(Series in Medical Physics and Biomedical Engineering) Francis a. Duck_ a.C Baker_ H.C Starritt-Ultrasound in Medicine-CRC Press (1998)

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)

PART1THEPHYSICSOFMEDICALULTRASOUND

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

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

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

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:

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

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).

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:

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

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.

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.

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

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

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,

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/ .

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

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

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

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

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

StepanishenPRandBenjaminKC1982ForwardandbackwardprojectionofacousticfieldsusingFFTmethodsJ.Acoust.Soc.Am.7180312

WeightJPandHaymanAJ1978ObservationsofthepropagationofveryshortultrasonicpulsesandtheirreflectionbysmalltargetsJ.Acoust.Soc.Am.63396404

WellsPNT1977BiomedicalUltrasonics(London:Academic)

WilliamsEGandMaynardJD1982NumericalevaluationoftheRayleighintegralforplanarradiatorsusingtheFFTJ.Acoust.Soc.Am.72202030

ZemanekJ1971BeambehaviourwithinthenearfieldofavibratingpistonJ.Acoust.Soc.Am.4918191

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

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

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.

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

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

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

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

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

(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

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

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

memoryandmoreCPUtime.TheinclusionofpulsedwaveformsrequiresmorefrequencycomponentsintheinitialspectrumwhichagainrequiresmorememoryandmoreCPUtime.TherectangularcodeisanotherorderofmagnitudebiggerinitsrequirementsformemoryandCPUtime.TheresultsofCahillandBaker(1997b)requiredabout500MBphysicalmemoryandtookoftheorderof40hoursCPUperrunonaDECAlpha8400computer.Somesavingsincomputereffortweremadebyintroducinganartificiallyhighabsorptionfactorforthehighestharmonicswhichcanreducethetotalnumberofharmonicsrequiredinthesolution.FouriertransformmethodsalsoenabledconsiderableCPUtimesavingsbycalculatingthenonlinearinteractionsinthetimedomain.Spatialresolutionsometimeshastobetradedforsavingsinruntime:themorecloselyspacedthegridpointsinthefinitedifferencescheme,themorememoryandCPUtimethatisrequired.

ApartfromtheBergencodeanumberofothermethodsofsolutionarealsofeasible.ChristopherandParker(1991)havedemonstratedanonlinearmodelwhi

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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)