Br. J. Cancer (1982) 45, Suppl. V, 140

ACOUSTIC CAVITATION: A POSSIBLE CONSEQUENCEOF BIOMEDICAL USES OF ULTRASOUND

R. E. APFELFrom the Department of Applied Mechanics, Yale University, P.O. Box 2159, New Haven,

CT 06520, U.S.A.

Summary.-Those concerned with acoustic cavitation often use different measuresand nomenclature to those who employ ultrasound for medical purposes. Afterillustrating the connections between the two, acoustic cavitation phenomena aredivided into two classes: (1) relatively moderate amplitude changes in the bubblesize that occur during each acoustic cycle, as with rectified diffusion and resonantbubble motion, and (2) rather dramatic changes in the bubble radius that occur inone cycle. It is seen that pulse-echo diagnostic equipment can excite the dramaticchanges whereas continuous wave therapeutic equipment will excite the slower, butno less important, changes. The ranges of the acoustic variables and material statesfor which these phenomena are possible are quantified. It is shown that whereasthe concept of an ultrasonic (energy) dose may be appropriate for the effects ofacoustically induced heating or resonant bubble motion, it is inappropriate whendiscussing the effects of the transient type of cavitation that can occur from short,high amplitude acoustic pulses.

ACOUSTIC cavitation can be defined asany observable activity involving a bubbleor population of bubbles stimulated intomotion by an acoustic field. Since thiscavitational activity can be viewed as adramatic concentration of acoustic energyresulting in localized high stresses, tem-peratures and/or fluid velocities, its bio-logical consequences should be understoodby those who are trying to either optimizeor minimize its effects.For the purposes of this conference I

prefer not to attempt to review the manyreview articles on the subject (e.g. Flynn,1964; Neppiras, 1980; Coakley & Nyborg,1978; Apfel, 1981a). Nor do I intend toreview the empirical evidence for theeffects of acoustic radiation on biologicaltissues. The book by Hussey (1975)reviews much of the work before 1975,and many of my colleagues at this con-ference are far more qualified to discussthe present status of work in this area.Rather I prefer to relate and quantify theparameters used in describing cavitationalactivity with those parameters most oftenassociated with equipment used for bio-

medical applications of ultrasound. ThenI shall consider the range of these para-meters for existing diagnostic and thera-peutic equipment. And finally I shalldiscuss the kinds of cavitational activitythat are possible in each case, and brieflyoutline a methodology for assessingthresholds for bioeffects.

Some parameters associated with cavitation,with ultrasound equipment, and some rela-tionships between the two

Tables I and II give parameters norm-ally associated with cavitation phenomenaand with medical ultrasonic equipment,respectively. What is immediately clear isthat the language used for one is ratherdifferent to that used for the other. Wecan find the useful relationships betweenthe two as follows:

At transducer:

ITD = W/7T(D2/4)In focal region:

IFC=GIITD = WA/A2F2

ACOUSTIC CAVITATION

p (peak at focus) =1P22 x 103 ipl/2(0); ip(O), in W/m2

or p estimated assuming uniform distribu-tion across transducer face and in focalregion

=22pcx I_'FTr xPRF

1i73x103 / WRxA \1/2FA rp x PRF

115f j 'xA 1/2 Win wattsF \-rpxPRF) ' for water (1)

Example 1: Pulse-echo system, nominalfrequency 2 MHz, 2 cycles per pulse(rpl1Ius),PRF= 1 kHz

P=6*46 x 107 Wl/2D/F, in Pa; It' in watts20 4 W1/2D/F, in bars; TF in milliwatts

If W-= 10 mW and D/F= 0.3, then p(t)-~19 bars=1-9 x 1.06Pa.

Example 2: Therapeutic system; f-750 kHz, continuous wave, unfocused.

p= 1P73 x 103 T )

195 x 10 W1/2 Ur in wattsD 'D in metres (2)

For D = 0 06 m and TV= 2 watts,

p=46xlx4Pa-0*46 bars

Whereas the ultrasonic exposure (energyTdose) is much lower for diagnostic equip-ment than for therapeutic equipment, thepeak acoustic pressures are much higherin the diagnostic equipment! As we shallsee, this is an important consideration inassessing the likelihood and type ofcavitation and its potential bioeffects.

Types of cavitation possible in differenttypes of sound fieldsWe can divide most cavitation phenom-

ena into two categories: (1) small-to-moderate amplitude bubble motion inwhich the change in the bubble radius inone acoustic cycle is significantly less than

TABLE I. Some parameters associated withacoustic cavitation

Acoustic variablesInstantaneous acoustic pressureAcoustic frequency, periodlAcoustic wavelengthIf pulsed:

Pulse widthPulse repetition frequency

T'hermodynamic state of materia lHydrostatic pressureTemperatureDissolved gas concentration

(fraction of saturation) orpartial pressure

Material propertiesDensityViscositySurface tensioniThermal conductivxitySound velocityCompressibilitySpecific heatVapour pressure (for liquids)

Bubble parameters(assuming spherical)

Radius, (initial size)Radius: max. size, min. sizeBubble interface velocityInternal pressureInternal temperature

Symbolp(t)f, TA

I-pPRF

PoY oCi

PC"

p/1akc

pPv

R(t), (Ko)R max, RminRPiY1i

the radius; (2) large amplitude bubblemotion in which the bubble grows in onecycle by an amount equal to or greaterthan the radius.

Small-to-moderate amplitudes.-In thiscase the bubble may be growing or col-lapsing or just oscillating about anequilibrium size. Non-condensible gaswithin the bubble is responsible for therelatively small amplitudes.Growth is possible over many acoustic

cycles by a process called rectified diffus-ion, in which slightly more gas diffusesinto the bubble upon expansion thanleaves during contraction. Thus, if theacoustic pressure is greater than somethreshold (see Lewin & Bjorno, 1981 andCrum, 1980 for critiques) the bubble willslowly grow. Richardson (1947) has shownthat at a radius of:

Rr fJP O[i±Xr(1

I

Xr =PoRr (3)

141

R. E. APFEL

TABLE II.-Some parameters associated with medical ultrasonic equipment*Energy related measures Symbol

'Total transmitted acoustic power WTemporal average intensity near transducer averaged ITD

Time over area of active transducer elementsaverages' Temporal average-spatial average at focus IF

Temporal average-spatial peak intensity measured I(O)X at focal point or last axial maximum

Temporal average intensity in the focal plane, averaged over the IO.5area in which the intensity exceeds 0*5 x I(O)

Spatial peak intensity averaged over the duration of the pulse Ip(O)(temporal peak-spatial peak)

Instantaneous temporal peak-spatial peak intensity ip(O)Spatial measures

Radius of beam profile in focal plane for which the Ro.5temporal average intensity = I(0)/2

Focal length of transducer FPhysical transducer diameter DPhysical transducer area A = TD2/4Effective transducer diameter Derr

r ~~~~~~~~A2Theor. intensity gain at focus If spherically focusingITheor. pressure gain at focus and D <F/2 Gp A

AFTemporal measuresNominal (centre) frequency fPulse width, defined in terms of time between first and last zero Tp

crossings in the wave form at which the instantaneous pressureis i of the peak pressure

Pulse repetition period (rate) PRP (PRF)* Some nomenclature taken from Carson et al., 1978.

the bubble will be in resonance (the"gas" spring oscillating against the effec-tive mass of the surrounding fluid, plus asurface tension effect). Here K is a calcu-able constant, which depends on heatconduction and which is close to one formicron-sized bubbles. For reference, reson-ant air bubbles in water have the followingradii for the given frequencies:f, MHz 0 5 1 2 4 8Br, /tm 6 3-2 1-8 0*95 0*55It is noteworthy that in this frequencyrange of medical ultrasonic equipment, theresonance radius is comparable to the sizeof biological cells, whereas it is a factor of300-500 smaller than the wavelength ofsound at the frequency of resonance.The threshold acoustic pressure ampli-

tude for rectified diffusion in the lowmegahertz range is on the order of tenthsof bars (1 bar= 10 Pa) for bubbles reason-ably close to resonance size and for 100%gas saturation. As the frequency goes up,

however, the threshold goes up; as an

example, the threshold is the same forbubbles approximately 50% of resonancesize at 0.5 MHz and about 80% ofresonance size at 2-5 MHz.For continuous wave excitation, a peak

acoustic pressure of 105 Pa (1 bar) corre-sponds to a plane wave intensity of close3 kW/m2, which is well within the rangeof ultrasonic equipment used for physicaltherapy. For a frequency of 750 kHz,bubbles in aqueous solution in the radiusrange of 2-5 ,um are likely to grow forintensities of 3 kW/m2 or greater. Therectified diffusion threshold will go up asthe fraction of the transducer on-time isreduced, as surface tension works tocollapse the bubble.Bubble growth by rectified diffusion is

not a particularly exciting process untilthe bubble reaches resonance size, result-ing in larger amplitude oscillations and avariety of consequences in the medium. Ishall not dwell on this subject sinceProfessor Nyborg (1982) elaborates onthis topic elsewhere in these proceedings.

142

ACOUSTIC CAVITATION

It is worth noting, however, that if abubble is already near resonance size, forthe given frequency of excitation, thenthe pressure amplitude for its excitation,even at megahertz frequencies is rathermodest (< 0.5 x 105 Pa), with viscousdamping tending to predominate overthermal and acoustic damping for theconditions obtaining with biomedical usesof ultrasound (Prosperetti, 1977). terHaar et al. (1982) present elsewhere inthese proceedings some evidence that theeffects of rectified diffusion at therapeuticlevels (, 104 W/m2) may occur in vivo.

Potential for cavitational damage frompulsed ultrasound.-Biological damagefrom short pulses of ultrasound can occurif the growth of a bubble during one cycleof tension (during the negative phase ofacoustic cycle) exceeds some definedcriteria (which will be discussed shortly).The growth depends not only on the peakacoustic pressure amplitude but also onthe time allotted for growth (which is lessthan a half acoustic period). The lower theacoustic frequency for a given pressureamplitude, the greater the bubble growthand the more violent the collapse.Two factors which inhibit bubble growth

are the inertia and the viscosity of themedium surrounding the bubble. Bothtend to impede the initial growth and forboth we can estimate a time constantwhich can be thought of as a time for thegrowth process to get started. If the sumof the inertial start-up time (t1) and theviscous start-up time (tv) is comparableto or greater than a significant fraction(say 0.2) of an acoustic period (r), then thebubble growth will never really get startedbefore the phase of the acoustic cycle hasgone positive and the bubble begins todecelerate prior to collapse. Restating thiscriteria using expressions given by Apfel(1981a, b), we have very rough estimatesof t, and tI summed as follows:

tv+tI= 4tp +2Ri /P <0-2,r 0- (4)

where AP is the time-averaged pressurewhile the driving pressure plus ambient

pressure is negative, =2(PA-Po)13, withPA the peak acoustic pressure and Po is theambient pressure. Also , is the dynamicviscosity, RI is the initial bubble radius, fis the frequency (= l/ r), and p is thedensity of the medium. In Fig. 1, we plotwith dots Equation (4) as Ri versus PA fordifferent frequencies. Here we choose ablood plasma-like environment with ,u0-002 N.s/m2, or about twice that of water.We note that as the bubble gets bigger,the inertia of the medium becomes con-siderable, and significant growth is seri-ously impeded; the possibility for a strongcavitation event is correspondingly dimin-ished.What do we mean by strong cavitation

PEAK INTENSITY (W/m2)

E

xacp

._

-6

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30Peak Accustic Pressure Amplitude (BARS 105 Pa)

FIG. 1. Curves for the given frequencydelineating the initial bubble size or smallerfor which transient cavitation is possiblefor a particular peak acoustic pressure.The A's represent transient cavitation pre-dictions not taking into account viscosityor the initial inertial effects. The dotsrepresent the constraint implied by viscousand initial inertial effects. A curve is drawnthrough either the A or the dot, dependingon which corresponds to a lower radius fora given pressure. The upper abscissa isgiven in terms of peak acoustic intensity.The assumed viscosity is 0-002 N.s/m2 orabout twice that of water.

143

R. E. APFEL

event? Here we are in a bit of a quandary.We could define a transient cavitationevent by the criterion that the velocity ofthe collapsing cavity approaches thevelocity of sound in the medium beforebeing decelerated by the gas trapped inthe cavity. This will occur, as shown byNeppiras & Noltingk (1951), if the bubblereaches a maximum size Rm equal to a

little more than twice the initial size(2.3 to be more precise). This "collapsevelocity" criterion would be reasonable ifit were thought that the stresses andvelocities thus generated were the cause ofsignificant biological damage. This criter-ion, from Equation 18 of Apfel (1981b), isplotted with A's for different frequenciesin Fig. 1. However, the criterion will notbe met if the startup criteria are not alsomet. Therefore, for each frequency a solidline is drawn through the dots and A'swhich satisfy both criteria (transientcavitation, given sufficient time for growthin a fraction of an acoustic cycle).Another possibility for a damage

criterion is based on energy. This "energycriterion" is related to the absolute sizereached by the bubble as contrasted withthe "collapse velocity" criterion whichdepends only on the maximum bubble sizerelative to the initial size. It stands toreason that the more potential energy

(- volume x pressure difference across bub-ble) stored in the fluid at its maximumsize, the greater the kinetic (and even-

tually thermal) energy given back upon

the bubble's collapse.One approach in considering the pos-

sibility of cavitational damage in bio-logical tissues in a pulsed ultrasonic fieldmight be to say that the "collapsevelocity" criterion is met and, in addition,the maximum bubble radius equals or

exceeds some absolute size criterion. Forindividual cell damage it does not seem

unreasonable initially to choose this abso-lute size as that of a cell. For example, letus take the effective radius of a cell to beabout 5 Kim; then we hypothesize thatthere will be damage to the cell if both the"collapse velocity" criterion is met and

the bubble grows to a radius of at least5 [m. Apfel (1981b) gives an approximateexpression (his Equation 16) which relatesthe maximum size a bubble will reach tothe acoustic pressure and frequency. Thiscan be written as follows:

fMHz R14'm ,3 (p 2) [I1+ 2(P- 1)/3]1/3 (5)p1'2

where fMHz is the frequency in MHz,Rp,m is the maximum radius in ,um, andP =PA/PO. For R = 5 pm, and f= 2 MHz,p_6.

In Fig. 2, we show the 2 MHz case fromFig. 1 with a vertical line at PA/PO=6,corresponding to a 5 ,um maximum radiuscriterion. The cross-hatched region repre-sents the radius-pressure combinations forwhich both velocity and energy criteriaare met. Note that PA= 6 x 105 Pa corres-

ponds to a 100 W/m2 time averagedintensity assuming 2 cycles per pulse anda pulse repetition frequency of 1 kHz.

Time-Averaged Intensity (W/m2) Assuming 2 Cyclesper Pulse and kHz Pulse Repetition Frequency

102 5 X 102 103 1.5 X 103

E

Peak Acoustic Pressure Amplitude (BARS i05Pa)

FIG. 2.-Transient cavitation possibilities at2 MHz, taken from Fig. 1. Also drawn is a

vertical line at pA/PO= 6. The hatchedregion gives the initial bubble radius-pressure combinations for which bothtransient cavitation is possible and thebubble will reach a radius of at least 5 ,um.The upper abscissa corresponds to thetime-averaged intensity assuming twocycles per pulse and a pulse repetitionfrequency of 1 kHz.

[T

144

ACOUSTIC CAVITATION

Although there appears to be no lowersize limit to the hatched region, there isa lower, surface tension-dependent, nuclea-tion threshold for the initiation of thegrowth of a gas bubble, which Blake(1949) formulated and which can also befound in Apfel (1981a). For acousticpressures in excess of 6 x 105 Pa, however,the threshold radius is below 0-1 pm, andtherefore does not appear on the graph.It is entirely reasonable at this stage ofthe discussion to challenge the suggestedcriteria, because the macroscopic effectsnormally associated with transient cavita-tion usually result from multiple bubbleeffects. That is, in a continuous wave fieldone small bubble collapsing will shatter,producing seed bubbles that will growduring the later cycles. This collection ofbubbles will go through this process often,as the photographs of Crum (1972) sug-gest, until the collection of bubbles is ofresonant size, at which point radiationforces push the collection away fromregions of high acoustic pressure.For the diagnostic ultrasound case, only

one growth and collapse phase is likelyand the total energy is extremely small.It is simple to show that a bubble thathas grown substantially to a 5 ,um radiushas a potential energy of about 300 MeV(million electron volts). Nevertheless, thescale we are talking about is of the orderof the size of a cell. What must be deter-mined, therefore, is the appropriate scalefor damage that results in acceptablerisks. The vertical line in Fig. 2 can be

thought of as a line-cursor which can bemoved horizontally, depending on our bestestimate of reasonable damage criteria.A few additional comments on high

amplitude growth.-Figs 1 and 2 refer to aviscosity of 0-002 N. s/M2. Thresholds forbubble activity in tissues of much higherviscosity will be even higher, making itlikely that the only concern for damagedue to cavitation will be with aqueousenvironments. For instance, for a viscosityof 002 N s/m2 (a factor of 10 greater thanconsidered in Figs 1 and 2) a pressuregreater than 1.5 x 106 Pa would be re-quired for the possibility of transientcavitation.Our discussion has also left out absorp-

tion, which will tend to diminish cavita-tion, and focusing and standing waves,which may tend to enhance it.

CONCLUSION

Table III summarizes the cavitationaleffects that are possible in aqueous mediafrom biomedical ultrasound ranging frompulsed to continuous and from moderateto high acoustic pressure amplitudes.Whereas viscous effects can eliminate thepossibility of transient cavitation frompulse-echo ultrasound, it is probably lessimportant for rectified diffusion. Andwhereas the concept of energy dose maybe relevant when discussing heating dueto ultrasound, or the effects of resonantbubble motion, it is less appropriate fortransient cavitation produced by short

TABLE III.-Types of cavitation posu&ble for seed bu,bble in aqueous environment

Moderate pressure ampl;Therapeutic and dopplerdevices(PA< 1-0 atm)

Higher pressure ampl.

(PA > 3 atm)

1.2.

Continuous wave(or significant fraction of on-time)

Bubble growth by rectified diffusionMechanical activity produced byresonant bubble oscillations

1. Dynamic cavitation2. Bubble break-up and new growth of

several bubbles-collectivecavitational effects

3. Significant mechanical activity

Pulsed

none. for short duty cycle

Significant bubble growth andcollapse possible in one acousticcycle if pressure is great enoughand if frequency is low enough(see Figs 1 and 2). Time-averagedspatial peak intensities can beas low as 100 W/m2

145

146 R. E. APFEL

bursts of high pressure amplitude ultra-sound.For the future, the question of the

existence of seed bubbles in mammaliantissues requires a definitive answer. More-over, experimentalists seeking to deter-mine whether or not there are bioeffectsfrom ultrasound must take care to knowthe acoustic field, which means choosingequipment and geometrical configurationswhich are most conducive to knowing theinstantaneous pressure as a function ofposition, as well as the various integratedenergy measures, the time histories,temperature, and the state of the medium(gas saturation, solution properties, etc.).

This work is supported in part by the PhysicsProgram of the U.S. Office of Naval Research. Ithank E. A. Neppiras for his advice on this generalsubject.

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Ultrasonics. Ed. P. Edmonds. Methods of Experi-mental Physics Series, Ed. Marton. New York:Academic Press.

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CARSON, P. L., FISCHELLA, P. R. & OUGHTON, T. V.(1978) Ultrasonic power and intensities produced

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COAKLEY, W. T. & NYBORG, W. L. (1978) Cavita-tion: dynamics of gas bubbles; applications. InUltrasound: Its Applications in Medicine andBiology. Ed. F. J. Fry. New York: Elsevier.

CRUM, L. A. (1972) Velocity of transient cavities inan acoustic stationary wave. J. Acoust. Soc. Am.,52, 294.

CRUM, L. A. (1980) Measurement of the growth ofair bubbles by rectified diffusion. J. Acoust. Soc.Am., 68, 203.

FLYNN, H. (1964) Physics of acoustic cavitation inliquids. In Physical Acoustics. Ed. W. P. Mason.New York: Academic Press. p. 57.

HuSSEY, M. (1975) Diagnostic Uttrasound. Glasgow:Blackie.

LEWIN, P. A. & BJ0RNo, L. (1981) Acoustic pres-sure amplitude thresholds for rectified diffusionin gaseous microbubbles in biological tissue. J.Acoust. Soc. Am., 69, 846.

NEPPIRAS, E. A. & NOLTINGK, B. E. (1951) Cavita-tion produced by ultrasonics: theoretical condi-tions for the onset of cavitation. Proc. Phys. Soc.,B64, 1032.

NEPPIRAS, E. A. (1980) Acoustic cavitation, Phys.Rep., 61, 160.

NYBORG, W. L. (1982) Ultrasonic microstreamingand related phenomena. Br. J. Cancer, 45, Suppl.V, 156.

PROSPERETTI, A. (1977) Thermal effects and damp-ing mechanisms in the forced radial oscillations ofgas bubbles in liquids. J. Acoust. Soc. Am., 61, 17.

RICHARDSON, J. M. (1947) As quoted in footnote 14of Briggs, H. B., Johnson, J. B. & Mason, W. P.(1947) Properties of liquids at high sound pressure.J. Acoust. Soc. Am., 19, 664.

TER HAAR, G., DANIELS, S., EASTAUGH, K. C. &HILL, C. R. (1982) Ultrasonically induced cavita-tion in vivo. Br. J. Cancer, 45, Suppl. V, 151.