OS-2-1-010 Fifth International Symposium on Cavitation (CAV2003)Osaka, Japan, November 1-4, 2003

PROGRESS IN MODELING AND SIMULATION OFSHOCK WAVE LITHOTRIPSY (SWL)

Michel Tanguay Tim ColoniusDepartment of Mechanical Engineering Department of Mechanical Engineering

California Institute of Technology California Institute of TechnologyPasadena, California, 91125 Pasadena, California, 91125

michel@caltech.edu colonius@caltech.edu

ABSTRACTPast research in shock wave lithotripsy (SWL) has

shown that cavitation plays an essential role in thecomminution of kidney stones. To provide a bet-ter understanding of the role of cloud cavitation dy-namics in SWL, the flow in the focal region of alithotripter was modeled using an ensemble averagedtwo-phase flow model for the bubbly mixture com-bined with a high-order accurate shock capturingtechnique. The domain and initial conditions usedin the numerical model reflect the appropriate di-mensions and intensity of a Dornier HM3 electro-hydraulic lithotripter. The impact of factors suchas the size and number of bubble nuclei in the liq-uid, the intensity of the shock wave and the pulserate frequency (PRF) on the cavitating flow field isanalyzed. Conclusions regarding the impact of theseparameters on the potential for stone comminutionare also presented.

INTRODUCTIONSeveral aspects of the treatment of kidney stones

using SWL are not fully understood. Although theimportance of of cavitation in stone comminution hasbeen firmly established, see [1-2] as well as 3 pre-sented elsewhere in these proceedings, it has not beenpossible in the past to obtained detailed evolution forthe generated cavitation field. Because of short andlong term side effects of SWL, there is an extensive re-search effort to find optimized treatment parameterssuch as intensity and pulse rate frequency. In orderto achieve this goal, it is crucial to understand the

specific characteristics of the cavitation cloud whichpromote stone comminution.

At the present stage, the role of numerical model-ing has been constrained into predicting the pressurein the field of a lithotripter and the response of a sin-gle bubble to the pressure field [4-5]. However, theseapproaches are valid only in the limit of vanishingvoid fraction and cannot represent coupled interac-tions between the pressure and cavitation field. Aspresented in our earlier work [6-7], the two-phase con-tinuum model is able to represent some of the com-plex interactions occurring in the focal region of alithotripter. The present study is a continuation ofthis work and focuses on the interactions within thecollapsing bubble cloud and their importance with re-gards to stone comminution. By post-processing thenumerical results using a more complex bubble modelwhich includes gas diffusion, we were able to calculatean approximate pulse firing rate which would corre-spond to the conditions implemented.

MODELING

Physical considerationsAs mentioned previously, the objective of this work

is to accurately model the behavior of an electro-hydraulic lithotripter. The general features relatedto the generation and propagation of the shock wavein a lithotripter are depicted in figure 1. It shouldbe noted that the generation of a negative pressure(tensile stress region) behind the reflector shock wave(see bottom of figure 1) has not been noted in past

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OS-2-1-010 Fifth International Symposium on Cavitation (CAV2003)Osaka, Japan, November 1-4, 2003

experimental observations, most likely because of itssmall amplitude. However, this region is clearly visi-ble in our current work and has been validated usinga wave propagation model [8].

F2

Focal pointF1

Focal pointF1

Focal point

Sphericalshock wave

Incident wave

Incident waveIncident wave

Reflector

Negative pressurePositive pressure

Reflected waveReflected wave

Reflector

Reflected waveReflected wave

Edge waveEdge wave

Initial shock wave generation

Reflection of shock wave by ellipsoidal surface

Generation of tensile stress behind refected wave and by the edge wave

Figure 1: Shock wave propagation in an electro-hydraulic lithotripter

The geometry of the domain used in the presentwork is shown in figure 2. These dimensions cor-respond to the dimensions of the Dornier-HM3, acommercial electro-hydraulic lithotripter, and theCaltech-EHL, which is a research lithotripter basedon the HM3 [5]. Typical measurements for the pres-sure at the focal point for two Caltech-EHL are [5]:

APL GALCITPmax(MPa): 29.9± 4.7 35.3± 5.4Pmin(MPa): −11.5± 0.3 −8.7± 0.4Duration (µs): 5-6 5-6

The magnitude of the tensile stress in the focal regionis well below the vapor pressure of the liquid andgenerate cavitating bubbles in its wake.

F2

Focal pointF1

Focal point

228mm

80mm

79.58mm

114mm98.36mm

Figure 2: Dimensions of the reflector used in thisstudy

Relationship between bubble growth and dura-tion

From the work of Lord Rayleigh [9], the time fora vapor filled spherical cavity to collapse can be de-rived under specific assumptions. Following the sameassumptions, the time required for an impulsivelyforced cavitation nuclei to grow and collapse (tc) canbe found to be twice the Rayleigh collapse time.

tc = 1.83Rmax

√ρ

P∞ − Pv(1)

This simple results predicts a linear relationship be-tween the maximum radius of a bubble and its life-time. In figure 3, this relationship is compared toresults obtained from using the Rayleigh-Plesset andGilmore bubble model with a Church pressure wave-form [10] which is the typical model used to approxi-mate the pressure at the focal point of a lithotripter.It is interesting to note that these single bubblenumerical results correspond almost exactly to theRayleigh collapse relationship. However, when com-pared to measurements of the bubble size and dura-tion in the focal region of a lithotripter [11], we findthat the actual bubbles behave a significant longerlifetime for their size (see figure 3).

The discrepancy between the actual behavior ofbubbles in a lithotripter and the single bubble nu-merical model is not caused by a limitation in thesingle bubble model but rather, as it will be shownin this work, by collective interaction in the bubblecloud.

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OS-2-1-010 Fifth International Symposium on Cavitation (CAV2003)Osaka, Japan, November 1-4, 2003

Maximum bubble radius (mm)

Tim

eto

colla

pse

t c(µ

s)

0 1 2 30

100

200

300

400

500

600

700

Rayleigh-PlessetGilmoretwice Rayleigh collapse time

Free field lithotripterSokolov et al.[11]

Dual-pulse lithotripterSokolov et al. [11]

Figure 3: Relationship between time to collapse andmaximum bubble radius

Role of pulse rate frequencyExperimental studies have been conducted on the

role of pulse rate frequency and have concluded thathas a measurable impact on the effectiveness of thetreatment [12]. Frequencies of 1 and 2Hz (whichare typical rates used by clinicians) was found tobe less effective in stone comminution than a rate of1/2Hz. Bailey et al. [13] have hypothesized that thefrequency dependence is due to a shielding effect ofthe stone by the bubble cloud. During the growth ofthe bubble cloud, non-condensible gas dissolves intothe bubble increasing its equilibrium radius. Depend-ing on the time delay between the arrival of the nextshock wave, all or part of the trapped gases can dis-solve back into solution. For a high PRF, bubbleswill tend to grow to large sizes (assuming no bubblefission) and interfere to a greater extent with the fo-cusing of the shock wave. As discussed below, thiseffect is in fact observed in our simulation with largebubble nuclei.

Numerical modeling of the two-phase mixtureThe numerical model for the continuum bubbly

flow used in this work is based on the work of Zhangand Prosperetti [14-15]. In this approach, the flowproperties are average over all possible bubble loca-tion and states. In order to compute some of theensemble average integrals, the bubble field is as-

sumed to vary smoothly. The equations are thenfurther simplified by assuming a low void fraction.The resulting equations show a strong similarity1 tothe equations presented in the work of Biesheuvel andvan Wijngaarden [16].

The derivation of the two-phase mixture equationsrequires the introduction of a model for the bubbledynamics. The present work follows the procedureshown in [15] in which the local perturbations in theliquid due to bubble motion are assumed to be in-compressible. However, in order to provide a morerealistic behavior, the model describing the bubbledynamics was corrected for compressibility (Gilmoremodel [17]).

The ensemble averaged equations are valid for apolydisperse bubble field. However, the cost of com-puting the evolution of a distribution of bubble statefor all grid location is prohibitively expensive. There-fore, a monodisperse bubble distribution was used(〈f(R)〉 = f(〈R〉)).

NUMERICAL IMPLEMENTATIONThe numerical implementation of our model fol-

lows closely from our previous work [6-7] and only abrief overview will be presented here.

The implementation of the ellipsoidal reflector wassimplified by using a prolate-spheroidal coordinatesystem in part of the domain (see figure 4). Thereflective and axisymmetric boundary condition wasimplemented following the work of Mohseni and Colo-nius [18]. The numerical implementation of the ap-proximately non-reflective boundary conditions wasbased on the work of Thomson [19] and connected byadding an absorbing buffer layer based on the workof Colonius et al. [20].

In order to achieve an accurate representation ofthe propagation of the shock wave, a Weighted Es-sentially Non-Oscillatory (WENO) fifth-order shockcapturing scheme was implemented. Implementa-tion details can be found in the work of Jiang andShu [21]. The time integration of the flow field wasdone using a third-order Total Variation Diminishing

1The equations used for the bubbly mixture in this workthe same as the one derived in [16] with the addition of a termproportional to the void faction βD. They are presented indetail in [8].

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OS-2-1-010 Fifth International Symposium on Cavitation (CAV2003)Osaka, Japan, November 1-4, 2003

Mesh stretching and filteringMesh stretching and filtering

Non-reflective boundary

No

n-r

efl

ect

ive

bo

un

da

ry

Reflector

Axisymmetry boundary condition F1F2

Figure 4: Representation of the numerical grid

(TVD) Runge-Kutta scheme [22]. The discretizedODE’s describing the evolution of the bubble fieldwere integrated separately within each time step us-ing a fifth-order Kaps-Rentrop adaptive time march-ing algorithm.

Initial conditions

Since the shock wave generation by the spark gapis beyond the scope of this work, the simulation isinitialized with the shock wave fully formed at somedistance from F1. This approach follows in the workof Averkiou and Cleveland [23] where the initial shockwave is taken to be of triangular profile. Because ofmass conservation constraints, this waveform requiresthe implementation of a distributed mass source atthe first focal point. This mass source is a simplifiedmodel of the vapor cavity initially generated by thefiring of the electrode and is discussed in [7-8].

In addition to the uncertainty related to the initialshock wave, the conditions of the cavitation nucleiin the liquid are not known a priori. The numberof bubble nuclei in suspension can be approximatedfrom high speed images of the bubble field after thepassage of the shock wave. In the work of Sokolovet al. [11], a number density of approximately 70bubbles/cm3 was reported. However, the simulationresults presented in this work used a more conserva-tive range of 0 to 40 bubbles/cm3. The initial size ofthe cavitation nuclei cannot be determined directlyfrom observations, but is specified and validated aposteriori. In the course of this work, sizes rangingfrom 3 to 50 µm were investigated and the resultscompared to empirical observations.

Damage and bubble energyIn order to identify optimal conditions for stone

comminution, a method of quantifying damage mustbe determined. Unfortunately, no simple model isavailable that could approximate the damage a bub-ble can inflict on a surface. Alternatively, the energyreleased during the collapse of a bubble can be calcu-lated. Although it is not possible to determine howmuch energy is released in the form of a micro-jet asopposed to a spherical shock wave or to establish aprecise correlation with surface damage, we are as-suming that the most energetic bubble collapse willresult in the most stone damage.

In post-processing the simulation data, the energyreleased by a bubble was computed using a slightlydifferent bubble model. We consider the Herringmodel [24]:[

1− 2R

c

]RR +

32

[1− 4

3R

c

]R2 =

pB − p∞(t)ρ

+R

c

d

dt

(pB − p∞(t)

ρ

),

(2)

where pB is the pressure at the surface of the bub-ble. This model is nearly identical to the Gilmoremodel discussed earlier and has a similar behavior.However, the Herring model can be integrated onceto yield:

12

(1 + 3

R

c

)R3R2 =

∫pB

ρR2dR +

∫R3R

ρc

dpB

dRdR

−1ρ

∫ [p∞ +

R

c

dp∞dt

]R2dR + constant.

(3)

The R/c dependence on the left hand side of theabove equation was found to be negligible in casesrelevant to this work. Similarly, the second term onthe right hand side was also found to be negligible.The above equation was therefore simplified to:

12R3R2 ≈

∫ R

Ro

pB

ρR2dR

−1ρ

∫ R

Ro

[p∞ +

R

c

dp∞dt

]R2dR,

(4)

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OS-2-1-010 Fifth International Symposium on Cavitation (CAV2003)Osaka, Japan, November 1-4, 2003

where:

Kinetic energy:12R3R2

Bubble potential energy:[∫

pB

ρR2dR

]R

Initial bubble energy:[∫

pB

ρR2dR

]Ro

Work done by liquid:

1ρ

∫ R

Ro

[p∞ +

R

c

dp∞dt

]R2dR

The difference between the bubble energy (internaland kinetic) and the work done by the liquid afterthe bubble collapse is equal to the energy released.

Rectified diffusion

More elaborate bubble models which can accountfor the contribution of heat and mass transfer in thebubble are too computationally expensive to be usedin our lithotripter simulations. However, in order tobe able to analyze the relationship between the sizeof bubble nuclei and PRF, a post-processing calcu-lation was performed using a more elaborate bubblemodel in conjunction with the pressure history mea-sured at the focal point of the simulation. A correc-tion accounting for the internal heat and mass dif-fusion was introduced based on the work of Prestonet al. [25]. The bubble exterior was considered in-compressible and isothermal for the purpose of cal-culating the diffusion of non-condensible gas and itsimplementation followed from the work of Plesset andZwick [26] and, Eller and Flynn [27]. It is importantto note that the results shown here were based onthe assumption that the liquid was degassed down to100 Torr and the non-condensible gas is air. Changesto the actual value of concentration and/or compo-sition of dissolved gases will impact the predictionsfor the PRF. Given the pressure field for a typicaldecoupled simulation (zero bubble number density),the required PRF to maintain the bubble equilibriumradius is shown in figure 5.

0.00

5.00

10.00

15.00

20.00

25.00

30.00

0 10 20 30 40 50 60 70

Equilibrium radius (µm)

PR

F (

Hz)

Figure 5: Relationship between equilibrium bubbleradius (Ro) and pulse repetition frequency for a sin-gle bubble forced by a typical pressure measurementfrom our simulation.

RESULTS

Free field lithotripterUsing the numerical model discussed here, the

pressure and cavitation fields for a Caltech-EHLelectro-hydraulic lithotripter were computed for thefree field case (no stone). The predicted pressure atthe focal point F2 is compared in figure 6 to an ex-perimental measurement obtained using a membranehydrophone. The numerical calculations were per-formed for two cases: one with zero density of bub-ble nuclei (decoupled approach), and the other witha density of bubble nuclei of 20 bubbles/cm3. It is ofinterest to note that the decoupled approach providesa better match with the measurements than the casewith No = 20 bubbles/cm3. For this case, the pres-ence of a small pressure rise at the tail of the wave iscaused by the distortion of the edge wave as it passesthrough the cloud of expanding bubbles [8]. Thisfeature of the pressure wave is not usually seen inreported pressure measurements of electro-hydrauliclithotripters. The absence of this bubble—wave inter-action can be partly explained by the typical exper-imental protocol for this measurement. In order toincrease the lifetime of the pressure transducer, mea-surements are usually carried out in clean degassedwater (low nuclei density). A second peak is some-

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OS-2-1-010 Fifth International Symposium on Cavitation (CAV2003)Osaka, Japan, November 1-4, 2003

times observed [13] but a more detailed analysis ofexperimental results is needed before a more in-depthcomparison can be made.

Time (µs)

Pres

sure

(MPa

)

180 182 184 186-20

-10

0

10

20

30

40

50

60 Typical measurement, Caltech-EHL (APL) Data courtesy of M.R. Bailey, Center for Industrial and Medical Ultrasound, APL, University of Washington, Seattle

Numerical model, Ro = 20 µm No = 0 bubbles/cm3 No = 20 bubbles/cm3

Figure 6: Comparison between numerical and exper-imental pressure measurement at F2.

Another source of validation for the present modelis the comparison of the size and shape of the calcu-lated bubble cloud with empirical observation. Sucha comparison is shown in figure 7 where appears tobe in a good agreement with the observations.

As a means of finding appropriate estimates for theinitial size and density of bubble nuclei, the values forthe time to collapse tc and maximum bubble radius atthe focal point were compared to experimental obser-vations. As seen in figure 8, the relationship betweenRmax and tc is dependent on the initial conditionschosen. For cases with low or zero number density,the simulation falls very close to the Rayleigh collapserelationship. Although larger number density simu-lations came closer to the experimental observations,no set of initial conditions were found to match ob-served Rmax and tc simultaneously. This discrepancymay be due to the low void fraction approximation inthe formulation of the ensemble averaged equations.A model capable of representing higher order inter-actions between the two phases may be able bridgethis gap.

As seen in figure 8, the interaction between thebubble and pressure field have a significant impact

66 mm

Integrated void fraction

ExperimentRmax 0.5 mmtc 340 31 µs-+

Rmax 0.76 mmtc 321µs

Time 360 µs

Gilmore

Max void 2.94%

Figure 7: Comparison between numerical and ob-served bubble cloud in the focal region.

on the overall behavior of the bubble cloud. The im-portance of this interaction is also made manifest inthe collapse of the bubble cloud. Figure 9 presentsthe pressure and void fraction contours during thecollapse of the bubble cloud. Pressure waves gen-erated by the collapsing bubbles at the edge of thecloud propagate inward and precipitate the collapseof inner bubbles. This type of collective collapse issimilar to earlier work on the collapse of sphericalbubble clouds [28-29-30-31]. This phenomenon is also

Rmax (mm)

t c (µ

s)

0.5 0.6 0.7 0.8 0.9

100

150

200

250

300

350

Experimental observations

Rayleigh collapse relationship

Increasing void fraction

Increasing bubble-bubble

interaction

Figure 8: Relationship between Rmax and tc for var-ious simulations compared to experimental observa-tions

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OS-2-1-010 Fifth International Symposium on Cavitation (CAV2003)Osaka, Japan, November 1-4, 2003

shown in figure 10 where the interactions within thebubble cloud caused an 9% increase in growth and a115% increase in the lifetime of a bubble at its center.

Posi

tion

(mm

)

-30-20-1001020300

5

10

-30-20-1001020300

5

10

-30-20-1001020300

5

10

-30-20-1001020300

5

10

-30-20-1001020300

5

10

-30-20-1001020300

5

10

-30-20-1001020300

5

10

-30-20-1001020300

5

10

-30-20-1001020300

5

10

-30-20-1001020300

5

10

-30-20-1001020300

5

10

-30-20-1001020300

5

10

-30-20-1001020300

5

10

-30-20-1001020300

5

10

-30-20-1001020300

5

10

-30-20-1001020300

5

10

-30-20-1001020300

5

10

Void fraction contour levels (%): 0.0 , 0.8 , 1.6 , 2.4 , 3.2 , 4.0

Pressure (MPa):-0.108 -0.050 0.009 0.126 0.185 0.244 0.302 0.361 0.420 0.478

time = 411 µs

time = 416 µs

time = 421 µs

time = 426 µs

time = 431 µs

time = 436 µs

time = 441 µs

time = 446 µs

time = 456 µs

time = 461 µs

time = 466 µs

time = 471 µs

time = 476 µs

time = 481 µs

time = 486 µs

time = 491 µs

time = 451 µs

Figure 9: Pressure field (color contours) and voidfraction (black lines) in the focal region during col-lapse of the bubble cloud

A direct benefit of cloud cavitation over single bub-ble cavitation is shown in figure 11 where the energyof a bubble at F2 is compared for both cases. Theenergy released is estimated with equation 4 and themeasured pressure at the focal point from simulation.The impact of the pressure waves observed in figure9 on the collapsing bubbles at the center of the cloudalmost tripled the amount of energy released.

Time (µs)

Bubb

leR

adiu

s(m

m)

200 300 400 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Bubble number density: 20 #/cm3Bubble number density: 0 #/cm3

Figure 10: Impact of number density on bubble ra-dius at F2 for Ro = 20µm

Time (µs)

Bubb

leen

ergy

(µJ)

200 300 400 5000

5

10

15

20

25

30

35

40

45

Bubble number density: 20 #/cm3Bubble number density: 0 #/cm3

Energy released at collapse

Energy released at collapse

Figure 11: Impact of number density on bubble en-ergy at F2 for Ro = 20µm

The impact of the pulse rate frequency on the pres-sure field is illustrated in figure 12. Keeping the bub-ble number density constant, an increase in the PRFtranslates into an increase in the initial and maxi-mum void fraction in the focal region. Because ofthe higher void fractions encountered, the shock wave

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OS-2-1-010 Fifth International Symposium on Cavitation (CAV2003)Osaka, Japan, November 1-4, 2003

cannot focus as accurately and shielding occurs.

PRF (Hz)

Peak

pres

sure

(MPa

)

100 10122

23

24

25

26

27

28

N=0N=10N=20

Figure 12: Peak pressure at the focal point as func-tion of pulse rate frequency

Figure 13 presents a comparison of the energy re-leased by collapsing bubble for a wide range of cases.The energy levels have been normalized with respectto the limiting case of zero number density and verysmall bubble nuclei size. Larger values of bubblenumber density provide significant increase in the en-ergy release. A measure of uncertainty in the numer-ical results was obtained by varying an arbitrary pa-rameter in the artificial compression method (ACM)of the WENO scheme used here (further details canbe found in [8]). The data presented in figure 13 and12 can be interpreted as follows:

• for the decoupled case, the energy released in-creases monotonically with PRF,

• for PRF lower than 20 Hz, the energy releasedis increased by the presence of bubble cloud in-teraction,

• increasing the PRF results in an increase in Ro.For PRF lower than 1 Hz, this translates into anincrease in void fraction, cloud interaction andenergy release,

• as the PRF and Ro is further increased, shield-ing effects due to higher void fraction decreasethe pressure amplitude at focus and the energyabsorbed by the bubble.

Pulse rate frequency (Hz)

Nor

mal

ized

bubb

leen

ergy

atco

llaps

e

100 101 102

0.5

1

1.5

2

2.5

3

3.5

N=0N=10N=20

Figure 13: Normalized energy released at bubble col-lapse as a function of pulse rate frequency

Lithotripter with artificial stone

Following the same procedure as the free field case,the impact of cloud interactions on the bubble col-lapse can be observed for cases with a 6.25 mm di-ameter artificial stone located at the focal point. Fig-ure 14 presents the energy released by a bubble at F2

in the presence of an artificial stone as a function ofPRF. It should be noted that the normalization anduncertainty in figure 14 is the same as in figure 13. Itis interesting to note that the same trends noticed infigure 13 appear in figure 14 and that the impact ofcloud activity on the energy released is even greaterin this case.

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OS-2-1-010 Fifth International Symposium on Cavitation (CAV2003)Osaka, Japan, November 1-4, 2003

Pulse rate frequency (Hz)

Nor

mal

ized

bubb

leen

ergy

atco

llaps

e

100 101 102

1

2

3

4

5

6

7

N=0N=20

Figure 14: Normalized energy released at bubble col-lapse as a function of pulse rate frequency with arti-ficial stone present

CONCLUSIONSIn this paper, we have presented some recent re-

sults using the ensemble averaged two-phase flowapproach to predict cloud cavitation in SWL. Theagreement between the numerical results from thepresent model and experimental results has been im-proved over our past work. Using the energy analysispresented here, we were able to observe how the col-lapse of the bubble cloud can generate a substantialincrease in the energy available for bubble collapse.Moreover, by post-processing simulation results us-ing a more complex bubble model with gas diffusion,initial simulation conditions were related to the pulserate frequency. Based on the results presented here,the following conclusions can be drawn:

• the discrepancy in the ratio of tc/Rmax betweendirect observation and single bubble models canbe explained by bubble interactions in the cloud,

• subjected to the same pressure history, the en-ergy released increases with the bubble equilib-rium size (PRF),

• for non-zero bubble number density, the col-

lapse of the bubble cloud generates pressure wavewhich enhances the energy released by the innerbubbles by up to a factor of three,

• higher void fractions deflect part of the shockwave energy away from the focal point and pre-vent inner bubbles from acquiring as much en-ergy,

• based on the previous observations, a PRF ofapproximately 1 Hz provides conditions for themaximum release of energy by a bubble at thefocal point and presumably inflicts the maximumstone damage for a given number density.

It should be kept in mind that the values for the PRFare dependent on the initial fraction of gas dissolvedin the liquid. Furthermore, it should be noted that,if present, bubble fission would increase the numberdensity and decrease the equilibrium size. There issome evidence that this in fact takes place [32]. Atthis stage, this phenomena cannot be introduced intothe model and any comparison of the PRF resultswith experimental observations should be made forcases with a stable density of bubble nuclei.

ACKNOWLEDGMENTSThis work was supported by NIH under a Program

Project Grant PO1 DK43881. We would like to thankDr. Andrew Evan and all the PPG investigators. Inparticularly, we thank Dr’s Robin Cleveland, MikeBailey and Yura Pishchalnikov for their commentsand experimental data.

REFERENCE

[1] L.A. Crum. Cavitation microjets as a contrib-utory mechanism for renal calculi disintegrationin ESWL. Journa of Urology, 140:1587–1590,1988.

[2] A.J. Coleman and J.E. Saunders. A review ofthe physical-properties and biological effects ofthe high amplitude acoustic fields used in extra-corporeal lithotripsy. Ultrasonics, 31(2):75–89,1993.

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OS-2-1-010 Fifth International Symposium on Cavitation (CAV2003)Osaka, Japan, November 1-4, 2003

[3] M.R. Bailey, L.A. Crum, O.A. Sapozh-nikov, A.P. Evan, J.A. McAteer, R.O.Cleveland, and T. Colonius. Cavitation inshock wave lithotripsy. In Proceedings ofthe 5th International Symposium on Cav-itation, Osaka, Japan, 2003. publishedelectronically at http://iridium.me.es.osaka-u.ac.jp/cav2003/index.html : paper OS-2-1-006.

[4] M.R. Bailey, D.T. Blackstock, R.O. Cleveland,and L.A. Crum. Comparison of electrohydrauliclithotripters with rigid and pressure-release ellip-soidal reflectors. ii. cavitation fields. Journal ofthe Acoustical Society of America, 106(2):1149–1160, 1999.

[5] R.O. Cleveland, M.R. Bailey, N. Fineberg,B. Hartenbaum, M. Lokhandwalla, J.A. McA-teer, and B. Sturtevant. Design and characteri-zation of a research electrohydraulic lithotripterpatterned after the Dornier HM3. Review of Sci-entific Instruments, 71:2514–2525, 2000.

[6] M. Tanguay and T. Colonius. Numerical sim-ulation of bubbly cavitating flow in shock wavelithotripsy. In Proceedings of the Fourth Inter-national Symposium on Cavitation, CaliforniaInstitute of Technology, Pasadena, California,2001.

[7] M. Tanguay and T. Colonius. Numerical sim-ulation of bubbly cavitating flow in shock wavelithotripsy. In Proceedings of FEDSM’02 ASMEFluid Engineering Division Summer Meeting,Montreal, Canada, July 2002.

[8] M. Tanguay. Computation of bubbly cavitatingflow in shock wave lithotripsy. PhD thesis, Cal-ifornia Institute of Technology, 2003.

[9] Lord Rayleigh. Pressure developed in a liquidduring the collapse of a spherical cavity. Philo-sophical Magazine, 34:94–98, 1917.

[10] C.C. Church. A theoretical study of cavita-tion generated by an extracorporeal shock wavelithotripter. Journal of the Acoustical Society ofAmerica, 86(1):215–227, 1989.

[11] D.L. Sokolov, M.R. Bailey, and L.A. Crum. Useof a dual-pulse lithotripter to generate a local-ized and intensified cavitation field. Journal ofthe Acoustical Society of America, 110(3):1685–1695, 2001.

[12] R.F. Paterson, D.A. Lifshitz, J.E. Lingeman,A.P. Evan, B.A. Connors, N.S. Fineberg, J.C.Williams, and J.A. McAteer. Stone fragmen-tation in shock wave lithotripsy is improved byslowing the stock wave rate: Studies with a newanimal model. Journal of Urology, 168:2211–2215, 2002.

[13] M.R. Bailey, R.O. Cleveland, L.A. Crum, andT.J. Matula. Personal communication.

[14] D.Z. Zhang and A. Prosperetti. Averaged equa-tions for inviscid disperse two-phase flow. Jour-nal of Fluid Mechanics, 267:185–219, 1994.

[15] D.Z. Zhang and A. Prosperetti. Ensemble phase-averaged equations for bubbly flows. Physics ofFluids, 6(9):2956–2970, 1994.

[16] A. Biesheuvel and L. van Wijngaarden. Two-phase flow equations for a dilute dispersion ofgas bubble in liquid. Journal of Fluid Mechanics,148:301–318, 1984.

[17] R.F. Gilmore. The growth or collapse of a spheri-cal bubble in a viscous compressible liquid. Tech-nical Report Rep. No. 26-4, California Instituteof Technology, 1952.

[18] K. Mohseni and T. Colonius. Numerical treat-ment of polar coordinate singularities. Journalof Computational Physics, 157(2):787–795, 2000.

[19] K.W. Thompson. Time dependent boundaryconditions for hyperbolic systems. Journal ofComputational Physics, 68:1–24, 1987.

[20] T. Colonius, S. Lele, and P. Moin. Bound-ary conditions for direct computation of aero-dynamic sound generation. AIAA Journal,31(9):1574–1582, 1993.

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OS-2-1-010 Fifth International Symposium on Cavitation (CAV2003)Osaka, Japan, November 1-4, 2003

[21] G.S. Jiang and C.W. Shu. Efficient implementa-tion of weighted ENO schemes. Journal of Com-putational Physics, 126(1):202–228, 1996.

[22] S. Gottlieb and C.-W. Shu. Total variation di-minishing Runge-Kutta schemes. Mathematicsof Computation, 67(221):73–85, 1998.

[23] M.A. Averkiou and R.O. Cleveland. Modelingof an electrohydraulic lithotripter with the KZKequation. Journal of the Acoustical Society ofAmerica, 106(1):102–112, 1999.

[24] C. Herring. Theory of the pulsations of the gasbubble produced by an underwater explosion.Technical Report 236, OSRD, 1941.

[25] A. Preston, T. Colonius, and C.E. Brennen. Asimplified model of heat transfer effects on thedynamics of bubbles. In ASME/FED FourthInternational Symposium on Numerical Meth-ods for Multiphase Flow, Montreal, Quebec,Canada, July 2002.

[26] M.S. Plesset and S.A. Zwick. A nonsteadyheat diffusion problem with spherical symmetry.Journal of Applied Physics, 23(1):95–98, 1952.

[27] A. Eller and H.G. Flynn. Rectified diffusion dur-ing nonlinear pulsations of cavitation bubbles.Journal of the Acoustical Society of America,37(3):493–503, 1965.

[28] K.A. Mørch. On the collapse of cavity cluster inflow cavitation. In Proceedings of the First In-ternational Conference on Cavitation and Inho-mogenieties in Underwater Acoustics, volume 4,pages 95–100. Springer Series in Electrophysics,1980.

[29] K.A. Mørch. On the collapse of cavity clusterin flow cavitation. In Proceedings of the ASMECavitation and Polyphase Flow Forum, pages 1–10, 1981.

[30] Y.-C. Wang and C.E. Brennen. The noise gen-erated by the collapse of a cloud of cavitation

bubble. In ASME/JSME Symposium on Cav-itation and Gas-Liquid Flow in Fluid Machin-ery and Devices, volume FED 226, pages 17–29,1995.

[31] G.E. Reisman, Y.-C. Wang, and C.E. Brennen.Observations of shock waves in cloud cavitation.Journal of Fluid Mechanics, 355:255–283, 1998.

[32] O.A. Sapozhnikov, V.A. Khokhlova, M.R. Bai-ley, J.C. Williams, J.A. McAteer, and R.O.Cleveland. Effect of overpressure and pulse rep-etition frequency on cavitation in shock wavelithotripsy. Journal of the Accoustical Societyof America, 2002.

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