Adaptive finite element methods for the solution o

AdaptivefiniteelementmethodsforthesolutionofinverseproblemsinopticaltomographyWolfgangBangerthDepartmentofMathematics,TexasA&MUniversity,CollegeStationTX77843-3368,USAE-mail:[e-mail]Radiology,BaylorCollegeofMedicine,HoustonTX77030,USAE-mail:[e-mail]stodetermineaspatiallyvariablecoefficientintheinteriorofabodyfrommeasurementsoflightfluxesattheboundary.Likeinmanyotherapplicationsinbiomedicalimaging,computingsolutionsinopticaltomographyiscomplicatedbythefactthatonewantstoidentifyanunknownnumberofrelativelysmallirregularitiesinthiscoefficientatunknownlocations,forexamplecorrespondingtothepresenceoftumors.Torecoverthemattheresolutionneededinclinicalpractice,onehastousemeshesthat,ifuniformlyfine,wouldleadtointractablylargeproblemswithhundredsofmillionsofunknowns.Adaptivemeshesarethereforeanindispensabletool.Inthispaper,wewilldescribeaframeworkfortheadaptivefiniteelementsolutionofopticaltomographyproblems.Ittakesintoaccountallstepsstartingfromtheformulationoftheproblemincludingconstraintsonthecoefficient,outerNewton-typenonlinearandinnerlineariterations,regularization,andinparticulartheinterplayofthesealgorithmswithdiscretizingtheproblemonasequenceofadaptivelyrefinedmeshes.Wewilldemonstratetheefficiencyandaccuracyofthesealgorithmsonasetofnumericalexamplesofclinicalrelevancerelatedtolocatinglymphnodesintumordiagnosis.AMSclassificationschemenumbers:65N21,65K10,35R30,49M15,65N50,78-05Submittedto:InverseProblems1.IntroductionFluorescenceenhancedopticaltomographyisarecentandhighlyactiveareainbiomedicalimagingresearch.ItattemptstoreconstructinteriorbodypropertiesusinglightintheredandinfraredrangeinwhichbiologicaltissuesarehighlyscatteringAdaptivefiniteelementmethodsforopticaltomography2butnotstronglyabsorbing.Itisdevelopedasatoolforimaginguptodepthsofseveralcentimeters,whichincludesinparticularimportantapplicationstobreastandcervixcancerdetectionandstaging,lymphnodeimaging,butalsoimagingofthebrainofnewbornsthroughtheskull.Comparedtoestablishedimagingtechniques,opticaltomographyusesharmlessradiation,andisacheapandfastimagingtechnique.Inaddition,itcanbemadespecifictoindividualcelltypesbyusingdyestargetedatcertainmolecules.Theideabehindopticaltomography[4]istoilluminatethetissuesurfacewithaknownlasersource.Thelightwilldiffuseandbeabsorbedinthetissue.Byobservingthelightfluxexitingthetissuesurface,onehopestorecoverthespatialstructureofabsorptionandscatteringcoefficientsinsidethesample,whichinturnisassumedtocoincidewithanatomicalandpathologicalstructures.Forexample,itisknownthathemoglobinconcentration,bloodoxygenationlevels,andwatercontentaffectopticaltissueproperties,allofwhicharecorrelatedwiththepresenceoftumors.Asaresultdiffuseopticaltomography(DOT)canuniquelyimagetheinvivometabolicenvironment[21,33,34,65].However,DOThasbeenrecognizedasamethodthatishardtoimplementbecauseitdoesnotproduceaverylargesignaltonoiseratio.Thisfollowsfromthefactthatarelativelysmalltumor,oronethatdoesnothaveaparticularlyhighabsorptioncontrast,doesnotproducemuchdimmingofthelightintensityonthetissuesurface,inparticularinreflectiongeometry.AnotherdrawbackofDOTisthatitisnon-specific:itdetectsareasofhighlightabsorption,butdoesnotdistinguishthereasonsforabsorption;forexample,itcannotdistinguishbetweennaturallydarktissuesascomparedtoinvadingdarkdiseasedtissues.Duringthelastdecade,anumberofapproacheshavethereforebeendevelopedthatattempttoavoidthesedrawbacks.Oneisfluorescence-enhancedopticaltomography,inwhichafluorescentdyeisinjectedthatspecificallytargetscertaintissuetypes,forexampletumororlymphnodecells.Thepremiseisthatitisnaturallywashedoutfromtherestofthetissue.Ifthedyeisexcitedusinglightofonewavelength,thenwewillgetafluorescentsignalatadifferentwavelength,typicallyintheinfrared,fromareasinwhichdyeispresent(i.e.wherethetumororalymphnodeis).Sincethesignalisthepresenceoffluorescentlight,notonlyafaintdimmingoftheincidentlightintensity,thesignaltonoiseratiooffluorescenceopticaltomographyismuchbetterthaninDOT.Itisalsomuchbetterthan,forexample,inpositronemissiontomography(PET)becausedyescanbeexcitedmillionsorbillionsoftimespersecond,eachtimeemittinganinfraredphoton.Inaddition,thespecificityofdyescanbeusedformolecularlytargetedimaging,i.e.wecanreallyimagethepresenceofdiseasedcells,notonlysecondaryeffects.Inthepast,fluorescencetomographyschemeshavebeenproposedforpre-clinicalsmallanimalimagingapplications[28]aswellasfortheclinicalimagingoflargetissuevolumes[23,25,27,35,48,49,51,56–58,60,62,64,67].TypicalfluorescenceopticaltomographyschemesemployiterativeimagereconstructiontechniquestodetermineAdaptivefiniteelementmethodsforopticaltomography3thefluorescenceyieldorlifetimemapinthetissuefromboundaryfluorescencemeasurements.Asuccessfulclinicallyrelevantfluorescencetomographysystemwillhavethefollowingattributes:(i)rapiddataacquisitiontominimizepatientmovementanddiscomfort,(ii)accurateandcomputationallyefficientmodelingoflightpropagationinlargetissuevolumes,and(iii)arobustimagereconstructionstrategytohandletheill-posednessintroducedbythe