Finite‐Amplitude Effects on Ultrasound Beam Patterns in Attenuating Media

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Finite-amplitude effects on ultrasound beam patterns in attenuating media ClarenceR. Reilly") and KevinJ. Parker Departmentof Electrical Engineering, and Center for BiomedicalUltrasound, University of Rochester, Rochester, New York 1462 7

(Received21 November1988; accepted for publication14 August 1989 ) Someproblems relevant to medicalultrasonics are addressedthrough experimental measurementsof focused,pure-tone beam patterns under quasilinear conditions where significantnonlinearities are manifested. First, measurements in water provide a comparisonof thebeam patterns of thefundamental and nonlinearly generated harmonics against recent theoreticalpredictions of others.The radialbeamwidths, presence and spacing of sidelobes, axialdistances to peakpressures, focal shock parameter, time-domain waveform asymmetry, and post-focalfalloff of the fundamentalthrough fifth harmonicsare discussedrelative to variousmodels under preshock conditions (or < 1). Second,the focusedsources are placedin a moreattenuating fluid to mimicthe behaviorof thesefields in tissue.The changesin beam characteristicsare examinedrelative to measurementsat the sameintensities in water, and relativeto theoreticalpredictions. The resultssuggest that, givena knownlinear(low- intensity)focused beam pattern in water,guidelines can be followedto predictthe beam patternof the fundamentaland higherharmonics at higherintensities in water, and then in attenuatingmedia suchas tissue. PACS numbers:43.25.Jh, 43.80.Sh, 43.30.Qd

INTRODUCTION The resultsof thisstudy provide practical guidelines for Finite-amplitude effectsin medical ultrasonicscan be extrapolatingfrom a known linear focusedbeam to the more potentiallyuseful as in thecase oflithotripsy, or candegrade complicatedcase of finite-amplitudepropagation at higher systemperformance as in the caseof conventionalimaging intensities,and throughtissue. instruments. Since the works of Carstensen and colleagues,•-3 much attention has been given to nonlinearwave propagationin ultrasonicfields. 4-•5 Parallel researchhas I. MATERIALS been underway in underwater acoustics, where the in- fluencesof finite-amplitudeeffects have beenstudied under Three, 1-in.-diam lead zirconate transducers with fun- differentconditions. •6-2ø At present,a varietyof theoretical damentalfrequencies of0.59, 2.25, and 3.38 MHz wereused approachesand predictionscan be appliedto the caseof to produce quasicontinuouswave beams. The 0.59-MHz medicalultrasound imaging, where frequencies typically transducer was driven at its third (1.75 MHz) and fifth rangefrom 2-10 MHz, transduceraperatures (when piston (2.95 MHz) harmonics, while the other transducers were or annular array sourcesare used) are a few centimetersin driven at their fundamentalfrequencies. The transducers diameter,and focal lengths can range from lessthan 3 cm to were simultaneouslymounted in a tank, and concaveLucite over15 cm, depending on application.However, experimen- lenseswere placed 2 mm above each tranSducer toprovide tal verificationof theory,especially in moreattenuating me- focusing.The lenseswere designedto have a nominalfocal dia, is limited. point of 8 cm in water. Attenuation in the lens resultedin This paper presentsexperimental studies of focused someapodization of the pistonsources. 2•'22 pure-tonebeam patterns within a rangeof parametersger- A calibrated polyvinylidenefluoride (PVDF) needle- mane to medicalultrasound imaging. The fundamental type hydrophone (Medicoteknisk Institut, Denmark) was beampatterns and harmonic beam patterns in preshockcon- usedas a receiver.The hydrophonehad a 1-mm active ele- ditions(shock parameter cry< 1 at the focus)are compared ment and was known to have a nearly flat frequencyre- with theoreticalpredictions of others.Included are particu- sponsefrom 1 to 12 MHz. The hydrophonesensitivity was lar featuressuch as the beamwidthof the harmonics,the determined from total power measurementsand hydro- presenceand locationof harmonicsidelobes, the on-axisfo- phone voltagesat the sourcesto be on the order of 0.022 calpeak of eachharmonic, the apparentshock parameter cr MPa/mV (16kW/cm2/V2).A fluidmedium was sought to at the focalpoint, the asymmetryof the pressurewaveform test existing theory on the effect of attenuation on finite- the post-focalfalloff of harmonics,and the effectsof attenu- amplitudewaves. The ideal mediumwould have a velocity ation on theseparameters. andB/A closeto water,and an attenuationof 0.02--0.05Np/ cm/MHz in the frequencyrange of 1-10 MHz. A suitable Currentaddress: Indianapolis Center for AdvancedResearch, Inc., In- mediumwas developedusing 49.6 mdof a 0.25 molar mix- dianapolis,IN 46104. ture of 50% mono-and 50% dipotassiumphosphate; 1.5 md

2339 J. Acoust.Soc. Am. 86 (6),December 1989 0001-4966/89/122339-10500.80 @ 1989Acoustical Society of America 2339 Karo corn syrup; and 48.9 m(isopropyl rubbing alcohol (90% pure). Alcohol was addedto the water/phosphate/ Acquisition Controller sugarmixtui'e to reducethe soundspeed and attenuation, System DataHMotor , and to stabilizethe medium againstbacterial growth. It is IBM known that the water and alcoholundergo a complexinter- Dig itizing Coordinate action.The additionof alcoholto waterapparently increases Oscilloscope System I PC-XT'"'IEEE z the relative amount of water in the bound versusfree state.23 Computer y

In general,the B/A of a water/alcohol mixture increases Spectrum x IEEE rapidlyfor alcoholvolume fractions between 10% and 50%, Analyzer and then startsto level off. The B/A of predominantlywater and T-butanol mixture rangedfrom 5-11 in Sehgalet al.'s I- study.23 Other 50% alcoholsolutions such as methanol and isopropanolhave measured B/A valueshigher than that of Function water.24 The differencein the nonlinearparameter of our Needle-type Generator attenuatingmedium, assuming B /A -- 6-10 (/3 = 4-6) rela- Hydrophone tive to water (/3 = 3.6), hasthe effectof increasingthe effective shockparameter (or harmonicpressure) by a factor of yrfsignal 1.1-1.7 (0.8-4.6 dB) over the same conditions in water. 50 dB Lucite Lens The frequency-dependentabsorption of our attenuating Amplifier medium was determined by radiation force insertion loss Transducer measurementsto be 0.012f1'9 Np/cm withfin MHz (Fig. 1). Although the frequencydependence (n = 1.9) is high- FIG. 2. Automated harmonic data-acquisitionsystem. er22 than that oftissues (n = 1.2-1.3), thenearly square-law functionof frequencyin the attenuatingfluid enablescom- parisonagainst other theoreticalresults. The velocityof the rubber absorber and a Sartorius microbalance.2• Total out- attenuatingmedium was determinedto be 1501 m?s using put power from continuous-wavemeasurements was divid- the time of flight of an unfocused10-MHz nominalfrequen- ed by sourcearea to determine spatial-averagedintensity cy pulse.The attenuatingmedium Was recovered after each and pressure.A linear increaseof pressurewith voltagewas day of useand testedfor a changein the acousticproperties observed,and the resultswere, in somecases, extrapolated to (attenuation and velocity) over time. The attenuationand higher voltages. velocityvaried by + 10% and + 3% (from 1501m/s up to Field pressuredistributions were determinedusing the 1549 m/s), respectively,over the courseof a week. The PVDF hydrophone.The 75-W hydrophonecable was con- changein attenuationis within the rangeof uncertaintyof necteddirectly to a Tektronix 7D20 digitizing oscilloscope theexperimental technique. 21 (40-MHz sampling rate; Zin •-1 MW;Cin = 20 pF) as shownin Fig. 2. A low-outputimpedance amplifier from the II. PROCEDURES oscilloscopewas used as a preamplifierfor the Tektronix The spatial-averagesource intensity Io through the lens 496P programmablespectrum analyzer. Harmonic beam was determinedfrom radiation force measurementsusing a patternswere obtained through computer control of the signal generator,spectrum analyzer, and steppingmotors on a three-axiscoordinate system (Velmex Inc., 0.01-mm preci- sion). The Tektronix FG5010 function generatorwas pro- grammedto senda burstof 100cycles of a specifiedfrequen- cy) every 2 ins. With the spectrum analyzer running asynchronouslyon maximum hold, the peak valuesof the / / first five harmonicsat eachpoint within the field (41 points / / per scanline) werestored in an array after a specifiedacqui- / / / sitionduration (usually between8 and 12 s). The point-to- / / point (incremental)distances for the axial and radial scans / at eachfrequency are givenin Table I. The incrementalradi-

TABLE I. Incremental scan distances.

Frequency(MHz) Axial increment(mm) Lateral increment(mm)

i t.oi i z.oi i 31.o ! 4.0! i s.oi i 61.o i 7.0i i 6o• 1.75 2.80 0.30 FREQUENCY (MHz) 2.25 1.75 0.25 FIG. 1. Frequency-dependentattenuation of water/alcohol/phosphate/su- 2.94 1.40 0.20 gar mixture.Data points(,) werefitted to the function0.012f 1'9 (---), 3.38 1.20 0.20 wheref is in MHz. The standarddeviation of the absorptiondata is shown.

2340 J. Acoust. Soc. Am., Vol. 86, No. 6, December 1989 C.R. Reilly and K. J. Parker: Finite-amplitudeeffects 2340 TABLE II. Spatial-averagesource intensity (Io). linearshock at thefocus for eachfrequency, while remaining in the linearoperating region of the electronicamplifier and Low intensity High intensity Generator ' Generator sources.Source intensities are given in Table II. The axis of Frequency voltage Io(W/cm2) voltage Io(W/cm2) the transducerwas determined from lateralscans in orthog- onal planes.Axial scanswere then performedat the two 1.75 0.1 0.049 0.55 1.52 intensity levelsto determine the focal distancefor each har- 2.25 0.1 0.087 0.5 2.24 monic. The focal distance was defined as the distance from 2.94 0.2 0.070 0.6 0.92 3.38 0.1 0.071 0.4 1.18 the centerof the lensto the centerof the regionwhere the peak amplitudevaried by lessthan - 0.5 dB. Lateral beam patternswere obtainedat the focusof the fundamentaland the secondharmonic at the four sourcefrequencies and two al distance was chosen to include some minor lobes of the intensitylevels. beam (only part of a sidelobeis evidentat 1.75 MHz) while In order to distinguishbetween linear superpositionof retainingan adequateresolution. transmittedharmonics and nonlinearlygenerated harmon- To observechanges in the harmoniccontent of the field icsin the fluid media,waveforms and spectralcontent at the with intensity, scansat "low"- and "high"- power levels acousticsource were measuredusing the hydrophone.The were performed.The powerspectrum at low powermainly second harmonic at the source was found to be between consistedof linear radiationfrom the transducer.The high- -- 25 and -- 30 dB belowthe fundamentalat all frequencies powerlevel was chosen to approachthe upperlimit of quasi- and power levels.

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FIG. 3. Axial andlateral beam patterns in waterfor the2.25-MHz sourceat FIG. 4. Axial andlateral beam patterns in waterfor the 2.25-MHz sourceat •0=0.087 W/cm •. The first (--), second (---), third (-'-), Io = 2.24W/cm •, The first (--), second(---), third (-'-), fourth(---), fourth (---), and fifth (...) harmonicsare shown. and fifth (...) harmonics are shown.

2341 J. Acoust.Soc. Am., Vol. 86, No. 6, December1989 C.R. Reillyand K. J. Parker:Finite-amplitude effects 2341 III. RESULTS suredat high intensityat the focus,are givenin Fig. 9 for The axial and lateral harmonicbeam patterns (normal- water and Fig. 10 for the attenuatingfluid. The higherthan ized to the peak fundamentalamplitude) in water for the expectedstrength of the sixth through tenth harmonicsin 2.25-MHz low and high intensitiesare givenin Figs. 3 and 4. Fig. 9(b) is related to the "resonance"of the probe-cable- In Fig. 3 (low power), the magnitudesof the third amplifiercircuit which increasesgain of frequenciesabove throughfifth harmonicsare barely abovethe noisefloor at 12 MHz (Ref. 10). approximately - 55 to - 60 dB. In Fig. 4 (high power), the beam patternsof all harmonicsare clearly shownwith diffractive sidelobes.Since the hydrophoneactive element IV. DISCUSSION has a 1-mm diameter, fine structures such as the fourth and A. Radial beamwidths fifth harmonic sidelobes are not resolved. High-power beam patternsfor all four frequenciesare As a result of their theoretical derivation, Du and Brea- shownin Figs. 5-8, wheredirect comparisonsof the results zeale8 predictedthat the Gaussian beamwidth for the second in water and attenuating fluid are made. The attenuation harmonicis simply l/v7 times that of the fundamental effectsare more significantat higherfrequencies and higher beamwidth,and a 1/x/-•relationship for higherharmonic harmonics,as expected.Even the highly attenuatedbeam beamwidthswas suggested. This is in contrastwith 1/n rela- patternsshow strong diffractive effects of sidelobesand axial tionship,which would resultfrom the linear propagationof nulls. the nth harmonic from a broadband source. The more traditional examplesof time- and frequency- To test this predictionunder our experimentalcondi- domainrepresentations of onewaveform (2.25 MHz), mea- tions, the -6-dB harmonic beamwidths in water and the

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FIG. 5. A comparisonof theaxial and lateral beam patterns in waterand the FIG. 6. A comparisonof theaxial and lateral beam patterns in waterand the absorbingfluid for the 1.75-MHzsource at Io -- 1.52W/cm 2. The first five absorbingfluid for the 2.25-MHz source at Io -- 2.24W/cm 2. The first five harmonicsin water (--) and in the absorbingfluid (---) are shown. harmonicsin water (--) and in the absorbingfluid (---) are shown.

2342 J. Acoust.Soc. Am., Vol. 86, No. 6, December1989 C.R. Reillyand K. J. Parker:Finite-amplitude effects 2342 (a) (a) ß 00 .00

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FIG. 7. A comparisonof the axialand lateral beam patterns in waterand the FIG. 8. A comparisonof the axialand lateral beam patterns in waterand the absorbingfluid for the 2.94-MHz sourceat Io = 0.92W/cm 2.The harmon- absorbingfluid for the 3.38-MHz sourceat Io = 1.18W/cm •. The firstthree icsin water (--) and in the absorbingfluid (---) areshown. Four harmonics harmonicsin water (--) and in the absorbingfluid (---) are shown. are shownfrom the axial beampattern in water, whereasthree harmonics are shownfrom the beampattern in absorbingfluid.

harmonicsidelobes are referredto as "fingers."27'33 In our experimentalresults, the hydrophone1-mm active element attenuatingmedium at the higherintensities are plottedver- smears much of the fine detail, but the second-harmonic sustheory for the 1.75-and 3.38-MHz beamsin Figs. 11 and sidelobedoubling can clearly be seenat 2.25 MHz (Fig. 4). 12. The harmonic beamwidth calculations were based on At 4.5-mm radial distance,the fundamentalbeam pattern 1/x/-fftimes the measured fundamental beamwidth. The the- includes the main lobe and one sidelobe,while the second- ory is generallya goodpredictor of the non-Gaussianexperi- harmonic beam pattern includesthe main lobe and three mental beamwidthsat all frequencies. sidelobes.Some evidence of this pattern is evidentin the We note also that only small or insignificant(•0.5 third-harmonicpattern. Thus our experimentalresults agree mm) changesin measuredbeamwidths were observedas a with the prediction of fingers by Hamilton and col- functionof power,or when comparingresults in water ver- leagues.17,18,25-27 susattenuating fluid.

B. Harmonic sidelobes C. Axial peak focal distances In recentyears, some groups have performed numerical Will the fundamentaland secondand higherharmonics solutionsto the so-calledKZK equationto generatebeam all reachaxial peaks (focal points) at the samedistance? The patternsof focusedpiston sources. •7'•8'25-27 Their results highlyfocused beams studied by Lucasand Muir,•9 and predictthat the second-harmonicbeam pattern has twice as Hart andHamilton 26 appear to havecoincident axial peaks. many sidelobesas the fundamentalcomponent, while the However, in the caseof Gaussian focusedbeams studied by third harmonichas three times as many, and so on. The extra Du and Breazeale,8 the second-harmonicaxial peakcan be

2343 J. Acoust. Soc. Am., Vol. 86, No. 6, December 1989 C.R. Reilly and K. J. Parker: Finite-amplitudeeffects 2343 ' t

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distalto the fundamentalfocus, and this is alsofound to be D. Focal point shock parameter • the casein earlierexperimental work of Smithand Beyer 28 The shockparameter c is a usefulconcept for nondif- andGould et al. •9 In general,our data show that each of th• fracting beams as it describesthe relative strength of the harmonics reach their axial peaks at increasingly distal harmonicmagnitudes and the dependenceof tr on positionis points comparedto the fundamental,as shown in Fig. 13. The influenceof attenuationwas not strongenough to significantly reversethe trend over the rangeof frequenciesstudied. In eachcase, the geometricfocal point is 80 mm from the lens base, while the fundamental axial peaks appeared around 50-70 mm at the four differentfrequencies studied, as expectedfrom lineartheory. 3ø'3• We notethat a linear E 4.0 superpositionof harmonicsemanating from the source(not -r •--. generatednonlinearly within the medium) would also exhibit increasingfocal lengthswith higher frequencies,ap- •' 3.0 r• proachingthe geometric focal point of 80mm in theideal r• case of infinitesimal wavelengths.However, in the linear • 2.0 case,as the f number (focal length to sourcediameter) is lowered (to produce tighter focusing), the result is closer proximity of axial peaks from different pure-tone excita- tions.The stronginfluence of this diffractiveeffect partially ß O0 I l.O 2•0 3•0 4.0' 5•0 explainsthe convergenceof axial peaksin Lucasand Muir's H^RHONIœ NUHBER andHart andHamilton's work with lowfnumber 0e/2 tight- FIG. 11. The --6-dB harmonic beamwidths for 1.75-MHz source at ly focused)beams, as comparedto the distal peaksin our Io = 1.52W/cm 2 comparedto 1/xf• theoryin (a) water:experimental experiments(approximately f/3 ) and Du and Brezeale'sre- (V1),theory (--); and (b) absorbingfluid: experimental (A), theory('"). port on higherfnumber (more weakly focused)beams.

2344 J. Acoust.Soc. Am., Vol. 86, No. 6, December 1989 C.R. Reillyand K. J. Parker:Finite-amplitude effects 2344 3.0 8O

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H^RHONIC NUHBER 1.75 MHz 2.25 MHz 2.94 MHz 3.38 MHz

FIG. 12. The --6-dB harmonic beamwidths for 3.38-MHz source at FIG. 13. The locationof axial peaks(focal point) for the fundamentaland Io = 1.18W/cm 2 comparedto 1/x/• theoryin (a) water:experimental higher harmonicsfor the four frequenciesstudied, in water and in the at- (E3), theory (---)-and (b) absorbingfluid: experimental(A). Since the tenuatingmedium. For eachcase, shown left to right, are the locationof the fundamentalbeamwidth is the samein the attenuatingfluid and in water, fundamental (black bar), then the second, third, fourth, and fifth harmon- the 1/x/-•theory curves are identical. • ics, respectively.Missing bars indicateinsufficient signal above noise.

a simple,one-dimensional relation for plane and spherical and gainmeasurements to estimatecr using Eq. (2). Next, an waves.32'33 In focusedbeams, the adaptation of a shockpa- ad hocestimate of crwas obtained by comparingthe harmon- rameteris questionablebecause of the rapidly changinghar- ic magnituderatios at the focusto the theoreticaldes•:rip- monic phasesand magnitudeswithin eventhe -- 3-dB fun- tion. For example, if the secondthrough fifth harmonics damentalfocal region.Nonetheless, the conceptof crat the were approximately -- 8, -- 12, -- 15, and -- 19 dB.down fundamentalfocal point.has been employedby others to from the fundamental, this would be labeled as cr = 1 sinceit characterize shock in focused ultrasonic beams. Dalecki •4 matchesthe theoretical•'32harmonic ratio for cr= 1. (In the appliedthe relationshipfor convergingspherical waves: preshockregime, or<1, the harmonicsare a monotonically incri•asingfunction of or,so thereis no ambiguity.) In cases cr=/3ekR ln(R/r), ( 1) where all harmonicsdid not match a singlecr theoretical where e is the Mach number (source velocity/speed of result,an estimatewas obtained by comparingeach harmon- sound), k is the fundamentalwavenumber, R is the effective ic againstthe fundamental,evaluating the cr for that ratio, radius of the sphericalsource, and r is the distanceback and averagingthe resultsß over all harmonics.The resultsare toward the sourcefrom the geometricfocal point. To avoid givenin Table III. the nonphysical,nondiffractive singularity at r = 0, we ap- The shockparameters from sphericalwave theory with plied a simplegeometric argument where the minimum r a minimum r are lessthan thosepredicted from Gaussian valuewas taken to be roughlythe axial distanceat which the theory. The theoreticalestimates of orat 1.75 MHz are much pressurewas 0.5 dB belowmaximum. In comparison,Ba- lower than observed. The estimates of cr from the measured con4'5assumed a Gaussian beam shape and derived a differ- harmonic ratios at 2.25 and 2.94 MHz fall between the theo- ent expressionfor the focal shockparameter: retical values.Both sphericalwave and Gaussiantheories seemto overestimateexperimental values of crat 3.38 MHz. cr--o),6'P•F[ln(G+x/G2-- 1)/pc3x/G2-- 1],. (2) In summary,only a rough agreementis found between where Pm is the maximum pressureat the focus, G is the publishedtechniques for estimatingcr at the focusand our small signalpressure gain, and F is the focal distance.Our measuredexperimental harmonic magnitudes.This is not experimentalcomparison used measuredvalues of axial surprisingsince all three techniquesuse different sets of as- beamwidthsto estimatecr from Eq. ( 1), thenused pressure sumptions,and sincethe harmonicsdo not peakcoincident-

TABLE III. Theoreticalshock parameters for sphericallyconverging and Gaussiansources.

Frequency I0 R (cm) r ( cm) O'sphericai Pm(MPa) Gain cro.... Jan O'H..... icest

1.75 1.52 5.16 1.4 0.16 0.92 4.22 0.28 0.6 2.25 2.24 5.67 0.9 0.40 2.56 8.11 0.74 0.6 2.94 0.92 5.20 0.56 0.37 2.04 8.45 0.68 0.5 3.38 1.18 6.90 0.78 0.62 2.773 10.45 1.234 0.5

2345 J. Acoust. Soc. Am., Vol. 86, No. 6, December 1989 C.R. Reilly and K. J. Parker: Finite-amplitudeeffects 2345 ly with the fundamental,small axial shiftswill producedif- TABLE IV. Harmonic attenuationat high intensity. ferent relative magnitudesand therefore different ad hoc estimates of c. Measuredatten. (dB) Atten.= ctfn(dB) aef1'9 (dB) Fre- quency 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd E. Asymmetry ratio 1.75 2.8 3.9 3.9 1.5 3.0 4.5 1.5 5.6 12.1 Bacons comparedexperimental and theoreticalvalues 2.25 2.8 4.7 7.1 2.7 5.4 8.1 2.7 10.1 21.8 of the ratio of peak positivepressure to peak negativepres- 2.94 4.0 7.9 11.5 4.2 8.4 12.6 4.2 15.6 33.9 sures,P+/P_ of time-domainwaveforms in water and an 3.38 7.2 14.7 20.3 7.1 14.2 21.2 7.1 26.5 57.3 attenuatinggel phantom.The conditions(3.5 MHz, f/4 focus) were similar to those in our measurements at 3.38 MHz. Baconfound that in water with the focalshock parameter 0.2 < rr < 1.0, the asymmetryratio rangedbetween 1.1- medium of this kind, it hasbeen shown that for rr>>1, Fay's 1.9. The increasedpeak positive pressureresults from a solution34 simplifies to showattenuation of thenth harmon- phaseshift betweenthe fundamentaland harmonics,5'•ø ic ase-,•z, not ase-,•2z. Thisis becausethe fundamental which is causedby diffractionand dispersionmechanisms. servesas an extended virtual sourceof the secondharmonic, In the attenuating gel, Bacon found the asymmetry ratio sothe decayis not expectedto be the sameas in linear propa- reducedto between1.0-1.05 for thesame range of shock' gationthrough a 10ssymedium. Although the experimental parameters.Our measuredtime domainasymmetry ratios in studiespresented herein are not plane-wavecases, the at- water and in the attenuatingfluid are plotted in Fig. 14 for tenuationof harmonicsgenerally increased as aœn, (linearly three sourcefrequencies at high intensities.At 2.25 MHz, withfrequency), as opposed to afn2, (quadraticallywith there is essentiallyno changein asymmetryratio from water frequency).as would be expectedfrom the smallsignal loss in to the attenuatingfluid. At the two higher frequencies,2.94 our attenuatingmedium. Table IV showsthe measureddB and 3.38 MHz, thereare largerdifferences between asymme- differencebetween the harmonicmagnitudes in water and in try ratios, as attenuation effectsbecome stronger. Our re- the attenuatingmedium as measuredat the fundamentalfo- sultsat 3.38 MHz are in agreementwith thosepresented by cus. (The second- and third-harmonic measured differences Bacon. have been adjusted for an estimated 2.3-dB difference betweenthe fl of the two media.) Also givenare predictions F. Attenuation loss of harmonics of theloss assuming aœn and ad el'9 dependences over the In linearpropagation, the effectsof attenuationon a distancefrom the sourceto the focal point. The linear-with- weaklyfocused beam would be to first approximation: frequencyresults are a good match to the data. In soft tis- -- O:jr/2Z sues,22 where the small signalattenuation coefficients in- (z) l - IPW (z)le , (3) creaseas fl.2 or fl.3, we wouldstill expectthe harmonic where P•ln and PW,, denotenth harmonicpressure in the lossesto follow naf, orf 1'ø,since the nonlinearly generated attenuatingmedium and in water, respectively,as a function harmonics always decay less rapidly than a small signal of axial distancez. In Eq. (3) a frequency-squareddepen- waveof thesame frequency. 35 denceof attenuationis assumed: a(n) --o•frt2 withaf the Anothertest of thisconcept is the'decay of harmonics absorptioncoefficient at the reference(fundamental) fre- beyondthe focalregion. The slopesof the harmonicsin dB/ quency.For plane-wavenonlinear propagation through a cm, distal to the focus, were measured in water and in the attenuatingfluid. For linear propagation,the expecteddif- ferencein dB/cm would be equalto the small signalattenu- ationcoefficient at anyfrequency, which increases asfl'9 or nearly quadratically in our fluid. However, comparisons showedthat the axial falloffincreased roughly as fl.O, or WATER 1.8 linearly with frequency, for the casesstudied (Table V). 2.0• AI'rENUATED Thus it appearsas though quasilinear beam measurements in n 1.6 4- • 1.4

0 1.2 _ .:- • i" ' TABLE V. Post-focalaxial decay (dB/cm)' Differencesin H20 and at- •. 0.8 tenuatingfluid at high intensities.

Predicted Predicted • 0.4 '. Measured a = •l•f/• •1•= •l•{lf1'9 • '. Fre- quency 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd

2.25 2.94 3.38 1.75 0.3 0.4 0.4 0.3 0.6 0.9 0.3 1.1 2.4 2.25 0 1.7 2.3 0.5 1.0 1.5 0.5 1.9 4.0 FREQUENCY (MHz) 2.94 0.3 1.1 1.7 0.9 8.4 1.8 0.9 3.3 7.2 FIG. 14. Asymmetryratios measuredat focal points,in water and in the 3.38 0.8 2.5 3.3 1.1 2.2 3.3 1.1 4.1 8.9 attenuatingfluid, for the threehighest frequencies at high intensities.

2346 J. Acoust. Soc. Am., Vol. 86, No. 6, December 1989 C. Fl. Reilly and K. J. Parker: Finite-amplitudeeffects 2346 water can be used to predict resultsin tissues,where the matedusing the resultsof Sec.IV A, B, and D. The location smallsignal attenuation of the fundamentalis appliedto the of axial peakscan be crudely estimatedusing the resultsof fundamental, twice to the secondharmonic, and so forth. Sec.IV C. Then, the changesproduced by propagationin a As a finalnote, Saito et al.2ø described the ratio of funda- lossymedium can be estimated using the simplea•n results mental to secondharmonic as beingrelatively constantbe- ofSec. IV F. Thisshould be useful in predictingfinite-ampli- yond the focal region in their study. However, in all cases tude effectsof medicalultrasound equipment in tissuesand reportedherein, the secondand higher harmonicshad in- other related applications. creasinglysteeper slopes (by 1-3 dB/cm) comparedto the fundamentalfalloff beyond the focus. ACKNOWLEDGMENTS

The insight,encouragement, and supportof Professor E. L. Carstensen and Professor D. T. Blackstock are sincere- V. CONCLUSIONS ly appreciated.This work was supportedin part by NIH Focused,radially symmetric, ultrasoundbeams in the Grant No. CA39241. low-MHz medical ultrasound band have been studied to determine finite-amplitudeeffects in water and in an attenuating fluid. Comparisonshave been made against theoretical and experimentalresults of others.Specific results can be summarized as follows. ( 1 ) The -- 6-dB beamwidths of the harmonics decrease •T. G. Muir and E. L. Carstensen,"Prediction of nonlinearacoustic effects at biomedicalfrequencies and intensities,"Ultrasound Med. Biol. 6, 345- as 1/xf•, wheren is the harmonicnumber, in accordance 357 (1980). with the theoreticalresults for Gaussianbeams obtained by 2E. L. Carstensen,W. K. Law, N. D. McKay, and T. G. Muir, "Demon- Du and Breazeale. 8 strationof nonlinearacoustical effects at biomedicalfrequencies and intensities,"Ultrasound Med. Biol. 6, 359-368 (1980). (2) The sidelobesof the harmonicbeam patterns exhibit 3E.L. Carstensen,N. D. McKay, D. Dalecki,and T. G. Muir, "Absorption n timesthe numberof lobesfound in the radial beampattern of finite amplitude ultrasoundin tissues,"Acoustics 51(1), 116-123 of the fundamental,confirming the theoreticalprediction of (1982). fingersby theTjottas, Hamilton, and colleagues. •8'25'•7 4D. R. Bacon,"Finite amplitude distortion of pulsed fields used in diagnos- tic ultrasound," Ultrasound Med. Biol. 10(2), 189-195 (1986). ( 3 ) The axial peaks(or focalpoints) of harmonicsare 5D. R. Bacon,"Finite amplitudepropagation in acousticbeams," Ph.D. distal to the fundamental focus, in accordancewith the re- thesis,University of Bath (1986). sultsexpected from linear diffractiontheory. 6G. Du and M. A. Breazeale,"The ultrasonicfield of a Gaussiantrans- ducer," J. Acoust. Soc. Am. 78, 2083-2087 (1985). (4) Attenuationdoes not significantly change the 1 7G. Du and M. A. Breazeale,"Harmonic distortionof a finite amplitude beamwidthrelation or fingerpatterns. Attenuation causesa Gaussianbeam in a fluid," J. Acoust. Soc. Am. 80, 212-216 (1986). small shift in the focal peakstowards the source;however, 8G.Du andM. A. Breazeale,"Theoretical description of a focussedGaus- the harmonicsstill haveaxial peaksdistal to the fundamen- sian ultrasonic beam in a nonlinear medium," J. Acoust. Soc. Am. 81, 51- 57 (1987). tal peak. 9K. J. Parker, "Observationof nonlinearacoustic effects in a B-scanimag- (5) The ratio of harmonicsat the fundamental focusin ing instruments,"IEEE Trans.Sonics Ultrason. SU-32( 1), 4-8 (1985). water can be roughly estimatedfrom c calculationsusing •øK. J. Parker and E. M. Friets, "On the measurementof shockwaves," IEEE UFFC 34(4), 454-460 (1987). sphericalconverging wave [Eq. ( 1) ] or modifiedGaussian •D. Rugar,"Resolution beyond the diffraction limit in theacoustic micro- [Eq. (2) ] approaches. scope:A nonlineareffect," J. Appl. Phys.56(5), 1338-1346 (1984). (6) The asymmetry ratio of the time-domain wave- •2H. C. Starritt, M. A. Perkins,F. A. Duck, and V. F. Humphrey,"Evi- formsat the focusdrops from approximately1.8 in water to dencefor ultrasonicfinite amplitudedistortion in muscleusing medical equipment,"J. Acoust.Soc. Am. 77, 302-306 (1985). 1.2in the attenuatingmedium at 3.35MHz, in generalagree- •3W.Swindell, "A theoreticalstudy of nonlineareffects with focussedultra- ment with Bacon's results. soundin tissues:An AcousticBragg peak," Ultrasound Med. Biol. 11( 1), (7) The loss of harmonicsin the attenuating medium 121-130 (1985). (comparedto water)is proportionalto afnz, insteadof the •4D.Dalecki, "Nonlinear absorption of focussedultrasound and its effect on lesionthresholds," MS thesis,University of Rochester(1985). nearlyafn2z frequency dependence of small signals in our •SM.E. Haran andB. D. Cook,"Distortion of finiteamplitude ultrasound lossyfluid. This resultis especiallysignificant, since it paral- in lossymedia," J. Acoust.Soc. Am. 73, 774-779 (1983). lels earlier plane-waveand sphericallydiverging wave re- •6S.I. Aanonsen,T. Barkve,J. N. Tj4tta, and S. Tj4tta, "Distortionand sultsand providesa simplerule for extrapolatingbeam pat- harmonicgeneration in thenearfield of a finiteamplitude sound beam," J. Acoust. Soc. Am. 75, 749-768 (1984). ternsmeasured in waterto beampatterns in lossymedia such •7M. F. Hamilton, "Fundamentalsand applicationsof nonlinearacous- as tissues. tics,"in NonlinearWave Propagation in Mechanics-AMD,edited by T. W. It is possibleto usedifferent approaches with numerical Wright, ASME Symposiumon NonlinearWave Propagation(ASME, methodsto calculatebeam patterns under a varietyof condi- New York, 1986), Vol. 77. •SM.F. Hamilton,J. N. Tj4tta, and N. Tj4tta, "Nonlineareffects in the tions. This experimentalwork and comparisonsprovide farfield of a directive sound source," J. Acoust. Soc. Am. 78, 202-216 guidelinesfor generalunderstanding and prediction of beha- (1985). viorsin the quasilinearregime. Specifically, for a givenradi- •9B.G. Lucasand T. G. Muir, "Fieldof a finite-amplitudefocusing source," J. Acoust. Soc. Am. 74, 1522-1528 (1983). ally symmetricsource, a linearbeam pattern can be calculat- •øS.Saito, B.C. Kim, and T. G. Muir, "Secondharmonic component of a ed by conventional means. Then, the magnitude of nonlinearlydistorted wave in a focussedsound field," J. Acoust.Soc. Am. harmonic, beamwidths, and sidelobepatterns can be esti- 82, 621-628 (1987).

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2348 J. Acoust.Soc. Am., Vol. 86, No. 6, December1989 C.R. Reillyand K. J. Parker:Finite-amplitude effects 2348