Determination of soundspeedin biological tissuesbased on frequency analysis of pulse response Hiroyuki HachiyaandShigeo Ohtsuki Precision andIntelligence Laboratory, TokyoInstituteof Technology, Nagatsuta, Midori-ku, Yokohama227,Japan Motonao
Tanaka
Research Institute for ChestDiseases andCancer,TohokuUniversity, Sendal980,Japan
Floyd Dunn Bioacoustics Research Laboratory, University ofIllinois,Urbana,Illinois61801
(Received28 October1991;acceptedfor publication6 May 1992)
Thesound speed in biological tissues provides important diagnostic andtreatment planning information. Conventional methods of sound-speed determination generally requirethat transducers makephysical contact withspecimens in orderto measure thickness andtravel timein thetimedomain. Thephysical contact maycause deformation andaffectbloodflow andthemeasurement of traveltimein thetimedomainmaybesensitive to waveform
distortion dueto tissue inhomogeneity andsurface roughness. A method fordetermination of thesound speed isproposed in whichthesound traveltimein thesample andthedifference in totaltraveltimefromthetransducer to therigidreflector dueto thepresence of thesample are estimated in thefrequency domain andwhichdoesnotrequire physical contact ofultrasonic probes to livingor freshly excised tissue specimens. Ultrasonic speed measurements in silicone rubberandacrylicresinspecimens verified themethod validity.Thestandard deviation of the measurements overa 10-X 10-mmareais lessthan4 m/s. Sound-speed distribution
measurements of porcine muscle arein agreement withpreviously published results. PACS numbers: 43.35.Wa, 43.80. Ev, 43.80.Cs
INTRODUCTION
The spatialdistributionof soundspeedis an important acousticparameterfor quantitativecharacterizationof living tissuesbecauseit is fundamentallyassociatedwith the mediumscatteringbehavior,and is thereforethe interaction mostlyobservedin clincialdiagnosis,and becauseit affects all otherultrasonicparameters(Dunn et al., 1969). Conventionaltechniques for measuringsoundspeedrequireknowledgeof the thicknessof the tissuespecimentandof the travel time of the soundtraversingthe tissue(Dunn and Goss, 1986). The mechanicalcontact required in making the thickness measurement
causes tissue deformation
and af-
fectsthe blood flow of living tissue,making it difficultto determinethe soundspeedin tissueswith high precision
specimen onanagarstage areestimated. Thesound speed in thespecimen isthendetermined withoutmechanical contact to thespecimen, andthuswithouteffectuponthethickness of thespecimen. A description of theprinciple of thismethod and results of measurements follow. I. THEORY
A. Determination of sound speed without mechanical contact
The tissuesampleis placedon an agarstagein a liquid mediumhavingsoundspeed Co.Theagarstage,asshownin Fig.1,isassembled upona rigidreflector surface. Transmitted ultrasoundreflectionsoccur at the front and rear facesof
thesampleandat therigidreflectorsurface. The agarconcentration of thegelstageisverylow,of theorderof 1% by timein thetimedomainisfurthercomplicated by waveform weight,andits acoustic properties areverynearlythatof (Goss et al., 1978, 1980). Also, measurementof the travel
distortiondue to the inhomogeneityof tissueand to roughnessof the interface,which reducesthe accuracyof the estimate.
In this paper,a new noncontactmethodfor measuring the sound-speed distributionin tissuesis proposed.Herein, both the reflectedultrasonicwavefrom the tissuespecimen, placedin water,andthe reflectedsignalspassingthroughthe propagationmediumof knownsoundspeedare analyzedin thefrequencydomain,viz., the traveltime throughthe tissue and the travel time differencedue to the placementof the 1564
water.Thusreflectionat thewater/agarinterfaceis negligible. Here, tsais the traveltime betweenthe front and rear interfacesof the sample;that is, the travel time of sound
passing through thethickness d ofthesample, andtwaisthe traveltimeof soundpassing throughthedistance d without thesample present. Thethickness d of thesample is,therefore, givenby
d = Ctsd -- Cotwd = CO( tsd-- At),
( 1)
where
J. Acoust. Soc.Am.92 (3),September 1992 0001-4966/92/091564-05500.80 ¸ 1992Acoustical Society ofAmerica
1564
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tim•
Id
2* b.d
FIG. 1. Principleof measurement.
11 (2* •d) I Fsl/Fwl
(2)
At= tsd--
is the difference in the travel time from the t:ransducer to the
reflector due to the placementof the sam)le on the agar
>freq.
stage.
The soundspeedc in the sampleis c = Co(1 -- At/tsa ).
(3)
FIG. 2. The estimationof travel time tsa.
From (1) and (3), d and c can be determ ned using mea-
suredvaluesof At, tsaand the known value )f Co.This calculation procedureis similar to the method1:roposedby Kuo et al. (1990). The soundspeedCoof the cc.uplingmedium, which is usuallywater or saline,can be dete:rminedby movto ing the ultrasound transmitter/receiver transducer knownpositionsand determiningthe travel time differences.
B. Estimation of travel time in the frequ• racy domain Travel time is often measured in the ti• ne domain. But,
as waveform distortion due to tissue inhom•
•geneityand surface roughnesslimits the accuracyof the estimate, for the proposedmethod,travel time ts• and tray{1 time difference At are estimatedin the frequencydomain. Travel time ts• and At are estimatedby analysisin the frequencydomainusingfastFourier transR•rm (FFT) techniques.A pulse wave with center angula] frequencycois transmitted
from the transducer and reflec ed waves are re-
ceivedby the sametransducer.The reflecteCwaveamplitude without the samplepresent,which consistsvf only the wave reflectedfrom reflectorsurface,R w(t), is • iven by R•(t)
=A•R(t--
2t•),
(4)
where tw is the travel time from the trans&]cer to the agar/ reflectorinterfacewithout the samplepre•Dentand R (t) is the waveformof the transmittedsignalfrmn the transducer [ Figs. 2 (a) and 3 (a) ]. Here, A•, is a factm involvingsignal attenuation and a reflection coefficient. F requency-depen.on of tissues are dent attenuationand sound-speeddispersi not includedin order to simplifythe formtdation. 1565
J. Acoust.Soc. Am., Vol. 92, No. 3, Sel:tember 1992
As shownin Figs.2(b) and 3(b), the reflectedwave
withthesample present ontheagarstage, Rs(t), isgivenby Rs(t) = Rsl(t) + Rs2(t), (5) Rsl(t) =AsR(t-- 2ts)+ BsR(t--2(ts+ tsa)), (6) as2(t) = Csa(t-- 2(t• -- At)), (7) wheret• isthetraveltimefromthetransducer to thewater/ sampleinterface, A• andB• aretheamplitudes of reflected wavesfrom the front and rear samplefaces,and C• is the amplitudeof thewavereflected at the agar/reflector interface. Here, R(t) is the transmittedsignaland R• is comprisedof two parts,R• andR•2. R• consists of wavesreflectedfrom the front and rear facesof the sampleand Rs2 consists of wavesreflectedfrom the reflector/agarinterface.
Traveltimetsais estimatedusingFouriertransforms of Rs• andR•, andthetraveltimedifference At isestimated using Fouriertransformsof Rs2andR•. Sincethe reflectoris locatedbeyondtheagarstagewith sufficient distance fromthe sample,Rs canbe resolvedinto Rs• andRs2. The Fouriertransformsof Rs• andRs2,viz. Fsl and.Fs2, respectively, are givenby Fs• (co) = AsF(co)exp( --jco2ts)
-+-BsF(co)exp[--jco2(ts-+-tsa)] = F(co)exp(--jco2t•)
X [A• + B• exp(--jco2t•a)], F•2(co)= C•F(co)exp[--jco2(t• -- At) ], Hachiya et al.' Soundspeedinbiological tissues
(8)
(9)
1565
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At
Imag.
1 + (Bs /A.
) exp(-j
w 2 t.a)
> Real
time
(c)
t•rg(Fs2) -t•rg½F ¾•
FIG. 4. The trajectoryof [1 -[-(Bs/As)exp(--jco2tsa)] in the complex plane.
As shownin Fig. 2(c), IFs/F I exhibitsperiodicvariation
..
with frequencyand takesa local maximumat the point of
co2tsd = 2•rk
(k = O,1,...),
that is,
f2tsd = k
(k = 0,1,...).
The frequencydifferenceAf betweenneighboringmax-
imumpositions of [Fs•/F• I corresponds to traveltimetsaas givenby
Af= 1/2tsa.
>œr•9.
(14)
SinceRs• involvesreflectionsfrom internalstructuresof tissue,Fs• containssomeperiodiccomponentsother than the directingcomponent.Then tsdis obtainedasa maximumof
FIG. 3. The estimation of the time differenceAt.
the inverseFouriertransformof [Fs•/F•,I overthe trans-
whereF(co)istheFouriertransform of thetransmitted signal R (t),
F(co) =
R(t)exp( -jcot)dt.
(t0)
ducer'sbandto separatethe effectof reflectionsfrom internal structures of tissue.
The traveltime differenceAt is estimatedusingFs2and F• asillustratedin Fig. 3. The phasedifferencebetweenFs2 and Fw is givenby
The Fourier transform of the reflected wave from the
arg(Fs2) - arg(F•) = arg(Fs2/Fw)
reflectorwithoutthesample,Fw(co)is givenby Fw(co)= AwF(co)exp(--jco2tw),
(lt)
whereA• istheamplitude of reflected wave.Using(8) and (11),
Fs•(co)/F•(co)= (As/A•)exp[jco2( t• - rs)] (12)
Figure 4 shows the trajectory of [1 4-(Bs/As) X exp( --jco2tsd ) ] in the complexplane. The amplitude IFs/F I is givenby
IFs• /FwI = x/[As-t-Bscos (co2tsd) ]2-4-Bs • sin 2(co2tsd)/A w
1566
J.Acoust. Soc.Am.,Vol.92,No.3, September 1992
( 13)
(15)
as shownin Fig. 3 (c). The travel time differenceAt, asgivenby
At = [arg(Fs2 ) -- arg(F• ) ]/( -- 2co),
X [ t + (Bs/As)exp(--jco2tsd ) ].
= x/As 2-•-Bs • -•-2AsBs COS(CO2tsd )/Aw.
= - 2coAt
(16)
isshownin Fig. 3(d). Thisrelationiswellknownandcanbe usedto measurephasevelocity(Sachseet al., t 978). Eliminating the effectof multipath propagationin inhomogeneoustissue,At isestimatedasa gradientof theregression line through zero by the least-squaremethodover the transducer'sbandwidthusingEq. (15). The travel time tsaand the time difference•t are estimatedin the frequencydomain. Hachiya eta/.'Sound speedinbiological tissues
1566
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II. EXPERIMENTAL
RESULTS
The soundspeeds of materialsof known )ropertieswith
TABLE I. Resultsof measurements of the siliconerubberand acrylicresin plate samples.
thicknesses of several millimeters were m• •sured by this
This method
methodin orderto evaluate thetechnique.;ilicone rubber (Shinetsu siliconeKE 108) andacrylicresi• sampleswere placedin a watertank28 cmlong,28 cm w de, and 21 cm deep. Water at 25.1 d-0.1 øC was used as :he bath (cou-
pling)medium.An ultrasound pulsewastr tnsmitted from the transducer to the sampleanda brassre.•ector,beyond the sample.The reflectedwavewasreceivedby sametransducer.The received signalwascapturedby a Hewlett Packard 5180 waveform recorder at a 20-MHz sataplingrate and transferred,via an HP-IB interface,to a computer for the time difference determination. The transdu :er used had a
3.5-MHz centerfrequency,was13mm in a 50-mmfocallength.The -- 3-dBsignal
and had was 2
MHz (2.5to4.5MHz). Thetriggerclockfo•thepulser was generated by dividingthe 20-MHz sampli rate of the
waveform recorder forcompletely synchron operation. The soundspeedof theliquidmedium
determined
by movingthetransducer knowndistances withaccuracy d-5 pm, andestimating thetimedifferencby analyzing thewavereflected fromthebrass reflector, i• thefrequency domain.Figure5 showsthe measured speed, asa function of thetemperature, for distilled'ater(opencircles)and0.9% saline(opentriangles). The lineindicatesthe calculated valueusingGreens
's em-
piricalequation(Greenspan and Tschie:g, 1959). The measured values agreewellwithGreenspan' equation within d- 3 m/s ( d- 1.5m/sovera rangebetwee•20and40 øC). The resultsof measurementof the sili•
rubber and
acrylicresinplatesamples areshownin T• )leI. The thicknessof the sampleswasalsomeasured a micrometer. The table entries(meansd- standard ) are the averages at 25measuring pointsuniformly istributed overa
10-X 10-mmarea.It isseenthattheul'
icallymeasured
Thickness measured
Soundspeed
Siliconerubber ( 1) Siliconerubber (2) Acrylic resin
Thickness
by micrometer
(m/s)
(mm)
(mm)
1008.0ñ 4.6 1008.9 _ 3.9 2758.3 ñ 4.9
4.37 _ 0.03 6.36 _ 0.03 10.27ñ 0.02
4.39 ñ 0.01 6.35 ñ 0.01 10.30ñ 0.01
valuesof thicknessagreewell with the micrometerdeterminations.
The temperaturedependence of soundspeedof silicone rubber was determined for 4-mm-thick specimensand shownin Fig. 6. The deviationof the sound-speed measurementsis lessthan 4 m/s. The dottedline is a regression line obtainedby the least-squares method. In the measurementsof the acrylic resin and silicone rubbersamples,the estimatedmaximum errorsof time differenceAt and traveltime tsawere,respectively,15 and 50 ns. For a 5-mm sampleof tissue,with an assumedsound speedof 1600m/s, the maximumerror in sound-speed estimation is calculatedto be about 8 m/s; an averageerror should be smaller.
Figure 7(b) showsthe two-dimensionalsound-speed distribution in a sampleof porcinemuscle[Fig. 7(a) ]. The distributionwasobtainedfor 121 measuredpointsby scanning a 20-X 20-mm area at 2-mm intervals.The contourinterval is 25 m/s. The sound speedswere found to be 1521.2 d- 12.0 m/s in the fatty tissuesand 1585.2d-17.1 m/s in the muscletissues.Thesevaluescomparefavorably with thosepublishedby other investigators(Goss et al., 1978, 1980). III. DISCUSSION
ødi eti l ]ed
wa•er-
Determinationof thetwo-dimensional sound-speed distributionin tissuesis importantin the evaluationof the rela-
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30
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35
4O
45 20
Temperature [deE]
25
3
40
45
Temperature [deg.7 FIG. 5. The measured soundspeedfor distilledwat½ r (open circles) and 0.9% saline(opentriangles). Thedottedlineindicatethe calculated value usingGreenspan-Tschiegg's empirical equation.
1567
J. Acoust. Soc.Am.,Vol.92, No.3, Sep'•mber
1992
FIG. 6. The temperature dependence of thesoundspeedof siliconerubber. The error barsrepresentthe standarddeviation. Hachiyaet al.' Soundspeedin biologicaltissues
1567
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PORCINE
MUSCLE
tissue,with an assumedsoundspeedof 1600 m/s, a 10-ns error of At resultsin a 5-m/s error of soundspeed,while a 10ns error of tsaresultsin a 0.2-m/s error of soundspeed.So the measurementof At is moreimportantthan the measurement of tsa.For precisemeasurement, At is estimatedfrom the gradientof the regression line throughzeroin the phase differenceof stablereflectedwavesfrom the rigid surfaceasa functionof frequency.By usingtheleast-square methodover the transducer's bandwidth, the estimation result is the
propagation timeof theprincipalpathin multipathsthrough inhomogeneous or dispersive medialike tissues.This procedure is formally a cross-correlation technique,but it is easy to get the time resolutionshorterthan the time intervalof samplingwithout interpolationin the time domainusinga function like sin (x)/x.
FIG. 7. (a) A sampleof porcinemuscle.(b) The two-dimensionalsoundspeeddistributionin a sampleof porcinemuscle.
tionshipofacoustic characteristics andphysiological statein normal and diseasedtissuessincethis parameterinvolves boththe inertialandelasticproperties. The publishedliterature containsmany reports of such measurements(Goss et al., 1978,1980).However,measurement methods requiring physicalcontactof, for example,the transducerto the tissue,introduceerrorsas the readilydeformedsofttissues contravenethe possibilityof a uniquesamplethickness.As the proposed methoddoesnot requirephysicalcontactby a rigid objectand therebyeliminatesthe needfor measuring specimenthickness,normal blood flow can continuewhile thetwo-dimensional sound-speed distributionisdetermined by scanningthe area of interestnormal to the direction of wave propagation. This methodrequiresdeterminationof the soundtravel time in the sample,tsa,and the differencein the travel time dueto thepresence of thesample,At. For a 5-mmsampleof
1568
J. Acoust. Soc. Am., Vol. 92, No. 3, September 1992
Argumentsoffs2 and Fw take 2•rjumps sincethe argument is in the range + vrto -- vt.But it is normally easyto follow becausetissueshave a soundspeedrelatively closeto that of water. In the caseof very dispersivetissuesor inhomogeneoustissuesin which soundpropagatesalong many paths,the slopeof the phasemay not be straight,so it may not be easyto trace the phase. The estimationof tsa also utilizes all the information contained in the waveform, by the analysisof the received signalin the frequencydomain,estimationresultsare largely independentof waveformdistortiondue to the roughnessof surfaceand internalstructure,allowingthe sound-speed distribution in thin specimensto be determined.Sinceestimation of travel time in the frequencydomainis carriedout using FFT's, the sound-speeddistribution can be determined in nearly real-time. The basicprincipleof sound-speed estimationin the frequencydomain is basedon the assumptionof plane-wave propagation.However,the experimentsreportedhere were carriedout usingveryweaklyfocusedwaves,suggesting that errorsso introducedare usuallynegligible. '
Dunn, F., Edmonds,P. D., and Fry, W. J. (1969). "Absorptionand Dispersionof Ultrasoundin BiologicalMedia, in BiologicalEngineering,edited by H. P. Schwan(McGraw-Hill, New York), Chap. 3, pp. 205-332. Dunn, F., and Goss, S. A. (1986). "Definition of Terms and Measurements
of AcousticalQuantities,"in TissueCharacterizationwith Ultrasound, editedby J.P. Greenleaf(CRC Press,BocaRaton), Chap. 1, pp. 1-13. Goss,S. A., Johnson,R. L., andDunn, F. (1978). "Compilationof empirical ultrasonicpropertiesof mammaliantissues,"J. Acoust.Soc.Am. 64, 423-457.
Goss,S. A., Johnson,R. L., andDunn, F. (1980). "Compilationof empirical ultrasonicpropertiesof mammaliantissuesII," J. Acoust.Soc.Am. 68, 93-108.
Greenspan,M., and Tschiegg,C. E. (1959). "Tablesof the speedof sound in water," J. Acoust. Soc. Am. 31, 75-76.
Kuo, I. Y., Mete, B., and Shung,K. K. (1990). "A novel methodfor the measurement of acousticspeed,"J. Acoust.Soc.Am. 88, 1679-1682. Sachse,W., and Pao,Y. (1978). "On the determinationof phaseand group velocitiesof dispersive wavein solids,"J. Appl. Phys.49, 4320-4327.
Hachiya eta/.: Soundspeed in biologicaltissues
1568
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Tanaka
Research Institute for ChestDiseases andCancer,TohokuUniversity, Sendal980,Japan
Floyd Dunn Bioacoustics Research Laboratory, University ofIllinois,Urbana,Illinois61801
(Received28 October1991;acceptedfor publication6 May 1992)
Thesound speed in biological tissues provides important diagnostic andtreatment planning information. Conventional methods of sound-speed determination generally requirethat transducers makephysical contact withspecimens in orderto measure thickness andtravel timein thetimedomain. Thephysical contact maycause deformation andaffectbloodflow andthemeasurement of traveltimein thetimedomainmaybesensitive to waveform
distortion dueto tissue inhomogeneity andsurface roughness. A method fordetermination of thesound speed isproposed in whichthesound traveltimein thesample andthedifference in totaltraveltimefromthetransducer to therigidreflector dueto thepresence of thesample are estimated in thefrequency domain andwhichdoesnotrequire physical contact ofultrasonic probes to livingor freshly excised tissue specimens. Ultrasonic speed measurements in silicone rubberandacrylicresinspecimens verified themethod validity.Thestandard deviation of the measurements overa 10-X 10-mmareais lessthan4 m/s. Sound-speed distribution
measurements of porcine muscle arein agreement withpreviously published results. PACS numbers: 43.35.Wa, 43.80. Ev, 43.80.Cs
INTRODUCTION
The spatialdistributionof soundspeedis an important acousticparameterfor quantitativecharacterizationof living tissuesbecauseit is fundamentallyassociatedwith the mediumscatteringbehavior,and is thereforethe interaction mostlyobservedin clincialdiagnosis,and becauseit affects all otherultrasonicparameters(Dunn et al., 1969). Conventionaltechniques for measuringsoundspeedrequireknowledgeof the thicknessof the tissuespecimentandof the travel time of the soundtraversingthe tissue(Dunn and Goss, 1986). The mechanicalcontact required in making the thickness measurement
causes tissue deformation
and af-
fectsthe blood flow of living tissue,making it difficultto determinethe soundspeedin tissueswith high precision
specimen onanagarstage areestimated. Thesound speed in thespecimen isthendetermined withoutmechanical contact to thespecimen, andthuswithouteffectuponthethickness of thespecimen. A description of theprinciple of thismethod and results of measurements follow. I. THEORY
A. Determination of sound speed without mechanical contact
The tissuesampleis placedon an agarstagein a liquid mediumhavingsoundspeed Co.Theagarstage,asshownin Fig.1,isassembled upona rigidreflector surface. Transmitted ultrasoundreflectionsoccur at the front and rear facesof
thesampleandat therigidreflectorsurface. The agarconcentration of thegelstageisverylow,of theorderof 1% by timein thetimedomainisfurthercomplicated by waveform weight,andits acoustic properties areverynearlythatof (Goss et al., 1978, 1980). Also, measurementof the travel
distortiondue to the inhomogeneityof tissueand to roughnessof the interface,which reducesthe accuracyof the estimate.
In this paper,a new noncontactmethodfor measuring the sound-speed distributionin tissuesis proposed.Herein, both the reflectedultrasonicwavefrom the tissuespecimen, placedin water,andthe reflectedsignalspassingthroughthe propagationmediumof knownsoundspeedare analyzedin thefrequencydomain,viz., the traveltime throughthe tissue and the travel time differencedue to the placementof the 1564
water.Thusreflectionat thewater/agarinterfaceis negligible. Here, tsais the traveltime betweenthe front and rear interfacesof the sample;that is, the travel time of sound
passing through thethickness d ofthesample, andtwaisthe traveltimeof soundpassing throughthedistance d without thesample present. Thethickness d of thesample is,therefore, givenby
d = Ctsd -- Cotwd = CO( tsd-- At),
( 1)
where
J. Acoust. Soc.Am.92 (3),September 1992 0001-4966/92/091564-05500.80 ¸ 1992Acoustical Society ofAmerica
1564
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tim•
Id
2* b.d
FIG. 1. Principleof measurement.
11 (2* •d) I Fsl/Fwl
(2)
At= tsd--
is the difference in the travel time from the t:ransducer to the
reflector due to the placementof the sam)le on the agar
>freq.
stage.
The soundspeedc in the sampleis c = Co(1 -- At/tsa ).
(3)
FIG. 2. The estimationof travel time tsa.
From (1) and (3), d and c can be determ ned using mea-
suredvaluesof At, tsaand the known value )f Co.This calculation procedureis similar to the method1:roposedby Kuo et al. (1990). The soundspeedCoof the cc.uplingmedium, which is usuallywater or saline,can be dete:rminedby movto ing the ultrasound transmitter/receiver transducer knownpositionsand determiningthe travel time differences.
B. Estimation of travel time in the frequ• racy domain Travel time is often measured in the ti• ne domain. But,
as waveform distortion due to tissue inhom•
•geneityand surface roughnesslimits the accuracyof the estimate, for the proposedmethod,travel time ts• and tray{1 time difference At are estimatedin the frequencydomain. Travel time ts• and At are estimatedby analysisin the frequencydomainusingfastFourier transR•rm (FFT) techniques.A pulse wave with center angula] frequencycois transmitted
from the transducer and reflec ed waves are re-
ceivedby the sametransducer.The reflecteCwaveamplitude without the samplepresent,which consistsvf only the wave reflectedfrom reflectorsurface,R w(t), is • iven by R•(t)
=A•R(t--
2t•),
(4)
where tw is the travel time from the trans&]cer to the agar/ reflectorinterfacewithout the samplepre•Dentand R (t) is the waveformof the transmittedsignalfrmn the transducer [ Figs. 2 (a) and 3 (a) ]. Here, A•, is a factm involvingsignal attenuation and a reflection coefficient. F requency-depen.on of tissues are dent attenuationand sound-speeddispersi not includedin order to simplifythe formtdation. 1565
J. Acoust.Soc. Am., Vol. 92, No. 3, Sel:tember 1992
As shownin Figs.2(b) and 3(b), the reflectedwave
withthesample present ontheagarstage, Rs(t), isgivenby Rs(t) = Rsl(t) + Rs2(t), (5) Rsl(t) =AsR(t-- 2ts)+ BsR(t--2(ts+ tsa)), (6) as2(t) = Csa(t-- 2(t• -- At)), (7) wheret• isthetraveltimefromthetransducer to thewater/ sampleinterface, A• andB• aretheamplitudes of reflected wavesfrom the front and rear samplefaces,and C• is the amplitudeof thewavereflected at the agar/reflector interface. Here, R(t) is the transmittedsignaland R• is comprisedof two parts,R• andR•2. R• consists of wavesreflectedfrom the front and rear facesof the sampleand Rs2 consists of wavesreflectedfrom the reflector/agarinterface.
Traveltimetsais estimatedusingFouriertransforms of Rs• andR•, andthetraveltimedifference At isestimated using Fouriertransformsof Rs2andR•. Sincethe reflectoris locatedbeyondtheagarstagewith sufficient distance fromthe sample,Rs canbe resolvedinto Rs• andRs2. The Fouriertransformsof Rs• andRs2,viz. Fsl and.Fs2, respectively, are givenby Fs• (co) = AsF(co)exp( --jco2ts)
-+-BsF(co)exp[--jco2(ts-+-tsa)] = F(co)exp(--jco2t•)
X [A• + B• exp(--jco2t•a)], F•2(co)= C•F(co)exp[--jco2(t• -- At) ], Hachiya et al.' Soundspeedinbiological tissues
(8)
(9)
1565
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At
Imag.
1 + (Bs /A.
) exp(-j
w 2 t.a)
> Real
time
(c)
t•rg(Fs2) -t•rg½F ¾•
FIG. 4. The trajectoryof [1 -[-(Bs/As)exp(--jco2tsa)] in the complex plane.
As shownin Fig. 2(c), IFs/F I exhibitsperiodicvariation
..
with frequencyand takesa local maximumat the point of
co2tsd = 2•rk
(k = O,1,...),
that is,
f2tsd = k
(k = 0,1,...).
The frequencydifferenceAf betweenneighboringmax-
imumpositions of [Fs•/F• I corresponds to traveltimetsaas givenby
Af= 1/2tsa.
>œr•9.
(14)
SinceRs• involvesreflectionsfrom internalstructuresof tissue,Fs• containssomeperiodiccomponentsother than the directingcomponent.Then tsdis obtainedasa maximumof
FIG. 3. The estimation of the time differenceAt.
the inverseFouriertransformof [Fs•/F•,I overthe trans-
whereF(co)istheFouriertransform of thetransmitted signal R (t),
F(co) =
R(t)exp( -jcot)dt.
(t0)
ducer'sbandto separatethe effectof reflectionsfrom internal structures of tissue.
The traveltime differenceAt is estimatedusingFs2and F• asillustratedin Fig. 3. The phasedifferencebetweenFs2 and Fw is givenby
The Fourier transform of the reflected wave from the
arg(Fs2) - arg(F•) = arg(Fs2/Fw)
reflectorwithoutthesample,Fw(co)is givenby Fw(co)= AwF(co)exp(--jco2tw),
(lt)
whereA• istheamplitude of reflected wave.Using(8) and (11),
Fs•(co)/F•(co)= (As/A•)exp[jco2( t• - rs)] (12)
Figure 4 shows the trajectory of [1 4-(Bs/As) X exp( --jco2tsd ) ] in the complexplane. The amplitude IFs/F I is givenby
IFs• /FwI = x/[As-t-Bscos (co2tsd) ]2-4-Bs • sin 2(co2tsd)/A w
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J.Acoust. Soc.Am.,Vol.92,No.3, September 1992
( 13)
(15)
as shownin Fig. 3 (c). The travel time differenceAt, asgivenby
At = [arg(Fs2 ) -- arg(F• ) ]/( -- 2co),
X [ t + (Bs/As)exp(--jco2tsd ) ].
= x/As 2-•-Bs • -•-2AsBs COS(CO2tsd )/Aw.
= - 2coAt
(16)
isshownin Fig. 3(d). Thisrelationiswellknownandcanbe usedto measurephasevelocity(Sachseet al., t 978). Eliminating the effectof multipath propagationin inhomogeneoustissue,At isestimatedasa gradientof theregression line through zero by the least-squaremethodover the transducer'sbandwidthusingEq. (15). The travel time tsaand the time difference•t are estimatedin the frequencydomain. Hachiya eta/.'Sound speedinbiological tissues
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II. EXPERIMENTAL
RESULTS
The soundspeeds of materialsof known )ropertieswith
TABLE I. Resultsof measurements of the siliconerubberand acrylicresin plate samples.
thicknesses of several millimeters were m• •sured by this
This method
methodin orderto evaluate thetechnique.;ilicone rubber (Shinetsu siliconeKE 108) andacrylicresi• sampleswere placedin a watertank28 cmlong,28 cm w de, and 21 cm deep. Water at 25.1 d-0.1 øC was used as :he bath (cou-
pling)medium.An ultrasound pulsewastr tnsmitted from the transducer to the sampleanda brassre.•ector,beyond the sample.The reflectedwavewasreceivedby sametransducer.The received signalwascapturedby a Hewlett Packard 5180 waveform recorder at a 20-MHz sataplingrate and transferred,via an HP-IB interface,to a computer for the time difference determination. The transdu :er used had a
3.5-MHz centerfrequency,was13mm in a 50-mmfocallength.The -- 3-dBsignal
and had was 2
MHz (2.5to4.5MHz). Thetriggerclockfo•thepulser was generated by dividingthe 20-MHz sampli rate of the
waveform recorder forcompletely synchron operation. The soundspeedof theliquidmedium
determined
by movingthetransducer knowndistances withaccuracy d-5 pm, andestimating thetimedifferencby analyzing thewavereflected fromthebrass reflector, i• thefrequency domain.Figure5 showsthe measured speed, asa function of thetemperature, for distilled'ater(opencircles)and0.9% saline(opentriangles). The lineindicatesthe calculated valueusingGreens
's em-
piricalequation(Greenspan and Tschie:g, 1959). The measured values agreewellwithGreenspan' equation within d- 3 m/s ( d- 1.5m/sovera rangebetwee•20and40 øC). The resultsof measurementof the sili•
rubber and
acrylicresinplatesamples areshownin T• )leI. The thicknessof the sampleswasalsomeasured a micrometer. The table entries(meansd- standard ) are the averages at 25measuring pointsuniformly istributed overa
10-X 10-mmarea.It isseenthattheul'
icallymeasured
Thickness measured
Soundspeed
Siliconerubber ( 1) Siliconerubber (2) Acrylic resin
Thickness
by micrometer
(m/s)
(mm)
(mm)
1008.0ñ 4.6 1008.9 _ 3.9 2758.3 ñ 4.9
4.37 _ 0.03 6.36 _ 0.03 10.27ñ 0.02
4.39 ñ 0.01 6.35 ñ 0.01 10.30ñ 0.01
valuesof thicknessagreewell with the micrometerdeterminations.
The temperaturedependence of soundspeedof silicone rubber was determined for 4-mm-thick specimensand shownin Fig. 6. The deviationof the sound-speed measurementsis lessthan 4 m/s. The dottedline is a regression line obtainedby the least-squares method. In the measurementsof the acrylic resin and silicone rubbersamples,the estimatedmaximum errorsof time differenceAt and traveltime tsawere,respectively,15 and 50 ns. For a 5-mm sampleof tissue,with an assumedsound speedof 1600m/s, the maximumerror in sound-speed estimation is calculatedto be about 8 m/s; an averageerror should be smaller.
Figure 7(b) showsthe two-dimensionalsound-speed distribution in a sampleof porcinemuscle[Fig. 7(a) ]. The distributionwasobtainedfor 121 measuredpointsby scanning a 20-X 20-mm area at 2-mm intervals.The contourinterval is 25 m/s. The sound speedswere found to be 1521.2 d- 12.0 m/s in the fatty tissuesand 1585.2d-17.1 m/s in the muscletissues.Thesevaluescomparefavorably with thosepublishedby other investigators(Goss et al., 1978, 1980). III. DISCUSSION
ødi eti l ]ed
wa•er-
Determinationof thetwo-dimensional sound-speed distributionin tissuesis importantin the evaluationof the rela-
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Temperature [deE]
25
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40
45
Temperature [deg.7 FIG. 5. The measured soundspeedfor distilledwat½ r (open circles) and 0.9% saline(opentriangles). Thedottedlineindicatethe calculated value usingGreenspan-Tschiegg's empirical equation.
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J. Acoust. Soc.Am.,Vol.92, No.3, Sep'•mber
1992
FIG. 6. The temperature dependence of thesoundspeedof siliconerubber. The error barsrepresentthe standarddeviation. Hachiyaet al.' Soundspeedin biologicaltissues
1567
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PORCINE
MUSCLE
tissue,with an assumedsoundspeedof 1600 m/s, a 10-ns error of At resultsin a 5-m/s error of soundspeed,while a 10ns error of tsaresultsin a 0.2-m/s error of soundspeed.So the measurementof At is moreimportantthan the measurement of tsa.For precisemeasurement, At is estimatedfrom the gradientof the regression line throughzeroin the phase differenceof stablereflectedwavesfrom the rigid surfaceasa functionof frequency.By usingtheleast-square methodover the transducer's bandwidth, the estimation result is the
propagation timeof theprincipalpathin multipathsthrough inhomogeneous or dispersive medialike tissues.This procedure is formally a cross-correlation technique,but it is easy to get the time resolutionshorterthan the time intervalof samplingwithout interpolationin the time domainusinga function like sin (x)/x.
FIG. 7. (a) A sampleof porcinemuscle.(b) The two-dimensionalsoundspeeddistributionin a sampleof porcinemuscle.
tionshipofacoustic characteristics andphysiological statein normal and diseasedtissuessincethis parameterinvolves boththe inertialandelasticproperties. The publishedliterature containsmany reports of such measurements(Goss et al., 1978,1980).However,measurement methods requiring physicalcontactof, for example,the transducerto the tissue,introduceerrorsas the readilydeformedsofttissues contravenethe possibilityof a uniquesamplethickness.As the proposed methoddoesnot requirephysicalcontactby a rigid objectand therebyeliminatesthe needfor measuring specimenthickness,normal blood flow can continuewhile thetwo-dimensional sound-speed distributionisdetermined by scanningthe area of interestnormal to the direction of wave propagation. This methodrequiresdeterminationof the soundtravel time in the sample,tsa,and the differencein the travel time dueto thepresence of thesample,At. For a 5-mmsampleof
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J. Acoust. Soc. Am., Vol. 92, No. 3, September 1992
Argumentsoffs2 and Fw take 2•rjumps sincethe argument is in the range + vrto -- vt.But it is normally easyto follow becausetissueshave a soundspeedrelatively closeto that of water. In the caseof very dispersivetissuesor inhomogeneoustissuesin which soundpropagatesalong many paths,the slopeof the phasemay not be straight,so it may not be easyto trace the phase. The estimationof tsa also utilizes all the information contained in the waveform, by the analysisof the received signalin the frequencydomain,estimationresultsare largely independentof waveformdistortiondue to the roughnessof surfaceand internalstructure,allowingthe sound-speed distribution in thin specimensto be determined.Sinceestimation of travel time in the frequencydomainis carriedout using FFT's, the sound-speeddistribution can be determined in nearly real-time. The basicprincipleof sound-speed estimationin the frequencydomain is basedon the assumptionof plane-wave propagation.However,the experimentsreportedhere were carriedout usingveryweaklyfocusedwaves,suggesting that errorsso introducedare usuallynegligible. '
Dunn, F., Edmonds,P. D., and Fry, W. J. (1969). "Absorptionand Dispersionof Ultrasoundin BiologicalMedia, in BiologicalEngineering,edited by H. P. Schwan(McGraw-Hill, New York), Chap. 3, pp. 205-332. Dunn, F., and Goss, S. A. (1986). "Definition of Terms and Measurements
of AcousticalQuantities,"in TissueCharacterizationwith Ultrasound, editedby J.P. Greenleaf(CRC Press,BocaRaton), Chap. 1, pp. 1-13. Goss,S. A., Johnson,R. L., andDunn, F. (1978). "Compilationof empirical ultrasonicpropertiesof mammaliantissues,"J. Acoust.Soc.Am. 64, 423-457.
Goss,S. A., Johnson,R. L., andDunn, F. (1980). "Compilationof empirical ultrasonicpropertiesof mammaliantissuesII," J. Acoust.Soc.Am. 68, 93-108.
Greenspan,M., and Tschiegg,C. E. (1959). "Tablesof the speedof sound in water," J. Acoust. Soc. Am. 31, 75-76.
Kuo, I. Y., Mete, B., and Shung,K. K. (1990). "A novel methodfor the measurement of acousticspeed,"J. Acoust.Soc.Am. 88, 1679-1682. Sachse,W., and Pao,Y. (1978). "On the determinationof phaseand group velocitiesof dispersive wavein solids,"J. Appl. Phys.49, 4320-4327.
Hachiya eta/.: Soundspeed in biologicaltissues
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