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FULLY3D'03TUAM2–31ExtendedParallelBackprojection(EPBP)forArbitraryConeAngleandArbitraryPitch3DandPhase–Correlated4DCTReconstructionMarcKachelrie,WilliKalenderAbstract—RecentdevelopmentsinmedicalCTaimatfasterrotationspeedsandahighernumberofsimultane-ouslyacquiredslices.
TheseeortsarepushedfurtherbycardiacCTwhichiscurrentlythemostprominentspecial–purposeapplicationinCT[1].
Today,16-slicescannersarestate–of–the–art.
ButCTmanufacturershavealreadyan-nouncedscannerswithfarmoreslicesandsomeevenpro-moteprototypeswithupto256slices.
MedicalCTmustsupportthecircularscantrajectory(se-quencescan)andthespiraltrajectory(spiralscan).
Arbi-trarypitchselectionisofhighimportance.
Inanycase,thecompleteareaofthedetectoristobeexposedandeachmeasuredrayshouldcontributetotheimage.
Onlythen,optimizeddoseusagecanbeachieved.
Theserequirementscannotbefullledbycurrentre-constructionapproaches.
Exactcone–beamreconstruction,thatiscapableofreconstructinglargecone–angledata,can-notcopewitharbitrarypitchandphase–correlateddataseg-ments.
Onlyapproximatereconstructionapproacheshavethepotentialtohandleallrequirements.
Currently,theonlyknownapproachthatcanhandlephase–correlatedtruecone–beamdataisanextensiontotheAdvancedSingle–SliceRebinning(ASSR)algorithm[2],[3].
However,thisgeneralizedapproachASSRCIislimitedto32slices.
WehavethereforedevelopedanewapproximateFeldkamp–typealgorithm,theextendedparallelbackpro-jection(EPBP)[4].
Itsmainfeaturesareaphase–weightedbackprojectionandavoxel–by–voxel180normalization.
Therstfeatureensures3Dand4Dcapabilitieswithoneandthesamealgorithm,thesecondensures100%detectorusage(eachraycounts!
).
Thealgorithmisevaluatedusingsimulateddataofathoraxphantomandacardiacmotionphantomforscannerswithupto256slices.
Thestandardreconstructions(EPBPStd)areofexcellentqualityevenforasmanyas256slices.
Thecardiacrecon-structions(EPBPCI)areofhighqualityaswellandshownosignicantdeteriorationofobjectsevenfarothecenterofrotation.
SinceEPBPCIusesthecardiointerpolation(CI)phaseweightingthetemporalresolutionisequivalenttothatofthewellknownsingle–sliceandmulti–slicecardiacapproaches180CI,180MCI,andASSRCI,respectively,andliesintheorderof50msto100msforrotationtimesbetween0.
4sand0.
5s.
I.
IntroductionMEDICALcomputedtomographyiscurrentlyevolv-ingfasterthanever.
Increasedspatialresolution,de-creasedscantime,increasedtemporalresolution,decreasedpatientdose,andincreasedvolumecoveragearesomeoftheimportanttrendstomention.
Aslittleasveyearsago,single–slicespiralCTwasthestate–of–the–art.
Then,therst4–slicescannersbecameavailable.
Alreadyin2001,16–slicescannersstartedtore-InstituteofMedicalPhysics,UniversityofErlangen–N¨urnberg,Krankenhausstr.
12,91054Erlangen.
Correspondingauthor:PDDr.
MarcKachelrie,E–mail:marc.
kachelriess@imp.
uni–erlangen.
deplacethe4–slicemachines.
Apparently,thenearfuturewillshiftthenumberofsimultaneouslyscannedslicesto32,64andevenmore.
Slicethickness,andtherebyspatialresolu-tion,willcontinuetodecreasetofurtherimprovediagnosticaccuracy.
Atthesametime,doseutilizationwillincreasetokeeptheeectivepatientdoseatanacceptablelevel:dosemodulationtechniques,automaticexposurecontrolandimproveddetectormaterialswillhelptodoso[1].
Besidesimprovedspatialresolution,improvedcontrastresolutionandlowpatientdoseoneisfurtherinterestedinhighesttemporalresolutiontoallowimagingtheheart.
Thisisdoneusingshortrotationtimescombinedwithde-dicatedphase–correlatedreconstructionalgorithmsastheyareavailablesince1996/1997[5].
Therelevanceforimagereconstructionalgorithmsliesintheexpectedincreaseinconeangleandinthedemandforphase–correlatedreconstruction.
Neglectingthecone–angleofthescannerasdoneinall4–sliceandinallcom-mercialcardiac16–slicereconstructionalgorithmswillyieldunacceptableimageartifactsforfuturescanners.
Therearefastandecientcone–beamalgorithmswithacceptableimagequalityavailableforcone–beamspiralCT.
However,theydonotensure100%doseusageand/ortheydonotworkforarbitraryspiralpitch.
Further,noneofthemiscapableofcombiningthesedemandswithphase–correlatedimagingatwiderconeangles.
Tollthemissinggap,weproposeEPBP,anewapprox-imatecone–beamreconstructionthatallowsforarbitrarypitchandforphase–correlatedreconstructionandthaten-sures100%detectorusage.
DetailsofcardiacCTscanningandtherestrictionsonthemaximumpitchasafunctionofthepatient'sheartratecanbefoundinreferences[6],[7],[8].
Thephaseselectionmechanismsdescribedthere,namelythecardiointerpolation(CI)andthecardiodelta(CD)approachareusedforEPBPaswell.
EspeciallyEPBPCIthatusesCI,themostpromisingmulti–phaseweightingcurrentlyavailable,isevaluatedhere.
Inthispaper,wewilloutlinetheEPBPStdandtheEPBPCIalgorithmandgivesomedescriptiveexamples.
II.
SimulationsToevaluateournewapproachwehavesimulatedspiralcone–beamdatacorrespondingtothein–planegeometryofatypicalmedicalCTscanner(1160projectionsperro-tation,672detectorchannelsperdetectorrow,andafanangleΦ=52)usingadedicatedx–raysimulationtool(ImpactSim,VAMPGmbH,M¨ohrendorf,Germany).
Two2FULLY3D'03TUAM2–3phantomshavebeensimulated:thethoraxphantomde-scribedinthephantomdatabasehttp://www.
imp.
uni–erlangen.
de/forbildandthecardiacmotionphantomde-scribedin[7].
Thesimulatedscanprotocoluses0.
42srotationtime(143rpm),M*0.
75mmcollimationwithM=2msimul-taneouslyscannedsliceswherem=4,8,andapitchofp=0.
375.
Thetableincrementcanbefoundasd=MSp.
III.
GeometryA.
ScanGeometryThescangeometryassumedhereisafan–beamgeome-trywithcylindricaldetectorsandaspiralfocustrajectory.
TheEPBPapproachisbasedonarebinningtoparallelgeometry.
Othergeometries,suchasatdetectors,canbeeasilyincorporatedbymodifyingthecorrespondingrebin-ningandtransformequations.
Notethatinthelimitofp=0,thespiralreducestoacirculartrajectory.
EPBPcopeswithsequencedataaswellasitdoeswithspiraldata.
Thesourcetrajectoryisparameterizedbytheviewangleαass(α)=RFsinαcosα0+d001α2π.
(1)RFdenotestheradiusofthefocalspottrajectoryandddenotesthetableincrementperrotation.
Thecoordinatevectorofthedetectorelement(α,β,b)isgivenasr(α,β,b)=s(α)+RFDsin(α+β)cos(α+β)0+b001;βandbdenotethetransaxialandlongitudinaldetectorcomponents,respectively.
Torebinthetransaxialcomponentsofthecone–beamdatatoparallelgeometryweparameterizearaybyitsdis-tanceξtotheaxisofrotationandbyitsanglewithre-specttothenegativey–axis.
Thenormalformoftheray'sx–y–componentsisgivenasxcos+ysinξ=0.
Thisdenitionwaschosentohavethecentralraysforfan–beam(β=0)andforparallelbeam(ξ=0)coincidingforα=.
Therelationbetweenarayinfan–beamcoordinates(α,β)andaparallel–beamray(,ξ)isthefamiliartransform=α+βξ=RFsinβandα=+arcsinξ/RFβ=arcsinξ/RF.
(2)B.
PointProjectionForbackprojection,weneedtoknowthedetectorcoor-dinates(ξ,b)thatresultfromprojectingthepoint(x,y,z)froms(α)ontothecylindricaldetector.
Theradialcoordi-nateisgivenasξ=xcos+ysin.
Thelongitudinaldetectorcoordinatecanbecomputedus-ingtheintersectiontheorem.
ThetransaxialdistanceoftherespectivevoxeltothesourceisgivenasD2=(RFsinαx)2+(RFcosα+y)2=R2F2RFrsin(α)+r2with(x,y)=(rcos,rsin)or,equivalently,D=RFcosβ+η=R2Fξ2+ηwithη=ycosxsin.
Now,ndbbyscalingtheaxialdistancezdα2πfromDtoRFD:b=RFDD(zdα2π).
(3a)Andwendanotherrepresentationofξ:ξ=RFsinβ=RFxcosα+ysinαD.
(3b)IV.
ReconstructionTheextendedparallelbackprojectionalgorithmconsistsofthefollowingvesteps:-azimuthalrebinning:p(α,β,b)→p(,β,b),-longitudinalrebinning:p(,β,b)→p(,β,l),-radialrebinning:p(,β,l)→p(,ξ,l),-convolution:p(,ξ,l)→p(,ξ,l),-weightingandbackprojection:p(,ξ,l)→f(x,y,z).
A.
AzimuthalRebinningTheoriginalprojectiondatap(α,β,b)areconvertedfromfan–beamtofan–parallelgeometryusing(2)asfollows:p(,β,b)=p(α,β,b)withα=β.
B.
LongitudinalRebinningConvolvingspiraldatainthedetectorrowdirection(con-stantb)yieldsseverecone–beamartifacts.
AsindicatedbyASSR[2],SMPR[9],andexactcone–beamreconstruc-tion[10]theoptimaldirectionofconvolutionisthetangentds(α)/dα.
Toalignthefan–paralleldetectorrowswiththeoptimaldirectionofconvolutionalongitudinalrebinningisrequired.
Therefore,weareinterestedintherelationshipbetweenbandξwhenmovingalongds.
Using(3)onendsdbdξ=d(bD)d(ξD)=RFDdzRF(dxcosα+dysinα)=RFDd2πR2F;inthelaststep(1)wasusedtoinsertthecomponentsofds.
Now,wedeneanewlongitudinalvariablelasb=l+λξwithλ=dbdξ=dRFD2πR2Fsuchthatdl/dξ=0inthedirectionofds.
Then,dothelongitudinalrebinningp(,β,l)=p(,β,b)withb=l+λξ=lλRFsinβtoswitchtolasthenewindependentvariable.
FULLY3D'03TUAM2–33Wheneverbexceedsthedetectorlimitsbminandbmaxforsomeβweextrapolatebyrepeatingtheoutermostde-tectorrow.
Thevaluesthataremadeupbyextrapolationarerequiredduringconvolutionwhichoperatesoncom-pletedetectorrows,always.
Duringbackprojectiontheseextrapolatedpointsarenotaccessed;backprojectionratherrespectsthephysicaldetectorarea!
C.
RadialRebinningTheradialrebinningconvertstoequidistantparallelco-ordinates.
Weuse(2)tondp(,ξ,l)=p(,β,l)withβ=arcsinξ/RF.
D.
ConvolutionNow,convolutionofthedetectorrowsisperformedusingastandardconvolutionkernelk(ξ),as,forexample,theShepp–Logankernel.
p(,ξ,l)=p(,ξ,l)k(ξ)yieldstheconvolveddatapneededforbackprojection.
E.
WeightingandBackprojectionInthisstep,weregardthebackprojectionofaxedvoxel,sayonelocatedatr=(x,y,z).
LetVdenotethesetofviewanglesunderwhichrismeasured.
AssumeatemporalwindowTthatcomprisesallthatcorrespondtoalloweddata.
Forthestandardreconstruc-tionEPBPStd,alldataacquiredarealloweddataandthereforeT=R.
Forthereconstructionofcardiacdata,TcanbedenedbyspecifyingacardiacmotionphasecRthatcountsrelativetosomesynchronizationpeaksandaphasewidth0EPBPCIchoosescassmallasallowedbythecompletenesscondition(seebelow).
Otherdenitionsmayincludeabsolutetiminginformationortherestrictiontoonlyonetemporalintervaloflengthπ(single–phasere-construction),ortwointervals(bi–phasereconstruction).
RegardlessofwhatconventionisusedtodeneT,theintersectionI=V∩T,thatcomprisesallviewstobeused,mustbe180–complete:k(I+kπ)=R.
Now,assumeaweightingfunctionw()whosesupportequalsI,i.
e.
w(R\I)={0},andkw(+kπ)=0.
ThelastconditioncaneasilybeachievedbyusingpositiveweightsonIonly.
ForEPBPCIweuseamulti–triangularweightfunction:trianglefunctionslocatedoneachofI'sdisjunctintervals.
Bynormalizingwasw()=w()kw(+kπ)weachievekw(+kπ)=1anddw()=π.
Backprojectionf(x,y,z)=dp(,ξ,l)w()withξ=ξ(x,y,)=xcos+ysinα=β=+arcsinξ/RFl=b(x,y,z,α)λξthenyieldsthedesiredvoxelvalueat(x,y,z).
V.
ResultsFigure1showsthatimagequalityofthethoraxphantomisexcellentwithEPBP,evenforasmanyas256slices.
Asindicatedbytheribs,ASSR(whichisinfactdesignedforuptoaboutonly60slices[2])cannotcopewiththislargecone–angle;thesameappliestothehighlyrelatedAMPRalgorithmdenedinreference[11].
Fig.
1.
Thorax,scannedwith256*0.
75mmcollimationandd=72mmtableincrement.
Aheartrateof120min1wassimulatedforEPBPCI.
(0/500)ConsideringthatEPBPCIusesonlyafractionofthedataavailable(here,roughly25%),dependingonthelocalheartrateandonthereconstructionposition,theimagesarealmostasgoodastheEPBPStdreconstructions,apartfromtheincreasedimagenoise.
Theonlyexceptionisaslightvariationinthereconstructeddensityclosetothevertebrae.
Figure2showsreconstructionsofthecardiacmotionphantomfora16–sliceanda256–slicescanner.
Sincethe4FULLY3D'03TUAM2–3(a)16*0.
75mmcollimation(b)256*0.
75mmcollimationFig.
2.
TransaxialslicesandMPRsofthecardiacmotionphantomreconstructedwithvariousalgorithmsusingastandard16–slicescannerandawidecone–angle256–sliceCT.
ThesagittalMPRs(bottom)showadditionalδ–objectsusedtomeasureresolution.
(0/500)eldofviewshowsonlythecentralpartsofthepatient,thein–planeimagesoftheASSRapproachareacceptableevenforthe256–slicescanner.
However,with256–sliceASSRthemultiplanarreformations(MPRs)tendtobeblurredinthez–directionandfullwidthathalfmaximumFWHMzoftheslicesensitivityproleisincreasedsignicantlywhereasthein–planeresolutionFWHMxyisthesameasforthe16–slicecase.
EPBP,incontrast,behavesverywellforallsimulatedscanners(16,32,64,128and256slices).
Spatialresolu-tionisslightlyhigherthanforthesingle–slicerebinningalgorithms.
ImagenoiseincreasesforEPBPCIduetothephase–weighting.
Thisobservationisvalidforalltheothersimulatedgeometriesandheartrates(wehavelookedintofH=40min1140min1).
EPBPgenerallybehavesequaltoorbetterthanASSR.
Finally,gure3givesanexampleofreconstructedpa-tientdata.
Thedatashownarecorrelatedtothepatientmotionfunction,theso–calledkymogram,whichcandi-rectlybederivedfromtheacquiredrawdata[12].
ThestandardreconstructionsofASSRStdandEPBPStdarecomparableduetothelownumberofslices;thephase–correlatedEPBPCIimagesareofhighimagequalityandcorrespondtothegold–standardincardiacCTimaging.
VI.
DiscussionTheextendedparallelbackprojectionappearstobeade-quateformedicalCTimagereconstructioninallrespects.
EPBPimagequalityisequivalenttoexisting4–or16–slicestandardandcardiacalgorithmsforawiderangeofFULLY3D'03TUAM2–35Fig.
3.
ASSRandEPBP,12*0.
75mmcollimation,3.
375mmtableincrement.
Thephase–correlatedimagesarereconstructedrelativetothekymogrampeaksat0%and50%ofK-K,respectively.
(0/500)simultaneouslyscannedslices.
EvendatawithM=256slicesyieldsexcellentimagequality.
Forstandardrecon-structionsthisisnotsurprisingsinceEPBPStdissimilartootherFeldkampalgorithms(aslongastheseperformconvolutionalongthetangentdirection).
Forwideconeanglecardiacdata,wherenootherphase–correlatedcone–beamalgorithmisavailableyet,EPBPCIperformsverywellevenforobjectsfarotheisocenter(ribsingure1).
Feldkamp–typealgorithmsaresuperiortoASSR[2],AMPR[11],orSMPR[9]forlargeM.
Wideconeanglecar-diacCTiscurrentlyonlypossiblewithEPBP.
Itsuniqueweightingstrategythatassignsindividualdatarangestoeachvoxel,ensures100%datausageandthusthemax-imumdoseutilizationpossible.
ThefutureofmedicalsequentialandspiralCTwillcertainlyincludetheideaofphase–correlated/phase–weighted3DbackprojectionofEPBP–type,ormodicationsthereof.
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