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(Series in Medical Physics and Biomedical Engineering) Francis a. Duck_ a.C Baker_ H.C Starritt-Ultrasound in Medicine-CRC Press (1998)
Citation preview
Pagei
UltrasoundinMedicine
Pageii
UltrasoundinMedicine,TheThirdMayneordPhillipsSummerSchoolStEdmundHall,Oxford,June1997
Pageiii
UltrasoundinDedicine,TheThirdMayneordPhillipsSummerSchoolStEdmundHall,Oxford,June1997
1VictorHumphrey 8JeffBamber 15DarkoGroev 22JohnTruscott
2ElizabethMoore 9NadinePay 16SandyMather 23FrancisDuck
3BarryWard 10JimWilliams 17TimSpencer 24AndrzejJastrzebski
4MatthewReilly 11AndersOlsson 18MikeHalliwell 25FrankRakebrandt
5KatDixon 12BenKhoo 19PhilBurford 26PanagiotisTsiganos
6JimGreenleaf 13MalcolmSperrin 20MegWarner 27GarethPrice
7OsiyahPapayi 14ElvinNix 21TonyWhittingham 28KitHill
Pageiv
OtherbooksintheMedicalScienceSeries
ThePhysicsandRadiobiologyofFastNeutronBeamsDKBewley
MedicalPhysicsandBiomedicalEngineeringBHBrown,RHSmallwood,DRHose,PVLawfordandDCBarber
RehabilitationEngineeringAppliedtoMobilityandManipulationRACooper
PhysicsforDiagnosticRadiology,2ndeditionPPDendyandBHHeaton
LinearAcceleratorsforRadiationTherapy,2ndeditionDGreeneandPCWilliams
HealthEffectsofExposuretoLowLevelIonizingRadiationWRHendeeandFMEdwards(eds)
MonteCarloCalculationsinNuclearMedicineMLjungberg,SEStrandandMAKing(eds)
IntroductoryMedicalStatistics,3rdeditionRFMould
RadiationProtectioninHospitalsRFMould
RPLDosimetryRadiophotoluminescenceinHealthPhysicsJAPerry
ThePhysicsofConformalRadiotherapySWebb
ThePhysicsofMedicalImagingSWebb(ed)
ThePhysicsofThreeDimensionalRadiationTherapySWebb
DesignofPulseOximetersJGWebster
Pagev
MedicalScienceSerice
UltrasoundinMedicine
EditedbyFrancisADuck
RoyalUnitedHospital,BathandUniversityofBath,UK
AndrewCBakerChristianMichelsenResearchAS,Bergen,Norway
formerlyUniversityofBath
HazelCStarrittRoyalUnitedHospital,Bath,UK
BasedonInvitedLecturespresentedattheThirdMayneordPhillipsSummerSchool1997
sponsoredbyInstituteofPhysicsandEngineeringinMedicine
BritishInstituteofRadiologyInstituteofPhysics
BritishMedicalUltrasoundSociety
InstituteofPhysicsPublishingBristolandPhiladelphia
Pagevi
IOPPublishingLtd1998
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystemortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recordingorotherwise,withoutthepriorpermissionofthepublisher.MultiplecopyingispermittedinaccordancewiththetermsoflicencesissuedbytheCopyrightLicensingAgencyunderthetermsofitsagreementwiththeCommitteeofViceChancellorsandPrincipals.
TheEditorsandIOPPublishingLtdhaveattemptedtotracethecopyrightholdersofallmaterialreproducedinthispublicationandapologizetocopyrightholdersifpermissiontopublishinthisformhasnotbeenobtained.
BritishLibraryCataloguinginPublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN0750305932
LibraryofCongressCataloginginPublicationDataareavailable
SeriesEditors:
RFMould,Croydon,UKCGOrton,KarmanosCancerInstituteandWayneStateUniversity,Detroit,USAJAESpaan,UniversityofAmsterdam,TheNetherlandsJGWebster,UniversityofWisconsinMadison,USA
CoverillustrationcourtesyofAndrewBakerandMarkCahill
PublishedbyInstituteofPhysicsPublishing,whollyownedbyTheInstituteofPhysics,London
InstituteofPhysicsPublishing,DiracHouse,TempleBack,BristolBS16BE,UK
USOffice:InstituteofPhysicsPublishing,ThePublicLedgerBuilding,Suite1035,150SouthIndependenceMallWest,Philadelphia,PA19106,USA
TypesetinTEXusingtheIOPBookmakerMacrosPrintedinGreatBritainbyJWArrowsmithLtd,Bristol
Pagevii
TheMedicalScienceSeriesistheofficialbookseriesoftheInternationalFederationforMedicalandBiologicalEngineering(IFMBE)andtheInternationalOrganizationforMedicalPhysics(IOMP).
IFMBE
TheIFMBEwasestablishedin1959toprovidemedicalandbiologicalengineeringwithaninternationalpresence.TheFederationhasalonghistoryofencouragingandpromotinginternationalcooperationandcollaborationintheuseoftechnologyforimprovingthehealthandlifequalityofman.
TheIFMBEisanorganizationthatismostlyanaffiliationofnationalsocieties.Transnationalorganizationscanalsoobtainmembership.Atpresentthereare42nationalmembers,andonetransnationalmemberwithatotalmembershipinexcessof15000.Anobservercategoryisprovidedtogivepersonalstatustogroupsororganizationsconsideringformalaffiliation.
Objectives
Toreflecttheinterestsandinitiativesoftheaffiliatedorganizations.
Togenerateanddisseminateinformationofinteresttothemedicalandbiologicalengineeringcommunityandinternationalorganizations.
Toprovideaninternationalforumfortheexchangeofideasandconcepts.
Toencourageandfosterresearchandapplicationofmedicalandbiologicalengineeringknowledgeandtechniquesinsupportoflifequalityandcosteffectivehealthcare.
Tostimulateinternationalcooperationandcollaborationonmedicalandbiologicalengineeringmatters.
Toencourageeducationalprogrammeswhichdevelopscientificandtechnicalexpertiseinmedicalandbiologicalengineering.
Activities
TheIFMBEhaspublishedthejournalMedicalandBiologicalEngineeringandComputingforover34years.AnewjournalCellularEngineeringwasestablishedin1996inordertostimulatethisemergingfieldinbiomedicalengineering.InIFMBENewsmembersarekeptinformedofthedevelopmentsintheFederation.ClinicalEngineeringUpdateisapublicationofourdivisionofClinicalEngineering.TheFederationalsohasadivisionforTechnologyAssessmentinHealthCare.
Everythreeyears,theIFMBEholdsaWorldCongressonMedicalPhysicsandBiomedicalEngineering,organizedincooperationwiththeIOMPandtheIUPESM.Inaddition,annual,milestone,regionalconferencesareorganizedindifferentregionsoftheworld,suchastheAsiaPacific,Baltic,Mediterranean,AfricanandSouthAmericanregions.
TheadministrativecounciloftheIFMBEmeetsonceortwiceayearandisthesteeringbodyfortheIFMBE.ThecouncilissubjecttotherulingsoftheGeneralAssemblywhichmeetseverythreeyears.
Pageviii
ForfurtherinformationontheactivitiesoftheIFMBE,pleasecontactJosAESpaan,ProfessorofMedicalPhysics,AcademicMedicalCentre,UniversityofAmsterdam,POBox22660,Meibergdreef9,1105AZ,Amsterdam,TheNetherlands.Tel:31(0)205665200.Fax:31(0)206917233.Email:[email protected]:http://vub.vub.ac.be/~ifmbe.
IOMP
TheIOMPwasfoundedin1963.Themembershipincludes64nationalsocieties,twointernationalorganizationsand12000individuals.MembershipofIOMPconsistsofindividualmembersoftheAdheringNationalOrganizations.Twootherformsofmembershipareavailable,namelyAffiliatedRegionalOrganizationandCorporateMembers.TheIOMPisadministeredbyaCouncil,whichconsistsofdelegatesfromeachoftheAdheringNationalOrganizationregularmeetingsofCouncilareheldeverythreeyearsattheInternationalConferenceonMedicalPhysics(ICMP).TheOfficersoftheCouncilarethePresident,theVicePresidentandtheSecretaryGeneral.IOMPcommitteesinclude:developingcountries,educationandtrainingnominatingandpublications.
Objectives
Toorganizeinternationalcooperationinmedicalphysicsinallitsaspects,especiallyindevelopingcountries.
Toencourageandadviseontheformationofnationalorganizationsofmedicalphysicsinthosecountrieswhichlacksuchorganizations.
Activities
OfficialpublicationsoftheIOMParePhysiologicalMeasurement,PhysicsinMedicineandBiologyandtheMedicalScienceSeries,allpublishedbyInstituteofPhysicsPublishing.TheIOMPpublishesabulletinMedicalPhysicsWorldtwiceayear.
TwoCouncilmeetingsandoneGeneralAssemblyareheldeverythreeyearsattheICMP.ThemostrecentICMPswereheldinKyoto,Japan(1991),RiodeJaneiro,Brazil(1994)andNice,France(1997).ThenextconferenceisscheduledforChicago,USA(2000).TheseconferencesarenormallyheldincollaborationwiththeIFMBEtoformtheWorldCongressonMedicalPhysicsandBiomedicalEngineering.TheIOMPalsosponsorsoccasionalinternationalconferences,workshopsandcourses.
Forfurtherinformationcontact:HansSvensson,PhD,DSc,Professor,RadiationPhysicsDepartment,UniversityHospital,90185Ume,Sweden.Tel:(46)907853891.Fax:(46)907851588.Email:[email protected].
Pageix
CONTENTS.
ContributingAuthors xvii
Glossary xix
FrancisADuckIntroduction xxv
Acknowledgments xxx
References xxx
Part1ThePhysicsofMedicalUltrasound
1
VictorFHumphreyandFrancisADuck
1UltrasonicFields:StructureandPrediction
3
1.1CircularplaneSources 4
1.1.1PressureVariationontheAxis 6
1.1.2PressureVariationOfftheAxis 9
1.2PulsedFields 10
1.3FocusedFields 13
1.4SourceAmplitudeWeighting 15
1.5RectangularSources 17
1.6Conclusion 20
References 21
AndrewCBaker
2NonlinearEffectsinUltrasoundPropagation
23
2.1NonlinearPropagationinMedicalUltrasound 24
2.2ConsequencesofNonlinearPropagation 27
2.2.1ExperimentalMeasurements 27
2.2.2TheoreticalPredictions 31
2.2.3ClinicalSystems 34
References 36
FrancisADuck
3RadiationPressureandAcousticStreaming
39
3.1Radiationpressure 39
Pagex
3.2LangevinRadiationPressure,PLan 40
3.3RadiationStressTensor 43
3.3.1TheExcessPressure 43
3.4RayleighRadiationPressure,PRay 44
3.5AcousticStreaming 46
3.5.1MethodsofMeasuringAcousticStreaming 50
3.6ObservationsinVivoofRadiationPressureEffects 52
3.6.1Streaming 52
3.6.2ObservedBiologicalEffectsApparentlyRelatedtoRadiationPressure
52
3.7Discussion 53
References 54
JeffreyCBamber
4UltrasonicPropertiesofTissues
57
4.1BasicConcepts 57
4.1.1Attenuation,Absorption,ScatteringandReflection 57
4.1.2SpeedofSound 61
4.1.3Nonlinearity 61
4.1.4TransducerDiffractionField 61
4.1.5PulseEchoImaging,SpeckleandEchoTexture 62
4.1.6ReceiverPhaseSensitivity 64
4.2MeasurementMethods 64
4.2.1MeasurementoftheAbsorptionCoefficient 64
4.2.2MeasurementoftheAttenuationCoefficient 65
4.2.3MeasurementofSoundSpeed 68
4.2.4MeasurementofScattering 70
4.2.5MeasurementofNonlinearity 72
4.3UltrasonicPropertiesofTissues 73
4.3.1AbsorptionandAttenuation 73
4.3.2SoundSpeed 76
4.3.3Scattering 78
4.3.4Nonlinearity 83
References 83
Part2TechnologyandMeasurementinDiagnosticImaging
89
ThomasLSzabo
5TransducerArraysforMedicalUltrasoundImaging
91
5.1PiezoelectricTransducerElements 91
5.1.1ABasicTransducerModel 91
Pagexi
5.1.2TransducerElementsasAcousticResonators 93
5.1.3TransducerArrayStructures 95
5.1.4TransducerModels 96
5.1.5TransducerDesign 99
5.2Imaging 102
5.3BeamForming 103
5.4OtherImagingModes 108
5.5Conclusion 109
References 109
PeterNTWells
6CurrentDopplerTechnologyandTechniques
113
6.1TheDopplerEffect 113
6.2TheOriginoftheDopplerSignal 114
6.3TheNarrowFrequencyBandTechnique 116
6.3.1TheContinuousWaveDopplerTechnique 116
6.3.2ThePulsedDopplerTechnique 118
6.4FrequencySpectrumAnalysis 120
6.5DuplexScanning 120
6.6ColourFlowImaging 121
6.6.1BasicPrinciples 121
6.6.2AutocorrelationDetection 123
6.6.3OtherDopplerFrequencyEstimators 123
6.6.4TimeDomainProcessing 124
6.6.5ColourCodingSchemes 125
6.6.6ThreeDimensionalDisplay 126
6.7ContrastAgentsandSecondHarmonicImaging 126
References 127
TAWhittingham
7ThePurposeandTechniquesofAcousticOutputMeasurement
129
7.1WhyMeasureAcousticOutputs? 129
7.2UltrasoundDamageMechanismsandtheirBiologicalSignificance 129
7.2.1Heating 130
7.2.2Cavitation 131
7.3TrendsinAcousticOutputs 133
7.4RegulationsandStandards 134
7.4.1FDA510(k)Regulations 135
7.4.2AIUM/NEMAOutputDisplayStandard 135
7.4.3IEC61157 136
7.5IsThereaNeedforIndependentMeasurements? 137
7.6WhichOutputParametersshouldbeMeasured? 137
Pagexii
7.7TheNewcastlePortableSystemforAcousticOutputMeasurementsatHospitalSites
138
7.7.1TheHydrophoneandPreAmplifier 138
7.7.2VariableAttenuator,PowerAmplifierandPowerMeter 140
7.7.3Oscilloscope 141
7.7.4OscilloscopeCamera,PCandDigitisationTablet 141
7.7.5TheMeasurementTank 141
7.7.6TheHydrophonePositioningSystem 142
7.7.7TheProbeMountingSystem 142
7.7.8CalibrationandAccuracy 142
7.8TheNPLUltrasoundBeamCalibrator 142
7.9MeasurementofAcousticPower 143
7.10FindingWorstCaseValues 145
7.10.1WorstCaseIsptaofStationaryBeams,e.g.PulsedDopplerMode
145
7.10.2WorstCaseIsptaforScannedBeamModes,e.g.BMode 146
7.11Conclusions 146
References 147
Part3UltrasoundHyperthermiaandSurgery
149
JeffreyWHand
8UltrasoundHyperthermiaandthePredictionofHeating
151
8.1UltrasoundHyperthermia 151
8.1.1Introduction 151
8.1.2UltrasoundIntensity,AttenuationandAbsorption 152
8.1.3TransducersforHyperthermia 154
8.1.4HighIntensityShortDurationHyperthermia 163
8.2PredictionofHeating 165
8.2.1ThermalConduction 165
8.2.2Pennes'BioheatTransferEquation 166
8.2.3OtherApproachestoThermalModelling 168
8.3Summary 171
Acknowledgments 172
References 172
GailRterHaar
9FocusedUltrasoundSurgery
177
9.1MechanismsofLesionProduction 178
9.1.1ThermalEffects 178
9.1.2Cavitation 179
Pagexiii
9.2LesionShapeandPosition 179
9.3SourcesofUltrasound 179
9.4ImagingofFocusedUltrasoundSurgeryTreatments 182
9.4.1UltrasoundTechniques 182
9.4.2MagneticResonanceImaging 182
9.5ClinicalApplications 182
9.5.1Neurology 182
9.5.2Ophthalmology 183
9.5.3Urology 183
9.5.4Oncology 184
9.5.5OtherApplications 184
9.6Conclusion 184
References 184
MichaelHalliwell
10AcousticWaveLithotripsy
189
10.1PercutaneousContinuousWaveSystems 189
10.2ExtracorporeallyInducedLithotripsy 190
10.2.1TypesofPressureWaveTransducer 190
10.2.2PositioningSystems 191
10.2.3FieldMeasurement 192
References 196
Part4UltrasoundandBubbles
197
TimothyGLeighton
11AnIntroductiontoAcousticCavitation
199
11.1TheAcousticPropertiesoftheBubble 199
11.1.1StiffnessandInertia 199
11.1.2Resonance 200
11.1.3InertialCavitation 201
11.2TypesofCavitation 206
11.3TheImplicationsoftheOccurrenceofOneTypeofCavitationfortheOccurrenceofAnother
210
11.3.1AlterationoftheBubbleSizebyRectifiedDiffusion 210
11.3.2AlterationoftheAcousticPressureFieldattheBubblebyRadiationForces
212
11.3.3Nucleation 214
11.3.4PopulationEffects 214
11.4TheImplicationsoftheOccurrenceofOnetypeofCavitationforCausingChangetotheMedium
217
11.5Conclusion 219
References 219
Pagexiv
DavidOCosgrove
12EchoEnhancing(UltrasoundContrast)Agents
225
12.1NonBubbleApproaches 225
12.2MicrobubbleAgents 226
12.2.1History 226
12.2.2SafetyofContrastAgents 229
12.2.3BasicPrinciples 230
12.2.4ClinicalApplications 230
12.2.5QuantificationandFunctionalStudies 233
12.2.6NewUses:AgentsandTechniques 234
12.3Conclusion 236
References 236
GarethJPrice
13SonochemistryandDrugDelivery
241
13.1CavitationanditsEffects 243
13.2WhatCanUltrasounddoforChemists? 245
13.3BioEffectsandDrugDelivery 252
References 256
Part5ResearchTopicsinMedicalUltrasound
261
JamesFGreenleaf,RichardLEhman,MostafaFatemiandRajaMuthupillai
14ImagingElasticPropertiesofTissue
263
14.1Introduction 263
14.1.1ExogenousTransverseWaves:ImagingwithMRE 263
14.1.2StimulatedAcousticEmission:ImagingwithUSAE 264
14.2MagneticResonanceElastography(MRE) 264
14.2.1Theory 264
14.2.2Methods 265
14.2.3MREResults 266
14.3UltrasoundStimulatedAcousticEmission(USAE) 270
14.3.1TheoryofUSAE 270
14.3.2USAEResults 272
14.4Conclusions 275
14.4.1MRE 275
14.4.2USAE 276
Acknowledgments 276
References 276
ChristopherRHill
15TheSignaltoNoiseRelationshipforInvestigativeUltrasound
279
References 286
Pagexv
JohnGTruscottandRolandStrelitzki
16ChallengesintheUltrasonicMeasurementofBone
287
16.1Bone 288
16.2UltrasonicMeasurementsSuitableforBone 289
16.2.1SpeedofSound(SOS) 291
16.2.2Attenuation 292
16.2.3Problems 295
16.3EffectofStructureonBroadbandUltrasonicAttenuation 295
16.4ProblemsintheMeasurementofSpeedofSound 297
16.4.1TimeDomain(ZeroCrossingPointMeasurement) 297
16.4.2FrequencyDomainMeasurements 300
16.5Discussion 303
Acknowledgment 305
References 305
Index 307
Pagexvii
CONTRIBUTINGAUTHORS
DrAndrewCBakerChristianMichelsenResearchASFantoftvegen38,Postboks60315020BergenNorway(FormerlyDepartmentofPhysics,UniversityofBath,UK)[email protected]
DrJeffreyCBamberJointDepartmentofPhysicsTheRoyalMarsdenNHSTrustDownsRoadSuttonSurreySM25PTUKjeff@icr.ac.uk
ProfessorDavidOCosgroveDepartmentofRadiologyHammersmithHospitalDuCaneRoadLondonW120NNUKdcosgrov@rpms.ac.uk
DrFrancisADuckMedicalPhysicsDepartmentRoyalUnitedHospitalCombeParkBathBA13NGUKf.duck@bath.ac.uk
ProfessorJamesFGreenleafBiodynamicsResearchUnitDepartmentofPhysiologyandBiophysicsMayoFoundationRochesterMN55905USAjfg@mayo.edu
DrMichaelHalliwellMedicalPhysicsandBioengineeringBristolGeneralHospitalGuineaStreetBristolBS16SYUKmike.halliwell@bris.ac.uk
DrJeffreyWHandRadiologicalSciencesUnitDepartmentofImagingHammersmithHospitalDuCaneRoadLondonW120NNUKjhand@rpms.ac.uk
ProfessorChristopherRHillStoneyBridgeHouseCastleHillAxminsterDevonEX135RLUK(FormerlyPhysicsDepartment,RoyalMarsdenHospital,UK)
Pagexviii
DrVictorFHumphreyDepartmentofPhysicsUniversityofBathClavertonDownBathBA27AYUKv.f.humphrey@bath.ac.uk
DrTimothyGLeightonFluidDynamicsandAcousticsGroupInstituteofSoundandVibrationResearchUniversityofSouthamptonHighfieldSouthamptonSO171BJUKtgl@isvr.soton.ac.uk
DrGarethJPriceDepartmentofChemistryUniversityofBathClavertonDownBathBA27AYUKg.j.price@bath.ac.uk
DrHazelCStarrittMedicalPhysicsDepartmentRoyalUnitedHospitalCombeParkBathBA13NGUKh.c.a.starritt@bath.ac.uk
DrThomasLSzaboHewlettPackardImagingSystemsDivision3000MinutemanRoadAndover,[email protected]
DrGailterHaarJointDepartmentofPhysicsTheRoyalMarsdenNHSTrustDownsRoadSuttonSurreySM25PTUKgail@icr.ac.uk
DrJohnGTruscottCentreforBoneandBodyCompositionResearchInstituteofPhysicalScienceDepartmentofClinicalMedicineWellcomeWingLeedsLS13EXUKj.g.truscott@leeds.ac.uk
ProfessorPeterNTWellsMedicalPhysicsandBioengineeringBristolGeneralHospitalGuineaStreetBristolBS16SYUKpeter.wells@bris.ac.uk
DrTonyWhittinghamRegionalMedicalPhysicsDepartmentNewcastleGeneralHospitalWestgateRoadNewcastleUponTyneNE46BEUK
Pagexix
GLOSSARY
Thefollowingsummaryincludessubstantiallyallthedefinedsymbolsandacronymswhichhavebeenusedthroughoutthebook.Whereverpossiblethesamesymbolhasbeenuseduniquelyforaparticularquantity.Onthefewoccasionswhenthishasprovedtobeimpossible,localuseisidentifiedinthetext.Converselywhen,rarely,ithasbeennecessarytousethesamesymbolindifferentchaptersfordifferentquantities,thisisnotedinthelist.Onoccasionssubscriptshavebeenusedinthetexttodefinethematerial(wforwater,tfortissueandsoon).Thislevelofdetailisexcludedfromthelistofsymbols.Thelistalsoincludesseveralsymbolswhichhavebeenusedarbitrarilyasconstantswithinparticularequations.
A
a,b constantsinequationforbubblebehaviour
A backscatteredsignalamplituderadiatingarea
A(f) amplitudespectrum
ACF autocorrelationfunction
AL
acousticlossfactor
B
B/A nonlinearityparameter
B constantrelatingtemperaturerisetotime
BUA broadbandultrasonicattenuation
C
c acousticwavevelocity
ccc meanpulsevelocity
cD elasticstiffnessconstant(clamped)
cg acousticgroupvelocity
cp acousticphasevelocity
C0 capacitance(clamped)
D
d beamdiameter
piezoelectricelementthickness
d3 3dBfocalzonewidth
D radiationforce(drag)coefficient
effectivebeamdiameter
dielectricdisplacement
DIFF(z) diffractioncorrectionfactor
E
E0 energydensity
E timeaveragedenergydensity
Pagexx
Ep pulseenergy
EL electriclossfactor
F
f acousticfrequency
fD
Dopplershiftedfrequency
fopt optimalfrequencyforhyperthemia
fr resonantfrequency(bubble)
F radiationforce
FEM finiteelementmodelling
FUS focusedultrasoundsurgery
G
g gravitationalacceleration
g(x,y) amplitudebeamprofile
G geometricfactorforacousticstreaming
amplitudefocusinggain
G(t) magneticfieldgradient
H
h heightofliquidcolumn
depthofsphericaltransducersurface
apiezoconstant
h(x,y,z) pointspreadfunction
H forwardpropagationoperator
H*t backwardpropagationoperator
I
i
I electricalcurrent
acousticintensity
I(z) axialintensitydistribution
I,Ita timeaveragedintensity
Ipa plusaveragedintensity
Isp spatialpeakintensity
Isppa spatialpeakpulseaveragedintensity
Ispta spatialpeaktimeaveragedintensity
I.3
'derated'intensity
I0 sourceintensity
IR
intensityatthecentreofcurvatureofasphericalsource
K
k wavenumber
thermalconductivity
keff effectivethermalconductivity
kT electromechanicalcouplingcoefficient
K constantrelatingtothermalequilibrium
K timeaveragedkineticenergydensity
L
ld discontinuitylength
lv lengthofvessel
l3 3dBfocalzonelength
L perfusionlength
M
m mass
MI mechanicalindex
Pagexxi
MR magneticresonance
MRE magneticresonanceelastography
O
ODS OutputDisplayStandard
P
p acousticpressure
p complexpressure
p0 acousticpressureamplitudeatsourceorforplanewave
pc acousticpressureatpeakcompression
pr acousticpressureatpeakrarefaction
pf acousticpressureatthefocus
pn acousticpressureamplitudeatharmonicn
popt leastpeakrarefactionpressurecausinginertialcavitation
p() probabilitydensity
prf pulserepetitionfrequency
P excesspressure
PE excesspressure(Euleriancoordinates)
PL excesspressure(Lagrangiancoordinates)
Pi generalisedpropertyvalueoftissuei
PLan Langevinradiationpressure
PRay Rayleighradiationpressure
PII pulseintensityintegral
PPSI pulsepressuresquaredintegral
PSF pointspreadfunction
PVDF polyvinylidenefluoride
PZT leadzirconatetitanate
Q
Q heatflux
Qfactoratresonance
QBF heattermtoaccountforbloodflow
Qe electricalQ
R
r vesselradius
ra radiusofacircularsource
rz radialdistanceatdepthz
r(t) postionvectorofthemovingspin
R reflectioncoefficient
radiusofcurvature
RA radiationresistance
Ropt criticalbubbleradiusforinertialcavitation
R0 bubbleradius
RF radiofrequency
S
s specificheatcapacity
s( t) crosscorrelationfunction
S area
specklecellsize
S,Sij
radiationstress(tensor)
Si
generalisedsignalvalueassociatedwithtissuei
Pagexxii
SAR specificabsorptionrate
SNR signaltonoiseratio
T
t time
tp pulseperiod
T temperature
strain
T(x,y,z) tissuebackscatterimpulseresponse
TI thermalindex
TIB boneatfocusthermalindex
TIC cranialthermalindex
TIS softtissuethermalindex
TOA timeofarrival
TOF timeofflight
U
u particlevelocity
u0 particlevelocityamplitude(sinewave)
u complexparticlevelocity
USAE ultrasoundstimulatedacousticemission
V
streamingvelocity
wavevelocityofpiezoelectricmaterial
vectorvelocityofsourceorobserver
V volume
voltage
V timeaveragedpotentialenergydensity
W
w perfusionvolumeflowrate
W,WA acousticpower
WE electricalpower
W.3 'derated'acousticpower
W1 acousticpowerfroma1cmlengthofarray
Wv absorbedpowerperunitvolume
X
x,y dimensionsorthogonaltothebeamaxis
X reactance
XA radiationreactance
Xeq vesselthermalequilibriumlength
X6 6dBbeamwidth
z dimensionparallelwithacousticaxis
Z
zf focaldistance
Z,ZA acousticimpedance
Z(f) Fouriertransform
Z(f)* complexconjugateFouriertransform
ZT electricalimpedance
amplitudeattenuationcoefficient
a amplitudeabsorptioncoefficient
a0 amplitudeabsorptioncoefficientat1MHz
s amplitudescatteringcoefficient
Pagexxiii
nonlinearityparameter
ijKroneckerdelta
acousticMachnumberS dielectricconstant(clamped)
gyromagneticratio
Gol'dbergnumber
k,k0 adiabaticbulkcompressibility
wavelength
constantassociatedwiththermal
contactbetweenvesselandtissue
intensityattenuationcoefficient
a intensityabsorptioncoefficient
s intensityscatteringcoefficient
bs backscatteringcoefficient
ds differentialscatteringcoefficient
v kinematicviscosity
angleofrefraction
phaseoffset
(f) phasespectrum
( ) transversemagnetisationphase
shearviscosity
, 0 equilibriumdensity
standardvariation
shockparameter
m nonlinearpropagationparameter
( , ) differentialscatteringcrosssection
timeshift
0 peakdisplacementofspin
beamprofilephasefunction
tissueorientationangle
angularfrequency
solidangle
Pagexxv
INTRODUCTION
FrancisADuck
Theweatheroftheweekof7June1997wasalmostperfectinOxford.ThevenuefortheThirdMayneordPhillipsSummerSchool'UltrasoundinMedicine'hadbeenchosentobeStEdmundHall,Oxfordonlyonprecedent:thetwoearlierMedicalPhysicsSummerSchoolshadbeensuccessfullyheldthere.Intheevent,theweekturnedouttobeaveryspecialoccasionforthefortyorsolecturersandstudentswhoattendeduniqueandmemorable(frontispiece).ApartfromthelossofBarry'sglassesduringthepuntingexpedition,thememoriesoftheweekremainverypositive,aweekoflearningandcompanionship,andnewandrenewedfriendships.
ThisbookisoneoutcomeofthatSummerSchool.AllthelecturerswhocontributedtotheSchoolhavepreparedchapters,eachbasedaroundthetopicoftheirownlecture.InanumberofcasesthechapterhasbeenlimitedtothematerialcontainedwithinthelectureinOxford,whileotherauthorshaveextendedtheirmaterialtoincludedetailsmoreeasilypresentedinaprintedform.Moredetailsofthebook'scontentandstructurearedescribedbelow.Initially,however,ashortbackgroundtotheMayneordPhillipsMemorialTrustwillbegiven,sincewithoutitsestablishment,thisSummerSchool,andthisbook,wouldnothavebeencreated.
TheMayneordPhillipsMemorialTrustwasestablishedin1994,thefirstTrusteesrepresentingthethreefoundingbodies:theBritishInstituteofRadiology,theInstituteofPhysicsandtheInstituteofPhysicalSciencesinMedicine(nowtheInstituteofPhysicsandEngineeringinMedicine).OneoftheoriginalTrustees,ProfessorKitHill,isawelcomecontributortothisbook.TheTrustdeedidentifiedoneobjectiveas'thefurthering,forthebenefitofthepublic,theknowledgeandunderstandingofallaspectsandallapplicationsofmedicalphysicsandkindredsciences...bytheorganisationofeducationalmeetingstobecalledtheMayneordPhillipsSummerSchools'.InadditiontheTrusteesshould'arrangeforthepublicationeitherinfullorinpartofanysuchSchools'.TheTrusteesdecidedthattheThirdSchoolshouldtakethetopic'UltrasoundinMedicine'andtousetheSchoolandsubsequentpublicationtoexploreabroadranging
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studyofmedicalultrasound,includingultrasoundpropagation,interactionwithtissue,andanexplorationofanumberofcontemporaryinnovationsintheapplicationofultrasoundinmedicine.Giventhisbackground,itisclearthatthecontentofbothSchoolandpublicationwasandisrathernarrowerthanthetitlemightimply.Thefocusisspecificallyonthescienceandtechnology,thephysicsandtheengineering,ratherthanontheclinicalapplications.Thisisnottosaythatclinicalapplicationsareabsent,sinceitisthenatureofapplyingphysicstomedicinethatthelinkbetweenscientificandengineeringdevelopmentandclinicalapplicationmustbefirmlymade.Neverthelesstheemphasisalwaysremainsthus:todrawfromthebasicsciencesthoseaspectswhichrelatemostcloselytothechallengeofapplyingthissciencetoaparticularclinicalneed:andtoreviewagainsttheclinicalneednowsuccessfultechnologicalinnovationhasbeeninusingnaturalsciencetoimprovemedicaldiagnosisandtreatment.
WVMayneordandCESPhillipsweretwooutstandingpioneersoftheapplicationsofphysicstomedicine.Intheirnatureaspioneerstheybothhadastrongconcerntohelpandencourageyoungercolleaguestodeveloptheirowninterestsandexpertise.Thefollowingparagraphsbrieflysummarisetheirlivesandcontributionstomedicalphysics.Furtherdetailsmaybefoundelsewhere[1,2].
MajorPhillipshasbeendescribedasthefirstBritishmedicalphysicist.Bornin1871,hisearlyexperimentswithdischargetubesledtohisdescriptionofthe'Phillips'phenomenon',therotationofaluminousringinanelectricaldischargetubewithinastaticmagneticfield.In1897,hepublishedacompletebibliographyofXrayliterature,probablythelastoccasionwhenthiswasapossibility.HisworkonradiationstandardsduringthefirstdecadeofthetwentiethcenturyledhimtobecommissionedtopreparethreeradiumstandardsfortheRoentgenSociety.HebecamethephysicisttotheXRayCommitteeoftheWarOfficeduringthe191418war.HeworkedwiththeradiologistRobertKnoxashonoraryphysicistattheCancerHospital(nowtheRoyalMarsdenHospital)uptohisretirementin1927,duringwhichtimehehelpedtodevelopthescientificbasisofradiotherapy,handlingradioactivematerials,andradiationprotection.
ValMayneordwas22yearsoldwhenhegainedhisfirstjobasamedicalphysicistatStBartholomew'sHospitalin1924.VerysoonaftermovingtotheCancerHospitalonPhillips'retirement,Mayneordstartedmakingmajorcontributionstoradiationdosimetry.UnlikePhillips,whopublishedlittle,Mayneordwasaprolificwriter.Hisfirstbook[3],publishedbeforehis30thbirthday,remainsoneoftheclearestearlydiscussionsofthescientificrealitiesofmedicalradiationtherapyandprotection.Hisyear'ssecondmenttoCanadabytheUKgovernmentafterthewaronlyfiredhisenthusiasmonhisreturnfortheapplicationofphysicstoawiderangeofmedicalproblems.Perhapsevenmoreimportantthanhisownpersonalscientificachievementswashisbuildingupofadepartmentofphysicsappliedtomedicineatthe
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RoyalMarsdenHospital,whichearneditselfaninternationalreputationforworknotonlyinradiotherapy,butalsoinnuclearmedicine,diagnosticultrasoundandseveralotherareasofmedicalphysics.Heachievedthisbyunderstandingthatgoodmedicinemustbebasedongoodscience,andthatgoodphysicalscientistsneedastrongandstimulatingenvironmentinwhichtothrive.Thisisastruenowasitwasthen.
Ultrasoundhasbeenalatestarterinitsapplicationtomedicine.EvenduringtheperiodofvigorousgrowthinapplyingphysicstomedicalproblemswhichValMayneordandhiscontemporariesexperiencedfollowingthewar,ultrasoundstillhadasomewhatsecondaryplacetotheinnovationsinnuclearmedicine,diagnosticradiologyandradiotherapy.Interestingly,acoustics,thestudyofthescienceofsoundwaves,andtheknowledgeofpiezoelectricitybothsubstantiallypredatethediscoveryofXraysandofradioactivityduringthelastdecadeofthenineteenthcentury.PerhapsitwastheastonishingbreadthofLordRayleigh'sbook'TheTheoryofSound'[4],firstpublishedin1877,whichdiscouragedothersfromattemptinganydeeperstudy.Maybethedramaticoverturnofclassicaltheoriesofphysicsattheturnofthecenturycausedacousticstobecomeapoorrelationinphysics.Ormaybeitrequiredanappropriatepracticalobjectivetodrawtogetheracousticscienceandtransducertechnologytowardstheexploitationwhichcharacterisesmedicalultrasoundattheendofthetwentiethcentury.
TheCuriesrediscoveredpiezoelectricphenomenaincrystalsfollowingBecquerel'sworkearlierinthenineteenthcentury[5],whichwasitselfbasedonworkbyHauyinthelate1700s.TheCuries'workseemstohavegeneratedinterestmostlyamongscientistsratherthanpracticalpeople(bothRoentgenandKelvinshowedactiveinterestinthephenomenon).Nevertheless,theonlypracticaloutcomeoftheirobservationsofthereversepiezoelectriceffectofquartz[6]seemstobethe'Quartzpizolectrique'[7]whichwasusedsoeffectivelybyMarieandPierreCurieintheircarefulexperimentalstudiesoftheradioactivityofradium.ItwaslefttoLangevin,whohadpreviouslybeenastudentoftheCuries,toexploitthestrongpiezoelectricpropertiesofquartzasaresonantelectroacoustictransducerforunderwaterecholocationfordepthsoundingandsubmarinedetection[8].Whileacousticdepthsoundinghadbeensuggestedintheearlypartofthenineteenthcentury,itwasLangevin'sworkduringtheFirstWorldWarwhichestablishedtwokeyelementsforitssuccess.Firstly,herecognisedthecompromiserequiredintheselectionoftheoptimumfrequency,balancingresolutionagainstpenetration.Thisledhimtoidentifythepotentialadvantageofusingfrequenciesabovetheaudiblelimit(about20kHz)forunderwaterecholocation.ShortlyaftertheTitanicdisaster,LewisFryRichardsonhadtakenoutapatentfortheuseof100kHzsoundforthesamepurpose[9],butthedeviceseemsnevertohavebeenimplemented.Richardsonisbetterknownforhisbookonthemathematicalprediction
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ofweather.Langevinrealisedhoweverthattomakesuchadevicework,greatersensitivitywasrequired.Hissecondinnovationwastoexploittheelectronicmethodsalreadyavailableforradiocommunicationtodevelopresonantmodesoftransmissionandreception,sosubstantiallyenhancingtheoutputpower,andthedetectionsensitivityofhisquartztransducers.(Asafootnote,Langevinalsodescribesclearlythemeasurementofacousticpowerusingaradiationforcemethod.)Itliesoutsidethescopeofthisintroductiontotracefurtherthedevelopmentfromthisearlyuseofultrasonicechodetectiontotheextremesophisticationofmodernmedicaldiagnosticsystems.Muchoftheearlymedicalwork,duringthe1940sand1950s,hasbeenwelldescribedelsewhere[10,11].
Intheremainingparagraphsabriefoverviewwillbegivenofthechapterscontainedinthebook,andofthewayinwhichtheyrelatetooneanother.
Ultrasoundisrapidlybecomingtheimagingmethodofchoiceformuchofdiagnosticmedicine,andinsomespecialistareasithasallbutreplacedotherdiagnosticmethods.Ithasbeenestimatedthataquarterofallmedicalimagingstudiesworldwideisnowanultrasoundstudy[12].ThisissupportedbyUKDepartmentofHealthdatawhichsuggestthatinNHSTrustsinEnglandalone,over4millionultrasoundimagingstudieswerecarriedoutinoneyear,ofwhichabout1.7millionwereobstetricscans(1996/97figures).Eventhishugenumberexcludesthestudiescarriedoutinprimarycare(GPpractices)andintheprivatesector.Itrepresentsthreescansforeachlivebirth.Thisastonishinggrowthhasbeenencouragedinpartbecauseultrasoundisperceived,itmustbesaidwithgoodreason,tobeadiagnosticmethodwithnoriskoverhead.Theassertionthatmedicalultrasoundissafehasbecomeatruism,andmanyofthedevelopmentsindiagnosticultrasound,especiallythoseusingDopplermethodsanddescribedbyPeterWellsinChapter6,weremadepossiblebecauseofthisview.Thiswastrueinspiteoftheconsiderableincreasesinintensityandacousticpowerrequiredforsomeapplications,andthesearereviewedbyTonyWhittingham(Chapter7).UndoubtedlymanyoftheadvanceshavecomeaboutbecauseofthedevelopmentoftechniquesinarraytechnologywhicharedescribedbyTomSzaboinChapter5.ThesehaveallowedparalleloperationinDopplerandpulseechomodessoastofullyexploittheabilityofultrasoundtoimagebothstructure(throughpulseechoscanning)andbloodperfusion(throughDoppler).Miniaturisationofarraysnowgivesaccesstodeepstructuresbyarrayinsertionintotherectum,oesophagusandvagina,andeventhroughthevasculartreeasfarasthecardiacarteries.Thedevelopmentofveryhighfrequencyminiatureprobesisanexcitingdevelopmentareawhich,sadly,isoneactivitynotcoveredhere.Vascularultrasoundisbecomingfurtherenhancedbydevelopmentsinechoenhancingcontrastmaterials.DavidCosgroveintroducesthisrapidlygrowingclinicalareainChapter12.
Bycomparison,therapeuticandsurgicalusesofultrasound,someofwhichpredatedthediagnosticmethods[13],havedevelopedinparallelbuthave
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failedthusfartoexerttheimpactonmedicinethatwaspromisedfromtheearlywork.Perhapsthishasbeeninpartbecauseinsufficientemphasishasbeenplacedontheproperscientificdevelopmentofthesemethods,includingafullrecognitionofthecareneededinacousticdesign,andinestablishingamorecompleteunderstandingoftheinteractionbetweenultrasonicwavesandtissue.Fromrecentsuccessesinhyperthermiaandfocusedsurgery,discussedfullybyJeffHandinChapter8,andGailterHaarinChapter9,andalsotheuseofavarietyofultrasonicapproachestolithotripsydescribedbyMichaelHalliwellinChapter10,itmaybeconfidentlypredictedthatultrasoundwillindeedfindavaluableplaceinthesurgicalandtherapeuticarmouryofthefuture,perhapscomparablewiththatachievedalreadybydiagnosticultrasound.
However,strongandsuccessfulapplicationsonlydevelopfromastrongunderstandingofthebasicscience.Itissurprisinghowofteninsufficientattentionisplacedinstandardmedicalultrasoundtextsonthedifficultiesindescribingfullythepropagationofdiagnosticultrasoundpulsesthroughtissue.Eventhedescriptionofpulsepropagationinafocusednearfieldunderlinearconditionsinalosslessfluidposessomedifficulties,andthesearedescribedcarefullyinChapter1byVictorHumphrey.Therealityoffiniteamplitudeeffectsandnonlinearacousticpropagationisnowknowntobenotanesotericsideissuebutcentraltoallmeaningfuldiscussionsofmedicalultrasound.AndyBakerintroducessomeaspectsofthisdifficulttopicinChapter2,whileFrancisDuck'ssubsequentchapterdescribestwopracticalnonlinearphenomena,acousticpressureandacousticstreaming.Chapter4byJeffBambergivesacompleteoverviewofasubjectwhichiscentraltoalldiscussionsofultrasoundinmedicine,theacousticpropertiesoftissue.Attenuation,absorption,scattering,soundspeed,nonlinearparameter,andtheirfrequencydependenciesarealldescribed.
Anunderstandingofbiophysicalprocessesisimportantintheinterpretationoftissue/ultrasoundinteractions,bothforanunderstandingofultrasounddosimetryintherapyandsurgery,andinsafetydiscussions.Thermalprocesseshavealreadybeennoted(Chapter8)ashasstreaming(Chapter3).Thethirdprocessisacousticcavitation,whichisdescribedbyTimLeightoninChapter11.Oftenacousticcavitationseemstobeatopicontheboundariesofinterestinmedicalapplications,buttheuseofcontrastmaterials(Chapter12)hasbroughtanewinterestinthetopic,inadditiontoitsimportanceinultrasoundsurgeryandinsafety.Cavitationishowevercentraltotheuseofultrasoundinchemicalprocessing,andGarethPricedescribeshowthisscientificcousinmayinstructandilluminateapplicationsinmedicine,forexampleindrugdelivery(Chapter13).
ThepurposeoftheSummerSchoolwasnotonlytoreviewtopicswhicharepartofpresentclinicalpractice,butalsotoallowanexplorationofsomeresearchtopicswherenewdevelopmentsareactive.Itisrarelypossibletoseparatefullystateoftheartapplicationsfromnewdevelopments,but
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thelastthreechaptersinthebookeachrepresentinonewayoranothernewstepsforward.Linkingtheexcitingcapabilitiesforimagingpresentedbymagneticresonanceimaging,JimGreenleafandhiscolleaguesdescribemethodsofgreatoriginalityforstudyingthemechanicalpropertiesoftissue,whichhavefascinatingpotentialindiagnosticmedicine.KitHillreturnstoafundamentalissueindiagnosticultrasoundimaging,signaltonoiseratio.JohnTruscottreviewscriticallysomeofthemethodscurrentlybeingusedtoinvestigateboneusingultrasound,andsuggestsalternativemethodswhichmayhavethepotentialofimprovedprecision.
Allthechaptersinthisbookhavebeenpreparedwithaviewtobridgingthegapbetweenthetutorialtextswidelyavailableforsonographerandmedicaltraining,andbooksofacousticswhichcontainfewlinksbetweentheoreticalacousticsandtheapplicationsofultrasoundtomedicine.Somematerialwhichisverywelladdressedinstandardmedicalultrasoundtextshasbeendeliberatelyomittedforexamplethedescriptionofbasicpulseechoandimagingmethods,qualityassurancetestsusingphantoms,artefactgenerationandavoidanceandsoon.Readersaredirectedtowardsoneofmanytextswhichnowincludethismaterial.Thepresentbookisofferedasauniquecollectionofchapterscontainingwellreferencedmaterialofdirectrelevancetoanystudentwishingtoexploremedicalultrasoundatdepth.Wherespacelimitedthescopeofanychapter,amplereferencingwillallowtheseriousstudenttodiscoveramuchwiderbaseofknowledge.Itishopedthatthesepages,whichhavebeenpreparedwiththespiritofMayneordandPhillipsinmind,willservetoilluminateandinstructanywhowishtolearnatgreaterdepthofthescienceandtechnologyintheapplicationofultrasoundtomedicine.
Acknowledgments
Iwouldliketoaddmyheartfeltthankstoallthelecturersandauthorswhowerecajoledintotakingpartinthisenterprise,moreorlesswillingly.IalsoacknowledgetheenormoussupportgivenbyAndyBakerandHazelStarritt,coorganisersoftheSummerSchool,andcoeditorsofthisbook,whosewarmandsteadysupportwasessentialinthesuccessofbothprojects.AndfinallywewouldliketodedicatethebooktoallthosewhoattendedtheSummerSchoolwho,notknowingwhattheywerelettingthemselvesinfor,enjoyeditanyway.
References
[1]HillCRandWebbS1993TheMayneordPhillipsSummerSchools:BackgroundtotheSchoolsandShortProfilesoftheTwoPioneeringPhysicists(SuttonandLondon:InstituteofCancerResearchandRoyalMarsdenHospital)
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[2]SpiersFW1991WilliamValentineMayneordBiographicalMemoirsoftheRoyalSociety3734164
[3]MayneordWV1929ThePhysicsofXRayTherapy(London:Churchill)
[4]Rayleigh,Baron:StruttJW1877TheTheoryofSound(London:Macmillan)
[5]BecquerelAC1823Expriencessurledveloppementdel'lectricitparlapressionloisdecedveloppementAnnalesdeChimieetdePhysique22534
[6]CurieJandCurieP1881ContractionsetdilatationsproduitespardestensionslectriquesdanslescristauxhmiedresafacesinclinesCompteRenduAcad.Sci.Paris93113740
[7]CurieJandCurieP1893QuartzpizolectriquePhil.Mag.(5thSer.)363402
[8]LangevinP1924TheemploymentofultrasonicwavesforechosoundingHydrographicRev.IINo1,Nov1924,5791
[9]RichardsonLF1912ApparatusforwarningashipatseaofitsnearnesstolargeobjectswhollyorpartlyunderwaterUKpatent11125
[10]WhiteDN1976HistoricalsurveyUltrasoundinMedicalDiagnosisedDWhite(Kingston:Ultramedison)pp136
[11]LeviS1997Thehistoryofultrasoundingynaecology19501980UltrasoundMed.Biol.23481552
[12]WFUMB1997WFUMBNews4(2)UltrasoundMed.Biol.23followingp974
[13]BergmannL1938UltrasonicsandTheirScientificandTechnicalApplications(NewYork:Wiley)
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PART1THEPHYSICSOFMEDICALULTRASOUND
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Chapter1UltrasonicFields:StructureandPrediction.
VictorFHumphreyandFrancisADuck
Introduction
Pulsedultrasoundbeamssuchasthoseuseddailyduringmedicalexaminationshaveacousticstructuresofconsiderablecomplexity.Thisistrueevenwhenconsideringtheirpropagationsimplythroughanidealisedacousticallyuniformmediumwithnoloss.Propagationthroughtheacousticinhomogeneitiesofbodytissuesresultsinfurtheralterationsintheacousticfield,bothfromscatteringatsmallscaleandfromlargescaleinterfaceeffects.Thepurposeofthischapteristodiscussthefactorswhichcontroltheacousticbeamsusedformedicalapplications,andtodescribethesebeamsandthemethodsfortheirprediction.Thepropagationmodelsusedwillbelimitedtothosewherethebeamisassumedtobeofsufficientlysmallamplitudethatlinearassumptionsmaybemadeabouttheacousticwavepropagation.Whilethisisaninvalidassumptionforverymanymedicalultrasoundbeamsinpractice,itallowsinstructiveanalysestobedeveloped.SomeofthebeamcharacteristicswhichariseduetofiniteamplitudeeffectsaredescribedinChapter2.Thesecondbroadlimitationinthischapteristhatconsiderationislimitedtoalosslessliquidmediumwhichisacousticallyhomogeneous.Aswillbeseen,theuseofthesetwoassumptionsallowsthepropagationoftheultrasoundwavetobedescribedintermsofonlytwoacousticquantities,thewavenumberandtheacousticimpedance,togetherwithinformationaboutthesourcegeometry.Considerationofthesourceoftheultrasoundwave,thetransducer,isgiveninChapter5.
Forthemajorityofdiagnosticapplicationsofultrasoundtherangeoffrequenciesusedisquitenarrow,210MHz,wheretheparticularfrequencyisselectedinordertoachievethebestcompromisebetweenspatialresolutionanddepthofpenetration.Higherfrequenciesareonly
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usedinspecialisedapplicationssuchasophthalmology,skinimagingandintravascularinvestigations,reachingexperimentallyashighas100MHzormore.Frequenciesbelow2MHzareusedinDopplersystemsforfetalheartmonitoring,andthelowerpartoftheultrasonicspectrumisalsousedfortherapeuticandsurgicalapplicationssuchaslithotripsy(Chapter10)orhyperthermiaandfocusedultrasoundsurgery(Chapters8and9).Sonochemistry(Chapter13)andsomebioeffectsstudiesuseultrasonicfrequenciesbelow100kHzthischapterwillnotconsidertheparticularissuesassociatedwithbeamsusingsuchlongwavelengths.
Havingestablishedthesimplifyingassumptions,severalcomplicatingfactorsofparticularrelevanceinmedicalapplicationsofultrasoundareintroduced.Itisrecognisedthatitispulsedratherthancontinuouswaveultrasoundbeamswhichareofmostinterest.Inordertoachievegoodspatialresolution,theultrasoundpulsesarelessthan1mminlength.Thevelocityofsoundthroughsofttissues,ct,liesintherangeapproximately14501600ms1(seeChapter4),sotherangeofacousticwavelengths, (=ct/f),isabout0.15to0.75mm.Thepulsesthemselvesarethuscommonlylessthan1slongandconsistofveryfewacousticcycles:theyareallendandnomiddle.Thisfactresultsinsignificantdifferencesbetweenthebeamprofilesinsuchpulsedbeamsandthoseatasinglefrequencyfromthesamesource.Asecondimportantpracticalconsiderationisthatmedicalultrasoundpracticeusuallycouplesthetransducerdirectlytothetissuetobeinvestigated.Thismeansthatsignalsarereturnedfromthe'nearfield'ofthetransducer,inaregionwhichmaybestronglyinfluencedbythesizeandshapeofthetransduceritself.Theanalysisofthenearfieldisthereforeofsignificance.
Symmetryisofconsiderableimportanceinthestructureofacousticfields.Theanalysisofbeamswithcircularsymmetryhasbeenwelldevelopedintheliterature,andthiswillbedescribedbelow.Howeverthemajorityofmodernscannersdonotusecircularsourcesofultrasound,linear,curvilinearand'phased'arraysbeingalmostuniversallyused.Forthisreason,theanalysisofrectangularsourcesisveryimportant.Thefinalsignificantfactoristhatallpracticalmedicalultrasoundtransducersvaryinbothamplitudeandphaseovertheiraperture:thatistheyareapodisedandtheyarefocused.
Recognisingthesecomplexities,thischapterwillcommencewithasimpledescriptionofthebeamfromacircularplanepistonsourceofultrasound,andproceedtodescribethewayinwhicheachofthemorecomplexfeatureswhicharerelevanttothedescriptionofthestructureofmedicalultrasoundbeamshavebeenaddressed.
1.1CircularPlaneSources
Itiscommonfirsttoconsidertheacousticfieldgeneratedbyaplanecircular'piston'sourcewhichisvibratingwithasinusoidalmotiononlyinadirection
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Figure1.1.Asimplifiedaxialsectionoftheacousticfield
fromaplanecircularsinglefrequencysourcewhosediameterissignificantlygreater
thantheacousticwavelengthinthepropagatingmedium.
perpendiculartoitssurface.Theanalysisofthisfieldisincludedinmanytexts(Wells1977,Kinsleretal1982).Thisfieldhasparticularcharacteristicswhicharisefromtheveryspecifictemporalandspatialsymmetriesofthesource.Inpractice,thebeamsfrommanyphysiotherapytransducersapproximateinstructuretothefollowingdescription.
Thesimplestviewofthefieldofacirculartransducerwouldconsiderthefieldtobeaplanewaveofthesamediameterasthetransducerinthenearfield(theFresnelregion)andthentobeanexpandingsphericalwaveinthefarfield(Fraunhoferregion).Thisisshowninfigure1.1.
ThetransitionoccursattheRayleighdistancezR,where
whereaisthetransducerradius, isthewavelengthandk(=2 / )isthewavenumber.
Thismodelisconceptuallysimple,anduseful.Itispossible,forexample,tocalculatetheapproximateRayleighdistanceforatypical1MHzphysiotherapytransducerwitha=12.5mminwatertobeabout33cm,demonstratingthatforsuchatransducer,alltreatmentoccursinthenearfield.Neverthelessthissimplemodeldoesnotallowfordiffractionandinterferenceandfurtherdevelopmentisrequired.
Consideraplanar,circular,transducermountedinarigidbaffle(surface)andradiatingintoafluid.InordertocalculatethefieldduetothetransducerassumethateachsmallelementdSofthetransducersurfacevibratescontinuouslywiththesamevelocityunormaltothesurface(thex,yplane)where
TheneachelementdSgivesrisetoasphericalwavecontributinganelementalpressurecontributiondpataranger'of
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Figure1.2.Thegeometryforthecalculationofthepressurefieldp(r, ,t)atan
observerpointO,duetoaplanecircularpistonsource.
where 0isthedensityofthefluid.Theresultantfieldofthetransducercanbeevaluatedbyaddingupallofthecontributionsduetothesmallelements.Inthelimitthissummationbecomesanintegralandtheresultantpressurefieldp(r, ,t)isgivenby
Thesurfaceintegralisboundedbythecondition a,where istheradialpositionofthesurfaceelementdS(seefigure1.2).Theexpressioninequation(1.4)isoftenknownastheRayleighintegral.
Ingeneralitisonlypossibletofindsimpleclosedformsolutionstothisintegralforspecialsituations,thatisalongthesymmetryaxisofthetransducerandinthefarfield.Otherwisealternativenumericalstrategiesareneeded.Forexample,StepanishenandBenjamin(1982)andWilliamsandMaynard(1982)havedevelopedmethodsforthepredictionofacousticfieldsusingaspatialFouriertransformapproach,recognisingthatthefarfieldbeampatternistheFouriertransformoftheaperturefunction.Thisapproachcanbeofconsiderablevalueininvestigatingthefarfieldofnoncircularsources,suchastherectangularsourcesdiscussedinsection1.5.Inprinciplesuchmethodscanbefastandaccurate,providedthatthespatialgridisfineenoughtopreventspatialaliasing.ForthenearfieldthetimedomainnumericmethodsusedbyZemenek(1971)havebeenveryeffectiveinevaluatingthediffractionintegral.
1.1.1PressureVariationontheAxis
Theaxialpressurevariationmaybederivedfromequation(1.4).Geometricalconsiderationsgive:
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whered istheincrementalwidthofasurfaceannulusofradius .Substitutioninequation(1.4)gives
thatis,theintegrandisaperfectdifferential.Substitutingandevaluatingat =aand =0givestheaxialcomplexpressurep(r,0,t):
Thepressureamplitudeisthemagnitude(i.e.therealcomponent)ofp(r,0,t).Expressedinrectangularcoordinates,replacingrbyz,thedistancealongthebeamaxisperpendiculartothesource,theaxialvariationofthepressureamplitudep(z)is
Thisvariationisshowninfigure1.3,fromwhichitmaybeseenthattheaxialpressurevariationinthenearfieldischaracterisedbyaseriesofunequallyspacedpressuremaxima,withvalue2
r0cu0,separatedbylocalisedfieldnullswherethepressureiszero.Sincethepressureamplitudeatthesource,p0,is
r0cu0,thenearfield
pressuremaximahaveanamplitude2p0.Inthoseregionswherethesineisnegativethephaseofthepressurewaveisreversed.
Thepositionsofthemaximaandminimamaybecalculatedfromtheconditionsgivingthesinefunctioninequation(1.10)valuesof1(maxima)and0(minima).Thatis,
wherethepositionsofthemaximaarisewhenmisodd(m=1,3,5,...)andthepositionsofthenullsarewhenmiseven(m=2,4,6,...).Themost
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Figure1.3.Thecalculatedvariationofacousticpressureontheaxis
ofaplanecirculartransducerof38mmdiameterat2.25MHz.Thepressureisnormalisedtop0,thepressureatthesource.
distantmaximumfromthesourceiscommonlyreferredtoasthe'lastaxialmaximum'itsposition,zlam,maybecalculatedapproximatelybysettingm=1inequation(1.11),andassumingthataka,whenequation(1.10)reducesto
whereSistheareaofthesource.Equation(1.12)showsthattheaxialpressureinthefarfieldreduceswith1/(distance).
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Figure1.4.Contourplotofthenormalisedacousticpressure(p/p0)foracirculartransducerofradiusa=5 .
Theaxialdistanceisnormalisedtoa2/ .
1.1.2PressureVariationOfftheAxis
Atpositionsofftheaxisofsymmetrytheultrasoundpressurefieldhasconsiderablecomplexity.Acalculatedexampleofthenormalisedpressurefieldamplitudep/p0foracircularapertureofradiusa=5 isshowninfigure1.4,usinganumericalapproach(seeZemanek1971).Thecompletepressurefieldmaybethoughtofasbeingformedbyrotatingthisradialsectionaroundthezaxis,andconsistsofringsofhigheracousticpressurewhosenumberandradialfrequencyincreaseasthesourceisapproached.Itmayalsobeseenthatthe6dBbeamwidthatzlam(a2/ )isonlyabout0.4thatattheaperture,demonstratingthesocalled'selffocusing'ofaplanesource.
Thealternativerepresentationofthenearfieldpatternshowninfigure1.5emphasisesanalternativeapproachtotheanalysisoftheultrasonicnearfield,originallyusedbySchoch(1941).Heshowedthatthefieldcouldbeconsideredasaconvolutionbetweentwoparts,oneaplanewavepropagatingnormallyfromthesource,andtheotherawavefromitsboundary.Theinterferencebetweenthesetwowavesmaybeclearlyseeninthefieldpatternshowninfigure1.5.Thisviewofthefieldasbeingcomposedofaplanewaveandan'edgewave'isparticularlyusefulwhenconsideringthecharacteristicsofpulsedultrasonicbeams(seebelow).
InthefarfieldtheoffaxisacousticpressurepqmaybeexpressedintermsofitsdirectivityfunctionD
q:
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Figure1.5.Spatialdistributionofthenormalisedacoustic
pressure(p/p0)foracirculartransducerofradiusa=5 .
Theaxialdistanceisnormalisedtoa2/ .
andJ1isBessel'sfunctionofthefirstkind.J1(kasin )=0whenkasin =3.83,7.02,10.17,13.32etc.Thatisthefieldisformedofacentrallobe,andsidelobes.Theboundaryofthecentrallobeoccurswhere =sin1(0.61 /a).
1.2PulsedFields
Theimportanceofthedifferencesbetweenthesinglefrequencyandbroadbandpulsedbeamscannotbeoveremphasised.Severalauthorshavereviewedthestructureofpulsedfields(Friedlander1958,Harris1981a,Wells1977,Duck1980,KrautkramerandKrautkramer1990).Whilethefarfieldmainlobeisnotmuchaffected,thespatialvariationsinthepressurenearfieldbecomesmaller,andsidelobesmaydiminishinamplitudeandmerge.Inadditionthepressurepulsewaveform,anditsspectrum,varywithposition,includingthepotentialforpulsesplitting.Thesealterationsbecomeimportantoncethepulselengthreducestolessthansixcyclesofoscillation(KrautkramerandKrautkramer1990).Theyarethusimportantforallmedicalpulseechoultrasoundbeams,andalsoformanypulsedDopplerapplicationswhenshorterpulselengthsareused(DuckandMartin1992).
Oneapproachtotheanalysisofapulsedfieldistoconsideritasasummationofthecomponentfieldsofallthespectralcomponentscomprising
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thepulsespectrum(PapadakisandFowler1971).Foranysourceradius,thepositionsofnearfieldmaximaandminimadependonthewavenumberk(seeequation(1.11)),andhenceonthefrequency.Summationofallthespectralcomponentswillthusresultinasmearingofthelocalspatialvariationsinacousticpressure.Thisapproachhasbeenvaluableinitsabilitytointroduceattenuativelossintothecalculationofthepulsedacousticfield,withitsassociatedfrequencydependence.However,itisnecessarytogenerateasufficientsamplingofthefrequencydependentfieldstoavoiderrors,andgenerallythismethodmayonlybeexpectedtogivegoodapproximationsratherthanexactsolutionstothepredictionofthepulsedacousticfield.
Awidelyusedalternativemethodhasdevelopedfromtheanalysisofthetemporalimpulseresponseofasource.Thisallowsthepressurep(r,t)tobecalculatedfromtheconvolutionbetweenasourcefunctionandthepressureimpulseresponsefunctionh(r,t):
where*indicatesatemporalconvolution.Sinceh(r,t)isafunctiononlyofthesourceshape,andu0(t)isafunctionofthesourcevibrationonly,equation(1.15)givesapowerfulgeneralapproachtotheanalysisofavarietyofsourcegeometries,inadditiontothecircularpistonsourcewhichhasbeenconsideredsofar.FollowingStepanishen'soriginalpublicationsforthecircularpiston(Stepanishen1971,1974,Beaver1974),expressionsforh(r,t)foranumberofothersourcegeometrieshavebeenpublished:includingthoseforrectangularsources(LockwoodandWillette1973),shallowbowl(focused)sources(PenttinenandLuukkala1976a),andsourceswithavarietyofapodisingfunctions(Harris1981b).
Anexampleofacalculationofthepressurewavefrontinthenearfieldofaplanepistonsource,excitedusingasinglesinusoidalcycle,isshowninfigure1.6(a=8mm,f=4MHz,z=40mm).Thetimescaleisexaggeratedinordertoemphasisethepulsestructureacrossthebeam.Thepulsewaveformontheaxisconsistsoftwocomponentsseparatedintime.Thefirstisthatfromtheplanewavepropagatedfromthesource.Thesecondoccursfromtheconstructiveinterferenceoftheedgewave,andhasbeentermeda'replicapulse'.Itarrivesontheaxisatatimedelay
Itsphaseisinverted,anditsamplitudeisthesameasthatoftheplanewave.Offtheaxistherearetworeplicapulseswithloweramplitudes,becauseofincompleteconstructiveinterferencebetweentheedgewavecomponents.
Page12
Figure1.6.Calculatedpulsepressureprofileforacircularpistonsource,radius
8mm,vibratingwithonecycleat4MHz.c=1500ms1z=40mmanda2/
=170mm.
Intheregionoutsidetheprojectionofthesourcearea,theplanewavecomponentisabsent,andonlythetwoedgewavecomponentsexist.Inthisexampleitisonlyatthelateralboundariesofthebeamthatthereisoverlapbetweentheedgewaveandtheplanewave.Asthewavepropagates,sothedelaybecomessmallerbetweentheplanewavecomponentanditsreplicapulse(seeequation(1.16)).Interferencebetweenthetwocomponentscanonlyoccuratdistanceswherethetimedelay thasbecomelessthanthepulselength,andonlybeyondthisdistancedoesonaxisvariationinpulsepressureamplitudeappear.Experimentalobservationsofedgewavesandreplicapulseshavebeendemonstratedby,forexample,WeightandHayman(1978).
Othermethodsappropriatetothecalculationofbothpulsedandsinglefrequencybeamsarefiniteelement/boundaryelementmethods,andfinitedifferencemethods.Finitedifferencemethodshavefoundmosteffectiveapplicationinthepredictionofbeamswithinwhichfiniteamplitudeeffectsareimportant,givingrisetononlinearacousticbehaviour.Becauseoftheimportanceoftheseeffectsinmedicalultrasound,theyaredealtwithseparatelyinChapter2.
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Figure1.7.Geometryofasphericalfocusingsource.
1.3FocusedFields
Formanymedicalapplicationsofultrasoundtheneedarisestoreducethebeamwidth,ortoincreasethelocalpressureamplitude,orboth,fromthevalueswhichareeasilyachievedusingplanetransducers.Thisrequiresthebeamstobefocused,soachievingimprovedspatialresolutionforimaging,orintensitiesofsufficientmagnitudetodestroytissue(seeChapter9).UsefulanalyseshavebeenpublishedbyO'Neil(1949),Kossoff(1979)andLucasandMuir(1982).
Thesimplestmeansforfocusingabeamisbytheuseofasphericalcap,orbowltransducer(seefigure1.7).
TheamplitudegainGforabeamfromasourcewithradiusofcurvatureRcanbecalculatedapproximatelyby
Itiscommontoconsiderthreetypesoffocusedbeam,categorisedbytheirdegreeoffocusing.Theseare:
weakfocus:0
Page14
Figure1.8.Axialvariationofnormalisedacoustic
pressure(p/p0)insphericallyfocusedfields
withdifferentfocalgains(a)G=2and4(b)G=6,8,and10.Thedistanceisnormalised
tothegeometricfocallength,R.
onaxisminimamayoccurbeyondthefocus.Examplesofaxialprofileswithamplitudefocalgainsof2,4,6,8,and10areshowninfigure1.8(a)and(b).Theaxesarenormalisedtothesourcepressurep0andtotheradiusofcurvatureR.Theaxialvariationisgivenapproximatelyby
Page15
Atthegeometricfocus(infigure1.8wherez/R=1),thepressurepR0=G
p0.However,themaximumaxialpressurealwaysexceedsthisvalue,andthemaximum(i.e.
theacousticfocus)isalwaysreachedatapositionclosertothesourcethanthegeometricfocus.Furthermore,thisseparationbetweenthegeometricandacousticfocidecreasesasthefocusinggainincreases.Whileagainof10isassociatedwithanapproximate10%shiftinposition,foragainof4(usedinsomediagnosticsystems)theacousticfocusmayoccuronlyatabout0.7R.Ithasbecomeconventionalinpulseddiagnosticbeamstogivethefocallengthintermsoftheacousticfocus,wherethebeamintensityreachesamaximum(seeChapter7).Inspectionoffigure1.8a,balsodemonstratesthatthelengthofthefocalzonedecreasesasthegainandpR0/p0increases.TheradialpressurevariationinthefocalplanepR(y)isgivenapproximatelyby
whereyistheoffaxisdistance.pR(y)reducesto3dBofitsaxialvaluewheny=1.62R/ka.Inotherwordsthe3dBbeamwidthatthefocus,d3,is
Pulsedfocusedbeamsdifferfromsinglefrequencybeamsinamannercomparablewiththedifferencesforplanesources.Impulseresponsefunctionshavebeendevelopedforsphericalbowlsources(PenttinenandLuukkala1976a),forfocusinglenses(PenttinenandLuukkala1976b)andforconicalradiators(PattersonandFoster1982).Figure1.9demonstratesacomputedpulseprofileforacircularbowlsource,radiusofcurvature80mm,underconditionsotherwisesimilartothoseusedforfigure1.6.Theedgewavecomponentisstillvisible,buttheplanewaveisnowasphericalwaveconvergingtowardsthegeometricfocus.
1.4SourceAmplitudeWeighting
ThetheoreticaldevelopmentinalltheprecedingsectionshasassumedthatmovementofallelementsdSoverthesourcehasbeenofequalamplitude.Focusingcanbeconsideredsimplyasthealterationoftherelativephaseofthemovementoftheelements.Howevermostrealtransducersarenottrue'piston'sources,thatis,thereissomevariationinthesourceamplitudeoverthesourcearea.Thismaycomeaboutdeliberately,ashappenswithsomearraysforwhichweighting,orapodisation,isappliedacrossthearray.Itmayoccurbecauseofthephysicalmountingofthetransducer,forexampleiftheedgesarelessfreetomovethanthecentreofapiezoelectricelement,
Page16
Figure1.9.Calculatedpulsepressureprofileforacircularsphericalbowlsource,
radius8mm,vibratingwithonecycleat4MHz.c=1500ms1z=40mm.
Comparewithfigure1.6.
or,whenusingalens,fromthetransmissionlosswhichvariesfromthelensaxistoitsedge.Smallerscalevariationsacrossthesourceareamayalsooccur.Theseoccurbecauseoftheconstructionofanarrayandmayalsocharacterisethebehaviourofaphysiotherapytransducerdrivenatitsthirdharmonic.
Eachoftheseexamplesindicatesthatfieldsfromrealtransducersmaywelldifferfromthedescriptionderivedfromtheformaltheorysetoutinthischapter.Inpracticeitisquitecommontoapodiseanarrayusedinadiagnosticscanner,forexampleusingaGaussianamplitudeweightingfunction(DuandBreazeale1985).Figure1.10showstheoutcomeofGaussianapodisationforaplanesinglefrequencycircularsourcewitha=5 .Theeffectofapodisationistoreducethepressurevariationsinthenearfield,andtoreducethesidelobelevelinthefarfield.AmorecompletedescriptionoftheeffectsofradialweightingonpulsedfieldshasbeengivenbyHarris(1981b).
Foraplanecircularsourcetheeffectofapodisationcausedbyedgetetheringistoalterslightlythepositionofthelastaxialmaximum.Forthisreasonitiscommontoconsiderthesourceashavinganeffectiveradiusaeffsuchthatthemeasuredlastaxialmaximumliesat(aeff)2/ .
Page17
Figure1.10.Calculatedsinglefrequencyvariation
inpeakpressureamplitude,foraGaussiansourceapodisation
witha=5 :(a)axialpressurevariation(b)offaxispressure
variation.
1.5RectangularSources.
Theradialsymmetryassociatedwithbothplaneandsphericallyfocusedcircularsourceshasaparticulareffectonthebeamsproduced,particularlyalongthebeamaxis.Thisistruewhethersinglefrequencyorimpulsebehaviourisconsidered,andalsoforpracticalapplicationsusingbroadbandpulses.Anyalterationfromthecircularsymmetryofthesourceservestoalterthegeometricconditionsandhencethebeam.Ofparticularpracticalinterestformedicalapplicationsisthebehaviourofrectangularsourcesofultrasound,becauseoftheircommonuseinthearraysusedfordiagnosticimagingandassociatedDopplerapplications.ThedesignandfabricationofsucharraysisdescribedingreaterdetailinChapter5.Herewewillbeconcernedonlywiththetheoreticalconsiderationsoftheacousticbeamsgeneratedbysuchrectangularsourcesofultrasound.
Whilecircularlysymmetricbeamsmaybeanalysedintermsofonlytwo(rectangular)coordinates(y,z),anyothershapeofsource,includingrectangular,requiresthree(x,y,z)(figure1.11).Insimpletermsthebeamformation,bothintermsofthelengthofthenearfieldandthedivergenceinthefarfield,iscontrolledseparatelybythetwoorthogonaldimensionsofthesource,2aand2b.Ifa
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Figure1.11.Coordinatesystemforarectangularsource.
whichthereisoffaxisamplitudemodulationintheydirectionthaninthexdirection,andthebeamwilldivergemorestronglybeyondthisregion.Ifa=b,thesourcewillbehavesomewhatlikeacircularsourcewithradiusa,andtherewillbeamaximumontheaxisatz a2/ ,althoughthenearfieldamplitudemodulationwillnotbesopronounced.
Figure1.11showsanobservationpointOat(x,y,z)andasourceelementdS=dx0dy0at(x0,y0).Geometricconsiderationsgive
(r')2=z2+(xx0)2+(yy0)2
andusingabinomialexpansionr'canbeapproximatedby
Further,substitutingintheequationfortheRayleighintegral(equation(1.4))andreplacingthe1/r'termwith1/rwehave
Substitutingg=x0 (2/z )g0=b (2/z )h=y0 (2/z )h0=a (2/z )anddefiningtheaspectratiooftherectangleN=a/bsoh0=b/N (2/z )wehaveonaxis
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Figure1.12.Axialvariationinnormalisedacousticpressureamplitude(p/p0)forasquare
transducer.Comparewithfigure1.3foracircularsource.
Figure1.13.Axialvariationinnormalisedacousticpressure(p/p0)forarectangularsource
withaspectratio1:2.
Thetwointegralsontherighthandsideofequation(1.23)giveanamplitudedependencyonb2/z anda2/z .TherelativelocationsoftheacousticfeaturesinthebeamwilldependontheaspectratioN.
Thecalculatedaxialpressurevariationisshownforasquaretransducerinfigure1.12andforarectangularaperturewithaspectratioN=1:2infigure1.13.Pressuremaximaandminimavariationsarereducedforthesquaretransducer,andlargelyabsentfortherectangulartransducer.Inadditionthemaximumpressureamplitudeonaxisislessthan2p0,whichisreachedtheoreticallyontheaxisinthenearfieldofaplanecircularsource.
Figure1.14showsthepulsepressureprofileat40mmfromasquare
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Figure1.14.Calculatedpulsepressureprofilefor
asquarepistonsource,side2a=16mm,vibratingwithonecycleat4MHz.c=
1500ms1z=40mm.Compare
withfigure1.6.
source,a=8mm,operatingforasinglecycleat4MHz,calculatedusingtheimpulseresponsefunctionderivedbyLockwoodandWillette(1973).Theconditionsarethesameasthoseusedtocalculatetheprofileshowninfigure1.6foracircularsource,withwhichcomparisonmaybemade.Thisshowsthepresenceoffurtherreplicapulsesbeingcausedbytheedgewavesfromeachofthefouredgesofthesource.
Medicalultrasonicscannersnotonlyuserectangulartransducers,butapplyastigmatic(cylindrical)focusing.Theanalysisofthesefocusedpulsedfieldsmustbecarriedoutusingnumericalmethodssuchasthefinitedifferencemethodswhichwerementionedabove.
1.6Conclusion
Thischapterhasshownthatitispossibletodescribetheacousticpressurefieldstructurefromavarietyofsourcegeometries,usingappropriateapproximations.Theadventofpowerfulcomputershasenablednumericalmethodstobeappliedtopreviouslyintractableanalyses.Theparticularsymmetryassociatedwiththeacousticfieldfromanidealcircularpistonsourcevibratingwithasinglefrequencyisnotsharedbyothermorepractical
Page21
ultrasoundbeamsusedformedicalapplications.Apodisation,pulsingandtheuseofrectangularsourcesallservetosmoothoutthepressurevariations,resultinginbeamswithlowamplitudemodulationinthenearfield,andlowacousticsidelobeamplitudes.Thefullanalysisandpredictionofthefieldgeneratedbyapulsedrectangularsourcewithastigmaticfocusingandapodisation,propagatingnonlinearlythroughaninhomogeneousabsorbingandscatteringmedium,remainsachallenge.
References
BeaverWL1974SonicnearfieldsofapulsedpistonradiatorJ.Acoust.Soc.Am.5610438
DuGandBreazealeMA1985TheultrasonicfieldofaGaussiantransducerJ.Acoust.Soc.Am.7820836
DuckFA1980ThepulsedultrasonicfieldPhysicalAspectsofMedicalImagingedBMMooresetal(London:Wiley)
DuckFAandMartinK1992Exposurevaluesformedicaldevices,inUltrasonicExposimetryedsMZiskinandPLewin(BocaRaton:CRC)
FriedlanderF1958SoundPulses(Cambridge:CambridgeUniversityPress)
HarrisGR1981aReviewoftransientfieldtheoryforabaffledplanarpistonJ.Acoust.Soc.Am.701020
1981bTransientfieldofabaffledplanarpistonhavinganarbitraryvibrationamplitudedistributionJ.Acoust.Soc.Am.70186204
KinslerLE,FreyP,CoppensABandSandersJV1982FundamentalsofAcoustics3rdedition(NewYork:Wiley)
KossoffG1979AnalysisoffocusingactionofsphericallycurvedtransducersUltrasoundMed.Biol.535965
KrautkramerJandKrautkramerH1990UltrasonicTestingofMaterials4thedition(Berlin:Springer)pp8792
LockwoodJCandWilletteJG1973HighspeedmethodforcomputingtheexactsolutionforthepressurevariationsinthenearfieldofabaffledpistonJ.Acoust.Soc.Am.5373541
LucasBGandMuirTG1982ThefieldofafocusingsourceJ.Acoust.Soc.Am.72128996
O'NeilHT1949TheoryoffocusingradiatorsJ.Acoust.Soc.Am.2151626
PapadakisEPandFowlerKA1971Broadbandtransducers:radiationfieldandselectedapplicationsJ.Acoust.Soc.Am.5072945
PattersonMSandFosterSF1982AcousticfieldsofconicalradiatorsIEEETrans.SonicsUltrasonicsFreq.Contr.SU298392
PenttinenAandLuukkalaM1976aTheimpulseresponseandpressurenearfieldofacurvedultrasonicradiatorJ.Phys.D:Appl.Phys.9154757
1976bSoundpressurenearthefocalareaofanultrasoniclensJ.Phys.D:Appl.Phys.9192736
SchochA1941BetrachtungenuberdasSchallfeldeinerKolbenmembranAkust.Z.631826
StepanishenPR1971TransientradiationfrompistonsinaninfiniteplanarbaffleJ.Acoust.Soc.Am.49162938
1974AcoustictransientsinthefarfieldofabaffledcircularpistonusingtheimpulseresponseapproachJ.SoundVibr.32295310
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StepanishenPRandBenjaminKC1982ForwardandbackwardprojectionofacousticfieldsusingFFTmethodsJ.Acoust.Soc.Am.7180312
WeightJPandHaymanAJ1978ObservationsofthepropagationofveryshortultrasonicpulsesandtheirreflectionbysmalltargetsJ.Acoust.Soc.Am.63396404
WellsPNT1977BiomedicalUltrasonics(London:Academic)
WilliamsEGandMaynardJD1982NumericalevaluationoftheRayleighintegralforplanarradiatorsusingtheFFTJ.Acoust.Soc.Am.72202030
ZemanekJ1971BeambehaviourwithinthenearfieldofavibratingpistonJ.Acoust.Soc.Am.4918191
Page23
Chapter2NonlinearEffectsinUltrasoundPropagation
AndrewCBaker
Introduction
Inafluid,ultrasoundpropagatesaslongitudinalwavesofalternatecompressionsandrarefactions.Toafirstapproximationthewavetravelsataconstantspeed(c)andsoitsshaperemainsunchangedasitpropagates.Thislevelofapproximationcorrespondstothesimplestpossibleformofwaveequationandiswidelyapplicabletomanyacousticsystems(e.g.normalsoundlevelsinairandmostsonarsystemsinwater).Themethodsoflinearsystemstheoryareappropriatetothesolutionsofproblemsinthesefieldsandgreatuseismadeofmethodssuchassuperpositionandlinearscalingofsolutions.Theintroductionofafrequencydependentabsorptioncausesnogreatdifficultieseithersincethesystemislinear.Thelinearwaveequationdependsontwomainassumptions:firstlythattheparticlevelocity(u)ofthewaveisinfinitesimal(oratleastsmallcomparedtoc)andsecondlythatthepressuredensityrelationshipofthefluidislinear.
Iftheacousticamplitudeissufficientlyhighthenassumptionsoflinearityarenolongervalidandwillintroducesignificanterrors.Theresultingwavehascompressionalphasesthattravelataspeed(c+ u0),whichisfasterthanthespeedoftherarefactions(c u0) isaparametercharacterisingthenonlinearityofthemedium(thenonlinearityparameterisoftenexpressedasB/A=2( 1):measurementmethodsandtypicalvaluesaregiveninChapter4).Notethatthefiniteparticlevelocityandthenonlinearityofmediumbothproducethesameeffect.Thuswegetdistortionthatwillcauseawaveformthatisinitiallysinusoidaltobecomemorelikeasawtooth(figure2.1).Theamountofdistortionwillincreasewithdistancepropagatedandshocklikewaveformsarecommonlyencountered,withanabruptincreasefrompeaknegativepressuretopeakpositivepressureasthewavepassesanypoint.Intermsoffrequencycontent,thewaveform
Page24
Figure2.1.Initialwaveform(top, =0)
anddistortedwaveform(bottom,= /2).
distortionisequivalenttoharmonicgenerationatintegermultiplesoftheoriginalfrequency.Thusenergyispumpedtohigherfrequencieswheretheabsorptionlosseswillbehighercausing,amongotherthings,increasedintensitylosswhichcanleadtoenhancedheatingandstreamingeffects.Inarealbeamofultrasoundtherewillalsobediffractioneffectswhichinteractwithnonlinearityandabsorptiontofurthercomplicatematters.AgeneralhistoryofnonlinearityinfluidscanbefoundinthebookbyBeyer(1984).
2.1NonlinearPropagationinMedicalUltrasound
Theuseofultrasoundinmedicineonlybecomeprevalentduringthe1960sandalthoughnonlinearacousticeffectshavebeenknownaboutsincetheeighteenthcentury,itwas1980whenthefirstpapershighlightedtheimportanceofnonlineareffectsinmedicalultrasound.MuirandCarstensen(1980)andCarstensenetal(1980)discussedthepotentialforshockformation,enhancedabsorptionduetoharmonicgenerationandbeambroadening,causedbythetransferofenergyfromthemainlobeofthefundamentalbeamtohigherharmonics.Oneofthemeasuresofthestrengthofnonlinearitytheyusedwastheplanewaveshockparameter = kz,(=u0/c)istheacousticMachnumberwhereu0istheparticlevelocityamplitudeatthesource,k(=2 f/c)isthewavenumberandzisthedistance
Page25
thatthewavehastravelled.Avalueof =1indicatesthatashockisjuststartingtoform(i.e.averticaldiscontinuityisjustappearinginthepressurewaveform).Atthispointthedistancetravelled(z)isoftendenotedbytheplanewaveshockdistanceld=1/ k.Inthecaseofaplanewave,u0canbedeterminedfromtheacousticpressureamplitude,p0,usingtheplanewaveimpedancerelationu0=p0/p0cwherep0isthedensityofthemedium.When = /2thewaveisfullyshocked,withadiscontinuityfromthepeakpositivepressuretothepeaknegativepressure(figure2.1).Furtherdistortionleadstoreductionsofthepeakpositiveandnegativepressuresasmoreofthewavemovesintotheshockedregion.Notethat isproportionaltoboththeacousticpressure(p0)andthedistancetravelled(z)henceitispossibletodetermineexperimentallywhethernonlinearpropagationisoccurringinasystembynotingwhetherwaveformdistortiondecreasesaseitherdrivepressureand/orobservationdistanceisdecreased.Inwaterat20C, =3.5,henceifwetakeultrasonicparametersthataretypicalofcurrentimagingsystems(e.g.f=3.5MHz,p0=1MPaseeChapter7)wewillhaveaplanewaveshockdistanceld=43mm(assumingp=1000kgm3andc=1486ms1).Wewouldthereforeexpecttoobservenonlinearwaveformdistortionrelativelyeasilyatdistancesgreaterthanthis.Itshouldbenotedthatinclinicalsystemsfocusinganddiffractionwillalsoaffecttheshockformationdistancesoldshouldonlybeusedasaroughestimateofnonlineareffectsinclinicalbeams.
Thesituationissimilarinhumantissueswhere valuesaretypicallyintherange4to6(Duck1990)withthehighervaluesduetofattytissues.However,attenuationlossesarehigherintissuewhichtendstocounteractnonlineardistortion.ThisisindicatedbytheGol'dbergnumber =1/ld = k/ where isthelinearattenuationcoefficient(inneperm1).TheGol'dbergnumberaccountsforthefactthatabsorptioncounteractsnonlineargenerationsoitistheratioofthetwowhichisimportant.MethodsofmeasuringthenonlinearityparameterandthepossibilityofmappingittoforminvivoimageshavebeenreviewedbyBjrn(1986).
Figure2.2representsthenonlinearpropagationofa3.5MHz,500kPaplanewaveinwater(i.e.theGol'dbergnumber =38).Wecanseethatatzerorangeonlythefundamentalispresent.Theharmonicsbuildupwithdistanceandeventuallysettleinalmostconstantratiotothefundamental.Energyislostfromthefundamentalandispumpedintotheharmonics.Inthecaseofalinearwavewewouldexpectnoharmonicsandthefundamentaltoremainalmostconstantlinearabsorptionwouldonlyaccountforalossofafewpercentoverthissortofdistanceinwater.Arangeof85mmcorrespondsto =1forthiswaveanditcanbeseenthatthereisappreciablesecondandthirdharmoniccontent.Onlythefirstfiveharmonicsareplottedherealthoughmanymorewillbepresentespeciallyatlongerranges.Atarangeof133mmwehave = /2andappreciableenergyhasbeenlostfromthefundamentaltothehigherharmonics.
Page26
Figure2.2.Fundamentalandsecondtofifthharmonicsforanonlinearplanewaveinwater(f0=3.5MHz,
P0=500kPa, =38).
isausefulquantitywhentryingtoestimatethesignificanceofnonlinearityinagivensituationbutaplanewaverepresentsaratheridealisedcase.Inanattempttoincludetheeffectsoffocusing,Bacon(1984)proposedanonlinearpropogationparameter( m)whichtakesaccountofamplitudefocalgainG:
wherethefocalpressurepfisdefinedas(pc+pr)/2(pcandprarecompressionandrarefactionpressuremagnitudesatthefocaldistancezf),andfistheacousticfrequency.Thisequationcanbeusedalsotocalculate atthefocusupto = /2.Abovethisvalue mnolongerdependslinearlyonsourceamplitude,andultimatelyreachesasaturationvalueof2 .
Theinclusionofdiffractionandfocusinginthenonlinearproblemcausesphaseshiftsinwaveformsothatinsteadofresemblingasawtooth,anonlinearlydistortedultrasonicpulselooksmorelikethemeasurementsshowninfigure2.3.Thewaveformshownisarelativelylowamplitude2.25MHzpulsegeneratedbyaheavilydamped,shockexcitedtransducer.Thediffractivephaseshiftscausethetopbottomasymmetryofthedistortedpulse.
Diagnosticultrasoundtendstooperateovershorterdistancesthan600mmbutwithcorrespondinglyhigherdrivelevelsandfocusinggain,hencethedistortedwaveformshapeistypicalofthedistortedpulsesobservedfromclinicalsystems(DuckandStarritt1984).Eventhehighabsorptionoftissueisnotsufficienttosuppressnonlinearityhencesimilarwaveformdistortionandharmonicgenerationhavebeenobservedinbiologicaltissues(Starrittetal1985,1986).Anextensivesurveyoftheoutputofdiagnosticsystems(Ducketal1985)showedthatalmostallthesystemssurveyedwerelikelytobesubjecttononlinearpropagation.Amorerecentsurvey(Henderson
Page27
Figure2.3.Initialpulse(top)andnonlineardistortionofpulse(bottom)afterpropagating600mm
inwater.
etal1995)indicatedthattheacousticoutputlevelsofnewdiagnosticsystemshadincreasedconsiderablyandthusnonlineareffectsarenowevenmoresignificant.Othermedicalultrasoundsystemssuchaslithotriptersandhyperthermiasystemswillalsobesubjecttononlinearpropagationsincealthoughtheyusuallyhavelowerfundamentalfrequenciestheyalsohavehighacousticdrivelevels.
2.2ConsequencesofNonlinearPropagation
2.2.1ExperimentalMeasurements
Themostobviouspracticalconsequenceofnonlinearpropagationisthatanincreasedmeasurementbandwidthisnecessary.Thisistruebothforacousticfieldmeasurementswithhydrophones(seeChapter7)andforpulseechoimagingofharmonicbackscatterasisusedclinically,forexampleinharmonicimagingofcontrastmaterials(seeChapter12).Figure2.4showsthefrequencycontentofthepulsesinfigure2.3.Theinitialpulsehasitsenergyconcentratedaroundthecentrefrequencyofthetransducer(2.25MHz
Page28
Figure2.4.Initialspectrum(top)andspectrum
ofdistortedpulse(bottom).
inthiscase).Thegrowthofdistortionleadstopeaksatmultiplesofthecentrefrequencyuptoabout25MHzwherethehydrophonebandwidthstartstolimitthewidthofthespectrum.ThehydrophoneusedinthiscasewasaGECMarconibilaminarPVDF(polyvinylidenefluoride)membranedevicewhichhasasmoothresponseuptoitsmainresonanceatabout20MHz.Othervariantsofthisdevicehaveahigherresonantfrequencyprovidingevengreaterbandwidth(Bacon1982).Untilthemembranehydrophonewasdevelopeditwasdifficulttoobservethedistortedwaveformsproducedbymedicalultrasoundsystems.In1980itwasnotedthat'Althoughmicroprobeswithaflatresponseto10MHzhavebeenreported,theyaredifficulttoconstructandarenotcommerciallyavailable'(Carstensenetal1980).ThechoiceofhydrophoneisanimportantonewhendealingwithsuchdistortedwaveformsandfewdevicescancurrentlyapproachtheGECMarconimembraneintermsofthewidthandsmoothnessoftheiroperationalfrequencyrange.Inrecentyearsthoughtherehavebeensomeimprovementsinneedleprobehydrophonedesign,andthesedevicesusuallyhaveasignificantpriceadvantageovertheGECMarconimembranehydrophone.Neverthelessitisrecognisedthatthefrequencyresponsemaybequitevariable,particularlyatlowerfrequencies,andneedlehydrophonesneedto
Page29
beusedcriticallyifhighfidelitymeasurementsarerequired(Preston1991).
Severalfactorsareimportantwhenhandlingdistortedwaveformsofthistypeincluding:
2.2.1.1SystemBandwidth.
Thelossofhighfrequencycomponentsduetolimitedhydrophonebandwidth(orlimitedbandwidthinanypartofthesignalprocessingsystem)isparticularlynoticeableintheobservedpeakpositivepressurewhichcanappeartobesignificantlyreduced.Thechoiceofdigitisingfrequencymustalsobeappropriatetothesystembandwidth.Itiscommontousedigitisingfrequenciesof100MHzorhigherforthecaptureoftypicalmedicalimagingpulseswhichhavetheircentrefrequenciesinthelowMHzrange.Caremustalsobetakentoavoidaliasingindigitisers.
2.2.1.2HydrophoneCalibrationandFrequencyResponse
Thepresenceofnonlinearitymeansthatmeasurementscannotbescaledfromonedriveleveltoanother.Inalinearsystem,forexample,thedirectivityplotofanacousticbeamdoesnotchangewithdrivelevelundernonlinearconditionsitwillchangewithdrivelevel.Itisthereforeusefultoknowtheacousticpressureatwhichmeasurementsweremade.Thehydrophonecalibrationisalsoimportantinremovingwaveformartefactsduetohydrophoneresonances.Thefrequencyresponseofahydrophoneisusuallydeterminedbyoneormoreofitselectromechanicalresonanceswhichmaynotbeapparentinundistortedwaveforms.Adistortedwaveformhowevercanhavesufficientenergyathigherfrequenciestoexcitethehydrophoneresonanceshencetheobservedelectricalsignalisnotatruerepresentationofthepressurewaveformbeingmeasured.Itishardlyeverjustifiabletouseasinglefrequencycalibrationoverthebandwidthsrequired.
2.2.1.3HydrophoneSize
Theharmonicbeampatternsarenarrowerthanthefundamentalbeam(figure2.5).Thusahydrophonechosentobesmallenoughtoavoidspatialaveragingproblemsatthefundamentalfrequencymaywellberatherlargeincomparisonwiththehigherharmonicbeams.Thiswillleadtounderestimatesofthepeakvalues(Smith1989,ZeqiriandBond1992,Bakeretal1996).
2.2.1.4HydrophoneAlignment
Thenarrowerbeamwidthsoftheharmonicsmakehydrophonealignmentmorecritical.Theharmonicamplitudebeamwidthsvaryas wherenistheharmonicnumber(ReillyandParker1989).Thehigherharmonics,however,canprovideausefulguidetoalignmentsincethepeakpositivepressureissensitivetotheirpresence.
2.2.1.5HydrophoneLinearity
Theacousticpressuresaresufficientlyhighthatthehydrophoneitselfcangenerateharmoniccomponents.Themagnitudeofthesecomponentswillonlydependontheamplitudeof
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Figure2.5.Calculatedharmonicbeampatternsinfocalplaneoffocusedtransducer(f0=2.25MHz).
acousticfieldbeingmeasuredwhereasnonlinearityduetopropagationalsoaccumulateswithdistance.Itisthuspossibletodistinguishbetweenthesetwosourcesofnonlinearitybymovingthehydrophoneclosetotheultrasonicsourcewherethepropagationnonlinearityshouldbenegligible.Careisstillneededastheultrasonicsourcewilloftentransmitlowlevelsofharmonicdirectly.Theeffectofhydrophonenonlinearityanddirecttransmissionofharmonicsisnotusuallyserioussincethelevelsaresmallincomparisonwiththeharmoniclevelsgeneratedbynonlinearpropagation(Prestonetal1983).
2.2.1.6ChoiceofPropagatingMedium
Laboratorymeasurementsareinvariablymadeinwater,butthiscancreatedifficulties.Theabsorptionofultrasoundinwaterislow(relativetotissue)whichallowsagreaterdegreeofnonlineardistortiontooccurandhenceincreasedsignalbandwidth.Itisnotasimplemattertotranslatewaterbasedmeasurementstoinvivovalues.The'derating'procedurewhichiscommonlyusedinstandards(AIUM/NEMA1992)isbasedonassumptionsoflinearpropagation.ChristopherandCarstensen(1996)concludethatapplyingthelinearderatingfactortostronglyshockedmeasurementsinwatercanleadtosignificantunderestimatesofthepressurefieldintissue.
Theeffectofnonlinearpropagationcanalsobeobservedinmeasurementsofultrasonicpropertiessuchasabsorptioncoefficientwhichbecomedependentonthedrivelevelandmeasurementgeometry(Zeqiri1992,Wu1996).Bothofthesestudiesconcludedthatitisadvisabletominimisethetransmitterreceiverseparationandtokeeptheplanewaveshockparameter 0.1inordertoavoidsignificantnonlinearerrors.
Characteristicsforhydrophonesandguidanceformakingmeasurementsinthisfrequencyrangecanalsobefoundintherelevantinternationalstandards
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(IEC1987,1991,1993)andPreston(1991).OtherpracticalaspectsoftheuseofhydrophonesforexposuremeasurementaregiveninChapter7,section7.7.
2.2.2TheoreticalPredictions
Theoreticalmodelsforultrasoundpropagationareusefulinthedesignandanalysisofultrasoundsystems,especiallysinceinvivomeasurementsarenoteasytocarryout.Themaindifficultyinmodellingisthepresenceofnonlinearitywhichrulesoutmostofthemethodsthatareapplicabletolinearsystems.Thecumulativenatureofthedistortionwithdistanceanditsinteractionwithdiffractionandabsorptionmeanthatitisnotnormallypossibletocalculatetheamplitudeofasinglefieldpointatadistancefromthesourcewithoutcalculatingthefullfieldintheregionbetweenthefieldpointandthesource.Thusstraightforwardanalyticalsolutionscanonlybefoundforrelativelysimplegeometriesandingeneralitisnecessarytousecomputationallyintensivenumericalmethods.Inadditiontocalculatingtheacousticfield,thereisalsoarequirementtobeabletopredicteffectssuchasheating,streamingandcavitationsincethesearepotentialsourcesofbioeffectsandwilldependontheacousticfield.
Anumberofapproachestopredictingtheultrasonicfieldsofmedicalultrasoundsystemshavebeentriedbutthemethodthathasprobablyreceivedmostattentiontodateisafinitedifferencesolutiontoanapproximatenonlinearwaveequation.ThewaveequationisknownastheKhokhlovZabolotskayaKuznetsov(orKZK)equationanditaccountsfornonlinearity,absorptionanddiffraction(Kuznetsov1971).ThemostimportantassumptionintheKZKequation,inthiscontext,isthattheacousticenergypropagatesinafairlynarrowbeamthisisknownastheparabolicapproximationortheparaxialapproximation.Theparabolicapproximationisvalidforacousticsourceswhicharemanywavelengthsacrossandforfieldpointsthatarenottooclosetothesourceortoofaroffaxis.Forcircularsourcesofultrasoundweneed(kra)2>>1whereraisthesourceradiusandtheminimumaxialdistanceisra(kra/2)1/3(NazeTjttaandTjtta1980).Inpractice,formostweaklyfocuseddiagnosticbeams,theseconditionsdonotusuallyposeseriousdifficulties.AfinitedifferencesolutionfortheKZKequationwasdescribedbyAanonsen,Barkve,NazeTjttaandTjttaoftheUniversityofBergen,Norway(Aanonsenetal1984).NazeTjttaandTjttahavebeenresponsiblefordevelopingmuchofthemathematicalbackgroundinthefieldofnonlinearsoundbeamstheresultingnumericalsolutionsandcomputerprogramsarenowwidelyknownastheBergencode.TheapproachusedintheBergencodeistosubstituteaFourierseriesforthetimewaveformintotheKZKequationandsolvetheresultingsetofcoupleddifferentialequationsusingfinitedifferencemethods.
TheBergencodehasbeenappliedtoultrasonicsourcessimilartothosefoundinmedicalsystemsandhasprovedtobeareliablemodelofthebeam
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Figure2.6.ComparisonoftheKZKequation()withmeasurementsoffundamentalandsecond,thirdandfourthharmonics(+,x,*,)forafocusedultrasoundsourceinwater
(f0=2.25MHz,p0=68kPa,a=19mm).Theverticaldashed
lineindicatesthepositionofthefocalplane.
behaviourinwater.Planecircularsourcesofcontinuouswaveultrasoundhavebeenstudied(Bakeretal1988,TenCate1993,Nachefetal1995)aswellasfocusedsourcesasshowninfigure2.6(Baker1992,AverkiouandHamilton1995).
Figure2.6showsthatclosetothesourcetherearenoharmoniccomponents,onlythefundamental.Theharmonicsbuildupwithaxialrangewithvariousmaximaandminimamirroringthoseinthefundamentaluntilthefinalaxialmaximasettleatroughlyconstantlevels(approximately1/n)relativetothefundamental.TheKZKsolutiondoesnotshowtheexpectednearfieldoscillationsatveryshortranges,thisisaconsequenceofthestepsizeused.Smallerstepswouldhaveshownmoredetailatshortrangesinsteadweseetheaveragevalueofthesolutioninthatregion.
TheBergencodemayalsobeinitialisedwithapulsespectrumhencepulsedfieldssimilartodiagnosticsystemshavebeenexamined(BakerandHumphrey1992,Baker1991).Rectangulargeometrieshavealsobeenmodelled(Berntsen1990,Bakeretal1995).TheBergencodehasrecentlybeenappliedtoultrasoundsystemswithrectangulararrays(CahillandBaker1997a,b)anditwasfoundthatnonlinearitycaninteractwithdiffractiontocausetheregionofpeakintensitylosstomovefromtheacousticaxis(atlowdrivelevels)tooffaxislocationsathighdrivelevels(figure2.7).Thisshiftiscontrarytothepredictionsoflineartheories.Thefirstpartoffigure2.7showsthelinear(lowdrivelevel)pressurefieldofasquareapertureaswouldbe
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Figure2.7.Nonlineargenerationalongthediagonalofaplanesquareaperture(sidelength=20mm).Thebeampropagatesdownthepagefora
distanceof200mm(f0=2.25MHz).
measuredacrossitsdiagonaldiffractioneffectsarestrongestonthediagonalduetotheinteractionofedgediffractionfromthetwosides.Itcanbeseenthatinthelinearcasethepeakamplitudeoccursontheacousticaxisatthebottomoftheplot.Thesecondplotcorrespondstothefundamentalwhenthesourcepressureisincreasedto1MPa.Theregionofpeakfundamentalamplitudenowoccursoffaxisandmuchclosertothesource.Thesecondharmonicisstrongestwherethefundamentalisstrongestsoittoohasitspeakvaluesoffaxisnearerthesource.Thesecondharmonicalsoshowstwiceasmanyfringesacrossthebeamwhencomparedtothefundamentalthiseffectcanbeseeninfigure2.5.Thetenthharmonic(i.e.22.5MHz)hasasharplydefinededgetotheoffaxisregionandagainexhibitsamaximumamplitudeoffaxis.Thishasconsequencesforthepredictionofpotentialbioeffectssincetheintensitylossfromthebeamdeterminestheheatsourcedistributionforthermaleffectsandthedrivingforceforstreaming.
Computationalrequirementscanbecomeanissuefornonlinearmodelling.Thecontinuouswavecircularcaseatmoderatedrivelevelscanberunonapersonalcomputer(e.g.aPentiumbasedPC)inamatterofminutes.TheBergencode,however,worksinthefrequencydomainandifthedrivelevelisincreased,moreharmonicsareneededinthesolutionwhichrequiresmore
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memoryandmoreCPUtime.TheinclusionofpulsedwaveformsrequiresmorefrequencycomponentsintheinitialspectrumwhichagainrequiresmorememoryandmoreCPUtime.TherectangularcodeisanotherorderofmagnitudebiggerinitsrequirementsformemoryandCPUtime.TheresultsofCahillandBaker(1997b)requiredabout500MBphysicalmemoryandtookoftheorderof40hoursCPUperrunonaDECAlpha8400computer.Somesavingsincomputereffortweremadebyintroducinganartificiallyhighabsorptionfactorforthehighestharmonicswhichcanreducethetotalnumberofharmonicsrequiredinthesolution.FouriertransformmethodsalsoenabledconsiderableCPUtimesavingsbycalculatingthenonlinearinteractionsinthetimedomain.Spatialresolutionsometimeshastobetradedforsavingsinruntime:themorecloselyspacedthegridpointsinthefinitedifferencescheme,themorememoryandCPUtimethatisrequired.
ApartfromtheBergencodeanumberofothermethodsofsolutionarealsofeasible.ChristopherandParker(1991)havedemonstratedanonlinearmodelwhi