
,
1996
), pp.
549
570
.
7.
K.
Sarica
,
A.
Balat
,
A.
Erbagci
,
M.
Cekmen
,
M.
Yurekli
, and
F.
Yagci
, “
Effects of shock wave lithotripsy on plasma and urinary levels of nitrite and adrenomedullin
,”
Urol. Res.
31
,
347
351
(
2003
).
8.
S. F.
Graber
,
H.
Danuser
,
W. W.
Hochreiter
, and
U. E.
Studer
, “
A prospective randomized trial comparing 2 lithotriptors for stone disintegration and induced renal trauma
,”
J. Urol. (Baltimore)
169
,
54
57
(
2003
).
9.
R.
Gerber
,
U. E.
Studer
, and
H.
Danuser
, “
Is newer always better? A comparative study of 3 lithotriptor generations
,”
J. Urol. (Baltimore)
173
,
2013
2016
(
2005
).
10.
X. F.
Xi
and
P.
Zhong
, “
Improvement of stone fragmentation during shock wave lithotripsy using a combined EH/PEAA shock wave generator - In vitro experiments
,”
Ultrasound Med. Biol.
26
,
457
467
(
2000
).
11.
D. L.
Sokolov
,
M. R.
Bailey
, and
L. A.
Crum
, “
Use of dual-pulse lithotripter to generate a localized and intensified cavitation field
,”
J. Acoust. Soc. Am.
110
,
1685
1695
(
2001
).
12.
P.
Zhong
and
Y. F.
Zhou
, “
Suppression of large intraluminal bubble expansion in shock wave lithotripsy without compromising stone comminution: Methodology and in vitro experiments
,”
J. Acoust. Soc. Am.
110
,
3282
3291
(
2001
).
13.
Y. F.
Zhou
and
P.
Zhong
, “
Suppression of large intraluminal bubble expansion in shock wave lithotripsy without compromising stone comminution: Refinement of reflector geometry
,”
J. Acoust. Soc. Am.
113
,
586
597
(
2003
).
14.
M. F.
Hamilton
, “
Transient axial solution for the reflection of a spherical wave from a concave ellipsoidal mirror
,”
J. Acoust. Soc. Am.
106
,
102
112
(
1993
).
15.
A. J.
Coleman
,
M. J.
Choi
, and
J. E.
Saunders
, “
Theoretical predictions of the acoustic pressure generated by a shock wave lithotripter
,”
Ultrasound Med. Biol.
17
,
245
255
(
1991
).
16.
T.
Christopher
, “
Modeling the Dornier HM-3 lithotripter
,”
J. Acoust. Soc. Am.
96
,
3088
3095
(
1994
).
17.
M. A.
Averkiou
and
R. O.
Cleveland
, “
Modeling of an electrohydraulic lithotripter with the KZK equation
,”
J. Acoust. Soc. Am.
106
,
102
112
(
1999
).
18.
M.
Tanguay
and
T.
Colonius
, “
Numerical simulation of bubble cavitation flow in shock wave lithotripsy
,” in
Fourth International Symposium on Cavitation
,
CAV2001
,
Pasadena
,
2001
.
19.
M.
Tanguay
and
T.
Colonius
, “
Progress in modeling and simulation of shock wave lithotripsy (SWL)
,” in
Fifth International Symposium on Cavitation
,
CAV2003
,
Osaka, Japan
,
2003
.
20.
A. J.
Szeri
, “
Numerical modeling of shock wave focusing and bubble dynamics in an elastic tube
” (private communication).
21.
J. I.
Iloreta
,
A. J.
Szeri
,
Y. F.
Zhou
, and
P.
Zhong
, “
Wave propagation and shock formation in a shock wave lithotripter
,”
Phys. Fluids
(submitted).
22.
P.
Zhong
,
Y. F.
Zhou
, and
S. L.
Zhu
, “
Dynamics of bubble oscillation in constrained media and mechanisms of vessel rupture in SWL
,”
Ultrasound Med. Biol.
27
,
119
134
(
2001
).
23.
Y. S.
Lee
and
M. F.
Hamilton
, “
Time-domain modeling of pulsed finite-amplitude sound beams
,”
J. Acoust. Soc. Am.
97
,
906
917
(
1995
).
24.
R. O.
Cleveland
,
M. F.
Hamilton
, and
D. T.
Blackstock
, “
Time-domain modeling of finite-amplitude sound in relaxing fluids
,”
J. Acoust. Soc. Am.
99
,
3312
3318
(
1996
).
25.
C. C.
Church
, “
A theoretical study of cavitation generated by an extracorporeal shock wave lithotripter
,”
J. Acoust. Soc. Am.
86
,
215
227
(
1989
).
26.
S. L.
Zhu
and
P.
Zhong
, “
Shock wave-inertial microbubble interaction: A theoretical study based on the Gilmore formulation for bubble dynamics
,”
J. Acoust. Soc. Am.
106
,
3024
3033
(
1999
).
27.
R. H.
Cole
,
Underwater Explosions
(
Princeton University Press
,
Princeton
,
1948
).
28.
A. J.
Coleman
and
J. E.
Saunders
, “
A survey of the acoustic output of commercial extracorporeal shock wave lithotripters
,”
J. Acoust. Soc. Am.
15
,
213
227
(
1989
).
29.
R. O.
Cle veland
,
D. A.
Lifshitz
,
B. A.
Connors
,
A. P.
Evan
,
L. R.
Willis
, and
L. A.
Crum
, “
In vivo pressure measurements of lithotripsy shock waves in pigs
,”
Ultrasound Med. Biol.
24
,
293
306
(
1998
).
30.
M. R.
Bailey
,
D. T.
Blackstock
,
R. O.
Cleveland
, and
L. A.
Crum
, “
Comparison of electrohydraulic lithotripters with rigid and pressure-release ellipsoidal reflectors. II. Cavitation fields
,”
J. Acoust. Soc. Am.
106
,
1149
1160
(
1999
).
31.
A. C.
Baker
, “
Nonlinear pressure fields due to focused circular apertures
,”
J. Acoust. Soc. Am.
91
,
713
717
(
1992
).
32.
J.
Lighthill
,
Waves in fluids.
(
Cambridge University Press
,
Cambridge
,
1980
).
33.
T. S.
Hart
and
M. F.
Hamilton
, “
Nonlinear effects in focused sound beams
,”
J. Acoust. Soc. Am.
84
,
1488
1496
(
1988
).
34.
B.
Sturtevant
, “
Shock wave physics of lithotriptors
,” in
Smith’s Textbook of Endourology
, edited by
A. D.
Smith
 et al, (
Quality Medical
,
St. Louis
,
1996
). pp.
529
552
.
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