Chapter 7 v1.docx

Chapter 7 v1.docx

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Chapter7v1

ParallelImaging

Wehaveseeninthepreviouschapterthateachcoilinaphasedarrayimpartsachangeinthemagnitudeandphasetothesignalitreceivesfromeachpixel.Thesechangesdependontherelativelocationofthepixelcomparedtothecoil.Therefore,thecoilsinaphasearraycombineadditionalspatialinformationwiththesignalsreceived.Thestandardimagingtechniqueusesfieldgradientstocreatetimedomainsignalsthatcorrespondtothetwodimensionalk-spacedataofthedesiredimage.AfterthespinsareactivatedbytheRFpowerpulse,pulsedgradientsalongthephaseencodedirection（nominallythey-direction）presetthephaseofallofthespinscorrespondingtoaparticulardistancealongthey-directionink-space.Duringthefrequencyreadout,allofthecorrespondingx-directionkvaluesofthesignalarereadoutinarelativelyshorttime.Thetimeconsumingaspectofk-spacedataacquisitionresultsfromhavingtorepeatthisprocessforeachincrementinthey-directionofk-space.ThenetscantimecanbereducedifsometheinformationcontainedintheB1xy（r）mapsforeachcoilinthearraycanbeusedtoeliminatetheneedtocollectallofthek-spacedata.

SodicksonetalandPruessmanetaleachproposedamethodusingthecoilsensitivitymapstoreducethenumberofy-phaseencodingstepsneededforanimage.Sodicksonusedthecoilmapstofabricateadditionallinesofk-spacedatatofillbetweengapsinthedatathatheacquiredwithgradientencoding.HecalledhismethodSMASHforSiMultaneousAcquisitionofSpatialHarmonics.Hemanipulatedk-spacedatabeforetransformingittogetthefinalimage.Pruessmanreducedthek-spacedatabyundersamplinginthephaseencodedirection.Hereconstructedanimagefromeachcoilandthenmanipulatedtheimagedatatogethisfinalimage.PruessmancalledhismethodSENSEforSENSitivityEncoding.Bothmethodsareformsofageneralprocessthathasbeennameparallelimaging.

WewilllimitthetreatmentheretotheSENSEtechnique.Wewillstartwithasimplifiedversiontoillustratesomeoftheissuesthatmaybeobscuredbythemoreformalmathematicaltreatmentsgivenintheliterature.Ourmodeldealswithonlythemagnitudeofthesignalsproducedbyeachcoil.Thephasedifferencesinthesignalsareignored.Considerimagingaroundphantomwithaphasedarrayconsistingoftwocoils,oneoneachsideinthex-direction.

Fig.7.1

ThekthcoilhasasensitivityfactorgivenbyBk1xy（x）thatstartsatavalueof1.0atthecenterofthecoilandrollsoffasthedistanceincreasesacrossthefield-of-view（FOV）.

Fig.7.2

LetthesensitivityfactorforcoilkatlocationxjbegivenbyCkj

BeforemakingaSENSEimage,letusmakeasimplephasedarrayimageandusethesum-of-squarealgorithmtocombinetheimages.Thesignalreceivedbycoilkfromthevoxelatxjisgivenby

Skj=CkjMj+nkEqn7.1

wherenkisthenoise,whichhasthesamestandarddeviationσforbothcoils.DividingbothsidesofEqn7.1byCkjindicatesthatthenoisethataffectsMjhasastandarddeviationofσ/Ckj.Therefore,animageofMjwillhaveaSNRofCkjMj/σ.CombiningthedatafromthetwocoilswouldgiveavaluefortheSNRofthepixelatxjof

Eqn7.2

ConsideranumericalexampleusingthevaluesinFig.7.2,thesensitivitymatrixforthetwopixellocationsx1andx2is

Eqn7.3

SubstitutingthesenumbersintoEqn7.2gives

Eqn7.4

WewillusetheseSNRvaluestoseehowwelltheSENSEimageisreconstructedfromtheundersampleddata.

TheSENSEmethodreducesthescantimebyreducingtheFOVinthephaseencodedirectionbyafactoroftwo.Thesamesizedpixelsareobtainedbutonlyhalfasmanyy-encodingsareobtained.Theresultingimagefromeachcoilhaswraparound.

Fig.7.3

Thesignalfromthevoxellocatedatx1isnowwrappedaroundandaddedtothesignalfromthevoxelatx2.Therefore,thesignalpickedupbycoilkis

Sk=Ck1M1+Ck2M2+nkEqn7.5

ThishasastraightforwardsolutionifthedeterminantofCisnotzero.

Eqn7.6

wheretheextra0subscriptontheM’sindicatesthenoiselessoraveragevalue.Thetermsinvolvingthenoisesmustbeevaluatedbywayanensembleaverageofthesquareofthenoisevoltage.Thermsnoiseforeachpixellocationisgivenby

Eqn7.7

wheretheangularbracketsdenotetheensembleaverage.

Inthiscase,becauseonlyonehalfasmuchdatawasacquired,thermsvalueofthenoiseineachSkis21/2timestheσusedinEqn7.1forthefulldataacquisition.Thatis,=2σ2.

Eqn7.8

Thetermintheparenthesisrepresentsanamplificationofthethermalnoiseduetothemathematicalunfoldingofthealiasedimage.ThisquantityiscalledthegeometryfactorGbecauseitisstronglyinfluencedbytheconfigurationofthecoilsandthelocationofthepixelrelativetothecoils.IftheCmatrixhasonlydiagonalterms,eachcoilpicksupasignalfromthepixelnearesttothecoil.Thereisnomixingofthesignalsorthenoise.TheGfactorisaminimum.Forthatcase,thesignalandthenoiseinEqn7.6couldbecombinedtogivetheSNRofasinglecoil.Noteinthiscase,thereisnogainintheSNRforthesimplephasedarrayacquisitiongiveninEqn7.2.

Forcomparisonpurposes,wewillcalculationtheSNRforthetwopixelsrepresentedinFig.7.1usingthevaluesofCfromEqn7.3.Wewillassumetherearenoisecorrelations.

Eqn7.9

DividingthesevaluesbytheSNRgiveninEqn7.4obtainedbyacquiringthecompletek-spacedataandcombiningimageswiththesumofsquaresalgorithmgives

Eqn7.10

ThelastformoftheresultsemphasizestheeffectoftheGfactoronnoiselevel.Atpixellocation2,wherethereislessdifferenceinthesignalstrengthforthetwocoils,theGfactorisbigger.TheGfactorisevenlargeratthepointwherethetwosensitivityfactorsareequal.

TheGfactorisafunctionofcoilgeometry,thelocationofthepixelsthatarealiased,andnoisecorrelationmatrix.Ingeneral,theCmatrixwouldcontaincomplexnumbersthatrepresentedbothmagnitudeandphaseinformation.NoticethatinEqn7.8,apositivenoisecorrelationcoefficientwouldtendtodecreasethenoiseineachpixel.

Notethatthereareareasoftheimagewherethereisonlyonesignalpresentbecausethealiasedpixelliesoutsidethephantom.Forthesepoints,Eqn7.6shouldnotbeused.Otherwise,noiseintheairspacearoundthephantomwouldbeaddedtothereconstructedimage.

TheSENSEtechniquecanreducethescantimebysubstitutingpredeterminedinformationaboutthecoilconfigurationtoreplacethemissingk-spaceinformationineachscan.SometimemustbespentondeterminingtheB1fieldmapsforeachcoiloverthevolumetobeimagedinsubsequentscans.Timeisthensavedonallthefollowingscansaslongasthecoilshavenotbeenmoved.Inprinciple,theB1mapscouldbecalculatedifoneknewtheexactlocationofthecurrentcarryingelementsofeachcoil.Inaclinicalsituation,thisisnotpracticalbecausethecoilarrayplacementdifferswitheachpatient.VarioustechniqueshavebeendevisedtoobtainB1fieldmapsusingthescanner’simagingcapability.OnemethodassumesthattheB1fieldmapforaparticularsurfacecoilelementisproportionaltotheratioofthatcoil’simagedividedbyabodycoilimage.Thevariationsinthesignalintensityofbothimagesduetothetissuepropertieswillcancelout.TheratioisactuallyproportionaltotheB1valueforthesurfacecoildividedbytheB1valueforthebodycoil.Ofcourse,theotherassumptionmadewhenusingthismethodisthatthebodycoilhasauniformsensitivity.Ifthesurfacecoilisnotproperlydisabledduringthetransmitpulse,thebodycoilB1willbedistorted.ThesedistortionswillbetransferredtotheB1mapofthesurfacecoil.TheSENSEalgorithmmusthaveaccurateB1mapsforthesurfacecoilsinordertoavoidintroducingartifactsintheunfoldedimage.ThetwoimagesfortheB1maparetakenwithlowspatialresolutioninordertogetahighSNRforeach.AnynoiseintheB1mapwouldbetransferredtotheimagesthatusedthemap.TheB1mapisderivedfromreferencescanstakenbeforetheSENSEscanisexecuted.Ifthepatientismoved,anewsetofreferencescansisrequired.

GeneralSENSETreatment

Inthefollowingdiscussion,Iwillusematerialtakenfromthewebsitehttp:

//www.mr.ethz.ch/sense/createdbyPruessmanandassociates.

IhavechangesthenotationtoconformtonotationthatIusedinthesimplifiedtreatment.

Whenusinganimagingsequencethatemploysphaseencodinginmorethanonedirectionsuchas3-dimensionalimaging,thescantimecanbereducedbyareductionfactorR=R1R2,whereRiisthereductionofthephaseencodingintheithdirection.TheFOVoftheacquiredimageisreducedinbothdirections.

Fig.7.4

ThereducedFOVwithonepixelofinterestcanbetranslatedrepeatedlytoformarectangularmeshasshownwiththedottedlinesinFig.7.4.EachoftheadditionalpixelsthatliewithinthefullFOVwillbewrappedaroundandbemixedtogetherinthesignalpickedupbyeachcoil.LetnpbethenumberofpixelsthataresuperposedinthereducedFOVimages.Intheillustrationabove,np=4,whichalsoequalsthereductionfactorR.LetM（r）beavectorofthenpcomplexvaluesofthemagnetizationinthevoxelslocatedatthenppositionsrinthefullFOV.Assumethecoilarrayhasnccoilelements.Thecomplexnoiseineachcoilformsanncdimensionalcomplexvectorn.AnoisecorrelationmatrixNcanbecalculatedfrommeasurementsofthenoisefromtheindividualcoils.Nisanncxncsquarematrix.ThesensitivitymatrixC（r）ismadeupoftermsCjk,whichareproportional