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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,<
ni2>
=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