Mechanical characterization of microparticles by scattered ultrasound.

Mechanical characterization of microparticles by scattered ultrasound RonaldA. Roy•) andRobertE. Apfel Department ofMechanicalEngineering, YaleUniversity, Box2159 YaleStation,NewHaven,Connecticut 06520

(Received23 April 1989;accepted for publication11January1990)

A technique for determining thecompressibility anddensityof individualmicroparticles in suspension is described. Theparticles havediameters ontheorderof 10pro. Ultrasonictone burstsof 2-psdurationand30-MHz centerfrequency scatterfromindividualparticlesasthey traversethe confocalzoneof two transducers. The resultingscattered toneburstsaredetected at 90øand 180ø(backscattering). The receivedrf signalsaredemodulated, peakdetected, digitized,andstoredin computermemory.UsingRayleighscattering theory,the compressibility anddensityof a particlecanbecomputed givenknowledge of theparticlesize andhostfluidproperties. Resultsof experiments with latexmicrospheres arepresented and compared with calculations basedonlong-wavelength (Rayleigh)andelasticscattering theory. PACS numbers:43.20.Fn, 43.35.Bf, 43.35.Yb

INTRODUCTION

The task of performingrapid and quantitativemicroparticle analysishas attracted considerableattention in a numberof areas.Many industrialapplications,suchasmultiphaseflow monitoringand quality control,are well suited to microparticledetectiontechniques.Viability of very pure processes canbeestablished throughdetectorsthat are sensitive to particulatecontaminationor precipitation.In cases whereeitherliquid or solidparticleformationis part of the manufacturingprocess,particle analyzerscan yield realtime information useful to the control process,provide a mechanismfor qualitycontrol,or playa rolein productevaluation.

Biological applicationsfor cell characterizationand separationtechnologiesare numerousbecausechangesin optical and/or mechanicalpropertiesof cellsmay indicate

surementscan be performedby measuringthe densityof a suspension of a knownnumberof particlesof knowndensity, or by usingoptical techniquessuchas photon correlation

spectroscopy. • Density-gradient centrifugation provides informationregardingthe mean densityof a particlepopulation and facilitatesthe separationof the variousconstituents

of a mixedpopulation. 2 Measurements of the bulk elastic propertiesof microparticlesis limited primarily to acoustic techniquessuch as suspensionsound velocity measurements3 and acoustic levitation. 4

Single-particleproceduresoffer the advantageof being ableto establishdistributionsof propertiesfor a population of particles. In some cases,two or more measurement schemes canbesimultaneously employed,resultingin multi-

parameter characterization. 5Therearesomedisadvantages,

however.Single-particletechniquestendto beexpensiveand are susceptibleto signal-to-noiselimitations. Also, depending on the application, particle handling may prove probresolvingmixedcell populations into distincthomogeneous lematic.Individualparticlesizingcanbe accomplished optidisease. Researchers are often faced with the difficult task of

populations.Automatedcell analyzersandsorterscan processlargenumbersof cells;this facilitatesthe harvestingof largequantitiesof purepopulationsandpermitsoneto monitor or identify very small subpopulations within a heterogeneousparent distribution.

In this paper,we describea procedurethat is amenable to in vitro cell characterization. In the context of our work, "microparticlecharacterization"refersto the determination

of the volume,density,andadiabaticcompressibility of uniform, sphericalparticlesin suspension. Suspendedmicroparticlecharacterizationtechniquescan be subdividedinto two classes: multiple-particleand single-particle characterizers.

Multiple-particletechniques determinemeanproperties averagedover an ensembleof particles.Ensemblesizemea-

Currentaddress: NationalCenterforPhysical Acoustics, P.O.Box847, University,Mississippi38677.

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J. Acoust. Soc.Am.87 (6),June1990

cally via microscopyor forward laser scattering. 6 Electrozone sensing instruments 7providea convenient technique for sizing and counting large numbersof particles. Neither the opticalnor impedancetechniquearesensitiveto mechanicalproperties.Indeed, all of the commonly used proceduresfor measuringdensity and compressibilityare multiple-particletechniques. We havedevelopedan acousticscatteringtechniquefor quantitatively determiningthe density and adiabaticcompressibilityof large numbersof individual microparticlesin suspension.Relevant particle sizes, which for our work rangefrom4pm to about15pm in diameter,canbe adjusted by proper scalingof the acousticfrequency.Narrow-band, ultrasonictoneburstsscatteredfrom individualparticlesare detectedat two angles. Using either long-wavelengthor weak-scatteringtheory, we can calculate the mechanical propertiesof an individual particle, providedwe possess a priori knowledgeof the particle size. We can subsequently

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2332

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computemeanpropertiesfor a homogeneous populationor determinepropertiesfor subpopulations within a heterogeneous distribution.

The apparatusdescribedherein is a modificationof a

system developed byRoos.8'9Thepurpose of thispaperisto describethe techniqueand the underlyingtheory.We present resultsof experimentsperformedon polystyrenemicrospheres,whichwe useascalibrationparticles.Measurements made (usinga weak-scatteringapproximation) on biological particlessuchasChinesehamsterovarycellsandhuman granulocyteswill be presentedin a later paper.

B. Scattering from an elastic sphere in a fluid host

Supposenow that the scatterer is a uniform elastic sphere.A straightforward extensionof the normalmodeexpansiontechniqueyieldsthesolutionto theelasticscattering problem.One definescompressional and shearwavevelocities (c',c•) in termsof the Lam• constants(A ',t•') for the solid obstacle.Theseare givenby

c• = (/t'+ 21a')/p', c•2=la'/p'.

(3)

The displacementin the scatterer is the sum of both compressional and shearwavedisplacements. Two independentwaveequationsresult:scalarand vectorvelocitypotenI. SCATTERING FROM SPHERICAL PARTICLES: AN tial equationsfor the compressional and shearwavecompoOVERVIEW nents, respectively.Add a third boundary condition,the When a soundwave travelingthrougha homogeneous vanishingof the tangentialstressat the surfaceof the scatterer, and the subsequent calculationsare completelyanalomediumencountersa local inhomogeneity, a portionof the gousto the fluid case(the readermay chooseto referto the waveis scatteredin all directions.The resultingwaveis the excellent articlesby Faran,I I Hickling,•2andHay andBurlsuperposition of the incidentand scatteredwaves.The naing13for furtherdetails).The resultingexpression for the ture of the scatteredfield dependson the acousticalproperties of the scatterer and the host material, the acoustic fre-

quency,and the sizeand shapeof the scatterer.In the most generalcase,the host and the scattererwould admit the propagationof both compressional andshearwaves,the relevant boundaryconditionsbeingthe continuity of normal stress,normal velocity,tangentialstress,and tangentialvelocity acrossthe surfaceof the scatterer.For our purposes, we shall considerthe scatteringfrom both fluid and elastic spheressuspended in a fluid host.Thus the tangentialcomponentsof the velocityand stressfieldsvanish. A. Scattering from a fluid sphere in a fluid host

We considerfirst the caseof harmonicplane wavesof amplitudePoand angularfrequencycopropagatingthrough an infinite,inviscidfluid of densityp and compressibility •c. Thesewavesare incidentupona homogeneous, inviscidfluid

sphereof radiusa, densityp', and compressibility •c'.The stressfields outside and inside the scatterer satisfy linear compressional waveequations,wherethe relevantboundary conditionsare the continuity of normal stressand normal velocity acrossthe interface. Using eigenfunctionexpansions,Andersonløobtainedan exactsolutionfor the scattered wave. The farfield scatteredpressureamplitudePs is givenby

Ps(r) = [ (Po/r)eikr]•, for r>•a,

(1)

where

farfieldscattering strengthis13

-•- • (2m d-1)sin(•/m )exp( --i•lm )Pro (COS 0). m=0

(4)

An expression for the phaseanglefor the ruthnormalmode •/m is givenin Ref. 13. C. The long-wavelength approximation

Equations(2) and (4) are applicablesolutionsto the directscatteringproblem.However,the taskof inverting• to yieldthe scattererpropertiesgivenknowledgeof the host propertiesand the incidentand scatteredfieldsmustbe carried out numerically,an approachthat entailssubstantial computationaloverhead.We requirea readily computable solutionto the inversescatteringproblem,therebyfacilitating real-time data analysis.Consider,therefore,the case where the radius of the fluid scatterer is much less than the

wavelengthof soundin the host.To lowestorderin ka, only the m = 0 and 1 termsin the expansionare retainedand Eq. (2) reduces to

(I)= (1/4•) (k2V) (tSKd-tSpcos0),

(5a)

whereVis thescatterervolume.The functions& and6p are the compressibility and densitycontrastsgivenby

&:=(n:'-n:)/n:,

and 6p=3(p'-p)/(2p'+p).

(5b)

Equation(5) is the Rayleighresultfor long-wavelength scattering fromfluidspheres. 14Themonopole termdepends

the compressibilitycontrast and correspondsto a •___/•(- ])r•(2m +])P.,(cos 0) (2) on"breathing mode"oscillation,in whichthe spherepulsates ß

k

=o

(1 d-iCm)

In this expression,r is the radial distancefrom the centerof the scattererto the field point, k is the wavenumberin the hostfluid,thePmarethe Legendrepolynomials,and 0 is the scatteringangle (0 = 180ø for backscattering).The coefficientsCm,which resultfrom matchingthe boundaryconditions,aregivenin Ref. 10.Note that Pstakeson the form of a sphericalwave modulatedby an angulardistributionfunction (I), which is a function of co,0, a, and the scatterer and

radially.The dipoleterm,whichdepends onthedensitycontrast, resultsfrom the motion of the center of massof the scatterersalongthe axisof the incidentwave. A similarargumentcanbemadefor theelasticscatterer. To lowest order in ka, Eq. (4) also reducesto Eqs. (5), wherethe bulk compressibility of the solidis givenin terms of its Lain6 constantsby

•c'= (,4' d- 2/•'/3) -l.

(6)

host properties.We shall alsorefer to (I) as the "scattering strength."

In the Rayleighlimit, the particleistoo smallto supportany

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R.A. Roy and R. E. Apfel:Characterizationof microparticles

J. Acoust.Soc. Am., Vol. 87, No. 6, June 1990

2333


of the shearwave resonancemodes.Thus an elasticsphere canbe modeledby a fluid spherewith an effectivecompressibility givenby Eq. (6). Inverting Eqs. (5) to obtain the scattererpropertiesis straightforward. Let q•9oandq•18o bethescatteringstrengths measuredat 0 = 90øand 180ø,respectively.From Eq. 5 (a)

30 MHz Tone Burst

Generator

PmOW½ r l Backscattered [ana Peak Detector !

we solve for the contrasts:

plifier

Tone Burst

Preamp

•Sn' = 4rr(I)9o/(k

Nozzle Switch

and

To 90' Detection

{5/9 = 4r/'((I)90-(I)18o)/(k2I/3,

(7a)

Channel

which can be combinedwith Eq. (5b) to yield the particle properties:

tc'=tc(1 +6re), and p'=p[(3+•p)/(3--2•p)].

Jet Flow

(7b)

Con focal

Thus, in the long-wavelengthlimit, we can determine the compressibilityanddensityof the scattererby measuringthe scatteredsignalat 90øand 180ø (or any two anglesfor that matter) providedwe possess a priori knowledgeof the host compressibilityand densityaswell asthe scatterersize. II. EXPERIMENTAL

Region

Transmitter/180 øReceiver

90ø Receiver

(Channel A)

(Channel B)

Sink

APPARATUS

Figure 1 is a block diagram of the apparatususedto measurescatteringstrengths.Individual particlesare convectedthroughthe confocalregionof two submergedtransducersby a coaxialjet flow. High-frequency,narrow-band tone burststhat scatterfrom a particleare detectedat 180ø

FIG. 1. Block diagramof the microparticlecharacterizationapparatus ( 180*detectionchannelonly).

(channel A) and 90ø (channel B). The received echoes are

demodulated,peak detected,digitized, and storedin computer memory for subsequent processing.The traversalof a single particle (particle event) resultsin severalscattered echoes(scatteringevent). The objectiveis to determinethe particle scatteringstrength,which is the peak echo amplitude producedby the particleevent.

X-Y-Z

ISideViewI

X-Y-Z

Positioner

A. Transducers and particle handling Lucite Tank

We use matched, unbackedtransducersthat incorporate a 1.5-cm-diam,3.0-cm focal length, machinedstyrene lensbondedto a 0.95-cm-diam lithium niobatecrystal. The 3-dB beamwidthis about 275 •tm (measuredin the focal planeusinga 10¾tm-diamcylindricaltargetfashionedfrom

e

X-Y-Z

ssembly

Positioner

(transducer A)

a quartzfilament).The sidelobes are approximately15 dB down from the central maxima. We choose a 30-MHz

reso-

nancefrequencyin order to maximizeacousticsensitivity without severelycompromisingthe long-wavelengthscattering assumptionfor biologicalparticlesup to 10 •tm in diameter. 15(The valueofka abovewhichtheapproximation failsdependson the materialpropertiesof the scattererand the host.For example,the long-wavelength assumption fails at low ka for high-contrast scatterers.)Narrow-bandsignals are desirablesincethey facilitatebroadbandnoiserejection. A 2¾ts-longtoneburstyieldsa 500-kHz signalbandwidth

Alignment Microscope (transd_uc•r B)

1 mmmmm

X-Y-Z

Positioner

(jet assembly)

centered at 30 MHz.

Figure2 illustratesthe mechanicalsystemfor positioning the transducersand the jet flow assembly.Each transduceris held in placeby a Plexiglassupportarm that is affixed to an X-Y-Z micropositioningstage.A third X-Y-Z micropositioner is usedto orientthejet assemblysothat the particleflow passesthroughthe confocalregionof the two

FIG. 2. Transducermountingand alignmentsystem.

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R.A. Roy and R. E. Apfel: Characterizationof microparticles

d. Acoust.Soc. Am., Vol. 87, No. 6, June lgg0

ITop View

l/

AlignmentMicroscope (transducer A)

ll

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Needle Flow

Nozzle Flow

(syringepump)

(17.5cmhead)

SyringeFilter

fine streamof particlesdown the centerof the jet proper ( nozzleflow). The glassnozzle,whichwasdrawnfrom stock 3-mm tubing,hasan insidediameterat the exit of approximately270tim with a wall thicknessof 100tim. A 1-[glass reservoirprovides 17.5 cm of head, which yields a nozzle

flowrateof about0.017cm3/s.Thenozzleisapproximately

Glass Nozzle

cylindricalovera distanceof at least50 diameters,resulting in a fully developedPoiseuilleflow. The acousticconfocal regionis located0.37 cm downstreamfrom the nozzle tip. For a pure water jet, the Reynoldsnumber in the confocal region is about 100 based on the observedjet diameter ( • 400 tim) and centerlineflow velocity ( •24 cm/s). A sink collectsthe jet far downstreamof the confocalregion and depositsit into a reservoiron the floor. Suspended particlesare injectedinto the needleflow at a point far upstreamfrom the hub of the 30 gaugestainless steelneedle(i.d. = 150tim). The needleflow,whichis driven by a syringepump and passesthrougha 0.2-tim syringe

Rod

filter,hasa flowratethatistypically0.0002cm3/s.Byinject-

SupportArm

Insert Particles

30 GaugeNeedle SupportRod

30 G. NeedleTip

18.5mm

Not toScale i

11.9mm

Sink Flow

Sink

Reser voit

FIG. 3. Coaxialjet-flow assemblyusedto convectparticlesthroughthe acousticconfocalregion.

transducers.Both transducersand thejet assemblyare submergedin thehostliquid,whichishousedin a Plexiglastank suspended beneaththe mountingplate. To align the system,we employa 28-ttm-diamhollow

ing water-solubleink in placeof particles,we caneasilyvisualize the needle flow streamlines.These appear laminar alongthe entirelengthof thejet and havea diameterof about 40 tim in the planeof the transducers. The choice of nozzle and needle flow rate is determined

by experimentation. To estimatethe amplitudeof the scatteredsignal,we mustensurethat a sufficientnumberof scatteringeventsareproducedper particleevent.We selectflow ratesthat yield the fastestjet possible,while ensuringat least ten scatteringeventsper particlegivena pulserepetitionfrequency(PRF) of 4 kHz. Acousticscatteringfrom thejet is negligiblesincethe sameliquid is usedfor both thejet and the host.

B. Signal generation and detection

Figure 4 illustratesthe instrumentationusedto generate the transmitted

tone bursts as well as the sinusoids cos cot

glasssphereadhered to a 12-ttm-diamquartz filament stretchedacrossa Plexiglasbracket.Alignment consistsof first positioningthe target to maximize the backscattered signal,and thenmanipulatingtransducerB to maximizethe 90ø echo. The two transducersare thus confocallypositioned,with the target situatedin the centerof the confocal region.The target locationis "tagged"usingthe crosshairs of two filar micrometermicroscopes, after which the target is removedand the jet assemblyput in place.Using ink to visualizethe particle-carryingportion of the flow, the jet assemblyis positionedso that the particle flow streamlines passthroughthe confocalzone.This procedureis repeated prior to everyexperiment. Particlesstudiedin this work are convectedthroughthe confocalregionby the jet flow arrangementillustrated in Fig. 3. This systemservesthree purposes:(a) to constrain the particlepath to a very fine streamthat can be precisely positioned;(b) to control the averagenumber of particle eventsper second;(c) to regulatethe particle convection velocitythrough the confocalregion,which in turn determinesthe numberof scatteringeventsper particleevent. Particlesenter the jet via a needlepositionedslightly upstreamof thenozzlecontraction.The subsequent convergenceof the needleflow streamlinesresultsin an extremely

FIG. 4. Blockdiagramof the circuitrythat deliversamplifiedtoneburststo the transmitsignalinput of the T/R switch.

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R.A. Royand R. E. Apfel:Characterization of microparticles

J. Acoust.Soc.Am.,Vol.87, No.6, June1990

and sin cot,which are employedby the rf demodulatordescribedbelow.The 2¾ts-long, 20-mA currentpulseactivates the doublebalancedmixers (DBMs) that amplitudemodu-

Variable Attenuator

To T/R Switch

RF Power

Amplifier

...•0 mA SystemTime Base

To Oscilloscope ExternalTrigger cos COt

I I ]

14Way, 0øSplitter I

To Quadrature Demodulator

cosO)t • sin

I 2Way, 90 øSplitter

I Function Generator I I

2335


Transmit SignalChannel A.!"i•p' F Generation

A(t)sinq•

Transducer

[

A(t)cosq• S&H Gate

Sample& Hold Amplifier(34 dB)

BackDiodeNetwork

Sample& Hold Amplifier(34 dB)

I i

QuarterWave Transmission Line

2 Way, 0ø Splitter

i i Received Tone Burst

i

L2

A(t)cos(•t + q•)

i

•F I

To Channel B

i i

I

I

._l-L60mA

I

[ System Time Base I

'I

AnarogSquare

cos •t

I

LSI 11/23 Computer

I Emodulaor sin •t

I

I

I

with 12 bit ADC

14Way. 0ø splitter I 14Way. 0ø Splitter I Oscilloscope

I 2Way, 90 øSplitter I

I_l-L? •n^

I T/RSwitch

TriggerIn

I Function Generator I

DelayedTrig Out

System Time Base

Channel B Transducer

Demodulators +4.7dB

FIG. 6. Blockdiagramof the quadraturedemodulator andsignalreconstructioncircuitry.Only the channelA systemis shown(the channelB configuration is identical).

FIG. 5. Blockdiagramof theT/R switchandthereceivedsignalpreamplification circuitry.

late the continuous-wave input signal.This resultsin a 2-kts toneburstthat is amplifiedand fed into the transmitsignal input of the transmit/receive(T/R) switch. The T/R switch,which is shownin Fig. 5, is basedon a

designby Ridpath!6andexploitsPIN diodeandtransmission line technologyto provide rf switchingwithout mechanical relays. PIN diodes are high-speed,current-controlled devicesthat have typical "on" and "off" resistances of 0.75 12and 50 k12,respectively. In the transmit mode both PIN diodesare activatedby the 60-mA dc currentsignal.When turnedon, P2 effectively shortsthe outputof the transmissionline, which is reflected backto the transducersideasa high impedance.This results in an rf signalpath throughP 1 and into the transducer.The transmitmodeisolationbetweenthe poweramplifierand the preamplifierismeasuredto be 34.3dB, andthe lossfrom the power amplifierto the transduceris 2.7 dB. In the receivemode,the current signalis turned off and both P 1 and P2 ceaseto conduct, which isolates the trans-

ducer from the power amplifier and providesan rf signal path leadingfrom the output of the receivingtransducer, throughthe transmission line, andinto thepreamplifier.The receivemodelossfrom the transducerto the preamp is 1.2 dB. Back-to-backdiodepairshelp reducebroadbandamplifier noisethat leakspastP 1. Receivedtransducersignalamplitudesare on the orderof microvolts:thusverysmallnoise levelscan corrupt the signal. Also shownin Fig. 5 is the circuitryusedto preamplify 2336


the rf signalsfrom eachdetectionchannel.The channelA systemutilizes a DBM configuredas a current controlled attenuator,which servesasa backscatteredsignalgate.This gateis requiredto keepthe main bangleakagethroughthe T/R switchfrom saturatingthe channelA demodulator.In order to recoverthe 3.5-dB insertionlossthroughthe DBM aswell asthe 1.2-dBlossthroughtheT/R switch,we include an additional4.7 dB of gain.This yieldsan approximategain balance between the two channels.

C. Analog signal processing and computer interface

The task is to determine the amplitude of a scattered toneburstwithout any informationregardingits phase.We employa quadraturedetectionscheme,in which the scattered tone burst is treated as an rf carrier signalmodulated by a low-frequencyenvelopefunction, •(t)cos(cot + co),

wherecois the unknownphaseof the receivedsignal.The uncertaintyof the phaseis a consequence of the uncertainty in the particleposition;it is impracticalto constrainthe particle trajectorywell enoughto specifythe phaseto within a fractionof 2• rad. We ultimatelyseekthe amplitudeof the envelopefunction,•' = I(t)I, whichis proportionalto the scatteringstrength,(I). Figure 6 showsa blockdiagramof the quadraturedemodulator. Since each channel utilizes identical demodula-

tots, only oneis describedhere.The signalfrom the preamplifierstageis first split and thenmixedwith the sinusoids coscot and sincot. The resulting signals, •'(t)[cos

(2cotd-co)d-cos(co) ] and •'(t)(cos[2cotd- (cod-90")] d-sin(co)}-,arefedintoa pairoflinearphase, 500-kHzlowpassfilters.This yieldsthe detectedquadraturecomponents, R.A. Roy and R. E. Apfel: Characterizationof microparticles

2336


g(t) cos( c,o ) andg(t) sin( c,o ). Usingsampleandhold amplifiers,we peakdetectthe quadraturecomponents, producing the dc signalsg cos(c,o) andg sin(c,o).We thensquarethese signals,sumthe products,and take the squareroot of the sum.The resultingscatteredsignalamplitudeg is digitized into a 12-bitwordandstoredin computermemory.A pair of analog-to-digitalconversions(first channelA, then channel B) is triggeredby the oscilloscope delayed-triggeroutput

$i -q-n• = A fe(t -- Ti ) COS(cot + • ) + t/i.

inga stationary noiseprocess. Thedataacquisition program

A = VtGr(O)G• (0) ap= Gap,

(10)

The sequencef =f(zi), which we term the "combined transducerbeamprofile,"represents the normalizedspatial dependence of the productof the gain functionsand is assumedto bethe samefor all particleevents.Containedwithin the f is all the information concerningthe transducer beamwidths,the pulserepetitionfrequency,andthe particle convectionvelocity.The quantityA is the amplitudeof the pulse. Data aretakenin 2500-pointblocks,whichcorresponds largestechogeneratedin the courseof a particleevent.This maximumsignalconditionoccursasthe particlecrossesthe to 0.625 s of data at a PRF of 4.0 kHz. Also obtained are planeof thetransducers. We callA the eventamplitude.It is statisticsregardingthe dc offsetsand noisevariancesfor given in terms of the on-axisgain functions,the transmit eachchannel.Thesevaluesarecomputedby taking512 data pointswhilethejet isrunningwithoutparticles whileassum- signalvoltage,and the scatteringstrengthby then calculates the means and variances for both channels and writes these statistics into the header for each data file.

The programalsooffersa "previewmode,"in whicha 512-pointdata streamis displayedon a graphicsscreen, erased,and rewritten with a freshstream.This processcon-

tinuesuntil the operatordecidesto commencerecording data.Thisfeaturehelpsreducethedetectionof multipleparticle events,which resultwhenthe particleconcentrationis too high,a conditionwhichexistsimmediatelyafterinjection of particlesinto the needleflow.Eachexperimentinvolvesthe acquisition of at leastten blocksof data,which typicallycorrespond to about1000particleevents.

D. Digital signal processing When a receiverdetectsa scatteredtone burst, it pro-

ducesan electricalsignal.In theabsence of noise,theamplitude of the receivedsignalis givenby Vr = VtGr(z)GR (z) ap,

(8)

( 11)

whereG istermedthe detectionchannelgain.The scattering strength(and thereforethe particle properties)can be inferred from A, oncewe have determinedthe channelgain throughcalibration. For a weaklyscatteringparticle,the precisedetermination of A is complicatedby the presenceof noise.Roosderived a maximum likelihoodestimatorfor the event amplitude s

Am = •;f2(r/2) •;f •' --Z Nø/•2 2•;f,.•

(12)

whereNo/2 is the varianceof the noiseand•'i is the quadrature detectedamplitudeof the ith scatteredtone burst. To implementthe estimator,we take the sequence of • that makeup the recordeddata streamand computethe amplitudeestimateat eachtimestepusingEq. (12). Eachparticle event is representedby a singlepeak in the resultingse-

quence ofAm. Themaximum valuefora givenpeakApisthe maximum likelihood estimatefor the correspondingevent amplitudeA. Note that we requirea priori knowledgeof the noisevariancesand the combinedtransducerbeam profiles for each channel.We determineNo/2 prior to running an experimentby measuringthe scatteredsignalin the absence of particlesand, usinga techniquedescribedin Ref. 8, we obtainthef by measuringthe scatteringfrom a highlymonodispersepopulationof strongscatterers.

whereVristhereceivedvoltageamplitude,Vtisthetransmit voltageamplitude,andGr andGRarethegainfunctions for the transmittingand receivingchannels,respectively. The z axispasses throughthe centerof the confocalregionand is normal to the plane of the transducers(at which point z = 0). In writingEq. (8) weassume that particletrajectoriesdonotdeviatesignificantly fromthispath.The gainfunctions accountfor transducerfocusing,electromechanical E. Determination of particle properties sensitivity, attenuationin thehostliquid,andthegainof the In orderto characterizea populationof unknownpartipreamplification andanalogsignalprocessing circuitry. cles,we mustfirst calibrateby runningknown particlesand As theparticletraverses theconfocalregion,it produces the meanvalueof the measured Ap. Usingan a seriesof echoes.The ith detectedecho can be represented computing appropriate scattering theory and the calibration particle as a tone burst summed with zero-mean, white Gaussian properties, we then compute the mean scattering strength noise. This is written as (•). This resultsin the followingestimatefor the channel si -i- ni = (Vr)ie(t-- ri) cos(cot-i-•i) -i- ni, (9) gain: wheree( t - 7'i ) is a rectangulartoneburstenvelope(of unit amplitudeanddurationT), ri is the sampletime, and cois the tone burst center frequency,all of which we know a priori.Thec, oirepresent randomphaseshiftsthatresultfrom slightdeviationsin the particlepath. Sincewe are probing the particleat discretetime intervals,the spatialcoordinate of the scattereris treated as a discretevariable given by zi =riu, whereu is the particleconvection velocityin the confocalregion.CombiningEqs. (8) and (9) yields 2337

J. Acoust.Soc.Am.,Vol.87, No.6, June1990

Go = (Ao)/(apo),

(13)

where0 denotes thescattering anglefor thedetectionchannel underconsideration (we drop the subscript p for clarity).

We nowhavesufficient a prioriinformationto calculate theproperties of a population of particlesprovidedwepossessa relevantandinvertiblescatteringtheoryin additionto

knowledge of thehostfluidproperties andtheparticlesize R.A. Royand R. E. Apfel'Characterization of microparticles

2337


statistics.Consider,for example,a populationof Rayleigh scatterersof known mean volume.By combiningEqs. (7a) and (13), we canexpressthe meancompressibility and density contrastsin terms of the mean event amplitudes,the channelgains,and the meanparticlevolume,giving

(&) = 4•r•(,•o)/•o)/•z•

for bulk polystyrene.This providesjustificationfor adopting the publishedcompressional and shearwavespeeds for bulk polystyreneascharacteristicof PDVB spheresaswell. III. EXPERIMENTAL

•( •') )

and

cm2/dyn,whichis in remarkable agreement with thevalue

(14)

(6•O)-- 47T[ ((A90)/6•90) -- ((A,80)/(•,80)]/(k2{g)), which can be combinedwith Eq. (7b) to yield the mean particleproperties.In writing Eq. (14) we assumedthat

(&:) cr(V) - • asopposed to ( V- •). Thisisa goodapproximationprovidedthe particlesarenot broadlydistributedin volume.

PROCEDURE

A. Experiments with PDVB microspheres

To testthishypothesiswe measuredthe scatteringfrom individual suspensions of monodisperse(mean diameters rangedfrom 4.9 to 14.6/•m) PDVB spheres.We thencomputed,for eachsample,the meanparticlepropertiesaswell as a farfield, "relative" differentialscatteringcrosssection givenby

ao= lq,olV(rra --

(15)

Experimentalcrosssections werethencomparedto theoretical calculations based on the elasticpropertiesof bulk polyEffectiveimplementationof our particle characterizastyrene. Since the 4.9-/•m spheres werepreviouslycalibrated tion techniquerequirespreciseand accuratecalibration.In against a reliable standard (blood cells), they wereusedto additionto beingwell characterized,idealcalibrationparticalibrate the system. clesshouldbe spherical,monodisperse in size,chemically Table I givesthe meandiameterfor eachof the PDVB and osmoticallystable,easyto handle and store, inexpenpopulations employed,as well as the estimatederror and sive,plentiful,and shouldlast forever. precision of the mean.The manufacturer(Duke Scientific, A possiblecandidateis the human red bloodcell. Much Palo Alto, CA) sizes a 100-pointsampleusinga light microattentionhas beendevotedto the study of the mechanical scope calibrated against an NBS stagemicrometer.In addiproperties ofthesecells, 2-4andtheseproperties exhibitunition, the vendor runs the spheresthrough a Coulter model formity within a donorsampleas well as from onehealthy ZB particleanalyzer,which generatesdiameterand volume donorto the next. The meancellularvolumeis easilymeasuredusingautomatedcell sizersor bulk methods.Typical statisticsfor a 10 000-point sample.All diametersgivenin valuesfor meancellularvolume,compressibility, and den- Table I are the electricallydeterminedvalues,which we consity are 95 •m 3 (whichcorresponds to a 5.66-/•m-diam sideredto be moreprecisesincethe meaniscalculatedovera sphere),3.42X 10- 11cm2/dyn,and 1.099g/cm3, respec- much larger sampleof the parentpopulation.Statederrors correspondto the differencesbetweenthe opticallyand electively.However,therearenumerous drawbacks to usingred bloodcellsas calibrationparticles.They are nonspherical, trically determinedmeans.As a measureof precision,we theirproperties changewith age,theyexistin a limitedrange usedthe standarddeviationcomputedby the particleanaof sizes,and theyare chemicallyand osmoticallysensitive. lyzer. Listed in Table II are the estimatedmechanicalproperPolystyrene divinylbenzene (PDVB) microspheres ties for the PDVB spheres,usingthepolystyrenedatareportprovideuswith anotherchoiceof calibrationparticle.These ed in the literature. The stated PDVB densityerror represpheresare relatively monodisperse,come in a variety of sentsthedisagreement amongthepublishedvalues.Error in mean sizes,and are chemicallystableand osmoticallyinsenthe compressibility corresponds to the differencebetween sitive.The manufacturerprovidessizeand densityinformathe measurements performed by Roos, andthecomputation tion, which can also be measureddirectly using standard using the published polystyrene data. We lacksufficient intechniques. Usinga density-gradient method, 9 Roosfound formationto infer the error associatedwith the wavespeeds. thedensityof PDVB spheres to be 1.049g/cm3,whichis in Alsolistedin TableII arethe propertiesof distilled,degasseal close agreementwith the manufacturer'sdata for both water that we used as a host fluid. The density and sound PDVB andpolystyrene spheres ( 1.05g/cm3), andwithbulk velocity (from which we obtainthe compressibility)were polystyrenedensitydata reportedin the literature ( 1.05 g/ cm3). 14 Determiningthe elasticpropertiesof PDVB spheresis anothermatter.Compressional andshearwavevelocitiesfor TABLE I. Mean sizesfor eachPDVB suspension usedin this studyalong F. Calibration particles

bulkpolystyrene canbefoundin theliterature. 13'i7'i8 Using the reported values, C=2.380•105 cm/s and Cs= 1.100X 105cm/s,alongwiththepublished polystyrene densitydata,wecompute[ usingEqs.( 3) and (6) ] thecom-

pressibility ofbulkpolystyrene to be2.35X 10- I• cm2/dyn. Roos measuredthe acousticscatteringfrom 5-/•m-diam PDVB spheresin isotonicsaline(ka = 0.31, wherea is the radiusof the sphere),after havingcalibratedthe apparatus with humanredbloodcells.9 Usingthe Rayleighmodel,he

with estimatesof precisionand error. Mean diameter

(tim) 4.9 + 0.5 5.9 + 0.5 7.4 + 0.6 9.7 q- 0.2

10.5 q- 0.9 14.6 q- 0.4

Estimated

error

(%) + + + +

3.0 1.0 6.0 1.2

+__ 8.6 __0.2

computed a meanPDVB compressibility of 2.34X10-• 2338

d. Acoust.Soc. Am., Vol. 87, No. 6, June 1990

R.A. Roy and R. E. Apfel:Characterizationof microparticles

2338


TABLE II. Mechanicalproperties of PDVB spheres andthehostliquid. PDVB microspheres

Compressibility Density Compressional wavespeed Shearwavespeed

2.34X 10-II cm:/dyn 1.049g/cm3 2.380X l0scm/s 1.100X 10scm/s

1.5% and 4.0%. ___0.4% ___0.1%

Distilled water (temperature= 23 ñ 2 øC)

Compressibility Density

4.51X 10-II cm2/dyn 0.9976g/cm3

eventamplitudeby computingthe standarddeviationof the samplemeansthemselves. This valuegenerallyfell between

ñ0.7% ñ 0.07%

measuredusinga Mettier-Parr DMA 40 digitaldensitymeter and a Nusonics 6105 sonic solution monitor. Tabulated

errorscorrespondto measuredchangesin the hostproperties that resulted from variations in room temperature, which weretypically 23 _ 2 øC. When performingan experiment,calibrationparticles were run first. A dilute PDVB particle suspension was obtainedby dispersing onedropof thevendor'ssuspension into about0.5 mFof hostliquid, and introducedinto the apparatus by injecting•0.1 mFof this into the needleflow. Using thedataacquisitionprogramin thepreviewmode,we monitoredthe spheres astheycrossed theconfocalregion.When the particle event statisticsappeared acceptable (many events with few multiple peaks), data acquisitioncommenced.Usingelasticscatteringtheory [ Eq. (4) ], the measuredmeanevent amplitudesfor eachchannelwere combinedwith the PDVB and hostpropertiesto yieldthe channel gains. Becauseof the limited dynamicrangeof the system,we could not run all particlesizesat the sametransmitsignal level. We took data with the 4.9-, 5.9-, and 7.4-/•m-diam spheresat a transmitlevel of 40 V peak-to-peak,and then reducedto 15 V peak-to-peakin orderto run the 7.4-, 9.7-, 10.5-,and 14.6-/•mparticles.Using the 7.4-/•m results,we

When calculatingquantitiesof interest(crosssections, particleproperties,etc.) we propagateduncertaintiesby using a standardperturbationtechnique,where the relevant equationswere expandedin a Taylor seriesto first order in the perturbedparameters.It was assumedthat the fluctuationsin the eventamplitudemeasurements at differentangles were uncorrelated.The coefficientsin the expansion were calculated using the assumedphysical parameters (calibrationparticleand hostproperties,etc.) and the computed mean eventamplitudes. IV. EXPERIMENTAL

RESULTS

A. Particle properties

Plottedin Fig. 7 are the theoreticalscatteringcrosssectionsfor PDVB spheresin water versuska usingthe Rayleigh approximationas well as the exactsolutionfor elastic scatterers.Also plottedare the crosssectionscomputedusing the experimentalresults,wherethe error barsdenotethe precisionof the calculation.The closeagreementbetween the measuredcrosssectionsand thosepredictedby the exact

0.0015

0.0012

0.0003

were able to normalize the 15-V data to the 40-V data. From

0.0

0.2

0.4

thescattering dataforeachsphere size,wedetermined the mean eventamplitudesand generatedhistogramsof event amplitudestatistics.Usingthe Rayleighapproximation,the meanparticlepropertieswerecalculatedfor all of the PDVB samples. Finally, we computed the mean scattering strengthsfor each sample,from which the scatteringcross sectionsfollowedusingEqs. (2)-( 8 ).

0.6

0.8

Size Parameter(ka)

0.010

Elastic

ß

o...........

j_

•

mean for each block and found that the variation of the sam-

ple meanswasgenerallyequalto abouttwo standarderrors computedover all of the events.This is consistentwith observationsmadeby Roos,who attributedthe discrepancyto gradualfluctuationsof the positionof thejet. We therefore chose,as did Roos, to estimatethe precisionof the mean 2339


11

Rayleigh .................. 11_.

//

B. Estimates of precision

Experimentalprecisionwas basedon the precisionof thecalibrationparticlesizeestimateandthemeasuredprecisionsof the meaneventamplitudescomputedfrom both the calibrationdata and the unknown particle data. Every experimentyieldedtenblocksof data,eachof whichaccounted for roughly 100 particle events.We computedthe sample

1.0

0.000

0.0

0.2

0.4

0.6

0.8

1.0

Size Parameter(ka)

FIG. 7. A comparison of the measuredscatteringcrosssections for PDVB spheres in waterwithpredictions basedonelasticscattering theoryandthe Rayleighapproximation. R.A. Roy and R. E. Apfel:Characterizationof microparticles

2339


TABLE III. Measuredmean compressibilityand densityfor each PDVB suspension. The apparatuswascalibratedusingthe 4.9-pm-diamspheres usingRayleighscatteringtheory. Diameter

Compressibility

Density

(pm)

( X 10--II cm•-/dyn)

(g/cm3)

4.9 5.9 7.4 9.7 10.5 14.6

(ka (ka (ka (ka (ka (ka

= 0.30) =0.37) = 0.46) = 0.60) = 0.65) = 0.90)

2.34 + 0.09 2.33 +0.07 2.34 + 0.07 2.43 + 0.06 2.44 + 0.10 2.38 + 0.07

180øChannelHistogram

•

180 øChannel Histogram

ß

1.049 + 0.027 1.031 +0.024 0.991 + 0.020 0.904 + 0.015 0.895 + 0.020 0.692 + 0.010

elastictheorysuggests that the acousticscatteringapparatus did not bias the measurementsin such a way as to yield nonphysicalresults.Sincethey are referencedto a 4.9-pm calibrationparticlewhosepropertieswe presumeto know,

PeakEventAmplitude

17000

0.0

90øChannelHistogram

0.0

PeakEventAmplitude

5.9 I•m diameterparticles

17000

PeakEventAmplitude

17000

90' ChannelHistogram

0.0

PeakEventAmplitude

17000

14.6I•m diameterparticles

the cross-sectionmeasurementsare not absolute. However,

thereisgoodagreementbetweentheoryandexperimentover the entire rangeof sizesunder study. This self-consistency supportsour contentionthat the valueslisted in Table II adequatelydescribethe mechanicalpropertiesof PDVB spheres. Table III showsthe measuredcompressibilities anddensitiesfor eachsizesphereusingthe Rayleightheory to both calibrate and compute the mean particle properties.The compressibilities all agreewith eachotherto within the stated precision,but the densitiesexhibita significantreduction with increasingparticle size.This bias,which was observed

FIG. 8. Distributionsof measuredeventamplitudesfor 5.9- and 14.6-pmdiam PDVB spheresin water.

However,at 90øthe Rayleightheory succeeds in predicting PDVB behavior.Thus the 90ø-eventamplitudesare proportionalto the scatterervolume,providedthe particlesare narrowly distributedin densityand compressibility.Consider the sequenceof 90ø-eventhistogramsgivenin Fig. 9. These are distributionsof particle size computedfrom the cube previously byRoos, 9results fromthefactthattheRayleigh root of the measuredevent amplitude. The ability of the approximation overestimates the backscattered signal acousticscatteringapparatusto discriminatesize is clear strengthfor ka largerthan about0.4. This breakdownin the from this seriesof plots.Also evidentis the variationin the long-wavelengthassumptionleadsto a reductionin the comdistributionsfrom oneparticlesizeto the next. In Table IV puted densitycontrastwhich, for PDVB spheresin water, we presentthe computedstandarddeviationsof the sizediscorrespondsto a decreasein the computeddensity.Recall that the compressibility calculationsaremadeusingonly90ø scatteringinformation,wherethe Rayleighand elastictheo4.9 pm ries show relativelycloseagreement.This explainsthe absenceof a similar biasin the compressibilityvalues. The inabilityof Rayleightheoryto predictthescattering from PDVB sphereslargerthan about5 pm is of no conse0 quenceto the calibrationissue.All we requirefor calibration is a well-characterizedpopulationof monodisperse (in size) 10.5 pm 5.9 pm particlesanda solutionto the directscatteringproblem.The agreementbetweenpredictionsusingthe exactelastictheory and measurements performedwith PDVB spheressuggests that thesespherescanserveaseffectivecalibrationparticles for sizesup to ka = 1. 14.6 pm 7.4 pm B. Distributions

Representative distributionsof the particleeventamplitudesare givenin Fig. 8 for 5.9- and 14.6-pm-diammicrospheres.Note that althoughtheir volumesdifferby a factorof 15,thebackscattering from thesespheresappearsverysimilar. This serves to further

illustrate

the limitations

of the

Particle Size

Particle Size

Rayleighapproximation,whichpredictsa nearlylinearrelation betweenthe scatteredstrengthand particle volume.

FIG. 9. ChannelB (90ø) histogramsfor PDVB spheresin water.The horizontalaxiscorresponds to the cuberoot of the eventamplitude,whichfor PDVB is approximatelyproportionalto size.

2340



2340


TABLE

IV. Standard deviations of the measured PDVB

distributions

for

eachparticlesuspension. Also tabulatedare the samestatisticsmeasured with a Coulter counter (suppliedby the manufacturer). Diameter

Standard deviation

(/•m)

(channelA )

Standard deviation

(channelB )

Standard deviation

(Coultercounter)

4.9

9.2%

9.3%

12%

5.9 7.4 9.7 10.5 14.6

7.1% 5.9% 3.1% 4.4% 4.2%

7.3% 6.4% 3.0% 7.4% 3.4%

8.0% 8.1% 2.1% 8.7% 2.7%

becauseindependentknowledgeof the size of a particle is requiredin order to determineits compressibilityand density.Due to this limitation,it is impossiblefor usto establish independentlythe distributionsof particlepropertiesin a heterogeneous population.Effortsareunderway to incorporate individual particle sizing through the addition of an electrozonesensingorifice upstreamof the confocal re-

gion.•9 ACKNOWLEDGMENTS

We gratefullyacknowledge the assistance and adviceof Xucai Chen, Chaur Jian Hsu, and StephenWardlaw aswell asthe financialsupportof the National Institutesof Health, Grant R01-GM30419, and the Office of Naval Research.

tributionsfor eachparticlediameter,alongwith the Coulter analyzerstatisticsprovidedby the manufacturer.The statistics for the distributionsusingthe two techniquesare comparable.Sincethe Coulterdeviceis sensitiveonly to volume changes,thissuggests that thecompressibility anddensityof the PDVB spheresare narrowly distributedcomparedto their size. V. CONCLUSIONS

In thispaperwe haveaddressed the problemof quantitative microparticlecharacterizationusingscatteredultrasonictonebursts.This approachis uniquein that it brings the advantagesof a single-particleobservationtechniqueto the problemof density and compressibilitymeasurement. One can eithercomputemeanpropertiesover a population of particles,or displayhistogramsof scatteredsignalstatistics.

Technical problems associatedwith calibration have beenconsidered(in particularthe choiceof calibrationparticle). Although Rayleighscatteringtheory provedineffective at predictingthe backscatteringcrosssectionof PDVB microspheres, the experimentaldataexhibitcloseagreement with Hay and Burling'sexactsolutionfor elasticscatterers computedusingthe measuredmechanicalpropertiesof bulk polystyrene.Sincean invertiblescatteringtheory is not requiredfor calibration,wewill makeuseof thePDVB spheres asreferenceparticlesfor future experiments. Becauseof its sensitivityand particlehandlingcharacteristics, the instrument is well suited to in vitro mechanical

characterizationof biologicalparticles.With the exception of the maximum likelihood estimator (which can be bypassedgivenhighsignal-to-noise conditions),all signalprocessingis performedby analogcircuits,which makesthe apparatusamenableto real-time applications,such as processmonitoringor selectivesortingbasedon mechanicaldescriptors. In thispaperwe presentdistributionsof measuredevent amplitudes,but not of computedparticleproperties.This is

2341

J. Acoust. Soc. Am., Vol. 87, No. 6, June 1990

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2341


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