High-speed video microscopy and numerical modeling of bubble dynamics near a surface of urinary stone

High-speed video microscopy and numerical modeling of bubble dynamics near a

surface of urinary stone

Yuri A. Pishchalnikov

, William M. Behnke-Parks

, Kevin Schmidmayer

, Kazuki Maeda

, Tim Colonius

, Thomas W.

Kenny

, and

Daniel J. Laser

Citation:

The Journal of the Acoustical Society of America

146

, 516 (2019); doi: 10.1121/1.5116693

View online:

https://doi.org/10.1121/1.5116693

View Table of Contents:

https://asa.scitation.org/toc/jas/146/1

Published by the

Acoustical Society of America

High-speed video microscopy and numerical modeling of bubble

dynamics near a surface of urinary stone

Yuri A.

Pishchalnikov,

William M.

Behnke-Parks,

Kevin

Schmidmayer,

Kazuki

Maeda,

Tim

Colonius,

Thomas W.

Kenny,

and Daniel J.

Laser

Applaud Medical, Incorporated, 953 Indiana Street, San Francisco, California 94107, USA

Division of Engineering and Applied Science, California Institute of Technology, Pasadena,

California 91125, USA

Department of Mechanical Engineering, Stanford University, Stanford, California 94305, USA

(Received 4 March 2019; revised 21 June 2019; accepted 26 June 2019; published online 26 July

2019)

Ultra-high-speed video microscopy and numerical modeling were used to assess the dynamics of

microbubbles at the surface of urinary stones. Lipid-shell microbubbles designed to accumulate on

stone surfaces were driven by bursts of ultrasound in the sub-MHz range with pressure amplitudes

on the order of 1 MPa. Microbubbles were observed to undergo repeated cycles of expansion and

violent collapse. At maximum expansion, the microbubbles’ cross-section resembled an ellipse

truncated by the stone. Approximating the bubble shape as an oblate spheroid, this study modeled

the collapse by solving the multicomponent Euler equations with a two-dimensional-axisymmetric

code with adaptive mesh refinement for fine resolution of the gas-liquid interface. Modeled bubble

collapse and high-speed video microscopy showed a distinctive circumferential pinching during the

collapse. In the numerical model, this pinching was associated with bidirectional microjetting nor-

mal to the rigid surface and toroidal collapse of the bubble. Modeled pressure spikes had ampli-

tudes two-to-three orders of magnitude greater than that of the driving wave. Micro-computed

tomography was used to study surface erosion and formation of microcracks from the action of

microbubbles. This study suggests that engineered microbubbles enable stone-treatment modalities

with driving pressures significantly lower than those required without the microbubbles.

V

2019 Acoustical Society of America

https://doi.org/10.1121/1.5116693

[CCC]

Pages: 516–531

I. INTRODUCTION

Mechanical action of engineered microbubbles in response

to acoustic excitation is increasingly used in medical applica-

tions, such as for targeted destruction of biomineralizations.

–

To promote preferential action on a target, the bubble shell can

be engineered to facilitate accumulation of micr

obubbles on

the surfaces of targets.

To target and treat urinary stones,

Ramaswamy

et al.

proposed placing engineered microbubbles

in the urinary tract or kidney, where the bubbles can

accumu-

late on exposed surf

aces of stones (Fig.

). The accumulated

microbubbles then can be energized by one or more energy

sources to mechanically erode, pit, and fragment the stone.

With a pulsed laser energy source, a recent

in vitro

study

has shown that adding stone-surface-accumulating (SSA)

microbubbles increased the rate of erosion, pitting, and frag-

mentation of model stones by

70%.

Other studies have

shown that SSA microbubbles can be placed in the urinary

tract cystoscopically and energized by an extracorporeal

acoustic energy source.

The use of ultrasound with wide

beam widths (several centimeters) reduces the burden of pre-

cise image guidance (Fig.

) and, with a cystoscopic delivery

of SSA microbubbles, can be performed in diverse clinical

settings including the urologist office.

In shock wave lithotripsy, urinary stones are broken

with focused shock pulses with pressure amplitudes of

15–150 MPa.

Recent

in vitro

experiments have shown that

urinary stones can be broken by bursts of focused ultrasound

with center frequencies in a sub-MHz range and amplitudes

of several MPa.

Preliminary experiments with SSA micro-

bubbles suggest that model and urinary stones can be broken

by acoustic bursts with pressure amplitudes at the upper lev-

els of diagnostic ultrasound (1.2

6

0.2 MPa, 0.3–1 MHz).

these experiments, the stones were implanted in a porcine

model and treated by wide beam-width ultrasound with no

evidence of urothelium damage and renal parenchymal hem-

orrhage on histological and gross anatomical examination of

post-procedure ureters and kidneys.

These encouraging results suggest a need to better

understand the dynamics and the mechanical action of SSA

microbubbles at these driving conditions. In this work,

we used bursts of ultrasound with central frequencies of

0.42 MHz and amplitudes of 1.4

6

0.4 MPa. The dynamics

of microbubbles was studied using high-speed video micros-

copy at a frame rate of up to 10

fps, capturing com-

plete expansion-collapse cycles of the bubbles. The observed

geometry of bubbles at their maximum expansion was used

as the initial shape for the numerical modeling of the col-

lapse. For modeling, we solved the multicomponent Euler

equations using a two-dimensional (2 D)-axisymmetric code

with adaptive time step and mesh refinement algorithms for

fine resolution of the gas-liquid interface and shock fronts.

Electronic mail: yuri.pishchalnikov@applaudmedical.com

Present address: Department of Mechanical Engineering, University of

Washington, Seattle, WA 98195, USA.

516 J. Acoust. Soc. Am.

146

(1), July 2019

V

2019 Acoustical Society of America

0001-4966/2019/146(1)/516/16/$30.00

The use of a 2 D code instead of a one-dimensional (1D)

spherical model

–

was motivated by the present and previ-

ous observations showing that oscillations of microbubbles

can be highly nonspherical.

–

Hsiao and Chahine

modeled a shelled microbubble

subjected to one cycle of 2.5-MHz ultrasound at 1 MPa. The

modeling showed nonspherical deformations due to the pres-

ence of a nearby rigid boundary and, depending on the stand-

off distance, one of the following dynamics of the collapse:

(1) a single reentrant jet, (2) a ring-type reentrant jet, and (3)

a pinching of the bubble.

Here, we report experimental

observations of a circumferential pinching and the numerical

modeling showing that the pinching propels two microjets

directed away and toward the rigid boundary. When hitting

the boundary, the jet was either a single reentrant jet or a

ring-type jet, depending on the amount of gas modeled in the

bubble.

The present paper is organized as follows. Section

describes the experimental arrangement to record the

dynamics of microbubbles at the surface of urinary stones.

Section

III

presents experimental results showing that micro-

bubbles, driven by

1.5 MPa, expand up to about 60

min

diameter, and acquire the shape resembling an oblate spher-

oid. Section

describes the numerical modeling of the col-

lapse showing the circumferential pinching and the

formation of the two microjets (Sec.

IV E

). The pressure

generated by the collapsing microbubble at the rigid wall

(Sec.

IV F

) exceeded the driving pressure by two-to-three

orders of magnitude. Section

shows micro-computed

tomography of urinary stones before and after the action of

SSA microbubbles illustrating surface erosion and formation

of microcracks. The significance and limitations of this work

are discussed in Sec.

. The

Appendix

describes numerical

details, including an illustration of the adaptive mesh refine-

ment (AMR) algorithm.

II. MATERIALS AND METHODS

A. SSA microbubbles and urinary stones

SSA microbubbles (Applaud Medical, Inc.) were made

of perfluoroalkane gas (C

) encapsulated by lipid shells

engineered to accumulate on stone surfaces.

The lipid shells

incorporate polyethylene glycol structures and synthetic

pyrophosphate analogs, collectively functioning to minimize

interaction with the urothelium and facilitate accumulation

on stone surfaces. The size distribution of the microbubbles

was measured with a Coulter counter (Multisizer 4e Coulter

Cell Analyzer, Beckman Coulter, Indianapolis, IN) using an

aperture size of 30

m in diameter. The mean diameter of

the microbubbles was [mean

6

standard deviation (SD)]

1.19

6

0.04

These experiments were conducted with surgically

retrieved urinary stones. The chemical composition of stones

was determined by Fourier-transform infrared spectroscopy

and was mostly calcium oxalate monohydrate. Specifically,

the surface of the stone shown in the inset of Fig.

2(b)

con-

tained 90% calcium oxalate monohydrate, 5% calcium oxa-

late dihydrate, and 5% apatite; the interior contained 70%

calcium oxalate monohydrate, 10% calcium oxalate dihy-

drate, and 20% apatite.

To position the stone in the focus of a microscope, the

stone was glued with a small amount of a Loctite Super Glue

to a tip of 0.25-mm thin coverslip (clear vinyl plastic,

0.25 mm, VWR International, PA) as shown in the

inset of Fig.

2(b)

. The stone was then kept in water for several

weeks for hydration. The hydrated stone was positioned in the

water tank (Fig.

) and aligned at the focus of the microscope

using an XYZ-micrometer stage (Thorlabs Inc., NJ).

B. Test tank

The test tank was three-dimensionally (3 D) printed from a

thermoplastic material (acrylonitrile butadiene styrene) and cov-

ered with a waterproof coating (Marine Grade Epoxy 109

Medium, Tap Plastics, CA). For imaging with the inverted

microscope, the bottom of the tank had a glass port (75

1 mm microscope slide, VistaVision, VWR International,

LLC, Radnor, PA) glued along its edges to the bottom of the

tank (Fig.

). The tank was filled with six liters of water

(PURELAB Chorus 1 for Life Science Applications, ELGA,

Veolia Water Solutions and Technologies, UK) with an electri-

cal resistivity of 18.2 MOhm-cm and the ultrafiltration to parti-

clesizelessthan0.05

m. The water remained in the tank for

several days and was in equilibrium with atmospheric gases.

During the experiments, the temperature of the water slowly

increased from

Cto

C due to the heating by intense

light used for the high-speed imaging.

C. Light sources

We used both continuous and flashlight illumination. The

continuous lighting was provided with a fluorescence illumina-

tion system EXFO X-cite 120 (XE120, Photonic Solutions

Inc., Mississauga, Ontario, CA). This light source uses a 120-

W Metal Halide lamp coupled to a liquid lightguide. The end

of this lightguide was positioned at about 1 cm above the stone

and provided back light to the stone (Fig.

The side lighting was provided with a flashlamp

WRF300 (Hadland Imaging, LLC, Santa Cruz, CA). This

spark-discharge lamp produced a light pulse with the dura-

tion of about 10

s. The spark light was delivered through a

separate liquid lightguide illuminating the side of the stone

proximal to the incoming acoustic waves (Fig.

FIG. 1. (Color online) The concept of treating urinary stones with SSA micro-

bubbles driven by an extracorporeal source of ultrasound: (a) general view and

(b)–(d) zoomed-in view of a urinary stone in the ureter. Gas-filled microbub-

bles are introduced into the urinary tract and accumulate at the surface of the

urinary stone (b); the accumulated microbubbles are excited with ultrasound (c)

and erode, pit, and fragment the stone facilitating its passage (d).

J. Acoust. Soc. Am.

146

(1), July 2019

Pishchalnikov

etal.

517

D. High-speed video microscopy

Images were captured with a high-speed camera HPV-

X2 (Shimadzu, Kyoto, Japan) operated in one of two modes.

In the full-pixel (FP) mode, the camera captures 128 frames of

400- by 250-pixels at a rate up to 5 Mfps. In the half-pixel

(HP) mode, the camera captures every other pixel in a checker-

board pattern, interpolating the images into 400- by 250-pixel

frames and recording 256 frames at rates up to 10 Mfps.

Exposures were 110 ns at 5 Mfps and 50 ns at 10 Mfps.

Magnification was achieved using a microscope (Nikon

Eclipse TS100) with a 4

objective (4

/0.13 PhL DL, WD

16.4, Nikon Plan Fluor), a 2.5

projection lens (Nikon CF

PL2.5

), and a 34-cm extension tube (Fig.

). The numerical

aperture of the objective was 0.13 giving the diffraction-limited

depth of field of 43

6

m and lateral resolution of 2

6

0.5

This diffraction-limited lateral resolution of the objective was

considered in choosing the magnification of the projection lens

and the length of the extension tube. These optical elements were

chosen to have a camera resolution of 1

m/pixel.

In post-processing, recorded images were digitally

enlarged (3

by resampling with preserving details in

Adobe Photoshop), rotated, and cropped. The acoustic radia-

tion force was directed approximately from right to left in

these images (Fig.

Mm. 1

, and

Mm. 2

). The buoyant force

was directed into the image plane (Fig.

E. Driving acoustic field

Driving acoustic waves were generated with a custom-

made piezoelectric transducer (Sonic Concepts, Inc., Bothell,

WA) positioned at

9 cm from the proximal surface of the uri-

nary stone (Fig.

). The active element of the transducer was

made of a piezoelectric plate (72.3

30.3

3.18 mm) divided

into eight elements and connected in pairs. In these experi-

ments, each pair was powered by one of four power amplifiers

(AP-400B, ENI, USA). The frequency and duration of the

acoustic bursts were computer controlled with a specially

designed signal generator, allowing us to trigger the HS-

camera with TTL pulses while reproducing the frequency mod-

ulationusedintheclinic

and also to study other driving

regimes. In this work, the transducer was driven with a fre-

quency set of 400, 400, 433, and 433 kHz, generating acoustic

bursts with a center frequency of 416.5 kHz and

s dura-

tion of the envelope (Fig.

The spatial characteristics of the acoustic beam (Fig.

)

were measured using a needle hydrophone (HNR-0500, Onda

Inc., Sunnyvale, CA) with a diameter of the sensitive element of

0.5 mm. The acoustic field was scanned with a 2-mm step using

an XYZ-positioning system assembled with three motorized lin-

ear slides (X-LSM150A, Zaber Technologies, Vancouver, BC,

Canada). The motion of the positioning system was controlled

by a computer through the RS-232 link with a program wr

itten

in Python, which was also used to acquire hydrophone data

recorded by a digital oscilloscope (HDO

4024 Teledyne, LeCroy

Corp., NY). These measurements were conducted in a

20 cm tank with acrylic walls acoustically isolated

with sheets of absorptive material 1-cm thick (Aptflex F28,

Precision Acoustics, UK) providing echo reduction greater than

25 dB. The tank was filled with 14 liters of water degassed with

a pinhole degasser [ref JASA cain] to 1.1

6

0.5 mg/l (measured

with a dissolved oxygen meter DO 6

, Eutech Instruments,

Singapore). To prevent the damage of the Onda hydrophone,

the spatial characteristics of the acoustic beam were measured

at low amplitudes and are shown normalized on the maximum

pressure

max

found among the scans in the three orthogonal

planes (Fig.

). At the position of the stone (

90 mm), the

FIG. 2. (Color online) Experimental setup: (a) schematic diagram and (b) view in the tank. A urinary stone was positioned in the focus of an inverted mic

ro-

scope (dotted rectangle). The optical path from the stone to the high-speed camera (HS-camera) is shown in the diagram by arrows and dashed lines with t

glass elements (lenses and prisms) shaded in gray. Bubbles were driven with bursts of ultrasound produced by a piezoelectric transducer positioned a

t9cm

from the proximal surface of the urinary stone. The inset in (b) shows a photograph of the urinary stone (bottom right). More general views of the setup a

shown in POMA (Ref.

518 J. Acoust. Soc. Am.

146

(1), July 2019

Pishchalnikov

etal.

acoustic beam had the cross section with the

6-dB width of

30 mm and the height of

60 mm (solid contour lines, Fig.

Selected measurements were conducted with another

needle hydrophone (Y-104, Sonic Concepts, Inc., Bothell,

WA). In particular, this hydrophone was used to measure

pressure traces at the driving amplitudes (Fig.

) in the focus

of the microscope at the position of the stone (Fig.

). For

these measurements, the stone was replaced with the sensi-

tive tip of the Y-104 hydrophone—the 1.5-mm diameter

ceramic crystal enclosed in a 3-mm metal tube. As the diam-

eter of the tip was comparable with the acoustic wavelength

(

3.6 mm), the hydrophone’s sensitivity depended on its

orientation. The angular response of the hydrophone was

measured and taken into account, increasing the uncertainty

of pressure measurements to

30%.

The pressure traces recorded in the HS-camera test

tank at the position of the stone were combined with the

HS-camera images into multimedia frames using programs

written in LabVIEW (National Instruments, Austin, TX) and

converted into movies with the H.264-video format using

QuickTime Pro 7 (Fig.

Mm. 1

, and

Mm. 2

). In these

movies, representative hydrophone traces (dark blue) were

superimposed on several waveforms (light blue) showing

shot-to-shot variability. The vertical arrow shows the timing

of the HS-camera with the uncertainty of 0.2

s. Time

was positioned at the start of implosion of the largest solitary

bubble.

F. Micro-computed tomography

Micro-computed tomography (micro-CT) of urinary stones

(Fig.

) was used to assess the stone damage produced by the

SSA microbubbles. The micro-CT images were acquired with

a high-resolution X-ray tomography (MicroXCT-200, Xradia,

FIG. 3. (Color online) Spatial characteristics of the acoustic beam: (a) 3D Cartesian coordinate system with the orientation of

, and

axes for beam plots;

(b)–(d) normalized pressure amplitudes

max

in three orthogonal planes. The surface of the transducer was at

0. An

-scan at

90 mm (d) shows the

cross section of the beam at the position of the stone. Pressure contour lines at half amplitude (solid lines) show the

6-dB dimensions of the acoustic beam.

J. Acoust. Soc. Am.

146

(1), July 2019

Pishchalnikov

etal.

519

Inc., Pleasanton, CA) before and after the action of SSA

microbubbles.

III. EXPERIMENTAL RESULTS

Bubble dynamics at the surface of a urinary stone is

shown in movies

Mm. 1

and

Mm. 2

. The movie

Mm. 1

was

recorded with the HS-camera in FP mode at 5 Mfps and pro-

vides full resolution images although at the slower frame

rate than

Mm. 2

, which was recorded in HP mode capturing

every other pixel at 10 Mfps.

Mm. 1.

Bubble dynamics at the surface of a urinary stone

recorded in FP mode at 5 Mfps and 110-ns exposure.

This is a file of type “mov” (8.8 Mb).

Mm. 2.

Bubble dynamics at the surface of a urinary stone

recorded in HP mode at 10 Mfps and 50-ns exposure.

This is a file of type “mov” (9.6 Mb).

During the first two acoustic cycles in these movies

(

3.4

0.8

s) the microbubbles expanded to a larger

maximum size from the first to the second cycle, concomi-

tant with increasing driving pressure. The second collapse

(

0.8

s) produced a cluster of bubbles. Bubbles in clusters

varied in shape, size, and standoff distance, merging with

other bubbles. Here, we focus on the collapse of a solitary

bubble starting from its maximum expansion (

0 – 0.8

Mm. 1

and

Mm. 2

At maximum expansion [Fig.

4(a)

and

Mm. 1–Mm. 2

], the bubbles were nonspherical with a

cross-section resembling an ellipse truncated by the

stone [Fig.

5(a)

]. Average major and minor axes of 16

bubbles were measured to be 2

6

mand

6

m[Fig.

5(b)

]. The bubbles had an eccentric-

ity

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

of 0.63

6

0.08 and centers located

6

m from the stone. This standoff distance

was used to model the collapse of the largest bubbles

approximating their initial shape as an oblate spheroid

with dimensions of 2

mand2

IV. NUMERICAL MODELING

A. Governing equations

The collapse of a microbubble was modeled by solving

the multicomponent Euler equations with a 2D-axisymmetric

FIG. 4. (Color online) Three frames from movie

Mm. 1

showing a microbubble at the surface of a urinary stone at three moments in time: (a)

0—bubble is at its maximum expansion, and (b) and (c) during the collapse. Bottom panels show driving acoustic pressure. The upper plot shows an

enlargement of two acoustic cycles. The timing of the HS-camera is indicated by the vertical arrow pointing to a circle on the pressure trace. The circl

eis

bounded by two vertical cursors encompassing the 0.2-

s interval between the frames. These HS-camera frames were recorded in FP mode at 5 Mfps

with an exposure of 0.11

FIG. 5. (Color online) Geometry of microbubbles at their maximum expan-

sion: (a) approximation of bubble’s shape as an oblate spheroid; (b) mea-

surements of major 2

and minor 2

axes for 16 bubbles. The bubbles

had an eccentricity

of 0.63

6

0.08.

520 J. Acoust. Soc. Am.

146

(1), July 2019

Pishchalnikov

etal.

code. Specifically, the code models the gas and the liquid as a

two-phase compressible flow using the following system of

equations:

$

$

ðÞ

$

ðÞ

$

$

ðÞ

$

$

ðÞ

$

(1)

Here,

and

are the volume fractions of the gas and

the liquid; vector

is the velocity of the flow. The gas and the

liquid are governed by their equations of state

;

where

and

are the density, the internal energy, and the

pressure of phase

. The gas in the bubble obeys the ideal-gas

equation of state with

gas

1 (the ratio of specific heats

for C

is 1.07).

The water obeys the stiffened-gas equation

of state with

water

6.12 and

;

water

Pa.

The

gas-liquid mixture has the density

. Likewise,

the pressure

in the gas-liquid mixture is the sum of partial

pressures

and

The right-hand side of Eq.

(1)

describes the relaxation

of pressure between the phases, where

coefficient

determines the speed of relaxation (section

of the

Appendix

), and interfacial pressure

.Here

is the acoustic impedance of

the phase

with

being the speed of sound of the corre-

sponding phase.

Due to

, the total-energy equation of the mixture

is replaced by the internal-energy equation for each phase.

Nevertheless, the mixture-total-energy equation of the sys-

tem can be written in the usual form

$

ðÞ

Þ¼

;

(2)

where

¼ð

Þjj

and

are the total and internal

energies. Although Eq.

(2)

may appear redundant (as the

internal-energy equations are solved for both phases), it is

important to ensure the energy conservation and, hence, to

preserve a correct treatment of shocks.

The system of equations

(1)

was solved with a two-

substepapproach(section

of the

Appendix

)usingadap-

tive time steps and an AMR algorithm (section

of the

Appendix

). In comparison with the numerical modeling in

POMA,

the use of an improved system of equations

(1)

allows us to take into account expansion and compression

of each phase in mixture regions. This improvement and

the use of the AMR algorithm (section

of the

Appendix

)

allowed us to increase the accuracy of the results, includ-

ing the spatial and time resolution of gas-liquid interfaces

and shock fronts.

B. Initial and boundary conditions (BCs)

To reduce the computation time and needed resources,

the dynamics of the collapsing bubble was modeled with

the following simplifications. First, the model was axi-

symmetric, although the experiments showed some appar-

ent divergency from axial symmetry, in part, due to

the irregular surface of urinary stones [inset in Fig.

2(b)

Fig.

]. Second, the wave scattering by the irregular

surface of the urinary stone was not modeled in this work

and the stone was modeled as a rigid plane. Third, the

absolute pressure far from the bubble was assumed to be a

constant

The assumption of a constant pressure

was an

approximation as the driving acoustic pressure varied during

the growth-collapse cycle of the bubbles (Fig.

Mm. 1

, and

Mm. 2

). The time for bubbles to reach their maximum size

varied from bubble to bubble and increased as the bubbles

grew from cycle to cycle (

Mm. 1

and

Mm. 2

). Here, we

model the bubble that reached its maximum size at

which was about 45 degrees into the positive pressure phase

of the driving acoustic wave (

s, Fig.

Mm.

and

Mm. 2

). The time-average pressure of this “sinusoidal”

half-cycle during the time of the collapse (0

was about 0.7 of the amplitude of the driving acoustic wave.

In the experiments, the driving acoustic wave was a superpo-

sition of sine waves at two frequencies giving the free-field

pressure amplitude of 1.4

6

0.4 MPa (Fig.

Mm. 1

, and

Mm. 2

). The pressure at the urinary stone was higher

50%

due to constructive interference of the incident

and scattered waves, increasing the pressure amplitude to

2.1 MPa. This amplitude gave the time-average value for

the absolute pressure during the collapse (0

s) of

1.55 MPa. This time-average pressure was assumed to

be spatially uniform within the computational domain

(768

768

m, Fig.

), which was much smaller than

the spatial characteristics of the driving acoustic beam (e.g.,

6-dB zone of 3

6 cm, Fig.

The initial pressure in the bubble

depended on the

amount of gas and vapor in the bubble. To bracket the

range of possible behaviors, we modeled the collapse

with two initial gas pressures:

10 Pa and

100 kPa. The pressure

would occur in the bubble

polytropically expanded from a nucleus with an initial

radius of

1.2

m to the maximum size observed in

the high-speed image (

m, and

m, Fig.

) if vaporization, condensation, and gas

diffusion were negligible. The pressure

would occur

in the expanded bubble if vaporization, condensation,

and gas diffusion were infinitely fast. The pressure

overestimates the pressure and amount of gas in the bub-

ble, potentially cushioning the collapse. In the experi-

ments, the pressure in the expanded bubble (

0) was

between

and

The initial density of gas in the bubble was

,and

2kg

(density of air

10–35

C) for

. The initial density of water was

1000 kg

. BCs and the computational domain

(Fig.

) are described in section

of the

Appendix

J. Acoust. Soc. Am.

146

(1), July 2019

Pishchalnikov

etal.

521

C. Numerical movies of collapsing bubbles

Numerical results are shown in

Mm. 3

and

Mm. 4

Mm. 3.

Modeling of the collapse with the initial gas pressure

10 Pa. This is a file of type “mov” (9.4 Mb).

Mm. 4.

Modeling of the collapse with the initial gas pressure

100 kPa. This is a file of type “mov” (9.3 Mb).

The content of these movies is illustrated in Fig.

, show-

ing a representative frame from

Mm. 3

. In both movies, the top

left panels show pressure in color on a logarithmic scale rang-

ing from 1 kPa (blue) to 0.5 GPa (red). Pressures outside of this

range are shown either in dark red (

0.5GPa)orindarkblue

(

1 kPa). The color image is overlaid by the volume fraction

of gas shown in black and white, with an opacity function to

render translucent surfaces. The black shows regions of high

gas content. The opaqueness decreases with volume fraction

until the gas volume fraction is zero (100% liquid) depicted as

100% transparent.

The bottom left panels in Fig.

Mm. 3

, and

Mm. 4

show numerical schlieren

of mixture-density gradients

exp

ðÞ

gas

ðÞ

water

$

$

max

;

(3)

where

is a scaling factor for simultaneous visualization of

waves in both fluids (

gas

20 and

water

200).

Thick lines in these images suggest the position of the

bubble-water interface. Thin lines typically show moving

fronts of pressure waves. The different contrast of shock

fronts in these movies show that the bubble with smaller

amount of gas (

Mm. 3

) produced stronger shocks than

the bubble with the larger amount of gas (

Mm. 4

). The

difference in contrast of shocks is also seen in Figs.

8(a)–8(b)

showing the final stages of the collapse. These results support

the notion that the gas in the bubble cushions the collapse,

so that bubbles with smaller amount of gas produce stronger

collapses.

D. Numerical modeling and experimental observations

Both experimental [Fig.

7(a)

] and numerical [Fig.

7(b)

]

images showed a distinctive circumferential narrowing of

the bubble (

0.4 – 0.7

s), indicating greater bubble-wall

velocities toward the axis of symmetry of the bubble. Figure

7(c)

shows bubble-wall velocities

v

(squares) and

v

(circles) measured at the moving points 1 (square) and 2 (cir-

cle) indicated in the inset. Numerical velocities are shown

for both initial gas pressures by thin (

) and thick (

)

lines. Both bubbles had similar dynamics but the bubble

with smaller amount of gas (

) developed greater bubble-

wall velocities and collapsed faster than the bubble with

The averaged bubble-wall velocities measured in HS-camera

images during the final stage of the collapse (

0.6

squares and circles) were in between the numerical curves.

This behavior might be expected as in the experiments the

gas pressure at

0 was between

and

E. Circumferential pinching and microjets

Both experimental (dots) and numerical (lines) veloci-

ties show that the bubble-wall velocity directed toward the

axis of symmetry of the bubble

v

was greater than the

FIG. 6. (Color online) Frame

0.726

s from

Mm. 3

showing the moment when the jet hits the rigid wall (located at the left boundary, Fig.

). Left panel

shows pressure in color overlaid by the volume fraction of gas (top) and schlieren of mixture-density gradients (bottom). Right panel shows maximum p

ressure

at the rigid wall. For this figure, the movie frame was modified by enlarging the pressure plot, adding some annotations, and drawing the initial positio

n of the

bubble wall (

0, dashed ellipse).

522 J. Acoust. Soc. Am.

146

(1), July 2019

Pishchalnikov

etal.

velocity

v

directed toward the stone (Fig.

). This dispro-

portion led to the circumferential pinching of the bubble,

splitting it into two parts. The distal-to-stone part of the

bubble was smaller than the proximal part and collapsed

first (

Mm. 3

and

Mm. 4

,Fig.

). The collapse was intensi-

fied by a pressure surge with an amplitude of

1.23 GPa at

and

0.37 GPa at

. This pressure surge was pro-

duced by the converging flow of liquid when it collided at

the axis of symmetry of the bubble (0.719

sin

Mm. 3

and

0.763

sin

Mm. 4

). The collision formed two axial micro-

jets directed away and toward the stone. The reentrant jet

hitting the stone was either a single (

) or a ring-type

(

)jet(Fig.

). This jet had the velocity

v

jet

shown in

Fig.

7(c)

with the maximum of 4117 m/s at

and

1164 m/s at

F. Pressure at the rigid wall

The impact of the reentrant microjet on the stone pro-

duced a hydraulic shock with the water-hammer pressure of

0.65 GPa at

and

0.5 GPa at

seen as the first pres-

sure spikes at

0inFig.

8(c)

. This water-hammer pressure

was followed by a longer somewhat triangular pulse with the

stagnation pressure of

0.3 GPa at

and

0.2 GPa at

[

0, Fig.

8(c)

The impact of the microjet hitting the stone was intensi-

fied by the pressure waves radiated by the collapse of the

distal part of the bubble (0.721

sin

Mm. 3

and 0.769

sin

Mm. 4

, frames 3 in Fig.

). These pressure waves also

intensified the collapse of the main part of the bubble located

proximal to the rigid surface. This proximal bubble toroidally

collapsed (frames 4–6, Fig.

) producing pressure waves

at the stone with local maximums initially located at

6.2

mat

(0.748

Mm. 3

)and

mat

(0.798

Mm. 4

The pressure waves radiated by the toroidal collapse

converged toward the axis of symmetry of the bubble, pro-

ducing the overall maximum pressure at the stone at

(0.75

Mm. 3

). As the toroidal bubble collapsed from

the periphery radiating multiple waves (frame 6 in Fig.

Mm. 3

and

Mm. 4

), the location of the maximum pressure

was slightly off-axis and not seen in Fig.

8(c)

.Themaxi-

mum pressure determined along the entire surface of the

stone—rather than at specified locations as in Fig.

8(c)

—is

shown on the right panel in Fig.

Mm. 3

and

Mm. 4

The maximum pressure reached 6.88 GPa at

(0.75

Mm. 3

) and 1.32 GPa at

(

Mm. 4

, the greater amount of gas in the bubble cush-

ioned the collapse, so that the pressure waves radiated by

the toroidal collapse of the proximal bubble were not as

strong as at

(frame 6 in Fig.

0.798

Mm. 4

These pressure waves collapsed the daughter bubbles that

were produced by the distal bubble. In turn, the collapse

of these daughter bubbles radiated pressure waves that

drove the rebound of the bubble at the stone surface near

the axis. The collapse of this rebounding bubble produced

the overall maximum pressure at the stone surface at

(1.32 GPa at 0.843

Mm. 4

FIG. 7. (Color online) Collapse of a

microbubble at the surface of urinary

stone. (a) HS-camera sequence of

images recorded at 10 Mfps (

0.1

– 0.8

Mm. 2

). (b) Frames with

0.1-

s step from the numerical model-

ing with

(

0.1 – 0.8

Mm. 4

Bubble-wall profiles are also shown in

the inset of Fig.

8(c)

for both

(top)

and

(bottom). (c) Bubble-wall

velocities

v

(squares) and

v

(circles)

at points 1 and 2 (inset). Error bars

show measurement uncertainties (ver-

tical) and HS-camera exposure time of

50 ns (horizontal). Numerical veloci-

ties are shown by thin (

) and thick

(

) lines. At the end of the traces

(

0.72

sat

and

0.764

sat

), the converging flow of liquid

v

forms the axial microjets toward and

away from the stone (

Mm. 3

and

Mm.

) with jet velocities

v

jet

toward the

stone reaching 4117 m/s at

and

1164 m/s at

J. Acoust. Soc. Am.

146

(1), July 2019

Pishchalnikov

etal.

523

V. DAMAGE ON THE STONE SURFACE: REMOVAL

OF STONE MATERIAL AND FORMATION OF

MICROCRACKS

Micro-CT images of a urinary stone before and after the

action of SSA microbubbles showed removal of stone mate-

rial and formation of microcracks (Fig.

). Specifically,

arrows 1–8 point to places of removed material in the 3 D

rendered surface of the stone [Fig.

9(a)

], whereas arrows 10

and 11 point to the formation of microcracks visible in the

cross sections of the stone [Figs.

9(b)

and

9(c)

VI. DISCUSSION

The combination of HS-video microscopy and numerical

modeling employed here showed a pronounced difference

between the previously reported dynamics of initially spherical

bubbles

–

and the dynamics of the SSA microbubbles at

the surface of urinary stones. The SSA microbubbles expanded

nonspherically (

Mm. 1

and

Mm. 2

) and, at their maximum

expansion, acquired the shape similar to an oblate spheroid

[Figs.

4(a)

and

7(a)

]. Approximating the cross-section of bub-

bles as an ellipse truncated by the plane rigid surface [Fig.

5(a)

], we modeled the collapse of the microbubbles at the sur-

face of urinary stones. Both the numerical modeling and the

experimental observations showed a circumferential constric-

tion pinching the bubbles (Figs.

–

Mm. 1–Mm. 4

). The

pinching was absent in the collapse of the initially spherical

bubbles as shown in Fig.

in POMA.

Furthermore, whereas for the initially spherical bubbles

the axial jet originated from the distal surface of the bub-

ble,

–

the circumferential pinching of the SSA micro-

bubble split the bubble into two parts producing two axial

microjets directed away and toward the stone. The reentrant

jet hitting the stone was either a single (

Mm. 3

)ora

ring-type (

Mm. 4

) jet. The velocity of this jet

v

jet

reached 4117 m/s at

and 1164 m/s at

[Fig.

7(c)

], sup-

porting the notion that microjets can reach high subsonic or

even supersonic speeds.

The quantitative verification of the predicted velocities

of these microjets requires fine spatial (

m) and temporal

FIG. 8. (Color online) Final stages of

the collapse for two initial gas pres-

sures: (a)

10 Pa (

Mm. 3

) and (b)

100 kPa (

Mm. 4

). These movie

frames were cropped to the width of

18.5

m and show the following

moments: (1) last frame shown in Fig.

7(b)

before the collapse, (2) circumfer-

ential pinching split the bubble into

two parts producing a pressure surge

and forming two microjets directed

away and toward the stone, (3) distal

bubble collapsed radiating pressure

waves, (4) microjet hit the surface, (5)

the microjet diverged reaching periph-

ery of the proximal bubble, and (6) the

proximal bubble collapsed radiating

shock waves. Plot (c) shows pressure

vs time at four radial distances

along

the rigid surface. The inset shows

bubble-wall profiles for

(top) and

(bottom) at steps of 0.1

s from 0

to 0.7

s, i.e., to the first frame shown

in (a) and (b). As gas in the bubbles

was cushioning the collapse, the bub-

ble with

collapsed later and pro-

duced smaller pressure than the bubble

with

. Maximum pressures were

associated with shorter pulses of

1ns

and seen near the axis of symmetry of

the bubbles. Longer pulses had dura-

tion of

10–30 ns and were seen both

0 and at greater distances off-

axis. Within a 10-

m radius from the

axis, the peak pressure produced by the

bubbles exceeded 150 MPa, i.e., was

two orders of magnitude greater than

the driving pressure.

524 J. Acoust. Soc. Am.

146

(1), July 2019

Pishchalnikov

etal.

(

1 ns) resolutions (Fig.

Mm. 3

and

Mm. 4

), both of

which are experimentally difficult. The numerical frames

(Fig.

) are shown with frame rates up to one billion frames

per second. The fastest frame rate of our state-of-the-art

camera was 10

fps (

Mm. 2

). Although there are cam-

eras that can record a small number (8–15) of frames at rates

up to 200

fps, capturing the exact moment of collapse

is difficult even with laser-nucleated bubbles

and it is

practically impossible with the microbubbles at the surface

of urinary stones reported in this work. The main reason is

that the bubbles are never exactly the same, and tiny varia-

tions in bubble’s size shifts the moment of collapse, making

the capturing of the fine details during the short moment of

the collapse technically challenging. Some verification of

the numerical model can be provided by comparison with

other accepted models and analytical solutions (see section

of the

Appendix

). The comparison shows a good agree-

ment with the established models and analytical solutions,

supporting the trustworthiness of the present numerical

results.

It has been suggested that the velocity of microjets is

increased by the reflection of the driving wave from the solid

surface.

Here, we showed that microjets produced by SSA

microbubbles were also accelerated by the pressure surge

when the circumferential flow collides on the axis of symme-

try of the bubble (frame 2, Fig.

). Even though short-lived,

FIG. 9. (Color online) Micro-computed

tomography of a urinary stone before

and after the action of SSA microbub-

bles: (a) 3D rendering of the surface,

(b)–(c) 2D cross sections A and B.

Numbered arrows point to removal of

stone material (1–9, 12) and formation

of microcracks (10, 11).

J. Acoust. Soc. Am.

146

(1), July 2019

Pishchalnikov

etal.

525