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Algorithms for sequencing multileaf collimators

University of Florida Institutional Repository

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ALGORITHMSFORSEQUENCINGMULTILEAFCOLLIMATORS By SRIJITKAMATH ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2005

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Copyright2005 by SrijitKamath

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Dedicatedtomyparents,whohavealwaysencouragedmetobec omeascientist.

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ACKNOWLEDGMENTS Firstandforemost,myutmostgratitudegoestomyadvisor,S artajSahni,forthe immeasurablyvaluableguidanceandsupporthehasgivenmed uringtheperiodofmy graduatestudy.Heintroducedmetothischallengingcomput ationalprobleminmedical physics.Hehaspatientlyabsorbedallmyideas,proofreade verymathematicalanalysis thatIhaveperformed,andgentlynudgedmeinthemostpromis ingdirectiononeach occasion.Hisjudgementandexperiencehaveprovedtobecru cialinthesuccessofthis work.Icouldnothaveaskedforabetteradvisor.Ifeelthath eisoneofthebestcomputer scienceeducatorsintheworld.Ithasbeenaprivilegeandan honortoworkwithhim. IamverythankfultoMeeraSitharamfortheseveralinsights shehasprovidedon algorithmicproblemsthroughherteachinginandoutofclas sandforhergeneralenthusiasm forproblemsolvingthathasledtomanyabsorbingdiscussio ns.Servingasthecoordinator ofthealgorithmsandtheoryseminar,whichwasstartedwith herencouragementand initiative,hasalsobeenanexcellentlearningexperience forme. ThanksgotoJonathanLiforcollaboratingontheproblemsan dforprovidingclinical data.ThanksgoalsotoJatinderPalta,AnandRangarajanand SanjayRankafortaking thetimetoserveonmycommittee. HaejaeJung,AnujJain,AmitaMalik,MohitDhawan,HaibinLu ,JiminYinand WenchengLumadeCSE329agoodplacetowork.Intenseprepara tionforthedoctoral qualifyingexamwaspossiblethankstolatenightstudysess ionswithVijayManianand PompiDiplan.Iwouldliketothankallmyfriends(simplytoo manytoname!)fortheir greatcompanyandsupportduringthecourseofmygraduatest udy,withspecialthanksto fellowstudentsSubi,Manas,Anuj,Andrew,PranavandShant anu. Thisworkwouldnothavebeenpossiblewithouttheconstante ncouragementIhave receivedfrommyparentsandfromuncleNarayan,whohasalso providednumerousinsights ongraduatestudyandworkintheUnitedStates. iv

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................iv LISTOFTABLES .....................................vii LISTOFFIGURES ....................................viii ABSTRACT ........................................x CHAPTER1INTRODUCTION ..................................1 1.1ProblemDescription .............................1 1.2MLCModelsandConstraints ........................4 1.3PriorWork ..................................5 1.4DissertationOutline .............................7 2SEQUENCINGOFSEGMENTEDMULTILEAFCOLLIMATORS ......8 2.1Methods ....................................8 2.1.1DiscreteProle ............................8 2.1.2MovementofLeaves .........................8 2.1.3OptimalUnidirectionalAlgorithmforonePairofLeav es .....11 2.1.4Bi-directionalMovement .......................16 2.1.5AlgorithmUnderMaximumSeparationConstraintCondi tion ..20 2.1.6AlgorithmUnderInter-PairMinimumSeparationConst raint ...23 2.2Conclusion ...................................34 3SEQUENCINGOFDYNAMICMULTILEAFCOLLIMATORS ........36 3.1Methods ....................................36 3.1.1MovementofLeaves .........................36 3.1.2MaximumVelocityConstraint ....................38 3.1.3OptimalUnidirectionalAlgorithmforonePairofLeav es .....38 3.1.4MinimumSeparationConstraint ...................42 3.1.5Bi-directionalMovement .......................46 3.1.6AlgorithmUnderMaximumSeparationConstraintCondi tion ..51 3.1.7AlgorithmUnderInterdigitationConstraint ............55 3.2Conclusion ...................................65 4ELIMINATIONOFTONGUE-AND-GROOVEUNDERDOSAGE .......66 4.1AlgorithmwithInterdigitationandTongue-and-Groove Constraints ...67 4.1.1Tongue-and-GrooveUnderdosageEect ..............67 4.1.2Algorithms ...............................68 v

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4.1.3EcientImplementationoftheAlgorithms .............79 4.2ExperimentalValidation ...........................81 4.3ComparisonwithAlgorithmofQueetal.(2004)) .............82 4.3.1AnalysisoftheAlgorithmofQueetal.(2004) ...........83 4.3.2Results .................................87 4.4Conclusion ...................................87 5ALGORITHMSFORSPLITTINGLARGEFIELDS ...............90 5.1Introduction ..................................90 5.2FieldSplittingWithoutFeathering .....................90 5.2.1OptimalFieldSplittingforOneLeafPair .............90 5.2.2OptimalFieldSplittingforMultipleLeafPairs ..........97 5.3FieldSplittingwithFeathering .......................100 5.3.1SplittingaProleintoTwo .....................104 5.3.2SplittingaProleintoThree ....................105 5.3.3Tongue-and-grooveEectandInterdigitation ...........106 5.4Results .....................................106 5.5Conclusion ...................................109 6CONCLUSION ....................................110 REFERENCES .......................................111 BIOGRAPHICALSKETCH ...............................115 vi

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LISTOFTABLES Table page 4{1Comparisonofthenumberofsegments .....................83 4{2NumberofMUsandsegments ..........................88 4{3AveragenumberofMUsandsegments .....................88 5{1TotalMUsforveclinicalcases .........................108 vii

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LISTOFFIGURES Figure page 1{1Alinearaccelerator ................................2 1{2Amultileafcollimator ..............................3 1{3Inter-pairminimumseparationconstraint ...................4 1{4Crosssectionofleaves ..............................5 2{1Geometryandcoordinatesystem ........................9 2{2Discretizationofprole ..............................9 2{3LeaftrajectoryduringSMLCdelivery .....................10 2{4Obtainingaunidirectionalplan .........................12 2{5Aproleanditsplan ...............................13 2{6Minimumseparationconstraintviolation ....................15 2{7Bi-directionalmovement .............................17 2{8Bi-directionalmovementunderminimumseparationcons traint .......19 2{9Bi-directionalmovementundermaximumseparationcons traint .......20 2{10Obtainingaplanundermaximumseparationconstraint ...........21 2{11Maximumseparationconstraintviolation ...................21 2{12Obtainingaschedule ...............................24 2{13Obtainingascheduleundertheconstraint ...................25 2{14Eliminatingaviolation ..............................26 2{15Eliminatingaviolation ..............................30 2{16Intensityprolesofadjacentleafpairs .....................31 2{17Prolesviolatinginter-pairconstraint .....................32 3{1LeaftrajectoryduringDMLCdelivery .....................37 3{2Obtainingaunidirectionalplan .........................40 3{3Obtainingaunidirectionalplanwithminimumseparatio nconstraint ....44 3{4Minimumseparationconstraintviolation ....................44 viii

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3{5SMLCplan:feasible;DMLCplan:infeasible ..................47 3{6Bi-directionalmovement .............................48 3{7Bi-directionalmovementunderminimumseparationcons traint .......50 3{8Bi-directionalmovementundermaximumseparationcons traint .......52 3{9Obtainingaplanundermaximumseparationconstraint ...........52 3{10Maximumseparationconstraintviolation ...................53 3{11Obtainingaschedule ...............................56 3{12Obtainingascheduleundertheconstraint ...................57 3{13EliminatingaType1violation ..........................58 3{14EliminatingaType2violation ..........................59 4{1Tongue-and-grooveeect .............................67 4{2Counterexample ..................................70 4{3Obtainingascheduleunderthetongue-and-grooveconst raint ........71 4{4Tongue-and-grooveconstraintviolation:case1 .................72 4{5Tongue-and-grooveconstraintviolation:case2 .................73 4{6Obtainingascheduleunderboththeconstraints ...............77 4{7FilmmeasurementoftheAPeld ........................82 4{8Leafpositions ...................................85 4{9Worstcaseexample ................................86 5{1Splittingaprole(a)intotwo ..........................93 5{2TightupperboundforLemma32a .......................95 5{3TightupperboundforLemma32b .......................95 5{4TightlowerboundforLemma32c .......................96 5{5TightupperboundforLemma32c .......................97 5{6Fieldmatchingproblem .............................102 5{7Exampleofeldsplittingwithfeathering ...................103 5{8Comparisonoftheeldsplitline ........................107 ix

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ALGORITHMSFORSEQUENCINGMULTILEAFCOLLIMATORS By SrijitKamath August2005 Chair:SartajSahniMajorDepartment:ComputerandInformationScienceandEng ineering Indeliveringradiationtherapyforcancertreatment,itis desirabletodeliverhighdoses ofradiationtothetargettumor,whilepermittingalowdosa geonthesurroundinghealthy tissues.Inrecentyears,thedevelopmentofintensitymodu latedradiationtherapy(IMRT) hasmadethispossible.IMRTmaybedeliveredbyseveraltech niques.Thedeliveryof IMRTwithamultileafcollimator(MLC)requiresthedeliver yofradiationfromseveral beamorientations.Theintensityproleforeachbeamdirec tionisdescribedasaMLCleaf sequence,whichisdevelopedusingaleafsequencingalgori thm.Importantconsiderations indevelopingaleafsequenceforadesiredintensityprole includemaximizingthemonitor unit(MU)eciency(equivalentlyminimizingthebeam-onti me)andminimizingthetotal treatmenttimesubjecttotheleafmovementconstraintsoft heMLCmodel.Inthiswork, wepresentasystematicstudyoftheoptimizationofleafseq uencingalgorithmsandprovide rigorousmathematicalproofsofoptimizedleafsequencese ttingsintermsofMUeciency undermostcommonleafmovementconstraintsthatincludemi nimumandmaximumleaf separation,leafinterdigitationandtongue-and-groove. Wealsodevelopalgorithmstosplit largeintensitymodulatedeldsintotwoorthreesubelds. x

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CHAPTER1 INTRODUCTION 1.1ProblemDescription Theobjectiveofradiationtherapyforcancertreatmentist odeliverhighdosesof radiationtothetargettumor,whilepermittingalowdosage onthesurroundinghealthy tissues.Forexample,forheadandnecktumors,itisnecessa ryforradiationtobedelivered sothattheexposureofthespinalcord,opticnerve,salivar yglandsorotherimportant structuresisminimized.Inrecentyears,thishasbeenmade possibleduetothedevelopmentofconformalradiationtherapy.Inconformaltherapy, treatmentisdeliveredusinga setofradiationbeamswhicharepositionedsuchthatthesha peofthedosedistribution \conforms"withtheshapeofthetumor.Thisistypicallyach ievedbypositioningbeams ofvaryingshapesfromdierentdirectionssothateachbeam approximatelyirradiatesthe sectionofthetumorvisiblefromitsdirectionandavoidsth eorgansatriskinthevicinity ofthetumor. Intensitymodulatedradiationtherapy(IMRT)isthestateof-the-artinconformalradiationtherpy.IMRTpermitstheintensityofaradiationbeam tobevariedacrossatreatment area,therebyimprovingthedoseconformity.Radiationisd eliveredusingamedicallinear accelerator(Figure 1{1 ).Arotatinggantrycontainingtheacceleratorstructurec anrotate aroundthepatientwhoispositionedonanadjustabletreatm entcouch.DeliveryofIMRTis possiblebyseveraltechniques.Incompensator-basedIMRT ,thebeamismodulatedwitha preshapedpieceofmaterialcalledthecompensator(modula tor).Thedegreeofmodulation ofthebeamvariesdependingonthethicknessofthematerial throughwhichthebeamis attenuated.Thecomputerdeterminestheshapeofeachmodul atorinordertodeliverthe desiredbeam.Thistypeofmodulationrequiresthemodulato rtobefabricatedandthen manuallyinsertedintothetraymountofalinearaccelerato r.Intomotherapy-basedIMRT, thelinearacceleratortravelsinmultiplecirclesallthew ayaroundthegantryringtodeliver theradiationtreatment.Thebeamiscollimatedtoanarrows litandtheintensityofthe 1

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2 beamismodulatedduringthegantrymovementaroundthepati ent.Caremustbetakento ensurethatadjacentcirculararcsdonotoverlapandthereb ydonotoverdosetissues.This typeofdeliveryisreferredtoasserialtomotherapy.Amodi cationofserialtomotherapyis helicaltomotherapy.Inhelicaltomotherapy,thetreatmen tcouchmoveslinearly(continuously)throughtherotatingacceleratorgantry.Soeachtim etheacceleratorcomesaround, itdirectsthebeamonaslightlydierentplaneonthepatien t.InMLC-basedIMRTthe acceleratorstructureisequippedwithacomputercontroll edmechanicaldevicecalleda multileafcollimator(MLC,Figure 1{2 )thatshapestheradiationbeam,soastodeliver theradiationasprescribedbythetreatmentplan.TheMLCma yhaveupto120movable leavesthatcanmovealonganaxisperpendiculartothebeama ndcanbearrangedsoas toshieldorexposepartsoftheanatomyduringtreatment.Th eleavesarearrangedin pairssothateachleafpairformsonerowofthearrangement. ThesetofallowableMLC leafcongurationsmayberestrictedbyleafmovementconst raintsthataremanufacturer and/ormodeldependent. Figure1{1:Alinearaccelerator(thegureisfromhttp://w ww.lexmed.com/medical services/IMRT.htm) TherststageinthetreatmentplanningprocessinIMRTisto obtainaccuratethree dimensionalanatomicalinformationaboutthetumorandits surroundings.Thisisachieved

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3 Figure1{2:Amultileafcollimator(thegureisfromhttp:/ /www.lexmed.com/medical services/IMRT.htm) usingcomputedtomography(CT)andmagneticresonance(MR) imaging.Anidealdose distributionwouldensureperfectconformitytothetarget volumewhilecompletelysparing allothertissues.However,suchadistributionisimpossib letorealizeinpractice.Therefore, minimumdosetargetsfortumorsandtolerabledosesforcrit icalstructuresareprescribed andanobjectivefunctionthatmeasuresthequalityofaplan isdevelopedsubjecttothese dosebasedconstraints.Next,asetofbeamparameters(beam angles,proles,weights) thatoptimizethisobjectivearedeterminedusingacompute rprogram.Thismethodis called\inverseplanning"sinceresultantdosedistributi onisrstdescribedandthebest beamparametersthatdeliverthedistribution(approximat ely)arethensolvedfor.Itis tobenotedthatinverseplanningisageneralconceptandits implementationdetailsvary vastlyamongvarioussystems.Followingtheinverseplanni nginMLC-basedIMRT,the deliveryofradiationintensityproleforeachbeamdirect ionisdescribedasaMLCleaf sequence,whichisdevelopedusingaleafsequencingalgori thm.Importantconsiderations indevelopingaleafsequenceforadesiredintensityprole includemaximizingthemonitor unit(MU)eciency(equivalentlyminimizingthebeam-onti me)andminimizingthetotal treatmenttimesubjecttotheleafmovementconstraintsoft heMLCmodel.Finally,when theleafsequencesforallbeamdirectionsaredetermined,t hetreatmentisperformedfrom

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4 variousbeamanglessequentiallyusingcomputercontrol.I nthiswork,wedevelopoptimized leafsequencingalgorithmsforvariousMLCmodels. 1.2MLCModelsandConstraints Thepurposeoftheleafsequencingalgorithmistogeneratea sequenceofleafpositions and/ormovementsthatfaithfullyreproducethedesiredint ensitymaponcethebeamis delivered,takingintoconsiderationanyhardwareanddosi metriccharacteristicsofthedeliverysystem.ThetwomostcommonmethodsofIMRTdeliveryw ithcomputer-controlled MLCsarethesegmentalmultileafcollimator(SMLC)anddyna micmultileafcollimator (DMLC).InSMLC,thebeamisswitchedowhiletheleavesarei nmotion.Inother words,thedeliveryisdoneusingmultiplestaticsegmentso rleafsettings.Thismethodis alsofrequentlyreferredtoasthe\stepandshoot"or\stopa ndshoot"method.InDMLC thebeamisonwhiletheleavesareinmotion.Thebeamisswitc hedonatthestartof treatmentandisswitchedoonlyattheendoftreatment.The fundamentaldierence betweentheleafsequencesofthesetwodeliverymethodsist hattheleafsequencedenesa nitesetofbeamshapesforSMLCandtrajectoriesofopposin gpairsofleavesforDMLC. Inpracticalsituations,therearesomeconstraintsonthem ovementoftheleaves.The minimumseparationconstraintrequiresthatopposingpair sofleavesbeseparatedbyat leastsomedistance( S min )atalltimesduringbeamdelivery.InMLCsthisconstrainti s appliednotonlytoopposingpairsofleaves,butalsotooppo singleavesofneighboring pairs.Forexample,inFigure 1{3 L 1and R 1, L 2and R 2, L 3and R 3, L 1and R 2, L 2and R 1, L 2and R 3, L 3and R 2arepairwisesubjecttotheconstraint.Thecasewith S min =0 iscalledinterdigitationconstraintandisapplicabletos omeMLCmodels.Whereverthis constraintapplies,oppositeadjacentleavesarenotpermi ttedtooverlap. Figure1{3:Inter-pairminimumseparationconstraint

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5 InmostcommerciallyavailableMLCs,thereisatongue-andgroovearrangementat theinterfacebetweenadjacentleaves.Acrosssectionoftw oadjacentleavesisdepictedin Figure 1{4 .Thewidthofthetongue-and-grooveregionis l .Theareaunderthisregiongets underdosedduetothemechanicalarrangement,asitremains shieldedifeitherthetongue orthegrooveportionofaleafshieldsit. movement Leaf Radiation l Figure1{4:Crosssectionofleaves Maximumleafspreadforleavesonthesameleafbankisonemor eMLClimitation, whichnecessitatesalargeeld(intensityprole)tobespl itintotwoormoreadjacent abuttingsub-elds.ThisistruefortheVarianMLC(VarianM edicalSystems,PaloAlto, CA),whichhasaeldsizelimitationofabout15cm.Theabutt ingsub-eldsarethen deliveredasseparatetreatmentelds.Thisoftenresultsi nlongerdeliverytimes,poorMU eciency,andeldmatchingproblems. 1.3PriorWork Optimizationoftheleafsequencingalgorithmhasbeenthes ubjectofnumerousinvestigations(forexample,ConveryandRosenbloom1992,Bortf eldetal.1994a,Dirkxetal. 1998,Maetal.1998,XiaandVerhey1998,Siochi1999,Langer etal.2001,Luanetal. 2003,Chenetal.2004).TreatmentdeliverywithIMRTisnotv eryecientintermsofMU eciency,whichisdenedastheratioofdosedeliveredatap ointinthepatientwithan IMRTeldtotheMUdeliveredforthateld.TypicalMUecien ciesofIMRTtreatment

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6 plansare3to10timeslowerthanthoseforopen/wedgeeld-b asedconventionaltreatment plans.Therefore,totalbodydoseduetoincreasedleakager adiationreachingthepatient inanIMRTtreatmentisamajorconcern(Followilletal.1997 ,IntensityModulatedRadiationTherapyCollaborativeWorkingGroup2001).LowMUec iencyofIMRTdelivery negativelyimpactstheroomshieldingdesignbecauseofthe increasedworkload(Intensity ModulatedRadiationTherapyCollaborativeWorkingGroup2 001,Muticetal.2001).The MUeciencydependsonboththedegreeofintensitymodulati onandthealgorithmused toconverttheintensitypatternintoaleafsequenceforIMR Tdelivery.Itisthereforeimportanttodesignaleafsequencingalgorithmthatisoptima lforMUeciencytominimize totalbodydosetothepatient.Fordynamicbeamdeliverywhe redoserateisusuallynot modulated,analgorithmthatoptimizestheMUsettingatagi vendoseratealsooptimizes thetreatmenttime. Dynamicleafsequencingalgorithmswiththeleavesinmotio nduringradiationdelivery havebeendeveloped(ConveryandRosenbloom1992,Spirouan dChui1994),andlater modied(vanSantvoortandHeijmen1996,Dirkxetal.1998)t oeliminatethetongue-andgrooveunderdosageeects.Similarleafsequencingalgori thmshavealsobeendevelopedfor thesegmentalmultileafcollimator(SMLC)deliverymethod (Bortfeldetal.1994a,Bortfeld etal.1994b,Maetal.1998,XiaandVerhey1998,Que1999,Eng el2003,Kalinowski2003, Lietal.2003).Manyofthesestudiesdidnotconsideranylea fmovementconstraints. Suchleafsequencingalgorithmsareapplicableforcertain typesofMLCdesigns.Forother typesofMLCdesigns,notablytheSiemens(SiemensMedicalS ystems,Inc.,Iselin,NJ) MLCdesign(Dasetal.1998)andElekta(ElektaOncologySyst emsInc.,Norcross,GA) MLCdesign(JordanandWilliams1994),othermechanicalcon straintsneedtobetaken intoconsiderationwhendesigningtheleafsettingsforbot hdynamicandSMLCdelivery. Theminimumleafseparationconstraint,forexample,wasre centlyincorporatedintothe designofleafsequence(ConveryandWebb1998).Ageneralde scriptionandcharacteristics ofsomeMLCmodelscanbefoundinXiaandVerhey(2001).

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7 1.4DissertationOutline Inthiswork,wepresentasystematicstudyoftheoptimizati onofleafsequencing algorithms.Thedissertationisorganizedasfollows.Inch apter2,wepresentleafsequencingalgorithmsfortheSMLCbeamdeliveryandproviderigoro usproofsofoptimizedleaf sequencesettingsintermsofMUeciencyundervariousleaf movementconstraints.Practicalleafmovementconstraintsthatareconsideredinclud etheminimumleafseparation constraintandminimuminter-leafseparationconstraint( leafinterdigitationconstraint). Thequestionofwhetherbi-directionalleafmovementwilli ncreasetheMUeciencywhen comparedwithuni-directionalleafmovementonlyisalsoad dressed.Inchapter3,wedevelopleafsequencingalgorithmsforDMLCbeamdelivery.Al gorithmsarepresentedto sequenceleaveswithmaximumleafseparationconstraintan dtheleafinterdigitationconstraint.Inchapter4,westudytongue-and-grooveeectfor SMLC.Weprovideboundson themaximumextenttowhichtongue-and-grooveeectcanbee liminatedandgivenecessaryandsucientconditionsforaunidirectionalleafsequ encetoattainthebound.We thenpresentalgorithmsthatgenerateleafsequencesthate liminatethetongue-and-groove eectandoptionallysatisfytheinterdigitationconstrai nt.Wealsocompareouralgorithms toarecentlypublishedleafsequencingalgorithmthatalso eliminatestongue-and-groove underdosage.Theproblemofsplittinglargeintensitymodu latedeldsintotwoorthree subeldsisdiscussedinchapter5.Allouralgorithmsgener ateunidirectionalleafmovement schedulesandareprovedtobeoptimalinMUsforunidirectio nalschedules.

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CHAPTER2 SEQUENCINGOFSEGMENTEDMULTILEAFCOLLIMATORS Inthischapter,wepresentasystematicstudyoftheoptimiz ationofleafsequencingalgorithmsfortheSMLCbeamdeliveryandproviderigoro usproofsofoptimizedleaf sequencesettingsintermsofMUeciencyundervariousleaf movementconstraints.Practicalleafmovementconstraintsthatareconsideredinclud etheminimumleafseparation constraintandminimuminter-leafseparationconstraint( leafinterdigitationconstraint). Thequestionofwhetherbi-directionalleafmovementwilli ncreasetheMUeciencywhen comparedwithuni-directionalleafmovementonlyisalsoad dressed.Werstintroducethe notationthatwillbeusedintheremainderofthiswork. 2.1Methods 2.1.1DiscreteProle Thegeometryandcoordinatesystemusedinthisstudyaresho wninFigure 2{1 Weconsiderdeliveryofprolesthatarepiecewisecontinuo us.Let I ( x )bethedesired intensityprole.Werstdiscretizetheprolesothatweob tainthevaluesatsample points x 0 ;x 1 ;x 2 ; ... ;x m I ( x )isassignedthevalue I ( x i )for x i x
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9 Figure2{1:Geometryandcoordinatesystem Figure2{2:Discretizationofprole

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10 themotionoftheleftleaf.Theleftleafbeginsat x 0 andremainshereuntil I l ( x 0 )MUs havebeendelivered.Atthistimetheleftleafismovedto x 1 ,whereitremainsuntil I l ( x 1 ) MUshavebeendelivered.Theleftleafthenmovesto x 3 whereitremainsuntil I l ( x 3 )MUs havebeendelivered.Atthistime,theleftleafismovedto x 6 ,whereitremainsuntil I l ( x 6 ) MUshavebeendelivered.Thenalmovementoftheleftleafis to x 7 ,whereitremains until I l ( x 7 )= I max MUshavebeendelivered.Atthistimethemachineisturnedo .The totaltherapytime, TT ( I l ;I r ),isthetimeneededtodeliver I max MUs.Therightleafstarts at x 2 ;movesto x 4 when I r ( x 2 )MUshavebeendelivered;movesto x 5 when I r ( x 4 )MUs havebeendeliveredandsoon.Notethatthemachineisowhen aleafisinmotion.We makethefollowingobservations: Figure2{3:LeaftrajectoryduringSMLCdelivery 1.AllMUsthataredeliveredalongaradiationbeamalong x i beforetheleftleafpasses x i fallonit.Thegreaterthe x value,thelatertheleafpassesthatposition.Therefore I l ( x i )isanon-decreasingfunction. 2.AllMUsthataredeliveredalongaradiationbeamalong x i beforetherightleaf passes x i areblockedbytheleaf.Thegreaterthe x value,thelatertheleafpasses thatposition.Therefore I r ( x i )isalsoanon-decreasingfunction.

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11 FromtheseobservationswenoticethatthenetamountofMUsd eliveredatapointis givenby I l ( x i ) I r ( x i ),whichmustbethesameasthedesiredprole I ( x i ). 2.1.3OptimalUnidirectionalAlgorithmforonePairofLeav es Unidirectionalmovement. Whenthemovementofleavesisrestrictedtoonlyonedirection,boththeleftandrightleavesmovealongpositive x direction,fromlefttoright (Figure 2{1 ).Oncethedesiredintensityprole, I ( x i )isknown,ourproblembecomesthat ofdeterminingtheindividual intensityproles tobedeliveredbytheleftandrightleaves, I l and I r suchthat I ( x i )= I l ( x i ) I r ( x i ) ; 0 i m (2.1) Wereferto( I l ;I r )asthe treatmentplan (orsimply plan )for I .Onceweobtainthe plan,wewillbeabletodeterminethemovementofbothleftan drightleavesduringthe therapy.Foreach i ,theleftleafcanbeallowedtopass x i whenthesourcehasdelivered I l ( x i )MUs.Also,wecanallowtherightleaftopass x i whenthesourcehasdelivered I r ( x i ) MUs.Inthismannerweobtain unidirectionalleafmovementproles foraplan. Algorithm. FromEquation 2.1 ,weseethatonewaytodetermine I l and I r fromthe giventargetprole I istobeginwith I l ( x 0 )= I ( x 0 )and I r ( x 0 )=0;examinetheremaining x i sfromlefttoright;increase I l whenever I increases;andincrease I r whenever I decreases. Once I l and I r aredeterminedtheleafmovementprolesareobtainedasexp lainedinthe previoussection.TheresultingalgorithmisshowninFigur e 2{4 .Figure 2{5 showsaprole andthecorrespondingplanobtainedusingthealgorithm. Maetal.(1998)showsthatAlgorithmSINGLEPAIRobtainspla nsthatareoptimal intherapytime.TheirproofreliesontheresultsofSpiroua ndChui(1994),Steinetal. (1994)andBoyerandStrait(1997).Weprovideamuchsimpler proofbelow. Theorem1 AlgorithmSINGLEPAIRobtainsplansthatareoptimalinther apytime. Proof: Let I ( x i )bethedesiredprole.Let inc 1 ;inc 2 ;:::;inck betheindicesofthepoints atwhich I ( x i )increases.So x inc 1 ;x inc 2 ;:::;x inck arethepointsatwhich I ( x )increases(i.e., I ( x inci ) >I ( x inci 1 )).Let i = I ( x inci ) I ( x inci 1 ). Supposethat( I L ;I R )isaplanfor I ( x i )(notnecessarilythatgeneratedbyAlgorithm

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12 AlgorithmSINGLEPAIRI l ( x 0 )= I ( x 0 ) I r ( x 0 )=0 For j =1to m do If( I ( x j ) I ( x j 1 ) I l ( x j )= I l ( x j 1 )+ I ( x j ) I ( x j 1 ) I r ( x j )= I r ( x j 1 ) Else I r ( x j )= I r ( x j 1 )+ I ( x j ) I ( x j 1 ) I l ( x j )= I l ( x j 1 ) Endfor Figure2{4:Obtainingaunidirectionalplan SINGLEPAIR).Fromtheunidirectionalconstraint,itfollo wsthat I L ( x i )and I R ( x i )are non-decreasingfunctionsof x .Since I ( x i )= I L ( x i ) I R ( x i )forall i ,weget i =( I L ( x inci ) I R ( x inci )) ( I L ( x inci 1 ) I R ( x inci 1 )) =( I L ( x inci ) I L ( x inci 1 )) ( I R ( x inci ) I R ( x inci 1 )) I L ( x inci ) I L ( x inci 1 ). Summingup i ,weget P ki =1 [ I ( x inci ) I ( x inci 1 )] P ki =1 [ I L ( x inci ) I L ( x inci 1 )]= TT ( I L ;I R ). Sincethetherapytimefortheplan( I l ;I r )generatedbyAlgorithmSINGLEPAIRis P ki =1 [ I ( x inci ) I ( x inci 1 )],itfollowsthat TT ( I l ;I r )isminimum. Corollary1 Let I ( x i ) 0 i m beadesiredprole.Let I l ( x i ) ,and I r ( x i ) 0 i m betheleftandrightleafprolesgeneratedbyAlgorithmSIN GLEPAIR. I l ( x i ) and I r ( x i ) 0 i m deneoptimaltherapytimeunidirectionalleftandrightle afprolesfor I ( x i ) 0 i m Proof: FollowsfromTheorem 1 Intheremainderofthispaper,( I l ;I r )istheoptimaltreatmentplanforthedesired prole I Propertiesoftheoptimaltreatmentplan. Thefollowingobservationsaremade abouttheoptimaltreatmentplan( I l ;I r )generatedusingAlgorithmSINGLEPAIR. Lemma1 Ateach x i atmostoneoftheproles I l and I r changes(increases).

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13 Figure2{5:Aproleanditsplan

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14 Lemma2 Let ( I L ;I R ) beanytreatmentplanfor I (a) ( x i )= I L ( x i ) I l ( x i )= I R ( x i ) I r ( x i ) 0 ; 0 i m (b) ( x i ) isanon-decreasingfunction. Proof: (a)Since I ( x i )= I L ( x i ) I R ( x i )= I l ( x i ) I r ( x i ) ;I L ( x i ) I l ( x i )= I R ( x i ) I r ( x i ). Further,fromCorollary 1 ,itfollowsthat I L ( x i ) I l ( x i ) ; 0 i m .Therefore,( x i ) 0 ; 0 i m (b)Weprovethisbycontradiction.Supposethat( x n ) > ( x n +1 )forsome n; 0 nI l ( x n +1 )+( x n +1 )= I L ( x n +1 ). Thisisnotpossiblebecause I L isanon-decreasingfunction. Case2: I r ( x n )= I r ( x n +1 ) Now, I R ( x n )= I r ( x n )+( x n ) >I r ( x n +1 )+( x n +1 )= I R ( x n +1 ). Thiscontradictsthefactthat I R isanon-decreasingfunction. Case3: I l ( x n ) 6 = I l ( x n +1 )and I r ( x n ) 6 = I r ( x n +1 ) FromLemma 1 itfollowsthatthiscasecannotarise. Therefore,( x i )isanon-decreasingfunction. Theorem2 Iftheoptimalplan ( I l ;I r ) violatestheminimumseparationconstraint,then thereisnoplanfor I thatdoesnotviolatetheminimumseparationconstraint. Proof: Supposethat( I l ;I r )violatestheminimumseparationconstraint.Assumethatt he rstviolationoccurswhen I 1 MUshavebeendelivered.Fromtheunidirectionalmovement constraint,itfollowsthattheleftleafhasjustbeenposit ionedat x j (forsome j; 0 j m ) atthistimeandthattherightleafisat x k suchthat x k x j islessthanthepermissible minimumseparation.Figure 2{6 illustratesthesituation. Weprovethetheorembycontradiction.Let( I L ;I R )beaplanthatdoesnotviolate theminimumseparationconstraint.When j =0,( I l ;I r )hasaviolationattheinitial positioning x 0 oftheleftleaf.Sincetheleavesmoveinonlyonedirection, theviolation iswhen I 1 =0.When I 1 =0,theleftleafin( I L ;I R )isalsoat x 0 (becausetheleftleaf mustbeginat x 0 inallplans;otherwise I ( x 0 )=0).For( I L ;I R )nottohaveaviolation at I 1 =0,therightleafmustbegintotherightof x k ,sayatsomepoint p>x k (note

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15 Figure2{6:Minimumseparationconstraintviolation that p maynotbeoneofthe x i s).TheMUsdeliveredat x k bytheplan( I L ;I R )are I L ( x k ) I R ( x k )= I L ( x k ) I l ( x k )(Corollary1).Also, I l ( x k )= I ( x k )+ I r ( x k ) >I ( x k ) ( I r ( x k ) > 0).So( I L ;I R )deliversmorethan I ( x k )MUsat x k andsoisnotaplanfor I Thiscontradictstheassumptionon( I L ;I R ).Hence, j 6 =0. Supposethat j> 0.Now, I l ( x j ) >I l ( x j 1 ).Also, I L ( x j )= I l ( x j )+( x j )and I L ( x j 1 )= I l ( x j 1 )+( x j 1 ).Since( x j ) ( x j 1 )(Lemma 2 (b)), I L ( x j ) >I L ( x j 1 ). Therefore,theleftleafispositionedat x j atsometimeduringtheoncycleoftheplan ( I L ;I R ).LettheamountofMUsdeliveredwhentheleftleafarrivesa t x j in I L be I 2 .Let therightleafbeat x = p atthistime.Notethat p maynotbeoneofthe x i s.If p>x k ,then I R ( x k ) I 2 .Also,fromLemma 2 wehave I L ( x k )= I l ( x k )+( x k ) I l ( x k )+( x j 1 )= I l ( x k )+ I 2 I 1 >I l ( x k )+ I 2 I r ( x k )= I ( x k )+ I 2 .Therefore, I L ( x k ) I R ( x k ) >I ( x k ).This contradicts I L ( x k ) I R ( x k )= I ( x k )(since( I L ;I R )isaplanfor I ).Therefore, j cannot be > 0either.So,thereisnoplan( I L ;I R )thatdoesnotviolatetheminimumseparation constraint. Theseparationbetweentheleavesisdeterminedbythedier encein x valuesofthe leaveswhenthesourcehasdeliveredacertainamountofMUs. Theminimumseparation oftheleavesistheminimumseparationbetweenthetwoprol es.Wecallthisminimum

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16 separation S ud min .WhentheoptimalplanobtainedusingAlgorithmSINGLEPAIR is delivered,theminimumseparationis S ud min ( opt ) Corollary2 Let S ud min ( opt ) betheminimumleafseparationintheplan ( I l ;I r ) .Let S ud min bethemininmumleafseparationinany(notnecessarilyopti mal)givenunidirectionalplan. S ud min S ud min ( opt ) 2.1.4Bi-directionalMovement Inthissectionwestudybeamdeliverywhenbi-directionalm ovementofleavesispermitted.Weexplorewhetherrelaxingtheunidirectionalmov ementconstrainthelpsimprove theeciencyoftreatmentplan.Propertiesofbi-directionalmovement. Foragivenleaf(leftorright)movement proleweclassifyany x -coordinateasfollows.Drawaverticallineat x .Ifthelinecutsthe leafproleexactlyoncewewillcall x a unidirectionalpoint ofthatleafprole.Iftheline cutstheprolemorethanonce, x isa bi-directionalpoint ofthatprole.Aleafmovement prolethathasatleastonebi-directionalpointisa bi-directionalprole .Allprolesthatare notbi-directionalare unidirectionalproles .Anyprolecanbepartitionedintosegments suchthateachsegmentisaunidirectionalprole.Whenthen umberofsuchpartitionsis minimal,eachpartitioniscalleda stage oftheoriginalprole.Thusunidirectionalproles consistofexactlyonestagewhilebi-directionalprolesa lwayshavemorethanonestage. InFigure 2{7 ,theleafmovementprole, B l ,showsthepositionoftheleftleafas afunctionoftheamountofMUsdeliveredbythesource.Thele afstartsfromtheleft edgeandmovesinbothdirectionsduringthetherapy.Clearl y, B l isbi-directional.The movementproleofthisleafconsistsofstages S 1 ;S 2 and S 3 .Instages S 1 and S 3 the leafmovesfromlefttorightwhileinstage S 2 theleafmovesfromrighttoleft. x j isa bi-directionalpointof B l .TheamountofMUsdeliveredat x j is L 1 + L 2 .Instage S 1 ,when L 1 amountofMUshavebeendelivered,theleafpasses x j .Now,noMUisdeliveredat x j tilltheleafpassesover x j in S 2 .Further, L 2 MUsaredeliveredto x j instages S 2 and S 3 Thuswehave I l ( x j )= L 1 + L 2 .Here, L 1 = I 1 ;L 2 = I 3 I 2 x k isaunidirectionalpoint of B l .TheMUsdeliveredat x k are L 3 = I 4 .Notethattheintensityprole I l isdierent fromtheleafmovementprole B l ,unlikeintheunidirectionalcase.

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17 Figure2{7:Bi-directionalmovement Lemma3 Let ( I l ;I r ) beaplandeliveredbythebi-directionalleafmovementpro lepair ( B l ;B r ) (i.e., B l and B r are,respectively,theleftandrightleafmovementproles ) (a) I l isnon-decreasing. (b) I r isnon-decreasing. Proof: (a)Wheneverapoint x i ; 0 i m ,isblockedbythetheleftleaf,thepoints x 0 ;x 1 ;:::;x i 1 arealsoblocked.Itfollowsthat I l ( x i ) I l ( x j ) ; 0 j i m (b)Theproofissimilarto(a) FromLemma 3 wenotethatabi-directionalleafmovementprole B deliversanondecreasingintensityprole.Thisnon-decreasingintensi typrolecanalsobedelivered usingaunidirectionalleafmovementprole(Section 2.1.3 ).Wewillcallthisprolethe unidirectionalleafmovementprolethatcorrespondstoth ebi-directionalprole B and wewilldenoteitby U toemphasizethatitisunidirectional.Thuseverybi-direc tional leafmovementprolehasacorrespondingunidirectionalle afprolethatdeliversthesame amountofMUsateach x i asdoesthebi-directionalprole.

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18 Theorem3 TheunidirectionaltreatmentplanconstructedbyAlgorith mSINGLEPAIRis optimalintherapytimeevenwhenbi-directionalleafmovem entispermitted. Proof: Let B L and B R bebidirectionalleafmovementprolesthatdeliveradesir ed intensityprole I .Let I L and I R ,respectively,betheintensityprolesfor B L and B R FromLemma 3 ,weknowthat I L and I R arenon-decreasing.Also, I L ( x i ) I R ( x i )= I ( x i ) ; 1 i m .FromtheproofofTheorem 1 ,itfollowsthatthetherapytimeforthe unidirectionalplan( I l ;I r )generatedbyAlgorithmSINGLEPAIRisnomorethanthatof ( I L ;I R ). Incorporatingminimumseparationconstraint. Let U l and U r beunidirectional leafmovementprolesthatdeliverthedesiredprole I ( x i ).Let B l and B r beasetof bi-directionalleftandrightleafprolessuchthat U l and U r correspondto B l and B r respectively,i.e.,( B l ;B r )deliversthesameplanas( U l ;U r ).Wecalltheminimumseparation ofleavesinthisbi-directionalplan( B l ;B r ) S bd min Theorem4 S bd min S ud min forabi-directionalleafmovementprolepairandits correspondingunidirectionalprole.Proof: Supposethattheminimumseparation S ud min occurswhen I ms MUsaredelivered.Atthistime,theleftleafarrivesat x j andtherightleafispositionedat x k .Let B 0 l and U 0 l respectively,betheprolesobtainedwhen B l and U l areshiftedrightby S ud min Since U 0 l is U l shiftedrightby S ud min andsincethedistancebetween U l and U r is S ud min when I ms MUshavebeendelivered, U 0 l and U r touchwhen I ms unitshavebeendelivered. Therefore,thetotalMUsdeliveredby( U 0 l ;U r )at x k iszero.SothetotalMUsdeliveredby ( B 0 l ;B r )at x k isalsozero(recallthat U 0 l and U r ,respectively,arecorrespondingprolesfor B 0 l and B r ).Thisisn'tpossibleif B r isalwaystotherightof B 0 l (forexample,inthesituationofFigure 2{8 ,theMUsdeliveredby( B 0 l ;B r )at x k are( L 1 + L 2 ) ( L 01 + L 02 + L 03 ) > 0). Therefore B 0 l and B r musttouch(orcross)atleastonce.So S bd min S ud min Theorem5 Iftheoptimalunidirectionalplan ( I l ;I r ) violatestheminimumseparationconstraint,thenthereisnobi-directionalmovementplanthat doesnotviolatetheminimum separationconstraint.

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19 Figure2{8:Bi-directionalmovementunderminimumseparat ionconstraint Proof: Let B l and B r bebi-directionalleafmovementsthatdelivertherequired prole. Lettheminimumseparationbetweentheleavesbe S bd min .Letthecorrespondingunidirectionalleafmovementsbe U l and U r andlet S ud min betheminimumseparationbetween U l and U r .Also,let S min betheminimumallowableseparationbetweentheleaves.Fro m Corollary 2 andTheorem 4 ,weget S bd min S ud min S ud min ( opt )
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20 Since U 0 l is U l shiftedrightby S ud max andsincethedistancebetween U l and U r is S ud max when I ms MUshavebeendelivered, U 0 l and U r touchwhen I ms unitshavebeendelivered. Therefore,thetotalMUsdeliveredby( U r ;U 0 l )at x k iszero.SothetotalMUsdeliveredby ( B r ;B 0 l )at x k isalsozero(recallthat U 0 l and U r ,respectively,arecorrespondingprolesfor B 0 l and B r ).Thisisn'tpossibleif B r isalwaystotheleftof B 0 l (forexample,inthesituation ofFigure 2{9 ,theMUsdeliveredby( B r ;B 0 l )at x k are( L 01 + L 02 + L 03 ) ( L 1 + L 2 ) > 0). Therefore B 0 l and B r musttouch(orcross)atleastonce.So S bd max S ud max Figure2{9:Bi-directionalmovementundermaximumseparat ionconstraint 2.1.5AlgorithmUnderMaximumSeparationConstraintCondi tion Inthissectionwepresentanalgorithmthatgeneratesanopt imaltreatmentplanunder themaximumseparationconstraint.RecallthatAlgorithmS INGLEPAIRgeneratesthe optimalplanwithoutconsideringthisconstraint.Wemodif yAlgorithmSINGLEPAIR sothatallinstancesofviolationofmaximumseparation(th atmaypossiblyexist)are eliminated.Weknowthatbi-directionalleafprolesdonot helpeliminatetheconstraint. Soweconsideronlyunidirectionalproles.Algorithm. ThealgorithmisdescribedinFigure 2{10 Theorem7 AlgorithmMAXSEPARATIONobtainsplansthatareoptimalint herapytime, underthemaximumseparationconstraint.

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21 AlgorithmMAXSEPARATION 1.ApplyAlgorithmSINGLEPAIRtoobtaintheoptimalplan( I l ;I r ) : 2.Findtheleastvalueofintensity, I 1 ,suchthattheleafseparationin( I l ;I r )when I 1 MUsaredeliveredis >S max ,where S max isthemaximumallowedseparationbetween theleaves.Ifthereisnosuch I 1 ,( I l ;I r )istheoptimalplan;end. 3.Let x j and x k ,respectively,bethepositionoftheleftandrightleavesa tthistime (seeFigure 2{11 ).Relocatetherightleafat x 0k suchthat x 0k x j = S max ,when I 1 MUsaredelivered.Let I = I l ( x j ) I 1 = I 2 I 1 .Movetheproleof I r ,which follows x 0k ,upby I along I direction.Tomaintain I ( x )= I l ( x ) I r ( x )forevery x movetheproleof I l ,whichfollows x 0k ,upby I along I direction. GotoStep2. Figure2{10:Obtainingaplanundermaximumseparationcons traint Figure2{11:Maximumseparationconstraintviolation

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22 Proof: Weuseinductiontoprovethetheorem. Thestatementweprove, S ( n ),isthefollowing: AfterStep3ofthealgorithmisapplied n times,theresultingplan,( I ln ;I rn ),satises (a)Ithasnomaximumseparationviolationwhen I<>: I l ( x )0 x<>: I r ( x )0 x
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23 Also, TT ( I 00 l 1 ;I 00 r 1 )= TT ( I 0 l 1 ;I 0 r 1 ) II l 1 ( x j ) Thisleadstoacontradictionasinthepreviouscase. (c) I 0 l 1 ( x j )
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24 prolesdenedalongtheaxesoftheleafpairs.Eachleafpai r i thendeliverstheplan forthecorrespondingintensityprole I i ( x ).Thesetofplansofallleafpairsformsthe solutionset.Werefertothissetasthe treatmentschedule (orsimply schedule ).Inthis section,wepresentleafsequencingalgorithmsforSMLCwit handwithoutconstraints.The constraintscondideredare(i)minimumseparationconstra intand(ii)tongue-and-groove constraintand(optionally)interdigitationconstraint. Weusetheterm intra-pairminimumseparationconstraint torefertotheconstraint imposedonanopposingpairofleavesand inter-pairminimumseparationconstraint to refertotheconstraintimposedonopposingleavesofneighb oringpairs.Recallthat,in Section 2.1.3 ,weprovedthatforasinglepairofleaves,iftheoptimalpla ndoesnotsatisfy theminimumseparationconstraint,thennoplansatisesth econstraint.Inthissectionwe presentanalgorithmtogeneratetheoptimalscheduleforth edesiredproledenedovera 2-Dregion.Wethenmodifythealgorithmtogenerateschedul esthatsatisfytheinter-pair minimumseparationconstraint.Optimalschedulewithouttheminimumseparationconstrain t. Assumewehave n pairsofleaves.Foreachpair,wehave m samplepoints.Theinputisrepresentedasamatrix with n rowsand m columns,wherethe i throwrepresentsthedesiredintensityproleto bedeliveredbythe i thpairofleaves.WeapplyAlgorithmSINGLEPAIRtodetermin ethe optimalplanforeachofthe n leafpairs.Thismethodofgeneratingschedulesisdescribe d inAlgorithmMULTIPAIR(Figure 2{12 ). AlgorithmMULTIPAIRFor( i =1; i n ; i ++) ApplyAlgorithmSINGLEPAIRtothe i thpairofleavestoobtainplan( I il ;I ir )that deliverstheintensityprole I i ( x ). EndFor Figure2{12:Obtainingaschedule Lemma4 AlgorithmMULTIPAIRgeneratesschedulesthatareoptimali ntherapytime. Proof: Treatmentiscompletedwhenallleafpairs(whichareindepe ndent)delivertheir respectiveplans.Thetherapytimeoftheschedulegenerate dbyAlgorithmMULTIPAIR is max f TT ( I 1 l ;I 1 r ) ;TT ( I 2 l ;I 2 r ) ; ... ;TT ( I nl ;I nr ) g .FromTheorem 1 ,itfollowsthatthis therapytimeisoptimal.

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25 Optimalalgorithmwithinter-pairminimumseparationcons traint. Theschedule generatedbyAlgorithmMULTIPAIRmayviolateboththeintra -andinter-pairminimum separationconstraints.Iftheschedulehasnoviolationso ftheseconstraints,itisthedesired optimalschedule.Ifthereisaviolationoftheintra-pairc onstraint,thenitfollowsfrom Theorem 2 thatthereisnoschedulethatisfreeofconstraintviolatio n.So,assumethatonly theinter-pairconstraintisviolated.Weeliminateallvio lationsoftheinter-pairconstraint startingfromtheleftend,i.e.,from x 0 .Toeliminatetheviolations,wemodifythoseplans oftheschedulethatcausetheviolations.Wescantheschedu lefrom x 0 alongthepositive x directionlookingfortheleast x v atwhichispositionedarightleaf(say Ru )thatviolates theinter-pairseparationconstraint.Afterrectifyingth eviolationat x v withrespectto Ru welookforotherviolations.Sincetheprocessofeliminati ngaviolationat x v ,may attimes,leadtonewviolationsat x j ;x j
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26 ofAlgorithmMINSEPARATIONisapplied p timestotheinputschedule M .Notethat M = N (0). Figure2{14:Eliminatingaviolation Toillustratethemodicationprocessweuseanexample(see Figure 2{14 ).Tomake thingseasier,weonlyshowtwoneighboringpairsofleaves. Supposethatthe( p +1)th violationoccurswhentherightleafofpair u ispositionedat x v andtheleftleafofpair t;t 2f u 1 ;u +1 g ,arrivesat x u ;x v x u <>: I tlp ( x ) x 0 x
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27 Onemayalsoverifythatsince I tl 0 and I tr 0 arenon-decreasingfunctionsof x ,soalsoare I tlp and I trp p> 0. Lemma5 Let F =(( I 0 1 l ;I 0 1 r ) ; ( I 0 2 l ;I 0 2 r ) ;:::; ( I 0 nl ;I 0 nr )) beanyfeasiblescheduleforthedesiredprole,i.e.,aschedulethatdoesnotviolatetheintr a-orinter-pairminimumseparation constraints.Let S ( p ) ,bethefollowingassertions. (a) I 0 il ( x ) I ilp ( x ) 0 i n;x 0 x x m (b) I 0 ir ( x ) I irp ( x ) 0 i n;x 0 x x m S ( p ) istruefor p 0 Proof: Theproofisbyinductionon p 1.Considerthebasecase, p =0.FromCorollary 1 andthefactthattheplans ( I il 0 ;I ir 0 ) ; 0 i n ,aregeneratedusingAlgorithmSINGLEPAIR,itfollowsthat S (0)istrue. 2.Assume S ( p )istrue.SupposeAlgorithmMINSEPARATIONndsanextviola tion andmodiestheschedule N ( p )to N ( p +1).Supposethatthenextviolationoccurs whentherightleafofpair u ispositionedat x v andtheleftleafofpair t arrivesat x u ;x v x u
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28 ThesituationisillustratedinFigure 2{14 Sincethereisnominimumseparationviolationin F ,theleftleafofpair t passes x 0u onlyaftertherightleafofpair u passes x v ,i.e., I 0 tl ( x 0u ) I 0 ur ( x v )(2.4) Since S ( p )istrue, I 0 ur ( x v ) I urp ( x v )= I tl ( p +1) ( x 0u )(2.5) FromEquations 2.4 and 2.5 I 0 tl ( x 0u ) I tl ( p +1) ( x 0u )(2.6) Addingandsubtracting I 0 tl ( x 0u )to I 0 tl ( x ), I 0 tl ( x )= I 0 tl ( x 0u )+ I 0 tl ( x ) I 0 tl ( x 0u ) ; 0 x x m (2.7) Similarly, I tl ( p +1) ( x )= I tl ( p +1) ( x 0u )+ I tl ( p +1) ( x ) I tl ( p +1) ( x 0u ) ; 0 x x m (2.8) Since I tlp ( x )= I tl ( x ) ;x x 0u I tl ( p +1) ( x )= I tl ( x )+ I;x 0u x x m (2.9) FromEquations 2.8 and 2.9 ,weget I tl ( p +1) ( x )= I tl ( p +1) ( x 0u )+( I tl ( x )+ I ) ( I tl ( x 0u )+ I ) ;x 0u x x m = I tl ( p +1) ( x 0u )+ I tl ( x ) I tl ( x 0u ) ;x 0u x x m (2.10) SubtractingEquation 2.10 fromEquation 2.7 I 0 tl ( x ) I tl ( p +1) ( x )=( I 0 tl ( x 0u ) I tl ( p +1) ( x 0u ))+( I 0 tl ( x ) I tl ( x )) ( I 0 tl ( x 0u ) I tl ( x 0u )) ;x 0u x x m (2.11)

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29 FromEquations 2.6 and 2.11 I 0 tl ( x ) I tl ( p +1) ( x ) ( I 0 tl ( x ) I tl ( x )) ( I 0 tl ( x 0u ) I tl ( x 0u )) ;x 0u x x m (2.12) FromLemma 2 b, I 0 tl ( x ) I tl ( x ) I 0 tl ( x 0u ) I tl ( x 0u ) ;x 0u x x m (2.13) FromEquations 2.12 and 2.13 ,weget I 0 tl ( x ) I tl ( p +1) ( x ) ;x 0u x x m (2.14) (b)Somepriormodicationhasbeenmadetopair t 'splanfor x x 0u .Thereexists amodicationat x w suchthat I tlp ( x ) >I tl ( x )+ I;x w x x m ,andthere isno xx 0u Inthiscase(seeFigure 2{15 ), I tl ( p +1) ( x )= 8><>: I tl ( x )+ Ix 0u x j
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30 Figure2{15:Eliminatingaviolation Lemma6 Ifanintra-pairminimumseparationviolationisdetectedi nStep 5 ofMINSEPARATION,thenthereisnofeasiblescheduleforthedesiredp role. Proof: Supposethatthereisafeasibleschedule F andthatleafpair t hasanintra-pair minimumseparationviolationin N ( p ) ;p> 0.FromLemma 5 itfollowsthat (a) I 0 tl ( x ) I tlp ( x ) ;x 0 x x m (b) I 0 tr ( x ) I trp ( x ) ;x 0 x x m where I 0 and I areasinLemma 5 .However,fromtheproofofTheorem 2 itfollowsthat if I tlp and I trp haveaminimumseparationviolation,thennotreatmentplan ( I 0 tl ;I 0 tr )that satises(a)and(b)canbefeasible.Therefore,nofeasible schedule F exists. Example1 Weillustrateaninstancewhereaninter-pairminmumsepara tionviolation isdetectedinStep 5 ofMINSEPARATION.Figure 2{16 showstwointensityproles,to bedeliveredbyadjacentleafpairs(say t and t +1 ).Theplansfor I t ( x ) and I t +1 ( x ) are obtainedusingalgorithmMULTIPAIR.TheyareshowninFigur e 2{17 .Eachofthese plans( ( I tl ( x ) ;I tr ( x )) and ( I ( t +1) l ( x ) ;I ( t +1) r ( x )) )isfeasible,i.e.,thereisnointra-pairminimumseparation( S min =7 ).However,whenMINSEPARATIONisapplied(forsimplicity considerleafpairs t and t +1 inisolation),itdetectsaninter-pairminimumseparation violationbetween I ( t +1) l and I tr ,when I ( t +1) l arrivesat x =6 and I tr ispositionedat x =11

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31 Toeliminatethisviolation, I ( t +1) l ispositionedat x =4 (since 11 4=7= S min )andits proleisraisedfrom x =4 .Consequently I ( t +1) r isalsoraisedfrom x =4 resultinginthe plan ( I ( t +1) l 1 ( x ) ;I ( t +1) r 1 ( x )) .Thismodicationresultsinanintra-pairviolationforpa ir t +1 ,when I ( t +1) l 1 isat x =1 and I ( t +1) r 1 isat x =4 .FromLemma 6 ,thereisnofeasible schedule. Figure2{16:Intensityprolesofadjacentleafpairs For N ( p ) ;p 0andeveryleafpair j; 1 j n ,dene I jlp ( x 1 )= I jrp ( x 1 )= 0 ; jlp ( x i )= I lp ( x i ) I lp ( x i 1 ) ; 0 i m and jrp ( x i )= I rp ( x i ) I rp ( x i 1 ) ; 0 i m Noticethat jlp ( x i )givesthetime(inmonitorunits)forwhichtheleftleafofp air j stops atposition x i .Let jlp ( x i )and jrp ( x i )bezeroforall x i when j =0aswellaswhen j = n +1. Lemma7 Forevery j; 1 j n andevery i; 1 i m jlp ( x i ) max f jl 0 ( x i ) ; ( j 1) rp ( x i + S min ) ; ( j +1) rp ( x i + S min ) g (2.15)

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32 Figure2{17:Prolesviolatinginter-pairconstraint Proof: Theproofisbyinductionon p .Fortheinductionbase, p =0.Putting p =0into therightsideofEquation 2.15 ,weget max f jl 0 ( x i ) ; ( j 1) r 0 ( x i + S min ) ; ( j +1) r 0 ( x i + S min ) g jl 0 ( x i )(2.16) Fortheinductionhypothesis,let q 0beanyintegerandassumethatEquation 2.15 holdswhen p = q .Intheinductionstep,weprovethattheequationholdswhen p = q +1. Let t;u ,and x v beasiniteration p 1ofthe while loopofalgorithmMINSEPARATION. Followingthisiteration,only tlp and trp aredierentfrom tl ( p 1) and tr ( p 1) ,respectively.Furthermore,only tlp ( x w )and trp ( x w ),where x w = x v S min maybelargerthan thecorrespondingvaluesfollowingiteration p 1.Atallbutatmostoneother x value (wheremayhavedecreased), tlp and trp arethesameasthecorrespondingvalues followingiteration p 1. Since x v istherightleafpositionfortheleftmostviolation,thele ftleafofpair t arrives at x w = x v S min aftertherightleafofpair u arrivesat x v = x w + S min .Followingthe modicationmadeto I tl ( p 1) ,theleftleafofpair t leaves x w atthesametimeastherightleaf ofpair u leaves x w + S min .Therefore, tlp ( x w ) ur ( p 1) ( x w + S min )= urp ( x w + S min ). Theinductionstepnowfollowsfromtheinductionhypothesi sandtheobservationthat u 2f t 1 ;t +1 g

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33 Lemma8 Forevery j; 1 j n andevery i; 1 i m jrp ( x i )= jlp ( x i ) ( I j ( x i ) I j ( x i 1 )(2.17) where I j ( x 1 )=0 Proof: Weexamine N ( p ).Themonitorunitsdeliveredbyleafpair j at x i are I jlp ( x i ) I jrp ( x i )andtheunitsdeliveredat x i 1 are I jlp ( x i 1 ) I jrp ( x i 1 ).Therefore, I j ( x i )= I jlp ( x i ) I jrp ( x i )(2.18) I j ( x i 1 )= I jlp ( x i 1 ) I jrp ( x i 1 )(2.19) SubtractingEquation 2.19 fromEquation 2.18 ,weget I j ( x i ) I j ( x i 1 )=( I jlp ( x i ) I jlp ( x i 1 )) ( I jrp ( x i ) I jrp ( x i 1 )) = jlp ( x i ) jrp ( x i )(2.20) Thelemmafollowsfromthisequality. Noticethatoncearightleaf u movespast x m ,noseparationviolationwithrespect tothisleafispossible.Therefore, x v (seealgorithmMINSEPARATION) x m .Hence, jlp ( x i ) jl 0 ( x i ),and jrp ( x i ) jr 0 ( x i ) ;x m S min x i x m ; 1 j n .Starting withtheseupperbounds,whichareindependentof p ,on jrp ( x i ), x m S min x i x m and usingEquations 2.15 and 2.17 ,wecancomputeanupperboundontheremaining jlp ( x i )s and jrp ( x i )s(fromrighttoleft).Theremainingupperboundsarealsoi ndependentof p Letthecomputedupperboundon jlp ( x i )be U jl ( x i ).Itfollowsthatthetherapytimefor ( I jlp ;I jrp )isatmost T max ( j )= P 0 i m U jl ( x i ).Therefore,thetherapytimefor N ( p )is atmost T max = max 1 j n f T max ( j ) g Theorem8 ThefollowingaretrueofAlgorithmMINSEPARATION: (a)Thealgorithmterminates. (b)WhenthealgorithmterminatesinStep 5 ,thereisnofeasibleschedule. (c)Otherwise,theschedulegeneratedisfeasibleandisopt imalintherapytimeforunidirectionalschedules.

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34 Proof: (a)Asnotedabove,Lemmas 7 and 8 provideanupperbound, T max onthe therapytimeofanyscheduleproducedbyalgorithmMINSEPAR ATION.Itiseasy toverifythat I il ( p +1) ( x ) I ilp ( x ) ; 0 i n;x 0 x x m I ir ( p +1) ( x ) I irp ( x ) ; 0 i n;x 0 x x m andthat I tl ( p +1) ( x 0u ) >I tlp ( x 0u ) I tr ( p +1) ( x 0u ) >I trp ( x 0u ) Noticethateventhoughavalue(proofofLemma 7 )maydecreaseatan x i ,the I ilp and I irp valuesneverdecreaseatany x i aswegofromoneiterationofthewhile loopofMINSEPARATIONtothenext.Since I tl increasesbyatleastoneunitat atleastone x i oneachiteration,itfollowsthatthewhileloopcanbeitera tedatmost mnT max times. (b)FollowsfromLemma 6 (c)IfterminationdoesnotoccurinStep 5 ,thennominimumseparationviolationsremain andthenalscheduleisfeasible.FromLemma 5 ,itfollowsthatthenalscheduleis optimalintherapytimeforunidirectionalschedules. Corollary3 When S min =0 ,AlgorithmMinseparationalwaysgeneratesanoptimalfeasibleschedule.Proof: When S min =0,AlgorithmMinseparationcannotterminateinStep 5 because theStep 4 modicationnevercausestheleftleafofaleafpairtocross therightleafofthat pair.TheCorollaryfollowsnowfromTheorem 8 2.2Conclusion Inconclusion,wepresentedmathematicalformalismsandri gorousproofsofleafsequencingalgorithmsforsegmentalmultileafcollimationw hichmaximizeMUeciency. Theseleafsequencingalgorithmsexplicitlyaccountformi nimumleafseparationconstraint andleafinterdigitationconstraint.Wehaveshownthatour algorithmsobtainallfeasible

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35 solutionsthatareoptimalintreatmentMUs.Furthermore,o uranalysisshowsthatunidirectionalleafmovementisatleastasecientasbi-direc tionalmovement.Thusthese algorithmsarewellsuitedforcommonuseinSMLCbeamdelive ry.

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CHAPTER3 SEQUENCINGOFDYNAMICMULTILEAFCOLLIMATORS DeliveryusingDMLCisdierentfromthatusingSMLC.Thelea fpositionschange withrespecttotime.IntermsoftheMLCcontrolleritisthec hangeinpositionwithrespect tomonitorunitsdeliveredthatisimportant.Theinputsreq uiredaretheleafpositionsat variouscontrolpoints,thefractionalnumberofmonitorun itstobedeliveredateachcontrol point,andthetotalnumberofmonitorunitstobedeliveredf orthatbeam.Inthischapter, wepresentasystematicstudyoftheoptimizationofleafseq uencingalgorithmsforthe dynamicbeamdeliveryandproviderigorousproofsofoptimi zedleafsequencesettingsin termsofMUeciencyundervariousleafmovementconstraint s.Practicalleafmovement constraintsthatareconsideredincludetheleafinterdigi tationconstraint.Thequestionof whetherbi-directionalleafmovementwillincreasetheMUe ciencywhencomparedwith unidirectionalleafmovementonlyisalsoaddressed. 3.1Methods 3.1.1MovementofLeaves Inouranalysiswewillassumethat I ( x 0 ) > 0and I ( x m ) > 0andthatwhenthebeam deliverybeginstheleavescanbepositionedanywhere.Weal soassumethattheleaves canmovewithanyvelocity v v max v v max ,where v max isthemaximumallowable velocityoftheleavesandthattheleafaccelerationcanbei nnite.Theconsequencesof assuminginniteleafaccelerationarenegligible.Figure 3{1 illustratestheleaftrajectory duringDMLCdelivery.Intheexample,theleavesmovefromle fttoright.Let I l ( x i )and I r ( x i ),respectively,denotetheamountofMonitorUnits(MUs)de liveredwhentheleftand rightleavesleaveposition x i .Considerthemotionoftheleftleaf.Theleftleafbeginsat x 0 andremainshereuntil I l ( x 0 )MUshavebeendelivered.Atthistimetheleftleafleaves x 0 andismovedto x 1 ,whereitremainsuntil I l ( x 1 )MUshavebeendelivered.Theleftleaf thenmovesto x 3 whereitremainsuntil I l ( x 3 )MUshavebeendelivered.Atthistime,the leftleafismovedto x 5 ,whereitremainsuntil I l ( x 5 )MUshavebeendelivered.Thenit 36

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37 movesto x 6 ,whereitremainsuntil I l ( x 6 )MUshavebeendelivered.Thenalmovementof theleftleafisto x 10 .Theleftleafarrivesat x 10 when I max MUshavebeendelivered.At thistimethemachineisturnedo.Thetotaltherapytime, TT ( I l ;I r ),isthetimeneeded todeliver I max MUs.Therightleafstartsat x 0 andbeginstomoverightawaytillitreaches x 2 ;leaves x 2 when I r ( x 2 )MUshavebeendelivered;leaves x 4 when I r ( x 4 )MUshavebeen delivered,andsoon.Notethatthemachineisonthroughoutt hetreatment.AllMUs thataredeliveredalongaradiationbeamalong x i beforetheleftleafpasses x i fallonit. Similarly,allMUsthataredeliveredalongaradiationbeam along x i beforetherightleaf passes x i ,areblockedbytheleaf.SothenetamountofMUsdeliveredat apointisgiven by I l ( x i ) I r ( x i ),whichmustbethesameasthedesiredprole I ( x i ). ll 012345 I I l r I l I (x ) 65310 I (x ) l 678910 l I x x x x x x x x x x x x I (x )I (x ) I max I (x ) Figure3{1:LeaftrajectoryduringDMLCdelivery Theorem9 Thefollowingaretrueofeveryleafpairtrajectorythatdel iversadiscrete prole:(a)Theleftleafmustreach x 0 atsometime. (b)Therightleafmustreach x m atsometime. (c)Theleftleafmustreach x m atsometime. (d)Therightleafmustreach x 0 atsometime. Proof: (a)Supposethattheleftleafalwaysstaystotherightof x 0 ,then x 0 doesnot receiveanyMUs,contradictingourassumptionthat I ( x 0 ) > 0.

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38 (b)Similartothatof(a).(c)Iftheleftleafdoesn'treach x m (i.e.,itdoesn'tgototherightof x m 1 ),from(b),it followsthattheregionbetween x m 1 and x m receivesanon-uniformdistributionofMUs. Hencethediscreteprolecan'tbeaccuratelydelivered.(d)Similartothatof(c). 3.1.2MaximumVelocityConstraint Asnotedearlier,thevelocityofleavescannotexceedsomem aximumlimit(say v max ) inpractice.Thisimpliesthattheleafprolecannotbehori zontalatanypoint.From Figure 2{3 ,observethatthetimeneededforaleaftomovefrom x i to x i +1 is ( x i +1 x i ) =v max .IfistheruxdensityofMUsfromthesource,thenumberofMU sdeliveredin thistimealongabeamis ( x i +1 x i ) =v max .So, I l ( x i +1 ) I l ( x i ) ( x i +1 x i ) =v max = x=v max .Thesameistruefortherightleafprole I r 3.1.3OptimalUnidirectionalAlgorithmforonePairofLeav es Unidirectionalmovement. Whenthemovementofleavesisrestrictedtoonlyonedirection,boththeleftandrightleavesmovealongthepositi ve x direction,fromlefttoright (Figure 2{1 ).Oncethedesiredintensityprole, I ( x i )isknown,ourproblembecomesthat ofdeterminingtheindividual intensityproles tobedeliveredbytheleftandrightleaves, I l and I r suchthat I ( x i )= I l ( x i ) I r ( x i ) ; 0 i m (3.1) Ofcourse, I l and I r aresubjecttothemaximumvelocityconstraint.Wereferto( I l ;I r ) asthe treatmentplan (orsimply plan )for I .Onceweobtaintheplan,wewillbeableto determinethemovementofbothleftandrightleavesduringt hetherapy.Foreach i ,the leftleafcanbeallowedtopass x i whenthesourcehasdelivered I l ( x i )MUs.Also,wecan allowtherightleaftopass x i whenthesourcehasdelivered I r ( x i )MUs.Inthismanner weobtain unidirectionalleafmovementproles foraplan.Somesimpleobservationsabout theleafprolesaremadebelow.Theorem10 Ineveryunidirectionalplantheleavesbeginat x 0 andendat x m Proof: FollowsfromTheorem 9 andtheunidirectionalconstraint.

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39 Lemma9 Intheregionbetweeneachpairofsuccessivesamplepoints, say x i and x i +1 bothleafprolesmaintainthesameshape,i.e.,oneismerel yaverticaltranslationofthe other.Proof: Asexplainedpreviously,theinputproleisdiscretizedto asquarewave I .Since theproleof I ishorizontalbetweensuccessivesamplepointsandsinceit isequalto I l I r I l and I r musthavethesameshape.Forexample,iftheleftleafmovesa taconstant velocity v betweenpoints x i and x i +1 ,soshouldtherightleaf. Lemma10 Inanoptimalplan,bothleavesmustmoveat v max betweeneverysuccessive pairofsamplepointstheymoveacross.Proof: Supposethatinanoptimalsolutiontheleavesmovebetweenp oints x i and x i +1 atapossiblyvaryingvelocity v ( x ) v max .FromLemma 9 ,weknowthatbothleafproles arethesamebetween x i and x i +1 .Setting v ( x )= v max resultsinnewleafproleswhose dierenceremainsthesameasbefore(whichisthedesiredpr ole I )andtotaltherapytime islowered.Thisleadstoacontradiction. Corollary4 Inanoptimalplan,noleafstopsatan x thatisnotoneofthe x i s. Algorithm. FromEquation 3.1 ,weseethatonewaytodetermine I l and I r fromthe giventargetprole I istobeginfrom x 0 ;set I l ( x 0 )= I ( x 0 )and I r ( x 0 )=0;examinethe remaining x i stotheright;increase I l at x i whenever I increasesandbythesameamount (inadditiontotheminimumincreaseimposedbythemaximumv elocityconstraint);and similarlyincrease I r whenever I decreases.Thiscanbedonetillwereach x m .Sothe treatmentbeginswiththeleavespositionedattheleftmost samplepointandendswiththe leavespositionedattherightmostsamplepoint.Once I l and I r aredeterminedtheleaf movementprolesareobtainedasexplainedearlier.Noteth atwemovetheleavesatthe maximumvelocity v max whenevertheyaretobemoved.Theresultingalgorithmissho wn inFigure 3{2 .Figure 2{3 showsaprole I andthecorrespondingplan( I l ;I r )obtained usingAlgorithmDMLC-SINGLEPAIR. Maetal.(1998)showsthatAlgorithmDMLC-SINGLEPAIRobtai nsplansthatare optimalintherapytime.Theirproofreliesontheresultsof SpirouandChui(1994),Stein etal.(1994)andBoyerandStrait(1997).Weprovideasimple randdirectproofbelow.

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40 AlgorithmDMLC-SINGLEPAIRI l ( x 0 )= I ( x 0 ) I r ( x 0 )=0 For j =1to m do If( I ( x j ) I ( x j 1 )) I l ( x j )= I l ( x j 1 )+ I ( x j ) I ( x j 1 )+ x=v max I r ( x j )= I r ( x j 1 )+ x=v max Else I r ( x j )= I r ( x j 1 )+ I ( x j 1 ) I ( x j )+ x=v max I l ( x j )= I l ( x j 1 )+ x=v max Endfor Figure3{2:Obtainingaunidirectionalplan Theorem11 AlgorithmDMLC-SINGLEPAIRobtainsplansthatareoptimali ntherapy time.Proof: Let I ( x i )bethedesiredprole.Let0= inc 0 I ( x inci 1 ),assumethat I ( x 1 =0)).Let i = I ( x inci ) I ( x inci 1 ), i 0. Supposethat( I L ;I R )isaplanfor I ( x i )(notnecessarilytheplangeneratedbyAlgorithm DMLC-SINGLEPAIR).Since I ( x i )= I L ( x i ) I R ( x i )forall i ,weget i =( I L ( x inci ) I R ( x inci )) ( I L ( x inci 1 ) I R ( x inci 1 )) =( I L ( x inci ) I L ( x inci 1 )) ( I R ( x inci ) I R ( x inci 1 )) =( I L ( x inci ) I L ( x inci 1 ) x=v max ) ( I R ( x inci ) I R ( x inci 1 ) x=v max ) Notethatfromthemaximumvelocityconstraint I R ( x inci ) I R ( x inci 1 ) x=v max i 1.So I R ( x inci ) I R ( x inci 1 ) x=v max 0, i 1,and i I L ( x inci ) I L ( x inci 1 ) x=v max .Also,0= I ( x 0 ) I ( x 1 )= I ( x 0 ) I L ( x 0 ) I L ( x 1 ),where I L ( x 1 )=0. Summingup i ,weget P ki =0 [ I ( x inci ) I ( x inci 1 )] P ki =0 [ I L ( x inci ) I L ( x inci 1 )] k x=v max .Let S 1 = P ki =0 [ I L ( x inci ) I L ( x inci 1 )].Then, S 1 P ki =0 [ I ( x inci ) I ( x inci 1 )]+ k x=v max .Let S 2 = P [ I L ( x j ) I L ( x j 1 )],wherethesummationiscarriedoutoverindices j (0 j m ) suchthat I ( x j ) I ( x j 1 ).Thereareatotalof m +1indicesofwhich k +1donotsatisfy thiscondition.Sothereare m k indices j atwhich I ( x j ) I ( x j 1 ).Ateachofthese

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41 j I L ( x j ) I L ( x j 1 )+ x=v max .Hence, S 2 ( m k ) x=v max .Now,weget S 1 + S 2 = P mi =0 [ I L ( x i ) I L ( x i 1 )] P ki =0 [ I ( x inci ) I ( x inci 1 )]+ m x=v max .Finally, TT ( I L ;I R )= I L ( x m )= I L ( x m ) I L ( x 1 )= P mi =0 [ I L ( x i ) I L ( x i 1 )] P ki =0 [ I ( x inci ) I ( x inci 1 )]+ m x=v max = TT ( I l ;I r ).Hence,thetreatmentplan( I l ;I r )generatedby DMLC-SINGLEPAIRisoptimalintherapytime. Corollary5 Let I ( x i ) 0 i m beadesiredprole.Let I l ( x i ) and I r ( x i ) 0 i m betheleftandrightleafprolesgeneratedbyAlgorithmDML C-SINGLEPAIR. I l ( x i ) and I r ( x i ) 0 i m deneoptimaltherapytimeunidirectionalleftandrightle afprolesfor I ( x i ) 0 i m Proof: FollowsfromTheorem 11 IntheremainderofSection 3.1 ,( I l ;I r )istheoptimaltreatmentplangeneratedby AlgorithmDMLC-SINGLEPAIRforthedesiredprole I Propertiesoftheoptimaltreatmentplan. Thefollowingobservationsaremade abouttheoptimaltreatmentplan( I l ;I r )generatedusingAlgorithmDMLC-SINGLEPAIR. Lemma11 Atmostoneoftheleavesstopsateach x i Lemma12 Let ( I L ;I R ) beanytreatmentplanfor I (a) ( x i )= I L ( x i ) I l ( x i )= I R ( x i ) I r ( x i ) 0 ; 0 i m (b) ( x i ) isanon-decreasingfunction. Proof: (a)Since I ( x i )= I L ( x i ) I R ( x i )= I l ( x i ) I r ( x i ) ;I L ( x i ) I l ( x i )= I R ( x i ) I r ( x i ). Further,fromCorollary 5 ,itfollowsthat I L ( x i ) I l ( x i ) ; 0 i m .Therefore,( x i ) 0 ; 0 i m (b)Weprovethisbycontradiction.Supposethat( x n ) > ( x n +1 )forsome n; 0 nI l ( x n +1 ) x=v max +( x n +1 )= I L ( x n +1 ) x=v max Thisisnotpossiblebecause I L ( x n +1 ) I L ( x n )+ x=v max fromthemaximumvelocity constraint. Case2: I r ( x n +1 )= I r ( x n )+ x=v max (rightleafdoesnotstopat x n +1 ) Now, I R ( x n )= I r ( x n )+( x n ) >I r ( x n +1 ) x=v max +( x n +1 )= I R ( x n +1 )

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42 x=v max Thisisnotpossiblebecause I R ( x n +1 ) I R ( x n )+ x=v max fromthemaximumvelocity constraint. Case3: I l ( x n +1 ) 6 = I l ( x n )+ x=v max and I r ( x n +1 ) 6 = I r ( x n )+ x=v max (both leavesstopat x n +1 ) FromLemma 11 itfollowsthatthiscasecannotarise. Therefore,( x i )isanon-decreasingfunction. Corollary6 Let l ( x i )( r ( x i )) and L ( x i )( R ( x i )) ,respectively,denotetheamountof timeforwhichtheleft(right)leafstopsat x i inplans ( I l ;I r ) and ( I L ;I R ) .Then (a) L ( x i ) l ( x i ) (b) R ( x i ) r ( x i ) Proof: (a)Supposethat L ( x i ) < l ( x i ).Wehavethefollowingtwocases: Case1:Bothleavesmoveatthemaximumvelocitybetween x i 1 and x i in( I L ;I R ). Weget( x i ) < ( x i 1 )contradictingLemma 12 (b). Case2:In( I L ;I R ),theleavesdonottraveluniformlyatthemaximumvelocity between x i 1 and x i .Inthiscase,transformplan( I L ;I R )toaplan( I 0 L ;I 0 R )asfollows.Between x i 1 and x i movetheleavesatthemaximumvelocity.Theleavesnowarriv eat x i earlierthan theydidin( I L ;I R )byanamount i .Propagatethisdierencetotherightfrom x i sothat I 0 L ( x j )= I L ( x j ) i and I 0 R ( x j )= I R ( x j ) i j i .Notethatthistransformationpreserves thes,i.e., 0L ( x j )= L ( x j ).Also,theresultingleafproles, I 0 L and I 0 R ,stillformaplan for I .Let 0 ( x j )= I 0 L ( x j ) I l ( x j )= I 0 R ( x j ) I r ( x j ).Since 0L ( x i )= L ( x i ) < l ( x i )and sincetheleavesmoveatmaximumvelocityfrom x i 1 to x i in( I l ;I r )and( I 0 L ;I 0 R ),wehave 0 ( x i ) < 0 ( x i 1 )contradictingLemma 12 (b). (b)Similartoproofof(a). 3.1.4MinimumSeparationConstraint SomeMLCshaveaminimumseparationconstraintthatrequire stheleftandright leavestomaintainaminimumseparation S min atalltimesduringthetreatment.Notice thatintheplangeneratedbyAlgorithmDMLC-SINGLEPAIR,th etwoleavesstartand endatthesamepoint.Sotheyareincontactat x 0 and x m .When s min > 0,theminimum separationconstraintisviolatedat x 0 and x m .Inordertoovercomethisproblemwemodify

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43 AlgorithmDMLC-SINGLEPAIRtoguaranteeminimumseparatio nbetweentheleavesin thevicinityoftheendpoints( x 0 and x m ).Inparticular,weallowtheleftleaftobe initiallypositionedatapoint x 0 0 = x 0 s min andtherightleaftobenallypositioned at x m 0 = x m + S min .TheonlychangesmaderelativetoAlgorithmDMLC-SINGLEPA IR areforthemovementoftheleftleaffrom x 0 0 to x 0 andfortherightleaffrom x m to x m 0 Wedenethemovementoftheleftleaffrom x 0 0 to x 0 (andasymmetricdenitionforthe rightleaffrom x m to x m 0 )tobesuchthatitmaintainsadistanceofexactly S min from therightleafatalltimes.Oncetheleftleafreaches x 0 itfollowsthetrajectoryasbefore. Whilethismodicationresultsintheexactprolebeingdel iveredbetween x 0 and x m it alsoresultsinsomeundesirableexposuretotheintervals( x 0 0 ;x 0 )and( x m ;x m 0 ).Inthe remainderofthissectionwewillconsideranexposureofthi skindpermissible,provided theexactproleisdeliveredbetween x 0 and x m .Notethatformostaccelerators(Varian, Elekta)undesirableexposuretotheintervals( x 0 0 ;x 0 )and( x m ;x m 0 )canbeavoidedby positioningtheback-upjawsat x 0 and x m respectively.However,adicultyariseswhen thenumberofmonitorunitsdeliveredatthetimetheleftlea freaches x 0 inthisnewplan (callit( I 0 l ;I 0 r ))isgreaterthan I l ( x 0 ).Thiswouldpreventusfromusingtheoldplanfrom x 0 to x m ,sincetheleafcannotpass x 0 beforeitarrivesthere.Observehowever,thatif theleftleafweretoarriveat x 0 anyearlier,itwouldbetooclosetotherightleaf.Inthe discussionthatfollowsweshowthatinthisandothercasesw heretheoriginalplanviolates theconstraint,therearenofeasiblesolutionsthatdelive rexactlythedesiredprolebetween x 0 and x m ,whilepermittingexposureoutsidethisregion.Themodie dalgorithm,which wecallDMLC-MINSINGLEPAIR,isdescribedinFigure 3{3 .Notethatthetherapytime oftheplanproducedbyDMLC-MINSINGLEPAIRisthesameastha toftheplanproduced byDMLC-SINGLEPAIR.Therefore,themodiedplanisoptimal intherapytime. Theorem12 (a) S min =v max >I l ( x 0 ) or(b)Iftheplan ( I 0 l ;I 0 r ) generatedbyDMLCMINSINGLEPAIRviolatestheminimumseparationconstraint ,thenthereisnoplanfor I thatdoesnotviolatetheminimumseparationconstraint.Proof: Supposethat( I 0 l ;I 0 r )violatestheminimumseparationconstraint.Assumethatt he rstviolationoccurswhen I 1 MUshavebeendelivered.Sincethereisnoviolationwhenles s than I 1 MUsaredeliveredandsincetheleavesareeitherstationary ormoveatthemaximum

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44 AlgorithmDMLC-MINSINGLEPAIR1.ApplyAlgorithmDMLC-SINGLEPAIR2.Modifytheproleoftheleftleaffrom x 0 0 to x 0 andtherightleaffrom x m to x m 0 to maintainaminimuminterleafdistanceof S min .Callthisprole( I 0 l ;I 0 r ). 3.IfthenumberofMUsdeliveredwhentheleftleafarrivesat x 0 isgreaterthan I l ( x 0 ) thereisnofeasiblesolution.End.4.Elseoutput( I 0 l ;I 0 r ).Thereisafeasiblesolutiononlyif( I 0 l ;I 0 r )isfeasible. Figure3{3:Obtainingaunidirectionalplanwithminimumse parationconstraint velocity,atthetimeoftheviolation,itmustbethecasetha ttherightleafisstationary atasamplepoint(say x k )andtheleftleafismoving.Theviolationoccurswhenthele ft leafpasses x 0 = x k S min .Sincetheleftleafismoving, I 1 = I 0 l ( x 0 ) I 0 l ( x 0 )and( x k ) ( x 0 ),weget I 00 r ( x k ) >I 00 l ( x 0 ).Thereforethereisaminimumseparationviolationin( I 00 l ;I 00 r )whenthe theleftleafpasses x 0 I x Separation between the leaves k 0 mm' I 1 x 0' xxxx x' Figure3{4:Minimumseparationconstraintviolation

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45 Theseparationbetweentheleavesisdeterminedbythedier encein x valuesofthe leaveswhenthesourcehasdeliveredacertainamountofMUs. Theminimumseparationof theleavesistheminimumseparationbetweenthetwoproles .Wecallthisminimumseparation S ud min .WhentheoptimalplanobtainedusingAlgorithmDMLC-SINGL EPAIR isdelivered,theminimumseparationis S ud min ( opt ) Corollary7 Let S ud min ( opt ) betheminimumleafseparationintheplan ( I 0 l ;I 0 r ) .Let S ud min betheminimumleafseparationinany(notnecessarilyoptim al)givenunidirectionalplan. S ud min S ud min ( opt ) Theorem13 IfAlgorithmDMLC-MINSINGLEPAIRterminatesinStep3,then thereis noplanfor I thatdoesnotviolatetheminimumseparationconstraint. Proof: Let( I 00 l ;I 00 r )beafeasibleplan(i.e.,aplanthatdelivers I andsatisestheminimum separationconstraint).Let( I 0 l ;I 0 r )betheplanofStep2ofDMLC-MINSINGLEPAIR.From Corollary 5 ,itfollowsthat ^ I 00 r ( x 0 + S min ) ^ I 0 r ( x 0 + S min )+ I 00 r ( x 0 ),where ^ I r ( x )isthenumber ofMUsdeliveredwhentherightleafreaches x (notethat I r ( x )isthenumberofMUs deliveredwhentherightleafleaves x ).Since( I 00 l ;I 00 r )hasnominimumseparationviolation, I 00 l ( x 0 ) ^ I 00 l ( x 0 ) ^ I 00 r ( x 0 + S min ).Also,becauseDMLC-MINSINGLEPAIRterminatesin Step3, ^ I 0 l ( x 0 ) >I l ( x 0 ).Hence, I 00 l ( x 0 ) I 00 r ( x 0 ) ^ I 0 r ( x 0 + S min )= ^ I 0 l ( x 0 ) >I l ( x 0 )= I ( x 0 ). So,( I 00 l ;I 00 r )doesnotdelivertheproperdoseat x 0 andsocannotbefeasible. ComparisonwithSMLC. Kamathetal.(2003)discusstheoptimaltherapytime algorithmforSMLC.Theiralgorithmgeneratesanoptimalpl anthatsatisestheminimum separationconstraint,wheneveroneexists.Weprovetheex istenceofprolesforwhich therearefeasibleplansusingSMLC,butnofeasibleplansus ingDMLC. Lemma13 Lettheminimumseparationbetweentheleavesintheoptimal SMLCplan be S s min .Lettheminimumleafseparationintheplangeneratedbyalg orithmDMLCMINSINGLEPAIRbe S d min .Then S d min S s min Proof: Considerthedeliveryofaprole I bytheoptimalSMLCplanofKamathet al.(2003).Callthisplan( I s l ;I s r ).Let ^ I s l ( x )bethenumberofMUsdeliveredwhenthe leftleafarrivesat x usingtheplan I s l ^ I s r ( x ), ^ I 0 l ( x ),and ^ I 0 r ( x )aredenedsimilarly.Let l ( x k )= I s l ( x k ) ^ I s l ( x k )= I 0 l ( x k ) ^ I 0 l ( x k ). r ( x k )isdenedsimilarly.Notethat l ( x k ) > 0

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46 itheleftleafstopsat x k and l ( x k )givestheamountofMUsdeliveredwhiletheleftleaf isstoppedat x k .Let x i and x j j>i ,besuchthat S s min = x j x i and ^ I s l ( x i ) ^ I s l ( x i ) I s r ( x i )(3.2) Since I ( x i )= I s l ( x i ) I s r ( x i )= ^ I s l ( x i )+ l ( x i ) I s r ( x i )= ^ I 0 l ( x i )+ l ( x i ) I 0 r ( x i ), I 0 r ( x i )= ^ I 0 l ( x i ) ( ^ I s l ( x i ) I s r ( x i ))(3.3) Also,weseethat I 0 r ( x j )= I 0 r ( x i )+ P jk = i +1 r ( x k )+( j i ) x=v max = ^ I 0 l ( x i ) ( ^ I s l ( x i ) I s r ( x i ))+ P jk = i +1 r ( x j )+( j i ) x=v max (from( 3.3 )) > ^ I 0 l ( x i )+( j i ) x=v max (from( 3.2 )).So, S d min x j x i = S s min Thefollowingresultimmediatelyfollowsandcanbeeasilyv eried. Corollary8 AllprolesthathavefeasibleplansusingDMLChavefeasibl eplansusing SMLC.Thereexistprolesforwhichtherearefeasibleplans usingSMLC,butnofeasible plansusingDMLC. Figure 3{5 showstwoplansforanintensityprole.TheSMLCplanforthe proleisfeasible.ThecorrespondingDMLCplanobtainedusingAlgorith mDMLC-MINSINGLEPAIR isinfeasible.3.1.5Bi-directionalMovement Inthissectionwestudybeamdeliverywhenbi-directionalm ovementofleavesispermitted.Weexplorewhetherrelaxingtheunidirectionalmov ementconstrainthelpsimprove theeciencyoftreatmentplan.Propertiesofbi-directionalmovement. Foragivenleaf(leftorright)movement proleweclassifyany x -coordinateasfollows.Drawaverticallineat x .Ifthelinecutsthe leafproleexactlyoncewewillcall x a unidirectionalpoint ofthatleafprole.Iftheline cutstheprolemorethanonce, x isa bi-directionalpoint ofthatprole.Aleafmovement prolethathasatleastonebi-directionalpointisa bi-directionalprole .Allprolesthatare

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47 x I min S 0 x 0 x' Constraint violation I x Figure3{5:SMLCplan:feasible;DMLCplan:infeasible

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48 notbi-directionalare unidirectionalproles .Anyprolecanbepartitionedintosegments suchthateachsegmentisaunidirectionalprole.Whenthen umberofsuchpartitionsis minimal,eachpartitioniscalleda stage oftheoriginalprole.Thusunidirectionalproles consistofexactlyonestagewhilebi-directionalprolesa lwayshavemorethanonestage. InFigure 3{6 ,thebi-directionalleafmovementprole, B L ,showsthepositionofthe leftleafasafunctionoftheamountofMUsdeliveredbytheso urce.Themovementprole ofthisleafconsistsofstages S 1 ;S 2 and S 3 .Instages S 1 and S 3 theleafmovesfromleft torightwhileinstage S 2 theleafmovesfromrighttoleft. x j isabi-directionalpointof B L .Let I L betheintensityprolecorrespondingtotheleafmovementp role B L I L ( x ) givesthenumberofMUsdeliveredat x usingmovementprole B L .TheamountofMUs deliveredat x j is L 1 + L 2 .Instage S 1 ,when I 1 amountofMUshavebeendelivered,the leafpasses x j .Now,noMUisdeliveredat x j tilltheleafpassesover x j in S 2 .Further, I 3 I 2 MUsaredeliveredto x j instages S 2 and S 3 .Thuswehave I l ( x j )= L 1 + L 2 ,where L 1 = I 1 and L 2 = I 3 I 2 x k isaunidirectionalpointof B L .TheMUsdeliveredat x k are L 3 = I 4 .Notethattheintensityprole I L isdierentfromtheleafmovementprole B L unlikeintheunidirectionalcase. 4 x x L LL B SS S I I I I 3 L L L I j I (x ) j k 3 21 32 1 1 2 x I Figure3{6:Bi-directionalmovement Lemma14 Let I L and I R betheintensityprolescorrespondingtothebi-direction alleaf movementprolepair ( B L ;B R ) (i.e., B L and B R are,respectively,theleftandrightleaf

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49 movementproles).Let I ( x i )= I L ( x i ) I R ( x i ) 0 i m ,betheintensityproledelivered by ( B L ;B R ) .Then (a) I L ( x i +1 ) I L ( x i )+ x=v max (b) I R ( x i +1 ) I R ( x i )+ x=v max Proof: (a)Betweenthetime, t 1 ,theleftleafmovesrightwardfrom x i forthelasttime (sincetheleftleafendsat x m ,suchalastrightmovemustoccur)andtheleasttime t 2 t 2 >t 1 ,thattheleftleafreaches x i +1 (again,sincetheleftleafendsat x m ,sucha t 2 exists), x i +1 receivesatleast x=v max MUsthatarenotdeliveredto x i .Atallother timesduringthetherapy,whenevertheleftleafdoesn'tcov er x i ,italsodoesn'tcover x i +1 Hence,outsidethetimeinterval[ t 1 ;t 2 ],thenumberofMUsdeliveredto x i +1 isatleastas manyasdeliveredto x i .Therefore,fortheentiretherapy, I L ( x i +1 ) I L ( x i )+ x=v max (b)Theproofissimilartothatof(a). FromLemma 14 wenotethateverybi-directionalleafmovementprole( B L ;B R ) deliversanintensityprole( I L ;I R )thatsatisesthemaximumvelocityconstraint.Hence, ( I L ;I R )isdeliverableusingaunidirectionalleafmovementprol e(Section 3.1.3 ).Wewill callthisprolethe unidirectionalleafmovementprolethatcorrespondstoth ebi-directional prole .Thuseverybi-directionalleafmovementprolehasacorre spondingunidirectional leafprolethatdeliversthesameamountofMUsateach x i asdoesthebi-directional prole.Theorem14 TheunidirectionaltreatmentplanconstructedbyAlgorith mDMLC-SINGLEPAIRisoptimalintherapytimeevenwhenbi-directional leafmovementispermitted. Proof: Let B L and B R bebidirectionalleafmovementprolesthatdeliveradesir ed intensityprole I .Let I L and I R ,respectively,bethecorrespondingintensityprolesfor B L and B R .FromLemma 14 ,weknowthat I L and I R aredeliverablebyunidirectionalleaf movementproles.Also, I L ( x i ) I R ( x i )= I ( x i ) ; 1 i m .FromtheproofofTheorem 11 itfollowsthatthetherapytimefortheunidirectionalplan ( I l ;I r )generatedbyAlgorithm DMLC-SINGLEPAIRisnomorethanthatof( I L ;I R ). Incorporatingminimumseparationconstraint. Let U l and U r beunidirectional leafmovementprolesthatdeliverthedesiredprole I ( x i ).Let B l and B r beasetof

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50 bi-directionalleftandrightleafprolessuchthat U l and U r correspondto B l and B r respectively,i.e.,( B l ;B r )deliversthesameplanas( U l ;U r ).Wecalltheminimumseparation ofleavesinthisbi-directionalplan( B l ;B r ) S bd min S ud min istheminimumseparation ofleavesin( U l ;U r ). Theorem15 S bd min S ud min foreverybi-directionalleafmovementprolepair ( B l ;B r ) anditscorrespondingunidirectionalprole ( U l ;U r ) Proof: Supposethattheminimumseparation S ud min occurswhen I ms MUsaredelivered.Atthistime,theleftleafarrivesat x j andtherightleafispositionedat x k .Let B 0 l and U 0 l respectively,betheprolesobtainedwhen B l and U l areshiftedrightby S ud min Since U 0 l is U l shiftedrightby S ud min andsincethedistancebetween U l and U r is S ud min when I ms MUshavebeendelivered, U 0 l and U r touchwhen I ms unitshavebeendelivered. Therefore,thetotalMUsdeliveredby( U 0 l ;U r )at x k iszero.SothetotalMUsdeliveredby ( B 0 l ;B r )at x k isalsozero(recallthat U 0 l and U r ,respectively,arecorrespondingproles for B 0 l and B r ).Thisisn'tpossibleif B r isalwaystotherightof B 0 l (forexample,inthe situationofFigure 3{7 ,theMUsdeliveredby( B 0 l ;B r )at x k are( L 1 + L 2 ) ( L 01 + L 02 ) > 0). Therefore B 0 l and B r musttouch(orcross)atleastonce.So S bd min S ud min B B U x x I x I U U' B' r l r l l l 2 2 1 1 k j ms L L L' L' Figure3{7:Bi-directionalmovementunderminimumseparat ionconstraint

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51 Theorem16 Iftheoptimalunidirectionalplan ( I 0 l ;I 0 r ) violatestheminimumseparation constraint,thenthereisnobi-directionalmovementplant hatdoesnotviolatetheminimum separationconstraint.Proof: Let B l and B r bebi-directionalleafmovementsthatdelivertherequired prole. Lettheminimumseparationbetweentheleavesbe S bd min .Letthecorrespondingunidirectionalleafmovementsbe U l and U r andlet S ud min betheminimumseparationbetween U l and U r .Also,let S min betheminimumallowableseparationbetweentheleaves.Fro m Corollary 7 andTheorem 15 ,weget S bd min S ud min S ud min ( opt ) 0). Therefore B 0 l and B r musttouch(orcross)atleastonce.So S bd max S ud max 3.1.6AlgorithmUnderMaximumSeparationConstraintCondi tion Inthissectionwepresentanalgorithmthatgeneratesanopt imaltreatmentplan underthemaximumseparationconstraint.RecallthatAlgor ithmDMLC-SINGLEPAIR

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52 B L' L L' L l l B U x I U x x r r l l 2 2 1 1 k j ms I B'U' Figure3{8:Bi-directionalmovementundermaximumseparat ionconstraint generatestheoptimalplanwithoutconsideringthisconstr aint.WemodifyAlgorithm DMLC-SINGLEPAIRsothatallinstancesofviolationofmaxim umseparation(thatmay possiblyexist)areeliminated.Weknow(Theorem 17 )thatbi-directionalleafprolesdo nothelpeliminatetheconstraint.Soweconsideronlyunidi rectionalproles. Algorithm. ThealgorithmisdescribedinFigure 3{9 AlgorithmDMLC-MAXSEPARATION 1.ApplyAlgorithmDMLC-SINGLEPAIRtoobtaintheoptimalpl an( I l ;I r ) : 2.Findtheleastvalueofintensity, I 1 ,suchthattheleafseparationin( I l ;I r )when I 1 MUsaredeliveredis S max ,where S max isthemaximumallowedseparationbetween theleavesandtheleafseparationwhen I 1 + MUsaredeliveredis >S max ,forsome positiveconstant .Ifthereisnosuch I 1 ,( I l ;I r )istheoptimalplan;end. 3.FromLemma 10 itfollowsthatwhen I 1 MUsaredelivered,theleftleafisstoppedat some x j .Let x 0 bethepositionoftherightleafatthistime(seeFigure 3{10 ).Note that x 0 maynotbeoneofthesamplepoints x i j i m .Let I = I l ( x j ) I 1 = I 2 I 1 .Movetheproleof I r ,whichfollows x 0 ,upby I along I direction.To maintain I ( x )= I l ( x ) I r ( x )forevery x ,movetheproleof I l ,whichfollows x 0 ,up by I along I direction. GotoStep2. Figure3{9:Obtainingaplanundermaximumseparationconst raint Theorem18 AlgorithmDMLC-MAXSEPARATIONobtainsplansthatareoptim alintherapytime,underthemaximumseparationconstraint.

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53 I x II S After modification Before modification 21 max j xx' Figure3{10:Maximumseparationconstraintviolation Proof: Weuseinductiontoprovethetheorem. Thestatementweprove, S ( n ),isthefollowing: AfterStep(iii)ofthealgorithmisapplied n times,theresultingplan,( I ln ;I rn ),satises (a)Ithasnomaximumseparationviolationwhen I
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54 (a) I 0 l 1 ( x j )= I l 1 ( x j )= I 2 (1) Sincethereisnomaximumseparationviolationwhen I<>: I l ( x )0 x<>: I r ( x )0 xI l 1 ( x j ) Thisleadstoacontradictionasinthepreviouscase. (c) I 0 l 1 ( x j )
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55 contradictionthatthereisnosuchplan.Therefore S ( n +1)istruewhenever S ( n )is true.SincethenumberofiterationsofSteps(ii)and(iii)ofthea lgorithmisnite(foreach iteration,theleftleafmustbestationaryat x j andtherecanbeatmostoneiteration foreachsuch x j ),allmaximumseparationviolationswilleventuallybeeli minated. Whentheminimumseparationconstraintisalsoapplicable, wecanuseAlgorithm DMLC-MINSINGLEPAIRinplaceofAlgorithmDMLC-SINGLEPAIR inStep(i)ofAlgorithmDMLC-MAXSEPARATION.Notethatinthiscasetheminimu mleafseparationof theplanconstructedbyAlgorithmDMLC-MAXSEPARATIONis min f S ud min ( opt ) ;S max g FromTheorem 18 ,itfollowsthatAlgorithmDMLC-MAXSEPARATIONconstructs anoptimalplanthatsatisesboththeminimumandmaximumsepara tionconstraintsprovided that S ud min ( opt ) S min .Notethatwhen S ud min ( opt ) 0andthat I ( x m ) > 0.However,withmultipleleafpairs,therstandlast samplepointswithnon-zerointensitylevelscouldbedier entfordierentpairs.Hencewe willnolongermakethisassumption.Optimalschedulewithouttheinterdigitationconstraint. Forsequencingofmultipleleafpairs,weapplyAlgorithmDMLC-SINGLEPAIRtodet erminetheoptimalplan foreachofthe n leafpairs.Thismethodofgeneratingschedulesisdescribe dinAlgorithm DMLC-MULTIPAIR(Figure 3{11 ).Notethatsince x 0 x m arenotnecessarilynon-zerofor anyrow,wereplace x 0 by x l and x m by x g inAlgorithmDMLC-SINGLEPAIRforeachrow,

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56 where x l and x g ,respectively,denotetherstandlastnon-zerosamplepoi ntsofthatrow. Also,forrowsthatcontainonlyzeroes,theplansimplyplac esthecorrespondingleavesat therightmostpointintheeld(callit x m +1 ). AlgorithmDMLC-MULTIPAIRFor( i =1; i n ; i ++) ApplyAlgorithmDMLC-SINGLEPAIRtothe i thpairofleavestoobtainplan( I il ;I ir ) thatdeliverstheintensityprole I i ( x ). EndFor Figure3{11:Obtainingaschedule Lemma15 AlgorithmDMLC-MULTIPAIRgeneratesschedulesthatareopt imalintherapy time.Proof: Treatmentiscompletedwhenallleafpairs(whichareindepe ndent)delivertheir respectiveplans.Thetherapytimeoftheschedulegenerate dbyAlgorithmDMLC-MULTIPAIRis max f TT ( I 1 l ;I 1 r ) ;TT ( I 2 l ;I 2 r ) ; ... ;TT ( I nl ;I nr ) g .FromTheorem 11 ,itfollows thatthistherapytimeisoptimal. Optimalalgorithmwithinterdigitationconstraint. TheschedulegeneratedbyAlgorithmDMLC-MULTIPAIRmayviolatetheinterdigitationco nstraint.Notethatnointrapairconstraintviolationscanoccurfor S min =0.Sotheinterdigitationconstraintisessentiallyaninter-pairconstraint.Iftheschedulehasnointe rdigitationconstraintviolations, itisthedesiredoptimalschedule.Ifthereareviolationsi ntheschedule,weeliminate allviolationsoftheinterdigitationconstraintstarting fromtheleftend,i.e.,from x 0 .To eliminatetheviolations,wemodifythoseplansofthesched ulethatcausetheviolations. Wescantheschedulefrom x 0 alongthepositive x directionlookingfortheleast x v at whichispositionedarightleaf(say R u )thatviolatestheinter-pairseparationconstraint. Afterrectifyingtheviolationat x v withrespectto R u welookforotherviolations.Since theprocessofeliminatingaviolationat x v ,mayattimes,leadtonewviolationsinvolving rightleavespositionedat x v ,weneedtosearchafreshfrom x v everytimeamodication ismadetotheschedule.Wenowcontinuethescanningandmodi cationprocessuntil nointerdigitationviolationsexist.AlgorithmDMLC-INTE RDIGITATION(Figure 3{12 ) outlinestheprocedure.

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57 AlgorithmDMLC-INTERDIGITATION 1. x = x 0 2.While(thereisaninterdigitationviolation)do3.Findtheleast x v x v x ,suchthatarightleafispositionedat x v andthisright leafhasaninterdigitationviolationwithoneorbothofits neighboringleftleaves. Let u betheleastintegersuchthattherightleaf R u ispositionedat x v and R u hasaninterdigitationviolation.Let L t denotetheleftleafwithwhich R u hasan interdigitationviolation.Notethat t 2f u 1 ;u +1 g .Incase R u hasviolationswith twoadjacentleftleaves,welet t = u 1. 4.Modifythescheduletoeliminatetheviolationbetween R u and L t 5. x = x v 6.EndWhile Figure3{12:Obtainingascheduleundertheconstraint Let M =(( I 1 l ;I 1 r ) ; ( I 2 l ;I 2 r ) ;:::; ( I nl ;I nr ))betheschedulegeneratedbyAlgorithm DMLC-MULTIPAIRforthedesiredintensityprole.Let N ( p )=(( I 1 lp ;I 1 rp ) ; ( I 2 lp ;I 2 rp ) ;:::; ( I nlp ;I nrp ))bethescheduleobtainedafterStep 4 of AlgorithmDMLC-INTERDIGITATIONisapplied p timestotheinputschedule M .Note that M = N (0). Toillustratethemodicationprocessweuseexamples.Ther earetwotypesofviolationsthatmayoccur.CallthemType1andType2violationsan dcallthecorresponding modicationsType1andType2modications.Tomakethingse asier,weonlyshowtwo neighboringpairsofleaves.Supposethatthe( p +1)thviolationoccursbetweentheright leafofpair u ,whichispositionedat x v ,andtheleftleafofpair t;t 2f u 1 ;u +1 g InaType1violation,theleftleafofpair t startsitssweepatapoint xStart ( t;p ) >x v (seeFigure 3{13 ).Toremovethisinterdigitationviolation,modify( I tlp ;I trp )to( I tl ( p +1) ; I tr ( p +1) )asfollows.Welettheleavesofpair t startat x v andmovethematthemaximumvelocity v max towardstheright,tilltheyreach xStart ( t;p ).LetthenumberofMUsdelivered whentheyreach xStart ( t;p )be I 1 .Raisetheproles I tlp ( x )and I trp ( x ), x xStart ( t;p ), byanamount I 1 = ( xStart ( t;p ) x v ) =v max .Weget, I tl ( p +1) ( x )= 8><>: ( x x v ) =v max x v x
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58 I I I I I ulp urp tlp trp I tl(p+1) tr(p+1) I 1 xStart(t,p) v I x x Figure3{13:EliminatingaType1violation AType2violationoccurswhentheleftleafofpair t ,whichstartsitssweepfrom x x v passes x v beforetherightleafofpair u passes x v (Figure 3{14 ).Inthiscase, I tl ( p +1) isas denedbelow. I tl ( p +1) ( x )= 8><>: I tlp ( x ) x 0. Lemma16 I jrp ( xStart ( j;p ))=0 1 j m p 0 Proof: Theproofisbyinductionon p .Let T ( p )bethefollowingstatement: I jrp ( xStart ( j;p ))=0.

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59 I x I trp x I tr(p+1) I tlp ulp I II v 32 I urp I tl(p+1) Figure3{14:EliminatingaType2violation(closeparallel dottedandsolidlinesegments overlap,theyhavebeendrawnwithasmallseparationtoenha ncereadability) Forthebasecase, p =0.( I jl 0 ;I jr 0 )istheplangeneratedbyAlgorithmDMLCSINGLEPAIRanditsatisesthestatedproperty. Assumethat T ( p )istrue.Forthe( p +1)thviolation,wehavethefollowingtwocases: { The( p +1)thviolationisaType1violation. AType1modicationisapplied.Suchamodicationalwaysre sultsinchangingthestartpositionoftheleavesofpair t (asdenedinAlgorithmDMLCINTERDIGITATION)to x v (whichbecomes xStart ( t;p +1))and I tr ( p +1) ( x v )= 0.For j 6 = t I jr ( p +1) ( xStart ( j;p +1))= I jrp ( xStart ( j;p ))=0byinduction hypothesis. { The( p +1)thviolationisaType2violation. AType2modicationisapplied.Let t beasinAlgorithmDMLC-INTERDIGITATION.Supposethat xStart ( t;p )
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60 So,thecase xStart ( t;p )= x v cannotarise.For j 6 = t ,theplanisunchanged. So, I jr ( p +1) ( xStart ( j;p +1))= I jrp ( xStart ( j;p ))=0byinductionhypothesis. Corollary9 AType2violationinwhich I tlp ( x v )=0 cannotoccur. Proof: FromtheproofofLemma 16 ,itfollowsthatwheneverthereisaType2violation, xStart ( t;p ) 0. Lemma17 IncaseofaType1violation, ( I tlp ;I trp ) isthesameas ( I tl 0 ;I tr 0 ) Proof: Let p besuchthatthereisaType1violation.Let t u and v beasinAlgorithmDMLC-INTERDIGITATION.If( I tlp ;I trp )isdierentfrom( I tl 0 ;I tr 0 ),leafpair t was modiedinanearlieriteration(sayiteration q


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61 1.Considerthebasecase, p =0.FromCorollary 5 andthefactthattheplans ( I il 0 ;I ir 0 ) ; 0 i n ,aregeneratedusingAlgorithmDMLC-SINGLEPAIR,itfollowsthat S (0)istrue. 2.Assume S ( p )istrue.SupposeAlgorithmDMLC-INTERDIGITATIONndsane xt violationandmodiestheschedule N ( p )to N ( p +1).Supposethatthenextviolation occursbetweentherightleafofpair u ,positionedat x v ,andtheleftleafofpair t Wemodifypair t 'splanfor x x v ,toeliminatetheviolation.Allotherplansinthe scheduleremainunaltered.Therefore,toestablish S ( p +1)itsucestoprovethat I 0 tl ( x ) I tl ( p +1) ( x ) ;x x v (3.4) I 0 tr ( x ) I tr ( p +1) ( x ) ;x x v (3.5) Weneedproveonlyoneofthesetworelationshipssince I 0 tl ( x ) I 0 tr ( x )= I tl ( p +1) ( x ) I tr ( p +1) ( x ) ;x 0 x x m .Wenowconsiderpair t 'splanfor x x v .Notethat the( p +1)thviolationmayeitherbeaType1orType2violation.Wes howthat Equation 3.4 istrueinbothcases.This,inturn,impliesthat S ( p +1)istruewhenever S ( p )istrueandhencecompletestheproof.Notethatin( I tl ( p +1) ;I tr ( p +1) ),theleaves moveatmaximumspeedbetweenadjacentsamplepoints.So,it issucienttoshow Equation 3.4 forsamplepoints x v (a)The( p +1)thviolationisaType1violation. From S ( p )itfollowsthat I 0 ur ( x v ) I urp ( x v ).So,therightleafofpair u leaves x v noearlierin I 0 ur thanitdoesin I urp .Fromthisandthefactthat F satisesthe interdigitationconstraint,weconcludethatleafpair t cannotbeginitssweepat therightof x v .Thisobservationtogetherwiththefactthatin( I tl ( p +1) ;I tr ( p +1) ) theleavesmoveatthemaximumvelocityfrom x v to x 0 = xStart ( t;p )implies that ^ I 0 tl ( x 0 ) ^ I tl ( p +1) ( x 0 )and ^ I 0 tr ( x 0 ) ^ I tr ( p +1) ( x 0 ),where ^ I denotesanarrival time.Now,fromLemma 16 ,weget I 0 tr ( x 0 ) ^ I 0 tr ( x 0 ) ^ I tr ( p +1) ( x 0 )= I tr ( p +1) ( x 0 ). So I 0 tl ( x 0 )= I 0 tr ( x 0 )+ I t ( x 0 ) I tr ( p +1) ( x 0 )+ I t ( x 0 )= I tl ( p +1) ( x 0 ).Fromthisand thefactthattheleftleafofpair t movesatthemaximumvelocitybetween

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62 x v and x 0 ,itfollowsthatEquation 3.4 holdsforall x between x v and x 0 .To provethatEquation 3.4 holdsforallsamplepointstotherightof x 0 (andso holdsforall x between x 0 and x m ),considerasamplepoint x w thatistothe rightof x 0 .Let I 0 = I 0 tl ( x 0 ) I tl ( p +1) ( x 0 ) 0andlet I 1 beasinAlgorithm DMLC-INTERDIGITATION.Deneanewplan( I 00 tl ;I 00 tr )forleafpair t asbelow I 00 tl ( x )= 8><>: undefinedx<>: undefinedx
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63 (Lemma 12 byieldsEquation 3.9 onlyfor x x v and x isasamplepoint.From thisandthefactthattheleftleafmovesatmaximumvelocity in I tl between adjacentsamplepoints,wegetEquation 3.9 forall x x x v .) FromEquation 3.9 ,weget I 0 tl ( x ) I tl ( p +1) ( x ) I 0 tl ( x v ) I tl ( x v )+ I tl ( x ) I tl ( p +1) ( x ) ;x x v (3.10) FromthedenitionsofType1andType2modicationsandthew orkingofAlgorithmDMLC-INTERDIGITATION,itfollowsthat I tl ( p +1) ( x ) I tl ( x )= I tl ( p +1) ( x v ) I tl ( x v ) ;x x v (3.11) FromEquations 3.10 3.11 and 3.8 ,weget I 0 tl ( x ) I tl ( p +1) ( x ) I 0 tl ( x v ) I tl ( p +1) ( x v ) 0 ;x x v (3.12) Therefore, I 0 tl ( x ) I tl ( p +1) ( x ) ;x x v (3.13) Lemma20 FortheexecutionofAlgorithmDMLC-INTERDIGITATION (a) O ( n ) Type1violationscanoccur. (b) O ( n 2 m ) Type2violationscanoccur. (c)Let T max betheoptimaltherapytimefortheinputmatrix.Thetimecom plexityis O ( mn + n min f nm;T max g ) Proof: (a)ItfollowsfromLemma 17 thateachleafpaircanbeinvolvedinatmostone Type1violationaspair t ,i.e,thepairwhoseproleismodied.Hence,thenumberof Type1violationsis n (b)WerstobtainaboundonthenumberofType2violationsat axed x v .Let u t beas inAlgorithmDMLC-INTERDIGITATION.Notethat u ischosentobetheleastpossibleindex.Let u i bethevalueof u inthe i thiterationofAlgorithmDMLC-INTERDIGITATION at x v t i isdenedsimilarly.Let u maxi = max j i f u j g .If t i = u i 1,itispossiblethat u i +1 = t i = u i 1and t i +1 = u i 2.Notethatinthiscase, t i +1 6 = u i = u i +1 +1.Next,

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64 itispossiblethat u i +2 = u i 2and t i +2 = u i 3 (again t i +2 6 = u i 1= u i +2 +1).In general,onemayverifythat t i = u i +1ispossibleonlyif u maxi = u i .If t i = u i +1,then u i +1 t i = u i +1,sincetheviolationbetween u i and t i hasbeeneliminatedandnoproles withanindexlessthan t i havebeenchangedduringiteration i at x v .Itisalsoeasyto verifythat t i =1 ;u i =2 ) u i +1 u maxi ;u maxi +2 >u maxi .Fromthisand t i 2f u i +1 ;u i 1 g it followsthat u maxi + u maxi >u maxi .Weknowthat u max1 1.Itfollowsthat u max2 2, u max4 3, u max7 4andingeneral, u max( i ( i +1) = 2)+1 i +1.Clearly,forthelastviolation(say j th)at x v u maxj n andforthistobetrue, j = O ( n 2 ).SothenumberofType2violationsat x v is O ( n 2 ).Since x v hastobeasamplepoint,thereare m possiblechoicesforit.Hence,the totalnumberofType2violationsis O ( n 2 m ). (c)Sincetheinputmatrixcontainsonlyintegerintensityv alues,eachviolationmodication raisestheproleforonepairofleavesbyatleastoneunit.H ence,if T max istheoptimal therapytime,noprolecanberaisedmorethan T max times.Therefore,thetotalnumberof violationsthatAlgorithmDMLC-INTERDIGITATIONneedstor epairisatmost nT max Combiningthisboundwiththoseof(a)and(b),weget O ( min f n 2 m;nT max g )asabound onthetotalnumberofviolationsrepairedbyAlgorithmDMLC -INTERDIGITATION.By properchoiceofdatastructuresandprogrammingmethodsit ispossibletoimplementAlgorithmDMLC-INTERDIGITATIONsoastorunin O ( mn + n min f nm;T max g )time. NotethatLemma 20 providestwoupperboundsofonthecomplexityofAlgorithm DMLC-INTERDIGITATION: O ( n 2 m )and O ( n max f m;T max g ).Inmostpracticalsituations, T max
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65 (b)Whenthealgorithmterminates,nointerdigitationviol ationsremainandthenal scheduleisfeasible.FromLemma 19 ,itfollowsthatthenalscheduleisoptimalin therapytimeforunidirectionalschedules. 3.2Conclusion Wehavepresentedmathematicalformalismsandrigorouspro ofsofleafsequencing algorithmsfordynamicmultileafcollimationthatmaximiz eMUeciency.Theseleafsequencingalgorithmsexplicitlyaccountforleafinterdigi tationconstraint.Wehaveshown thatouralgorithmsobtainfeasiblesolutionsthatareopti malintreatmentMUs.Furthermore,ouranalysisshowsthatunidirectionalleafmove mentisatleastasecientas bi-directionalmovement.Thusthesealgorithmsarewellsu itedforcommonuseinDMLC beamdelivery.

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CHAPTER4 ELIMINATIONOFTONGUE-AND-GROOVEUNDERDOSAGE DeliveredofIMRTwithMLCinthestep-and-shootmodeusesmu ltiplestaticMLC segmentstoachieveintensitymodulation.Thesidesofeach leafofaMLChaveaprotruding tongueorastepononesidethattsintoasimilargrooveofth eadjacentleaf.Thisresults indierentradiologicalpathlengthsacrossdierentpart softheleaves.Galvinetal. (1993a)rstdescribedthatthedierentradiologicalpath lengthsmanifestthemselvesas varyingdosesinaplaneperpendiculartotheleafmotion.Th elowdoseregionbetweentwo adjacentleaveswasclassiedasthetongue-and-grooveee ct.InanIMRTtreatmentusing anMLC,thetongue-and-grooveeectoccurswhenthetongue, orthegrooveorbothfor themosttimeduringtreatmentdeliverycovertheoverlappi ngregionbetweentwoadjacent pairsofleaves.Aspointedoutbymanyinvestigators,theto ngue-and-groovearrangement alwaysresultsinunderdosagesofasmuchas10-25%inthetre atmenteldsinbothstatic anddynamicmultileafcollimation(Galvinetal.1993a,Gal vinetal.1993b,Chuietal. 1994,Mohan1995,Wangetal.1996,SykesandWilliams1998). Severalrecentpublications(vanSantvoortandHeijmen199 6,Webbetal.1997,ConveryandWebb1998,Dirkxetal.1998,XiaandVerhey1998)hav eshownthatthetongueand-grooveeectcanbesignicantlyreducedbysynchroniz ationoftheleaves.However, thecostofleafsynchronizationisusuallyanincreaseinth etotalnumberofsubelds andmonitorunits.vanSantvoortandHeijmen(1996)propose analgorithmtoeliminate tongue-and-grooveeectsforDMLCtreatmentplans.Althou ghtheynotethattheiralgorithmincreasesthenumberofmonitorunits,theydonotex aminetheoptimalityor suboptimalityoftheplanstheyobtain.Werecentlypublish edapaper(Kamathetal. 2003)thatgavemathematicalformalismsandrigorousproof sofleafsequencingalgorithms forsegmentalmultileafcollimation,whichmaximizeMUec iency.Weprovedthatourleaf sequencingalgorithmsthatexplicitlyaccountforminimum leafseparationobtainfeasible unidirectionalsolutionsthatareoptimal.Wenowextendth atworktodevelopalgorithms 66

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67 thatexplicitlyaccountforleafinterdigitationandtheto ngue-and-grooveeectandare optimalinMUeciencyforunidirectionalschedules.Wesho walsothatthealgorithmof vanSantvoortandHeijmen(1996)obtainsoptimaldynamicmu ltileafcollimationtreatment schedules. 4.1AlgorithmwithInterdigitationandTongue-and-Groove Constraints 4.1.1Tongue-and-GrooveUnderdosageEect Figure 4{1 showsabeams-eyeviewoftheregiontobetreatedbytwoadjac entleafpairs, t and t +1.Considertheshadedrectangularareas A t ( x i )and A t +1 ( x i )thatrequireexactly I t ( x i )and I t +1 ( x i )MUstobedelivered,respectively.Thetongue-and-groove overlaparea betweenthetwoleafpairsoverthesamplepoint x i A t;t +1 ( x i ),iscoloredblack.Letthe amountofMUsdeliveredin A t;t +1 ( x i )be I t;t +1 ( x i ).Ignoringleaftransmission,thefollowing lemmaisaconsequenceofthefactthat A t;t +1 ( x i )isexposedonlywhenboth A t ( x i )and A t +1 ( x i )areexposed. xx i-1ii+1 t+1 I t I t, t+1 AAA tt, t+1t+1 x I Figure4{1:Tongue-and-grooveeect Lemma21 I t;t +1 ( x i ) min f I t ( x i ) ;I t +1 ( x i ) g 0 i m 1 t
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68 4.1.2Algorithms Kamathetal.(2003)presentanalgorithmthatgeneratesasc hedulethatsatises inter-pairminimumseparationconstraint.Thescheduleis optimalintherapytime.However,itdoesnotaccountforthetongue-and-grooveeect.I nthissection,wepresenttwo algorithms.AlgorithmTONGUEANDGROOVEgeneratesminimum therapytimeunidirectionalschedulesthatarefreeoftongue-and-grooveund erdosageandmaybeusedfor MLCsthatdonothaveainterdigitationconstraint.Algorit hmTONGUEANDGROOVEIDgeneratesminimumtherapytimeunidirectionalschedule sthatarefreeoftongue-andgrooveunderdosagewhilesimultaneouslysatisfyingthein terdigitationconstraintandisfor MLCsthathaveaninterdigitationconstraint. Thefollowinglemmaprovidesanecessaryandsucientcondi tionforaunidirectional scheduletobefreeoftongue-and-grooveunderdosageeect s. Lemma22 Aunidirectionalscheduleisfreeoftongue-and-grooveund erdosageeectsif andonlyif, (a) I t ( x i )=0 or I t +1 ( x i )=0 ,or (b) I tr ( x i ) I ( t +1) r ( x i ) I ( t +1) l ( x i ) I tl ( x i ) ,or (c) I ( t +1) r ( x i ) I tr ( x i ) I tl ( x i ) I ( t +1) l ( x i ) 0 i m 1 t 0(4.1) Fromtheunidirectionalconstraint,itfollowsthat A t;t +1 ( x i )rstgetsexposedwhenboth rightleavespass x i ,anditremainsexposedtilltherstoftheleftleavespasse s x i .Further, ifaleftleafpasses x i beforeaneighboringrightleafpasses x i A t;t +1 ( x i )isnotexposedat

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69 all.So, I t;t +1 ( x i )=max f 0 ;I ( t;t +1) l ( x i ) I ( t;t +1) r ( x i ) g (4.2) where I ( t;t +1) r ( x i )=max f I tr ( x i ) ;I ( t +1) r ( x i ) g and I ( t;t +1) l ( x i )=min f I tl ( x i ) ;I ( t +1) l ( x i ) g From 4.1 and 4.2 ,itfollowsthat I t;t +1 ( x i )= I ( t;t +1) l ( x i ) I ( t;t +1) r ( x i )(4.3) Considerthecase I t ( x i ) I t +1 ( x i ).Supposethat I tr ( x i ) >I ( t +1) r ( x i ).Itfollowsthat I ( t;t +1) r ( x i )= I tr ( x i )and I ( t;t +1) l ( x i )= I ( t +1) l ( x i ).Nowfrom 4.3 ,weget I t;t +1 ( x i )= I ( t +1) l ( x i ) I tr ( x i )
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70 pair t +1exposesregion A t;t +1 ( x i ),orviceversa,whenever I t ( x i ) 6 =0and I t +1 ( x i ) 6 =0. Notethatifeither I t ( x i )or I t +1 ( x i )iszerothecontainmentisnotnecessary.Wewillrefer tothenecessaryandsucientconditionofLemma 22 asthe tongue-and-grooveconstraint condition .Schedulesthatsatisfythisconditionwillbesaidtosatis fythetongue-andgrooveconstraint.vanSantvoortandHeijmen(1996)presen tanalgorithmthatgenerates schedulesthatsatisfythetongue-and-grooveconstraintf orDMLC. XiaandVerhey(1998)claimthateveryschedulethatviolate stheinterdigitationconstraintalsoviolatesthetongue-and-grooveconstraint.W edemonstratewithacounterexamplethatthisisnotnecessarilythecase.Theintensityma trixshowninFigure 4{2 (a)can beexposedinasinglesegmentasshowninFigure 4{2 (b).Thesegmentisfreeoftongueand-grooveconstraintviolations,whileitclearlyviolat estheinterdigitationconstraint. 050000 500050 50 (a)(b) 0 0 00 0 00 0 Figure4{2:Counterexample.Theintensitymatrixshownin( a)canbetreatedusing asinglesegmentwith50MUsasshownin(b).Areasshadeddark arecoveredbyleft leavesandthoseshadedlightarecoveredbyrightleaves.Ar easnotshadedareexposed. Interdigitationconstraintviolationoccursthoughthere isnotongue-and-grooveviolation. Eliminationoftongue-and-grooveeect. NotethattheschedulegeneratedbyAlgorithmMULTIPAIRmayviolatethetongue-and-grooveconst raint.Iftheschedulehas notongue-and-grooveconstraintviolations,itisthedesi redoptimalschedule.Ifthereare violationsintheschedule,weeliminateallviolationsoft hetongue-and-grooveconstraint startingfromtheleftend,i.e.,from x 0 .Toeliminatetheviolations,wemodifythoseplans oftheschedulethatcausetheviolations.Wescantheschedu lefrom x 0 alongthepositive x directionlookingfortheleast x w atwhichthereexistleafpairs u t t 2f u 1 ;u +1 g thatviolatetheconstraintat x w .Afterrectifyingtheviolationat x w welookforother violations.Sincetheprocessofeliminatingaviolationat x w ,mayattimes,leadtonew

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71 violationsat x w ,weneedtosearchafreshfrom x w everytimeamodicationismadeto theschedule.However,wewillproveaboundof O ( n )onthenumberofviolationsthat canoccurat x w .Aftereliminatingallviolationsataparticularsamplepo int, x w ,wemove tothenextpoint,i.e.,weincrement w andlookforpossibleviolationsatthenewpoint. Wecontinuethescanningandmodicationprocessuntilnoto ngue-and-grooveconstraint violationsexist.AlgorithmTONGUEANDGROOVE(Figure 4{3 )outlinestheprocedure. AlgorithmTONGUEANDGROOVE 1. x = x 0 2.While(thereisatongue-and-grooveviolation)do3.Findtheleast x w x w x ,suchthatthereexistleafpairs u u +1,thatviolatethe tongue-and-grooveconstraintat x w 4.Modifythescheduletoeliminatetheviolationbetweenle afpairs u and u +1. 5. x = x w 6.EndWhile Figure4{3:Obtainingascheduleunderthetongue-and-groo veconstraint Let M =(( I 1 l ;I 1 r ) ; ( I 2 l ;I 2 r ) ;:::; ( I nl ;I nr ))betheschedulegeneratedbyAlgorithm MULTIPAIRforthedesiredintensityprole.Let N ( p )=(( I 1 lp ;I 1 rp ) ; ( I 2 lp ;I 2 rp ) ;:::; ( I nlp ;I nrp ))bethescheduleobtainedafterStep 4 of AlgorithmTONGUEANDGROOVEisapplied p timestotheinputschedule M .Notethat M = N (0). Toillustratethemodicationprocessweuseexamples.Toma kethingseasier,weonly showtwoneighboringpairsofleaves.Supposethatthe( p +1)thviolationoccursbetween theleavesofpair u andpair t = u +1at x w .Notethat I tlp ( x w ) 6 = I ulp ( x w ),asotherwise, either(b)or(c)ofLemma 22 istrue.Incase I tlp ( x w ) >I ulp ( x w ),swap u and t .Now, wehave I tlp ( x w )
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72 Thenewplanforpair t ,( I tl ( p +1) ;I tr ( p +1) )isasdenedbelow. If I ulp ( x w ) I tlp ( x w ) I urp ( x w ) I trp ( x w ),then I tl ( p +1) ( x )= 8><>: I tlp ( x ) x 0 x<>: I trp ( x ) x 0 x
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73 x w I trp I I tr(p+1) tl(p+1) tlp urp ulp I I I I x Figure4{5:Tongue-and-grooveconstraintviolation:case 2(closeparalleldottedandsolid linesegmentsoverlap,theyhavebeendrawnwithasmallsepa rationtoenhancereadability) Since( I tl ( p +1) ;I tr ( p +1) )diersfrom( I tlp ;I trp )for x x w thereisapossibilitythat N ( p +1)isinvolvedintongue-and-grooveviolationsfor x x w .Sincenoneoftheother leafprolesarechangedfromthoseof N ( p )notongue-and-grooveconstraintviolations arepossiblein N ( p +1)for x 0. Lemma23 Let F =(( I 0 1 l ;I 0 1 r ) ; ( I 0 2 l ;I 0 2 r ) ;:::; ( I 0 nl ;I 0 nr )) beanyunidirectionalschedulefor thedesiredprolethatsatisesthetongue-and-groovecon straint.Let S ( p ) ,bethefollowing assertions. (a) I 0 il ( x ) I ilp ( x ) 0 i n;x 0 x x m (b) I 0 ir ( x ) I irp ( x ) 0 i n;x 0 x x m S ( p ) istruefor p 0 Proof: Theproofisbyinductionon p 1.Considerthebasecase, p =0.FromCorollary 1 andthefactthattheplans ( I il 0 ;I ir 0 ) ; 0 i n ,aregeneratedusingAlgorithmSINGLEPAIR,itfollowsthat S (0)istrue.

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74 2.Assume S ( p )istrue.SupposeAlgorithmTONGUEANDGROOVEndsanextvio lationandmodiestheschedule N ( p )to N ( p +1).Supposethatthenextviolation occursbetweenleafpairs u and t t 2f u 1 ;u +1 g .Hence, I tlp ( x w ) I urp ( x w ) I trp ( x w ).So I 0 tl ( x w ) I tl ( p +1) ( x w ). Itremainstobeprovedthat I 0 tl ( x i ) I tl ( p +1) ( x i ), ww I 0 tl ( x v )
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75 I 00 tl ( x i )= 8><>: I tl 0 ( x i ) i<>: I tr 0 ( x i ) i
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76 Lemma24 Aschedulesatisesthetongue-and-groove-idconstrainti itsatisesthetongueand-grooveconstraintandtheinterdigitationconstraint Proof: Itisobviousthatthetongue-and-groove-idconstraintsub sumesthetongue-andgrooveconstraint.Ifaschedulehasaviolationoftheinter digitationconstraint, 9 i;t I ( t +1) l ( x i )
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77 respectively.AlgorithmTONGUEANDGROOVE-IDisshowninFi gure 4{6 .Allnotation usedinthealgorithmandtherelateddiscussionintheremai nderofSection 4.1.2 isalso thesameasthatusedinSection 4.1.2 andcorrespondsdirectlytotheusageinAlgorithm TONGUEANDGROOVE.AlgorithmTONGUEANDGROOVE-ID 1. x = x 0 2.While(thereisatongue-and-groove-idviolation)do3.Findtheleast x w x w x ,suchthatthereexistleafpairs u u +1,thatviolatethe tongue-and-groove-idconstraintat x w 4.Modifythescheduletoeliminatetheviolationbetweenle afpairs u and u +1. 5. x = x w 6.EndWhile Figure4{6:Obtainingascheduleunderboththeconstraints Lemma25 Let F =(( I 0 1 l ;I 0 1 r ) ; ( I 0 2 l ;I 0 2 r ) ;:::; ( I 0 nl ;I 0 nr )) beanyunidirectionalschedulefor thedesiredprolethatsatisesthetongue-and-groove-id constraint.Let S ( p ) ,bethefollowingassertions. (a) I 0 il ( x ) I ilp ( x ) 0 i n;x 0 x x m (b) I 0 ir ( x ) I irp ( x ) 0 i n;x 0 x x m S ( p ) istruefor p 0 Proof: Theproofisbyinductionon p 1.Considerthebasecase, p =0.FromCorollary 1 andthefactthattheplans ( I il 0 ;I ir 0 ) ; 0 i n ,aregeneratedusingAlgorithmSINGLEPAIR,itfollowsthat S (0)istrue. 2.Assume S ( p )istrue.SupposeAlgorithmTONGUEANDGROOVE-IDndsanext violationandmodiestheschedule N ( p )to N ( p +1).Supposethatthenextviolation occursbetweenleafpairs u and t t 2f u 1 ;u +1 g .AsintheproofofLemma 23 weonlyneedproveeitherEquation 4.8 orEquation 4.9 tocompletethisproof.We completetheproofforthefollowingthreecasesthatareexh austive. case1: I t ( x w ) 6 =0and I u ( x w ) 6 =0. TheremainderoftheproofforthiscaseisthesameasthatofL emma 23 case2: I t ( x w )=0. Inthiscase, I tlp ( x w )= I trp ( x w ).Since I ulp ( x w ) I urp ( x w ),wehave I ulp ( x w )

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78 I tlp ( x w ) I urp ( x w ) I trp ( x w ).ThemodicationprescribedbyEquation 4.7 is applicable.Notethatif I urp ( x w ) I trp ( x w )= I ulp ( x w ) I tlp ( x w ),Equation 4.6 isthesameasEquation 4.7 .Inparticular, I tr ( p +1) ( x w )= I trp ( x w )+ I urp ( x w ) I trp ( x w )= I urp ( x w )(4.13) Since I t ( x w )=0, I tr ( p +1) ( x w )= I tl ( p +1) ( x w )(4.14) FromEquations 4.13 and 4.14 I urp ( x w )= I tl ( p +1) ( x w )(4.15) Since F satisestheinterdigitationconstraint,theleftleafofp air t doesnot pass x w beforetherightleafofpair u passes x w .So, I 0 tl ( x w ) I 0 ur ( x w )(4.16) From S ( p )andEquation 4.15 ,weget, I 0 ur ( x w ) I urp ( x w )= I tl ( p +1) ( x w )(4.17) Equations 4.16 and 4.17 yield I 0 tl ( x w ) I tl ( p +1) ( x w ) 0(4.18) Lemma 2 bimplies, I 0 tl ( x ) I tl ( x ) I 0 tl ( x w ) I tl ( x w ) ;x x w (4.19) Subtracting I tl ( p +1) ( x )fromEquation 4.19 ,andrearrangingtermsweget I 0 tl ( x ) I tl ( p +1) ( x ) I 0 tl ( x w ) I tl ( x w )+ I tl ( x ) I tl ( p +1) ( x ) ;x x w (4.20) FromEquations 4.6 and 4.7 andtheworkingofAlgorithmTONGUEANDGROOVE-ID,itfollowsthat I tl ( p +1) ( x ) I tl ( x )= I tl ( p +1) ( x w ) I tl ( x w ) ;x x w (4.21)

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79 FromEquations 4.20 4.21 and 4.18 ,weget I 0 tl ( x ) I tl ( p +1) ( x ) I 0 tl ( x w ) I tl ( p +1) ( x w ) 0 ;x x w (4.22) Therefore, I 0 tl ( x ) I tl ( p +1) ( x ) ;x x w (4.23) case3: I u ( x w )=0. Theproofissimilartothatofcase2. 4.1.3EcientImplementationoftheAlgorithms Intheremainderofthissectionwewilluse`algorithm'tome anAlgorithmTONGUEANDGROOVEorAlgorithmTONGUEANDGROOVE-IDand`violation 'tomeantongueand-grooveconstraintviolationortongue-and-groove-id constraintviolation(dependingon whichalgorithmisconsidered)unlessexplicitlymentione d. Theexecutionofthealgorithmstartswithschedule M at x = x 0 andsweepstothe right,eliminatingviolationsfromtheschedulealongthew ay.Themodicationsappliedto eliminateaviolationat x w ,prescribedbyEquations 4.6 and 4.7 ,modifyoneoftheviolating prolesfor x x w .Fromtheunidirectionalnatureofthesweepofthealgorith m,itisclear thatthemodicationoftheprolefor x>x w canhavenoconsequenceonviolationsthat mayoccuratthepoint x w .Thereforeitsucestomodifytheproleonlyat x w atthe timetheviolationat x w isdetected.Themodicationcanbepropagatedtotherighta s thealgorithmsweeps.Thiscanbedonebyusingan( n m )matrix A thatkeepstrackof theamountbywhichtheproleshavebeenraised. A ( j;k )denotesthecumulativeamount bywhichthe j thleafpairproleshavebeenraisedatsamplepoint x k fromtheschedule M generatedusingAlgorithmMULTIPAIR.Whenthealgorithmha seliminatedallviolations ateach x w ,itmovesto x w +1 tolookforpossibleviolations.Itrstsetsthe( w +1)thcolumn ofthemodicationmatrixequaltothe w thcolumntorerectrightwardpropagationofthe modications.Itthenlooksforandeliminatesviolationsa t x w +1 andsoon.

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80 Theprocessofdetectingtheviolationsat x w meritsfurtherinvestigation.Weshow thatifonecarefullyselectstheorderinwhichviolationsa redetectedandeliminated,the numberofviolationsateach x w ,0 w m willbe O ( n ). Lemma26 Thealgorithmcanbeimplementedsuchthat O ( n ) violationsoccurateach x w 0 w m Proof: Theboundisachievedusingatwopassschemeat x w .Inpassonewecheck adjacentleafpairs(1 ; 2) ; (2 ; 3) ;:::; ( n 1 ;n ),inthatorder,forpossibleviolationsat x w .In passtwo,wecheckforviolationsinthereverseorder,i.e., ( n 1 ;n ) ; ( n 2 ;n 1) ;:::; (1 ; 2). Soeachsetofadjacentpairs( i;i +1),1 i
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81 Proof: (a)Lemma 27 providesapolynomialupperbound( O ( n m ))onthecomplexity ofAlgorithmsTONGUEANDGROOVEandTONGUEANDGROOVE-ID.Th eresult followsfromthis. (b)WhenAlgorithmTONGUEANDGROOVEterminates,notongueand-grooveviolationsremain.FromthisandLemma 23 ,itfollowsthattheschedulegeneratedby AlgorithmTONGUEANDGROOVEisoptimalintherapytimeforun idirectional schedulesfreeoftongue-and-grooveviolations. (c)WhenAlgorithmTONGUEANDGROOVE-IDterminates,notong ue-and-groove-id violationsremainandfromLemma 24 thenalschedulesatisesthetongue-andgrooveandinterdigitationconstraints.FromthisandLemm a 25 ,itfollowsthatthe schedulegeneratedbyAlgorithmTONGUEANDGROOVE-IDisopt imalintherapy timeforunidirectionalschedulesfreeofbothtypesofviol ations. Theorem21 TheschedulegeneratedbythealgorithmofvanSantvoortand Heijmen(1996) isfreeofinterdigitationandtongue-and-grooveconstrai ntviolationsandisoptimalintherapytimeforunidirectionalDMLCscheduleswiththisproper ty. Proof: SimilartothatofTheorem 20 (c). 4.2ExperimentalValidation ThealgorithmswerevalidatedonaVarian2100C/Dwith120-l eafMLC(Varian MedicalSystems,PaloAlto,CA).Theintensitymapsofa7-e ldheadandneckplan fromacommercialinversetreatmentplanningsystem(CORVU S5.0,NOMOSCorporation, Cranberry,PA)weresequencedusingAlgorithmMULTIPAIR,w hichoptimizestheMU eciency,andAlgorithmTONGUEANDGROOVE-ID,whichelimin atesthetongue-andgrooveeectandinterdigitation.Theintensitymapshavea bixelsizeof1cmx1cm anda20%intensitystep.Figure 4{7 showsthelmmeasurementoftheruencemaps oftheAPeld.Thetongue-and-grooveeectisreadilyseeni nFigure 4{7 (a),whileit iscompletelyeliminatedinFigure 4{7 (b)usingAlgorithmTONGUEANDGROOVE-ID. Table 4{1 comparesthenumberofsegmentsandtheMUecienciesofallt hreealgorithms. TheMUeciencyisdenedastheratioofthemaximumruenceof intensitymodulated eldperMUtotheruenceofanopeneldperMU.Comparedtothe leafsequenceswithno

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82 constraints,theconsiderationoftongue-and-groovecorr ectionincreasedboththenumberof segmentsandMUs,withanaverageincreaseof21%and19%,res pectively,forthe7intensity mapsconsideredhere.Withtheadditionaleliminationofin terdigitation,theincreaseswere 25%and24%,respectively.Examinationofallthesubeldso ftheleafsequencesgenerated withAlgorithmTONGUEANDGROOVE-IDveriedthatnointerdi gitationconstrainthas beenviolated. Figure4{7:FilmmeasurementoftheAPeld(eldID1inTable 1)ofaseven-eldheadand neckplan.Theoptimizedleafsequencesweregeneratedwith out(AlgorithmMULTIPAIR, (a))andwithtongue-and-groove-idcorrection(Algorithm TONGUEANDGROOVE-ID, (b)). 4.3ComparisonwithAlgorithmofQueetal.(2004)) Recentlyanewalgorithmtoeliminatetongue-and-groovee ectsinstepandshootdeliveryhasbeenproposed(Queetal.2004).ThealgorithmofQ ueetal.(2004)isdesignedto eliminatetongue-and-grooveeect.Althoughthisalgorit hmeliminatestongue-and-groove eectonall1000randommatricestriedinQueetal.(2004),n oproofthatthealgorithm eliminatestongue-and-grooveeectonallpossiblematric eshasbeenprovided.Further,it isnotknownwhetherornotthealgorithmofQueetal.(2004)m inimizestherapytime. WeanalyzethealgorithmofQueetal.(2004)andshowthatiti salwayssuccessfulin eliminatingtongue-and-grooveeect;thegeneratedleafs equenceisalsofreeofinterdigitation.Wealsoperformatheoreticalandexperimentalcomp arisonofthisalgorithmwith ouralgorithms(Kamathetal.(2004)).

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83 Table4{1:ComparisonofthenumberofsegmentsandMUecien cyof thethreeleafsequencingalgorithms(MULTIPAIR,TONGUEAN DGROOVEand TONGUEANDGROOVE-ID)for7intensitymapsofaheadandneckt reatmentplangeneratedfromacommercialtreatmentplanningsystem.Theperce ntincreasesinthenumberof segmentsandMUsforAlgorithmsTONGUEANDGROOVEandTONGUE ANDGROOVEIDwithrespecttoAlgorithmMULTIPAIRarealsoshown.Theav eragepercentincreases inthenumberofsegmentsare21%and25%,respectively.Thea veragepercentincreases inthenumberofMUsare19%and24%,respectively. FieldID 1 2 3 4 5 6 7 MULTIPAIR #ofSegments 11 8 13 15 14 10 10 MUEciency 0.47 0.63 0.40 0.35 0.37 0.40 0.51 TONGUEANDGROOVE #ofSegments 14 10 15 21 14 13 11 MUEciency 0.37 0.51 0.35 0.25 0.37 0.40 0.40 %Segment#increase 27 25 15 40 0 30 10 %MUincrease 26 24 15 37 0 0 28 TONGUEANDGROOVE-ID #ofSegments 14 11 16 21 14 14 11 MUEciency 0.37 0.47 0.33 0.25 0.37 0.37 0.37 %Segment#increase 27 38 23 40 0 40 10 %MUincrease 26 36 22 37 0 7 38 4.3.1AnalysisoftheAlgorithmofQueetal.(2004) InthissectionweanalyzethealgorithmofQueetal.(2004). Theyusethe`sliding window'methodproposedbyBortfeldetal.(1994b)togenera teatentativesegment.They thensearchthroughtherightleafpositionstodetermineth eleftmostrightleafposition andpositionallrightleavesatthatposition.Thisdenest herstsegmentoftheleaf sequence.Theresidualintensitymatrixiscalculatedandt heprocessisrepeated.To obtainthe`slidingwindow'leafsequenceforeachleafpair ,horizontallinesaredrawnat unitintensitylevelstointersecttheintensityprolefor thatleafpair.Theleftandrightleaf positionsaredeterminedfromtheseintersectionsandares ortedfromlefttorighttogive thenalunidirectionalleafsequence.Theprocessisrepea tedforallleafpairs.Forthecase wheretheintensitylevelsinthemapgeneratedbytheoptimi zerareintegers,itispossible toshowthatthealgorithmsofMaetal.(1998)andKamathetal .(2003)(Algorithms SINGLEPAIRandMULTIPAIRforoneandmultipleleafpairsres pectively)willyieldthe

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84 sameleafsequenceasthatobtainedusingthe`slidingwindo w'methodofBortfeldetal. (1994b). Notethatthediscreteintensityprolethatneedstobedeli vered, I ,isoutputfrom theoptimizer.Let n bethenumberofleafpairsand m bethenumberofsamplepointsfor eachleafpair(i.e.,foreachrowoftheprole).Wedenoteth erowsof I by I 1 ;I 2 ;:::I n .Let I t ( x i )denotethenumberofMUsthatneedtobedeliveredatsamplep oint i ( i thcolumn) ofleafpair t ( t throw). Lemma28 ThealgorithmofQueetal.(2004)generatesunidirectional schedules. Proof: Duringeachiteration,thenextsegmentgeneratedusingthe `slidingwindow' methodissuchthattheleftleavesarepositionedattheleft mostnon-zerosamplepoint (i.e.,theleast i suchthat I t ( x i ) >I t ( x i 1 ),where I t ( x 1 )=0)foreachrow t intheresidual matrix I .Therightleavesarepositionedattherstcolumnsofthere spectiverowswhere thereisadecreaseinintensityprole(i.e.,theleast j forwhich I t ( x j )
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85 I 4 x 5 x x x x x x x x x 1 8 7 6 3 2 0 Figure4{8:Leafpositions:Theleftleafwillbepositioned at x 2 ,i.e.,itwillshield x 0 and x 1 .Therightleafwillbepositionedat x 6 andwillshield x i i 6. Proof: Let I 0 tl ( x i )and I 0 tr ( x i ),respectively,bethenumberofMUsdeliveredwhentheleft andrightleavesofpair t pass x i intheschedulegeneratedbythealgorithmofQueetal. (2004).Intheschedulegenerated,allrightleavespasspoi nt x i ,0 i m (duringtheirleft torightmovement)afterexactlythesamenumberofmonitoru nits(MUs)aredelivered. So I 0 tr ( x i )= I 0 ( t +1) r ( x i ),0 i m ,1 t
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86 time,say 0r ( x i ),whichisequaltothemaximumofthetimesforwhicharightl eafstops at x i intheschedulegeneratedbyAlgorithmMULTIPAIR,i.e., 0r ( x i )=max j f jr ( x i ) g Thetherapytimefortheschedulegeneratedbythealgorithm ofQueetal.(2004)is therefore T 0 tng id = P mi =0 0r ( x i )= P mi =0 max j f jr ( x i ) g .Sinceeach jr ( x i ),0 i m 1 j
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87 4.3.2Results WeimplementedAlgorithmsTONGUEANDGROOVEandTONGUEANDG ROOVEID(Kamathetal.2004)andthealgorithmofQueetal.(2004). Forperformancecomparison,weusedtwoseparatedatasets.Therstsetconsistedo fthreeclinicalIMRTplans with7,5and7beams,respectively.Thersttwoplanshada20 %ruencestepandlastplan hada10%ruencestep.Table 4.3.2 givesthetotalMUsandnumberofsegmentsrequired foreachofthe19beamsinthe3clinicalplans.Onourclinica ldataset,thealgorithmof Queetal.(2004)generatedscheduleswith2-4timesasmanyM Usandsegmentsasdid thealgorithmsofKamathetal.(2004).Theseconddatasetco nsistedof100,000randomly generated15 15matrices.Theintensityvaluesinthesematriceswereran domintegers from0to10.TheaverageMUsandsegmentsforschedulesgener atedusingthethree algorithmsforthissetandtheirrespectivestandarddevia tionsareshowninTable 4.3.2 Onthisset,thealgorithmofQueetal.(2004)generatedsche duleswithabout2.5times asmanyMUsandsegmentsasdidthealgorithmsofKamathetal. (2004).Notethatin bothcasesthenumberofMUsandsegmentsintheschedulesgen eratedusingAlgorithm TONGUEANDGROOVE-ID(Kamathetal.2004)areonlyslightlyg reaterthaninthose generatedusingAlgorithmTONGUEANDGROOVE(Kamathetal.2 004). 4.4Conclusion Wehavedescribedmathematicalformalismandrigorousproo fsofleafsequencingalgorithmsforsegmentalmultileafcollimation,whichmaximiz eMUeciencywhilecompletely eliminatingthetongue-and-grooveunderdosage.Eventhou ghithasbeenshownthatfor amultipleeldIMRTplan( 5),thetongue-and-grooveeectontheIMRTdosedistributionisclinicallyinsignicant(Dengetal.2001)duetot hesmearingeectofindividual elds,yetitstillcanbeproblematicforasmallnumberofe ldsandforthepatientsetup withminimaluncertainty.Comparedtotheunconstrainedle afsequencingalgorithms,the presentedmethodsyieldleafsequences,whichdecreasesth eMUeciencyalittle.Butthey completelyovercometongue-and-grooveunderdosages.One ofthemethodsalsoeliminates leafinterdigitation.Mostimportantly,mathematicalpro ofsshowthatthesealgorithms areoptimalinMUeciencyforunidirectionalschedules.We havealsoprovedthatthe algorithmofQueetal.(2004)generatesschedulesthataref reeofthetongue-and-groove

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88 Table4{2:NumberofMUsandsegmentsgeneratedfor19clinic alintensitymodulatedbeamsfrom3IMRTplansusingalgorithmsA(Algorithmo fQueetal.2004), B(AlgorithmTONGUEANDGROOVE(Kamathetal.2004))andC(Al gorithm TONGUEANDGROOVE-ID(Kamathetal.2004)).Beams1-12havea 20%ruencestep, whilebeams13-19havea10%ruencestep. Beamnumber A B C MUsSegments MUsSegments MUsSegments 1 78038 28014 28014 2 52026 20010 22011 3 76033 30015 32016 4 84040 42021 42021 5 74032 28014 28014 6 78034 26013 28014 7 64029 26011 28011 8 150074 38019 42021 9 86043 24012 24012 10 150067 42020 42020 11 166078 42021 44022 12 84039 28014 28014 13 88078 28025 28024 14 1080102 30030 34033 15 107090 31027 32026 16 100090 34031 39036 17 89071 34029 34028 18 99075 31029 31030 19 101084 33030 33030 Table4{3:AveragenumberofMUsandsegmentsgeneratedover asetof100,000 random15 15matricesusingalgorithmsA(AlgorithmofQueetal.2004) B(AlgorithmTONGUEANDGROOVE(Kamathetal.2004))andC(Al gorithm TONGUEANDGROOVE-ID(Kamathetal.2004)).Therespectives tandarddeviations arealsoshown.Theintensityvaluesinthematriceswereran domlygeneratedintegersfrom 0to10. A B C MUsSegments MUsSegments MUsSegments Average 114.3111.6 47.545.7 48.246.4 StandardDeviation 6.16.0 3.43.0 3.53.0

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89 eectandinterdigitation.Ouranalysisshowsthatthealgo rithmofQueetal.generates schedulesthatmayrequireupto n timesthetherapytimerequiredbythatforanoptimal leafsequencefreeoftongue-and-grooveeectandinterdig itation,where n isthenumberof involvedleafpairs.Inexperimentswithclinicalandrando mlygenerateddatasetswend thatthealgorithmofQueetal.(2004)generatesschedulest hatrequire2to4timesthe therapytimerequiredbytheschedulesgeneratedbyouralgo rithms.

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CHAPTER5 ALGORITHMSFORSPLITTINGLARGEFIELDS 5.1Introduction Thedeliveryofabuttingsub-eldsthatresultfromthespli tofalargeeldoften resultsinlongerdeliverytimes,poorMUeciency,andeld matchingproblems.Dogan etal.(2003)pointoutthattheuncertaintiesinleafandcar riagepositionscauseerrorsin thedelivereddose(hotorcoldspots)alongthematchlineof theabuttingsub-elds.They observeddosedierencesofupto10%alongtheeldsplitlin ewhenthesplitlinecrossed throughthecenterofthetargetforalltheelds.Theproble mofdosimetricperturbation alongtheeldsplitlinehasbeenaddressedinseveralrecen tpublications(Wuetal.2000, Hongetal.2002,Doganetal.2003).Thesolutionsincludeda utomaticfeatheringofspliteldsbymodifyingthesplitlinepositionforeachgantrypo sition(Hongetal.2002,Dogan etal.2003)orbydynamicallychangingradiationintensity intheoverlapregionofthesplit elds.Noneoftheeldsplittingtechniquesreportedinthe literaturehasaddressedthe issueoftreatmentdeliveryandMUeciency.Webelievethat itisequallyimportantto addressthisissue. Ouroptimaleldsplittingalgorithmswithandwithoutfeat heringmaybeintegrated intoourpreviouslydevelopedleafsequencingalgorithmst ooptimallyaccountforinterdigitationandtongue-and-grooveeectofsomemultileafcoll imators.Weproviderigorous mathematicalproofsthattheproposedschemesforeldspli ttingareoptimalinMUeciency.Experimentalresultsshowthatouroptimaleldspl ittingalgorithmwithoutfeatheringreducestotalMUsbyupto26%onclinicalcasesandupto 63%onsyntheticcases comparedtoacommercialplanningsystemthatalsosplitse ldswithoutfeathering. 5.2FieldSplittingWithoutFeathering 5.2.1OptimalFieldSplittingforOneLeafPair Inthissectionwedeviateslightlyfromourearliernotatio nandassumethatthesample pointsare x 1 ;x 2 ; ... ;x m ratherthan x 0 ;x 1 ; ... ;x m .Allothernotationremainsunchanged. 90

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91 DeliveringaproleusingOneeld. Let I bethedesiredintensityprole.The problemofdeliveringtheexactprole I usingasingleeldhasbeenextensivelystudied. Maetal.(1998)providean O ( m )algorithmfortheproblemsuchthatthetherapytimeof thesolutionisminimized,where m isthenumberofsamplepoints.Kamathetal.(2003) alsodescribethealgorithm(AlgorithmSINGLEPAIR)andgiv eanalternateproofthatit obtainsaplan( I l ;I r )withoptimaltherapytimefor I ,where I l and I r denotetheleftand rightleafmovementproles,respectively.Theoptimalthe rapytimefor I isgivenbythe followinglemma.Lemma29 Let inc 1 ;inc 2 ;:::;incq betheindicesofthepointsatwhich I ( x i ) increases, i.e., I ( x inci ) >I ( x inci 1 ) .Thetherapytimefortheplan ( I l ;I r ) generatedbyAlgorithm SINGLEPAIRis P qi =1 [ I ( x inci ) I ( x inci 1 )] ,where I ( x inc 1 1 )=0 AlgorithmSINGLEPAIRcanbedirectlyusedtoobtainplanswh en I isdeliverable usingasingleeld.Let l betheleastindexsuchthat I ( x l ) > 0andlet g bethegreatest indexsuchthat I ( x g ) > 0.Wewillassumewithoutlossofgeneralitythat l =1.Sothe widthoftheproleis g samplepoints,where g canvaryfordierentproles.Assuming thatthemaximumallowableeldwidthis w samplepoints, I isdeliverableusingoneeld if g w ; I requiresatleasttwoeldsfor g>w ; I requiresatleastthreeeldsfor g> 2 w Thecasewhere g> 3 w isnotstudiedasitneverarisesinclinicalcases.Theobjec tiveof eldsplittingistosplitaprolesothateachoftheresulti ngprolesisdeliverableusinga singleeld.Further,itisdesirablethatthetotaltherapy timeisminimized,i.e.,thesum ofoptimaltherapytimesoftheresultingprolesisminimiz ed.Wewillcalltheproblemof splittingtheprole I ofasingleleafpairinto2proleseachofwhichisdeliverab leusing oneeldsuchthatthesumoftheiroptimaltherapytimesismi nimizedasthe S 2(single pair2eldsplit)problem.Thesumoftheoptimaltherapytim esofthetworesulting prolesisdenotedby S 2( I ). S 3and S 3( I )aredenedsimilarlyforsplitsinto3proles. Theproblem S 1istrivial,sincetheinputproleneednotbesplitandisto bedelivered usingasingleeld.Notethat S 1( I )istheoptimaltherapytimefordeliveringtheprole I inasingleeld.FromLemma 29 andthefactthattheplangeneratedusingAlgorithm SINGLEPAIRisoptimalintherapytime, S 1( I )= P qi =1 [ I ( x inci ) I ( x inci 1 )].

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92 Splittingaproleintotwo. Supposethataprole I issplitintotwoproles.Let j betheindexatwhichtheproleissplit.Asaresult,wegettw oproles, P j and S j P j ( x i )= I ( x i ),1 ij [ I ( x inci ) I ( x inci 1 )]= I ( x j )+ P inci>j [ I ( x inci ) I ( x inci 1 )](since S j ( x j 1 )=0 and S j ( x j )= I ( x j )).If I ( x j ) >I ( x j 1 ),thiscanbewrittenas S 1( S j )=( I ( x j ) I ( x j 1 ))+ P inci>j [ I ( x inci ) I ( x inci 1 )]+ I ( x j 1 )= P inci j [ I ( x inci ) I ( x inci 1 )]+ I ( x j 1 ).If I ( x j ) I ( x j 1 ), S 1( S j )= P inci>j [ I ( x inci ) I ( x inci 1 )]+ I ( x j )= P inci j [ I ( x inci ) I ( x inci 1 )]+ I ( x j ).Therefore S 1( S j )= P inci j [ I ( x inci ) I ( x inci 1 )]+min f I ( x j 1 ) ;I ( x j ) g .Byaddition, S 1( P j )+ S 1( S j )= P qi =1 [ I ( x inci ) I ( x inci 1 )]+min f I ( x j 1 ) ;I ( x j ) g = S 1( I )+ ^ I ( x i ). WeillustrateLemma 30 usingtheexampleofFigure 5{1 .Theoptimaltherapy timefortheprole I isthesumofincrementsinintensityvaluesofsuccessivesa mplepoints.However,if I issplitat x 3 into P 3 and S 3 ,anadditionaltherapytimeof ^ I ( x 3 )=min f I ( x 2 ) ;I ( x 3 ) g = I ( x 3 )isrequiredfortreatment.Similarly,if I issplitat x 4 into P 4 and S 4 ,anadditionaltherapytimeof ^ I ( x 4 )=min f I ( x 3 ) ;I ( x 4 ) g = I ( x 3 )isrequired. Lemma 30 leadstothefollowing O ( g )algorithmfor S 2. Algorithm S 2 (1)Compute ^ I ( x i )=min f I ( x i 1 ) ;I ( x i ) g ,for g w 2 w twoeldsareinsucient.SoitisusefultoapplyAlgorithm S 2onlyfor w
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93 x 5 x 6 P (x) 3 P (x) 4 S (x) 3 S (x) 4 x 4 31 P (x) 4 2 (a) I x x 1 x 2 x 3 x S (x) xxxx 3456 I(x ) 3 ^ (b) (c) (d) (e) 3 4 x 56 x x 4 I(x ) 4 I(x ) 4 ^ 4 x S (x) I(x ) 31 2 2 x I x 1 x 2 x 3 x I(x ) I(x ) I(x ) I 1 xx 3 I(x ) P (x) 3 I(x ) Figure5{1:Splittingaprole(a)intotwo.(b)and(c)showt heleftandrightproles resultingfromasplitat x 3 ;(d)and(e)showtheleftandrightprolesresultingfroma splitat x 4

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94 theprole I issplitintotwoasdeterminedbyAlgorithm S 2,theleftandrightprolesare deliveredusingseparateelds.Thetotaltherapytimeis S 2( I )= S 1( P j )+ S 1( S j ),where j isthesplitpoint. Splittingaproleintothree. Supposethataprole I issplitintothreeproles.Let j and k j
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95 improvesoptimaltherapytime.Foranupperboundontherati o,notethat S 1( I ) min f I ( x j 1 ) ;I ( x j ) g sinceatleastmin f I ( x j 1 ) ;I ( x j ) g MUsarerequiredtodeliver I So S 2( I ) 2 S 1( I ).TheexampleofFigure 5{2 showsthattheupperboundistight. Theprole I has2 w samplepoints,i.e.,ithasawidth2 w x .Soithastobesplit exactlyat x w +1 .Theresultingleftandrightproleseachhaveanoptimalth erapy timeequaltothatof I x w w I Figure5{2:TightupperboundforLemma32a (b) S 3( I )= S 1( I )+min f I ( x j 1 ) ;I ( x j ) g +min f I ( x k 1 ) ;I ( x k ) g ,where j and k areas inAlgorithm S 3.Clearly, S 3( I ) =S 1( I ) 1.Also, S 1( I ) min f I ( x j 1 ) ;I ( x j ) g and S 1( I ) min f I ( x k 1 ) ;I ( x k ) g .Therefore, S 3( I ) 3 S 1( I ).Onceagaintheupper boundistightasshownintheFigure 5{3 .Theproleshownhaswidth3 w x and needstobesplitat x w +1 andat x 2 w +1 .Eachoftheresultingproleshasoptimal therapytimeequalto S 1( I ). w w w x I Figure5{3:TightupperboundforLemma32b (c)From(a)and(b), S 3( I ) S 1( I )and S 2( I ) 2 S 1( I ).So S 3( I ) =S 2( I ) 0 : 5. S 3( I ) =S 2( I )=0 : 5onlyif S 3( I )= S 1( I )and S 2( I )=2 S 1( I ).Suppose

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96 that S 3( I )= S 1( I ).Thenthereexistindices j;k suchthatmin f I ( x j 1 ) ;I ( x j ) g + min f I ( x k 1 ) ;I ( x k ) g =0,i.e.,min f I ( x j 1 ) ;I ( x j ) g =0andmin f I ( x k 1 ) ;I ( x k ) g =0. Thisandthefactthat I ( x 1 ) 6 =0 ;I ( x g ) 6 =0impliesthattheprolehasatleast twodisjointcomponentsseparatedbyasamplepointatwhich thedesiredintensity iszero.Samplepointsinthetwodisjointcomponentscannot beexposedatthe sametimeandsotheredoesnotexistapoint x i suchthat I ( x i )= S 1( I ).So S 2( I )= S 1( I )+min g w 0 : 5.Figure 5{4 showsanexamplewheretheratiocanbemade arbitrarilycloseto0.5.Inthisexample, S 1( I )= I 2 .Theprolehasawidthof 2 w x andthereforeneedstobesplitat x w +1 .Theresultingproleseachhavean optimaltherapytimeof S 1( I )sothat S 2( I )=2 S 1( I ). S 3( I )= S 1( I )+2 I 1 andso S 3( I ) S 1( I )as I 1 0. I w x 1 2 I I jk x x ww x Figure5{4:TightlowerboundforLemma32c Toobtainanupperboundnotethatthebestsplitpointfor S 2(say x j )isalways apermissiblesplitpointfor S 3.Byselectingthisasoneofthetwosplitpointsfor S 3,wecanconstructasplitintothreeprolessuchthattheto taltherapytimeof prolesresultingfromthissplitis S 2( I )+min f I ( x k 1 ) ;I ( x k ) g ,where k isthesecond splitpointdeningthatsplit.Sincemin f I ( x k 1 ) ;I ( x k ) g S 1( I ) S 2( I ),thetotal therapytimeofthesplit 2 S 2( I ).So S 3( I ) =S 2( I ) 2.Theratiocanbearbitrarily closeto2asdemonstratedinFigure 5{5 .Onecanverifythatfortheprole I inthis example, S 3( I ) =S 2( I ) 2as I 1 0.

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97 I x x I 1 1 Figure5{5:TightupperboundforLemma32c Lemma 32 tellsusthattheoptimaltherapytimescanatmostincreaseb yfactorsof2and 3,respectively,asaresultofasplittingasingleleafpair proleinto2and3.Also,the optimaltherapytimeforasplitinto2canbeatmosttwicetha tforasplitinto3andvice versa.5.2.2OptimalFieldSplittingforMultipleLeafPairs Theinputintensitymatrix(say I )fortheleafsequencingproblemisobtainedusing theinverseplanningtechnique.Thematrix I consistsof n rowsand m columns.Each rowofthematrixspeciesthenumberofmonitorunits(MUs)t hatneedtobedelivered usingoneleafpair.Denotetherowsof I by I 1 ;I 2 ;:::;I n .Forthecasewhere I isdeliverableusingoneeld,theleafsequencingproblemhasbeenw ellstudiedinthepast.The algorithmthatgeneratesoptimaltherapytimeschedulesfo rmultipleleafpairs(Algorithm MULTIPAIR)appliesalgorithmSINGLEPAIRindependentlyto eachrow I i of I .Without lossofgeneralityassumethattheleastcolumnindexcontai ninganonzeroelementin I is 1andthelargestcolumnindexcontaininganonzeroelementi n I is g .If g>w ,theprole willneedtobesplit.Wedeneproblems M 1, M 2and M 3formulipleleafpairsasbeing analogousto S 1, S 2and S 3forsingleleafpair.Theoptimaltherapytimes M 1( I ), M 2( I ) and M 3( I )arealsodenedsimilarly. Splittingaproleintotwo. Supposethataprole I issplitintotwoproles.Let x j bethecolumnatwhichtheproleissplit.Thisisequivalent tosplittingeachrowprole

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98 I i ,1 i n ,at j asdenedforsingleleafpairsplit.Asaresultwegettwopro les, P j (left)and S j (right). P j hasrows P 1 j ;P 2 j ;:::;P n j and S j hasrows S 1 j ;S 2 j ;:::;S n j Lemma33 Suppose I issplitintotwoprolesat x j .Theoptimaltherapytimefordelivering P j and S j usingseparateeldsis max i f S 1( P i j ) g +max i f S 1( S i j ) g Proof: Theoptimaltherapytimeschedulefor P j and S j areobtainedusingAlgorithm MULTIPAIR.Thetherapytimesaremax i f S 1( P i j ) g andmax i f S 1( S i j ) g respectively.Sothe totaltherapytimeismax i f S 1( P i j ) g +max i f S 1( S i j ) g FromLemma 33 itfollowsthatthe M 2problemcanbesolvedbyndingtheindex j 1
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99 Algorithm M 3solvesthe M 3problem. Algorithm M 3 (1)Computemax i f S 1( P i j ) g +max i f S 1( M i ( j;k ) ) g +max i f S 1( S i k ) g for1
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100 splitismax i f S 1( P i j ) g +max i f S 1( M i ( j;k ) ) g +max i f S 1( S i k ) g max i f S 1( P i j ) g +2 max i f S 1( S i j ) g 2 M 2( I ).So M 3( I ) =M 2( I ) 2. Notethattheexamplesusedtoshowtightnessofboundsinthe proofofLemma 32 canalsobeusedtoshowtightnessofboundsinthiscase. Lemma 35 tellsusthattheoptimaltherapytimescanatmostincreaseb yfactorsof2and 3,respectively,asaresultofsplittingaeldinto2and3.A lso,theoptimaltherapytime forasplitinto2canbeatmosttwicethatforasplitinto3and viceversa.Thesebounds giveusthepotentialbenetsofdesigningMLCswithlargerm aximalaperturesothatlarge eldsdonotneedtobesplit.Tongue-and-grooveeectandinterdigitation. Algorithms M 2and M 3maybeextendedtogenerateoptimaltherapytimeeldswitheliminat ionoftongue-and-grooveunderdosageand(optionally)theinterdigitationconstrain tontheleafsequences.Kamathet al.(2004)presentalgorithmsfordeliveringanintensitym atrix I usingasingleeldwith optimaltherapytime,whileeliminatingthetongue-and-gr ooveunderdosage(Algorithm TONGUEANDGROOVE)andalsowhilesimultaneouslyeliminati ngthetongue-and-groove underdosageandinterdigitationconstraintviolations(A lgorithmTONGUEANDGROOVEID).Denotetheseproblemsby M 1 0 and M 1 00 respectively( M 2 0 M 2 00 M 3 0 and M 3 00 are denedsimilarlyforsplitsintotwoandthreeelds).Let M 1 0 ( I )and M 1 00 ( I ),respectively, denotetheoptimaltherapytimesrequiredtodeliver I usingtheleafsequencesgeneratedby thesealgorithms.Tosolveproblem M 2 0 weneedtodetermine x j where M 1 0 ( P j )+ M 1 0 ( S j ) isminimizedfor g w
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101 etal.2000).Toillustratetheproblemweuseanexample.Con siderthesingleleafpair intensityproleofFigure 5{6 a.Duetowidthlimitations,theproleneedstobesplit. Supposethatitissplitat x j .Furthersupposethatthelefteldisdeliveredaccurately andthattherighteldismisalignedsothatitsleftendispo sitionedat x 0j ratherthan x j Duetoincorrecteldmatchingtheactualproledeliveredm aybe,forexample,eitherof theprolesshowninFigure 5{6 borFigure 5{6 d,dependingonthedirectionoferror.In Figure 5{6 b,theregionbetween x 0j and x j getsoverdosedandisa hotspot .InFigure 5{6 d, theregionbetween x j and x 0j getsunderdosedandisa coldspot Onewaytopartiallyeliminatetheeldmatchingproblemist ousethe`feathering' technique(Wuetal.2000).Inthistechnique,thelargeeld isnotsplitatonesample pointintotwonon-overlappingelds.Insteadtheprolest obedeliveredbythetwoelds resultingfromthesplit,overlapoveracentral featheringregion .ThebeamsplittingalgorithmproposedbyWuetal.(2000)splitsalargeeldwithfea thering,suchthatinthe featheringregionthesumofthespliteldsequalsthedesir edintensityprole.Figure 5{7 a showsasplitoftheproleofFigure 5{6 withfeathering.Figures 5{7 cand 5{7 dshowthe eectofeldmatchingproblemonthesplitwithfeathering. Theextentofeldmismatches isthesameasthoseinFigures 5{6 band 5{6 d,respectively.Notethatwhiletheprole deliveredinthecasewithfeatheringisnottheexactprole either,thedeliveredproleis lesssensitive tomismatchcomparedtothecasewhenitissplitwithoutfeat heringasin Figure 5{6 .Inotherwords,thepurposeoffeatheringistolowerthemag nitudeof maximum intensityerror e inthedeliveredprolefromthedesiredproleoverallsamp lepointsin thejunctionregion. Inthissection,weextendoureldsplittingalgorithmstoi ncorporatefeathering.In ordertodoso,wedeneafeatheringschemesimilartothatof Wuetal.(2000).However, therearetwodierencesbetweenthesplittingalgorithmwe proposeandthealgorithmof Wuetal.(2000).First,ourfeatheringschemeisdenedforp rolesdiscretizedinspace andinMUsasistheprolegeneratedbytheoptimizer.Second ,thefeatheringschemewe proposedenestheprolevaluesinthefeatheringregion,w hichiscenteredatsomesample pointcalledthe splitpoint forthatsplit.Thusgivenasplitpoint,ourschemewillspec ify howtosplitthelargeeldwithafeatheringregionthatisce nteredatthatpoint.Thesplit

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102 x xx xx x' xx' e e (a) (b) I x'xx (c) (d) j jj jj jj I II Figure5{6:Fieldmatchingproblem:Theprolein(a)isthed esiredprole.Itissplitinto twoeldsat x j .Duetoincorrecteldmatching,theleftendofrighteldis positionedat point x 0j insteadof x j andtheeldsmayoverlapasin(c)ormaybeseparatedasin(d) In(c),thedottedlineshowstheleftproleandthedashedli neshowstherightprole. (b)showstheseprolesaswellasthedeliveredproleinthi scaseinbold.In(d),the leftandrighteldsareseparatedandtheirtwoprolestoge therconstitutethedelivered prole,whichisshowninbold.Thedeliveredprolesinthes ecases,varysignicantlyfrom thedesiredproleinthejunctionregion. e isthemaximumintensityerrorinthejunction region,i.e.,themaximumdeviationofdeliveredintensity fromthedesiredintensity.

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103 II x xx xx x' x I e e j x' j j j j j (d) (c) (b) xx I (a) Figure5{7:Exampleofeldsplittingwithfeathering:(a)s howsasplitoftheproleof Figure 5{6 withfeathering.Thedottedlineshowstherightpartofthel eftproleand thedashedlineshowstheleftpartoftherightprole.Thele ftandrightprolesare shownseparatelyin(b).(c)and(d)showtheeectofeldmat chingproblemonthesplit withfeathering.Theextentofeldmismatchesin(c)and(d) arethesameasthosein Figure 5{6 bandFigure 5{6 d,respectively,ie.,thedistancesbetween x j and x 0j arethe sameasinFigure 5{6 .Notethatthemaximumintensityerror e reducesinbothcaseswith feathering.

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104 pointtobeusedintheactualsplitwillbedeterminedbyaspl ittingalgorithmthattakes intoaccountthefeatheringscheme.Incontrast,Wuetal.(2 000)alwayschoosethecenter oftheintensityproleasthesplitpoint,astheydonotopti mizethesplitwithrespectto anyobjective. Westudyhowtosplitasingleleafpairproleintotwo(three )eldsusingourfeathering schemesuchthatthesumoftheoptimaltherapytimesofthein dividualeldsisminimized. Wewilldenotethisminimizationproblemby S 2 F ( S 3 F ).Theextensionofthemethods develpedforthemultipleleafpairsproblems( M 2 F and M 3 F )isstraightforwardandis thereforenotdiscussedseparately.5.3.1SplittingaProleintoTwo Let I beasingleleafpairprole.Let x j bethesplitpointandlet P j and S j be theprolesresultingfromthesplit. P j isa leftprole and S j isa rightprole of I .The featheringregionspans x j and d 1samplepointsoneithersideof x j ,i.e.,thefeathering regionstretchesfrom x j d +1 to x j + d 1 P j and S j aredenedasfollows. P j ( x i )= 8>>>><>>>>: I j ( x i )1 i j d d I j ( x i ) ( j + d i ) = 2 d e j d>>><>>>>: 01 i j d I j ( x i ) P j ( x i ) j dg w ) j g w + d .Theserangerestrictionson j leadtoanalgorithmfor the S 2 F problem.Algorithm S 2 F ,whichsolvesproblem S 2 F ,isdescribedbelow.Note

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105 thatthe P i sand S i scanallbecomputedinasinglelefttorightsweepin O ( d )timeateach i .SothetimecomplexityofAlgorithm S 2 F is O ( dg ). Algorithm S 2 F (1)Find P i and S i usingEquations 5.1 and 5.2 ,for g w + d i w d +1. (2)Splittheeldatapoint x j where S 1( P j )+ S 1( S j )isminimizedfor g w + d j w d +1. 5.3.2SplittingaProleintoThree Supposethataprole I issplitintothreeproleswithfeathering.Let j and k j>>><>>>>: I j ( x i )1 i j d d I j ( x i ) ( j + d i ) = 2 d e j d>>>>>>>>><>>>>>>>>>>: 01 i j d I j ( x i ) P j ( x i ) j d>>><>>>>: 01 i k d I j ( x i ) M ( j;k ) ( x i ) k d
PAGE 116

106 that g 2 w +3 d 1 j w d +1andthat g w + d k 2 w 3 d +2.Also, k j +1+2( d 1) w ) k j w 2 d +1.Usingtheserangesfor j and k ,wearrive atAlgorithm S 3 F ,whichcanbeimplementedtosolveproblem S 3 F in O ( dg 2 )time. Algorithm S 3 F (1)Find P j M ( j;k ) and S k usingEquations 5.3 5.4 and 5.5 ,for g 2 w +3 d 1 j w d +1, g w + d k 2 w 3 d +2and k j w 2 d +1. (2)Splittheeldattwopoints x j x k ,where S 1( P j )+ S 1( M ( j;k ) )+ S 1( S j )isminimized, subjectto g 2 w +3 d 1 j w d +1, g w + d k 2 w 3 d +2and k j w 2 d +1. 5.3.3Tongue-and-grooveEectandInterdigitation Thealgorithmsfor M 2 F and M 3 F maybefurtherextendedtogenerateoptimal therapytimeeldswitheliminationoftongue-and-grooveu nderdosageand(optionally)the interdigitationconstraintontheleafsequencesasisdone foreldsplitswithoutfeatheringin Section 5.2.2 .Thedenitionsofproblems M 2 F 0 ( M 3 F 0 )and M 2 F 00 ( M 3 F 00 ),respectively, forsplitsintotwo(three)eldsaresimilartothosemadein Section 5.2.2 forsplitswithout feathering. 5.4Results TheperformanceoftheAlgorithms M 2, M 3, M 2 F and M 3 F wastestedusing27 clinicalruencematrices,eachofwhichexceededthemaximu mallowableeldwidth w =14, with d =2forfeathering.Theruencematricesweregeneratedwitha commercialinverse treatmentplanningsystem(CORVUSv5.0,NOMOSCorp.,Sewic kley,PA)forveclinical cases.Algorithm M 2 F wasusedwhentheprolewidthwas 2 w 2 d +1=25and algorithm M 2wasusedwhenevertheprolewidthwas 2 w =28.Algorithms M 3 and M 3 F wereusedinallcases.TheoptimalMUsforthespliteldswer ecalculated assumingthatthespliteldsineachcasearedeliveredbyse quencingleavesusingAlgorithm MULTIPAIR.Table 5{1 displaystheresultingtotalMUsfortheeldsplitsobtaine dusing thefouralgorithms.AlsoshownarethetotalMUsobtainedus ingtheeldsplitlinesas givenbythecommercialtreatmentplanningsystem( C ( I )).TheMUsarenormalizedto giveamaximumpixelvalueof100ofaruencemap.Thepercentd ecreaseinMUsof min f M 2( I ) ;M 3( I ) g asaresultofoptimaleldsplittingover C ( I )isalsoshowninthelast

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107 column.MUreductionsofupto26%areseen.Inabout30%ofthe casesthereduction wasover20%.TheaveragedecreaseinMUsisfoundtobeabout1 1%forthe27ruence matrices.Notethattheapplicationofoptimalsplittingal gorithmswithfeatheringcan reduceMUascomparedtotheoptimalalgorithmswithoutfeat heringasaresultofthe reductioninintensityvaluesinthefeatheringregioninea cheldresultingfromthesplit. Weobservethat M 2 F ( I )
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108 Table5{1:TotalMUsforveclinicalcases Matrix( I ) Width C ( I ) M 2( I ) M 3( I ) M 2 F ( I ) M 3 F ( I ) %MUdecrease 1 15 280 280 330 290 335 0 2 15 310 310 350 318 350 0 3 16 300 260 340 260 310 13.3 4 16 400 300 370 290 353 25 5 16 350 350 380 360 394 0 6 16 340 310 310 325 382 8.8 7 16 390 310 360 310 338 20.5 8 16 350 320 340 320 357 8.6 9 16 400 300 370 310 365 25 10 16 440 350 390 322 398 20.5 11 16 320 320 360 310 362 0 12 16 400 300 340 280 360 25 13 17 380 280 320 285 323 26.3 14 18 280 240 260 240 280 14.3 15 20 320 320 380 320 375 0 16 22 400 360 400 360 400 10 17 22 320 320 360 300 360 0 18 24 540 480 520 480 500 11.1 19 24 540 500 500 490 500 7.4 20 24 460 420 460 420 425 8.7 21 24 520 520 540 525 545 0 22 24 560 520 520 505 505 7.1 23 25 360 360 380 340 380 0 24 26 560 440 460 430 21.4 25 29 520 480 445 7.7 26 29 580 440 440 24.1 27 32 560 480 470 14.3

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109 5.5Conclusion Wehavedevelopedalgorithmstosplitlargeintensity-modu latedeldsintotwoor threesub-elds.Suchawork-aroundneedstobeimplemented forMLCsthathavea maximumleafspreadlimitation,whichimposesaeldwidthl imitation.Wehavepresented algorithmsthatsplitlargeeldsintonon-overlappingsub -eldsalongoneortwocolumns. Alsopresentedarealgorithmsthatspliteldswithfeather ing.Featheringofsplitelds helpsreducetheeectoftheeldmatchingproblemthatoccu rsintheeldjunctionregion duetouncertaintiesinsetupandorganmotion.Wehaveshown thatouralgorithmsresult ineldsplitsforwhichtheMUeciencyisoptimal.Applicat ionofouroptimaleld splittingalgorithmswithoutfeatheringtoclinicaldatar educedtotalMUsbyupto26% andonsyntheticdataupto63%comparedtoacommercialplann ingsystemthatalsosplits eldswithoutfeathering.Wehavealsoshownthatouralgori thmscaneasilybeextended tospliteldsresultinginmaximalMUeciencywhentheMLCm odelissubjecttothe interdigitationconstraintand/orthetongue-and-groove eectistobeeliminated.

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CHAPTER6 CONCLUSION Wehavepresentedasystematicstudyofleafsequencingalgo rithmsformultileafcollimation.Algorithmsarepresentedforsequencingleavesw ithoutanyconstraints,with theintra-pairmaximumseparationconstraintandwiththei nter-pairminimumsepration constraintforSMLCandwithoutanyconstraints,withthein tra-pairmaximumseparation constraintandwiththeinterdigitationconstraintforDML C.Alsopresentedarealgorithms thateliminatethetongue-and-grooveunderdosage(andopt ionallytheinterdigitationconstraint)forSMLCandacomparisonofthesealgorithmswitha recentlypublishedalgorithm thatalsoeliminatesthetongue-and-grooveeect.Finally ,algorithmsaredeveloped,that splitalargeintensitymodulatedeldintotwoorthreesub elds.Wehaveshownthat allthesealgorithmsobtainfeasiblesolutionswheneverth eyexist.Further,thesolutions generatedarealwaysoptimalintherapytimeforunidirecti onalschedules.Thealgorithms developedareapplicabletosomeofthepopularcommerciall yavailabledeliverysystems. 110

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REFERENCES BortfeldT,BoyerA,SchlegelW,KahlerD,WaldronT1994aRea lizationandvericationofthree-dimensionalconformalradiotherapywith modulatedelds IntJRadiationOncologyBiolPhys 30 899-908 BortfeldTR,KahlerDL,WaldronTJandBoyerAL1994bX-raye ldcompensationwithmultileafcollimators IntJRadiationOncologyBiolPhys 28 723-30 BoyerALandStraitJP1997Deliveryofintensitymodulatedt reatmentswithdynamicmultileafcollimators ProcXIIIntConfonUseofComputersinRadiation Therapy (Madison,WI:MedicalPhysicsPublishing)p13-15 ChenD,HuX,LuanSandWangC2004Geometricalgorithmsforst aticleafsequencingproblemsinradiationtherapy IntJournalofComputationalGeometryand Applications 14 311-39 ChuiC-S,LoSassoTandSpirouS1994Dosecalculationforpho tonbeamswithintensitymodulationgeneratedbydynamicjawormultileafcolli mations MedicalPhysics 21 1237-44 ConveryDJandRosenbloomME1992Thegenerationofintensit y-modulatedelds forconformalradiotherapybydynamiccollimation PMB 37 1359-74 ConveryDJandWebbS1998Generationofdiscretebeam-inten sitymodulationby dynamicmultileafcollimationunderminimumleafseparati onconstraints PMB 43 2521-38DasIJ,DesobryGE,McNeelySW,ChengECandSchultheissTE19 98Beam characteristicsofaretrotteddouble-focussedmultilea fcollimator MedicalPhysics 25 1676-84 DengJ,PawlickiT,ChenY,LiJ,JiangSandMaC-M2001TheMLCt ongue-andgrooveeectonIMRTdosedistributions PMB 46 1039-1060 DirkxMLP,HeijmenBJMandvanSantvoortJPC1998Leaftrajec torycalculation fordynamicmultileafcollimationtorealizeoptimizedrue nceproles PMB 43 1171-84 DoganN,LeybovichLB,SethiAandEmamiB2003Automaticfeat heringofsplit eldsforstep-and-shootintensitymodulatedradiationth erapy PMB 48 1133-1140 111

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112 EngelE2003Anewalgorithmforoptimalmultileafcollimato reldsegmentation Preprint,FachbereichMathematik,UniRostockFollowillD,GeisPandBoyerA1997Estimatesofwhole-bodyd oseequivalentproducedbybeamintensitymodulatedconformaltherapy IntJRadOncolBiolPhys 38 667-672KalinowskiT2003Analgorithmforoptimalmultileafcollim atoreldsegmentation withinterleafcollisionconstraint Preprint,FachbereichMathematik,UniRostock GalvinJ,SmithA,andLallyB1993aCharacterizationofamul tileafcollimatorsystem IntJRadiatOncolBiolPhys 25 181-192 GalvinJM,ChenX-GandSmithRM1993bCombiningmultileafe ldstomodulate ruencedistributions IntJRadiatOncolBiolPhys 27 697-705 HongL,KaledA,ChuiC,LoSassoT,HuntM,SpirouS,YangJ,Amo lsH,Ling C,FuksZandLeibelS2002IMRToflargeelds:whole-abdomen irradiation IntJ RadiatOncolBiolPhys 54 278-89 IntensityModulatedRadiationTherapyCollaborativeWork ingGroup2001Intensitymodulatedradiotherapy:Currentstatusandissuesofinter est IntJRadiationOncologyBiolPhys 51 880-914 JordanTJandWilliamsPC1994Thedesignandperformancecha racteristicsofa multileafcollimator PMB 39 231-51 KallmanP,LindB,EklofAandBrahmeA1988Shapingofarbitra rydosedistributionsbydynamicmultileafcollimation, PMB 33 1291-300 KamathS,SahniS,LiJ,PaltaJandRankaS2003Leafsequencin galgorithmsfor segmentedmultileafcollimation PMB 48 307-324 KamathS,SahniS,LiJ,PaltaJandRankaS2004OptimalLeafSe quencingwith EliminationofTongue-and-GrooveUnderdosage PMB 49 N7-N19 LangerM,ThaiVandPapiezL2001Improvedleafsequencingre ducessegmentsor monitorunitsneededtodeliverIMRTusingmultileafcollim ators MedicalPhysics 28 2450-2458LiY,YaoJ,YaoD2003Geneticalgorithmbaseddeliverablese gmentsoptimization forstaticintensity-modulatedradiotherapy PMB 48 3353-74 LuanS,ChenD,ZhangL,WuXandYuCX2003Anoptimalalgorithm forconguringdeliveryoptionsofaonedimensionalintensity-modula tedbeam PMB 48 2321-38

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113 MaL,BoyerA,XingLandMaC-M1998Anoptimizedleaf-setting algorithmfor beamintensitymodulationusingdynamicmultileafcollima tors PMB 43 1629-43 MaL,BoyerAL,MaCM,andXingL1999Synchronizingdynamicmu ltileafcollimatorsforproducingtwo-dimensionalintensity-modulate deldswithminimumbeam deliverytime, IntJRadiationOncologyBiolPhys 44 1147-54 MohanR1995Fieldshapingforthree-dimensionalconformal radiationtherapyand multileafcollimation SeminarsinRadiatOncol 5 86-99 MuticS,LowDA,KleinE,DempseyJF,PurdyJA2001Roomshield ingfor intensity-modulatedradiationtherapytreatmentfacilit ies IntJRadiationOncology BiolPhys 50 239-46 QueW1999Comparisonofalgorithmsformultileafcollimato reldsegmentation MedicalPhysics 26 2390-96 QueW,KungJandDaiJ2004\Tongue-and-groove"eectininte nsitymodulated radiotherapywithstaticmultileafcollimatorelds PMB 49 399-405 SiochiS1999Minimizingstaticintensitymodulationdeliv erytimeusinganintensity solidparadigm IntJRadiationOncologyBiolPhys 43 671-80 SpirouSVandChuiCS1994Generationofarbitraryintensity prolesbydynamic jawsormultileafcollimators MedicalPhysics 21 1031-41 SteinJ,BortfeldT,DorschelBandSchegelW1994Dynamicx-r aycompensationfor conformalradiotherapybymeansofmultileafcollimation RadiotherOncol 32 163-7 SykesRJandWilliamsPC1998Anexperimentalinvestigation ofthetongueand grooveeectforthePhilipsmultileafcollimator PMB 43 3157-65 vanSantvoortJPCandHeijmenBJM1996Dynamicmultileafcol limationwithout \tongue-and-groove"underdosageeects PMB 41 2091-105 WangX,SpirouS,LoSassoT,SteinJ,ChuiCandMohanR1996Dos imetricvericationofintensitymodulatedelds MedicalPhysics 23 317-28 WebbS,BortfeldT,SteinJandConveryD1997Theeectofstai r-stepleaftransmissiononthe\tongue-and-grooveproblem"indynamicradioth erapywithamultileaf collimator PMB 42 595-602 WuQ,ArneldM,TongS,WuYandMohanR2000Dynamicsplitting oflarge intensity-modulatedelds PMB 45 1731-40 XiaPandVerheyLJ1998Multileafcollimatorleafsequencin galgorithmforintensity modulatedbeamswithmultiplestaticsegments MedicalPhysics 25 1424-34

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114 XiaPandVerheyLJ2001Deliverysystemsofintensity-modul atedradiotherapy usingconventionalmultileafcollimators MedicalDosimetry 26 169-77

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BIOGRAPHICALSKETCH SrijitKamathcompletedhishighschooleducationfromKend riyaVidyalaya,IIT Madras,India,in1996.In2000hereceivedaB.Tech.degreei ninformationtechnologyfromtheUniversityofMadras,India.Hethenmovedtothe UniversityofFlorida(UF) forgraduatestudies.SinceAugust2000,hehasbeenaResear chAssistantintheComputerandInformationScienceandEngineeringDepartment. HisworkatUFhasledto vejournalpapersandtwopatentapplications.HeearnedaM .S.in2002andwillreceive aPh.D.in2005,bothfromUF. 115


Permanent Link: http://ufdc.ufl.edu/UFE0011548/00001

Material Information

Title: Algorithms for sequencing multileaf collimators
Physical Description: Mixed Material
Language: English
Creator: Kamath, Srijit ( Dissertant )
Sahni, Sartaj ( Thesis advisor )
Ranka, Sanjay ( Thesis advisor )
Palta, Jatinder ( Reviewer )
Rangarajan, Anand ( Reviewer )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2005
Copyright Date: 2005

Subjects

Subjects / Keywords: Collimators (Optical instrument)
Computer and Information Science and Engineering thesis, Ph. D.
Dissertations, Academic -- UF -- Computer and Information Science and Engineering
Radiography, Medical -- Digital techniques

Notes

Abstract: In delivering radiation therapy for cancer treatment, it is desirable to deliver high doses of radiation to the target tumor, while permitting a low dosage on the surrounding healthy tissues. In recent years, the development of intensity modulated radiation therapy (IMRT) has made this possible. IMRT may be delivered by several techniques. The delivery of IMRT with a multileaf collimator (MLC) requires the delivery of radiation from several beam orientations. The intensity profile for each beam direction is described as a MLC leaf sequence, which is developed using a leaf sequencing algorithm. Important considerations in developing a leaf sequence for a desired intensity profile include maximizing the monitor unit (MU) efficiency (equivalently minimizing the beam-on time) and minimizing the total treatment time subject to the leaf movement constraints of the MLC model. In this work, we present a systematic study of the optimization of leaf sequencing algorithms and provide rigorous mathematical proofs of optimized leaf sequence settings in terms of MU efficiency under most common leaf movement constraints that include minimum and maximum leaf separation, leaf interdigitation and tongue-and-groove. We also develop algorithms to split large intensity modulated fields into two or three subfields.
Subject: field, IMRT, intensity, leaf, MLC, multileaf
General Note: Title from title page of source document.
General Note: Document formatted into pages; contains 125 pages.
General Note: Includes vita.
Thesis: Thesis (Ph. D.)--University of Florida, 2005.
Bibliography: Includes bibliographical references.
General Note: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0011548:00001

Permanent Link: http://ufdc.ufl.edu/UFE0011548/00001

Material Information

Title: Algorithms for sequencing multileaf collimators
Physical Description: Mixed Material
Language: English
Creator: Kamath, Srijit ( Dissertant )
Sahni, Sartaj ( Thesis advisor )
Ranka, Sanjay ( Thesis advisor )
Palta, Jatinder ( Reviewer )
Rangarajan, Anand ( Reviewer )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2005
Copyright Date: 2005

Subjects

Subjects / Keywords: Collimators (Optical instrument)
Computer and Information Science and Engineering thesis, Ph. D.
Dissertations, Academic -- UF -- Computer and Information Science and Engineering
Radiography, Medical -- Digital techniques

Notes

Abstract: In delivering radiation therapy for cancer treatment, it is desirable to deliver high doses of radiation to the target tumor, while permitting a low dosage on the surrounding healthy tissues. In recent years, the development of intensity modulated radiation therapy (IMRT) has made this possible. IMRT may be delivered by several techniques. The delivery of IMRT with a multileaf collimator (MLC) requires the delivery of radiation from several beam orientations. The intensity profile for each beam direction is described as a MLC leaf sequence, which is developed using a leaf sequencing algorithm. Important considerations in developing a leaf sequence for a desired intensity profile include maximizing the monitor unit (MU) efficiency (equivalently minimizing the beam-on time) and minimizing the total treatment time subject to the leaf movement constraints of the MLC model. In this work, we present a systematic study of the optimization of leaf sequencing algorithms and provide rigorous mathematical proofs of optimized leaf sequence settings in terms of MU efficiency under most common leaf movement constraints that include minimum and maximum leaf separation, leaf interdigitation and tongue-and-groove. We also develop algorithms to split large intensity modulated fields into two or three subfields.
Subject: field, IMRT, intensity, leaf, MLC, multileaf
General Note: Title from title page of source document.
General Note: Document formatted into pages; contains 125 pages.
General Note: Includes vita.
Thesis: Thesis (Ph. D.)--University of Florida, 2005.
Bibliography: Includes bibliographical references.
General Note: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0011548:00001


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AL;GO!. i iT li' FOR -UEN'CING MUUT:IILEAF ( OLL;IMATrORS


ByS

SREIJIT KAMAIITH


















AE~i 1. il iliON JPF F. UNTED TO THE :RADU!~.i i SCHOOL
OF T rii;T UNIVERSITY OF` FLOCRIDA INr PARTI-'`AL FU7LFil i
OF T iilT Ri'- UlIREMENTSI` FOR T ii;T D>EGREEI OF`
DO('C)TOR OF PHILOSOPHY


i~iii ii OFt i i:.01 i )A





































C


















Dedicated to : -. parents, who have\- awaiyvs encoura~gedl me to become a scientist.



















First and .. :, my utmost gratitude goes to my advisor, Sartaj Sahni, for the

immeasurably valuable guidance a~nd support, he has given me during the :1 >d of my

graduate study. Hle initroduecd mei to this challengingg computationa l i 1 ** in :

He ha~s patientlly absorbed all my ideas, proofread every mat(hem2t~ical1 analysis

that I have perforrmed, and gently nudged me in the most, i :. ..1
occasion. i i ~judgemnent and experience have proved to be crucial in the success of this

work. I could not have asked i : a. better a~dvisor. I feel t~hat he is one of the best computer

science educators in Ithe world. It has been a privilege and a~n honor to work wczith him.

Iamrr very th~ankfuli to Meec~cra. :: -- -- for the several insights she h~as provided on

algorithmic: problems through her teachingf in aind out of class and fo~r her general enthusiasm

El. ?! solving t~ha~t ha~s ledi to many absorbing discussions. Serving as t~he coordiinat~or

Sthe algorithms and '1: seminar, wh-ich was started with her encouragement and

initiative, ha~s also been an excellent learning : .. fr e

go to Joona~than Li for collaborating on the problems and for providing clinical

data. TIli .: 1 go also to Jlatindier P:alta, Anand IRanga: .1 and 'r y I Ranka for taking

the time to serve on myS committees.

I 1 7::: A?.::::. Jain, Amnita I i Mohit Dhhawtan, Haibin Lu, Jimin Yin aind

We~tncheng Lu made I '= : a gfoodi place to work. Intense preparation fo~r tIhe doctoral

q : i :: examm wa~s :i i thanks to latle night study sessions with VBl Z Mania~n a~nd

Pomnpi Diplan. I would like to thank all my friends ( '..:" too many to namee) their

great I :::i : and support during the course of ni-- graduate study, with? special thanks to

fellow students Subi, M~Ianas, .1==:i, Andrew, P:ranav andii I

Ti work would not have been possible without, the constant, encouragement. I have

received i ::: my parents aind i .. uncle N~a .:. who ha~s also proviided numerous insights

on gra~duatei study and wvor~k in the U~nited States.



















TABLE OF CONTENTS
page

ACK(NOWLEDGMENTS .......... . .. .. iv

LIST OF TABLES ......... . .. .. vii

LIST OF FIGURES ......... . .. viii

ABSTRACT ............. .......... .. x

CHAPTER

1 NTRODUCTION..........1

1.1 Problem Description ......... .. 1
1.2 MLC Models and Constraints . ..... .. 4
1.3 Prior Work ............. ..... .... 5
1.4 Dissertation Outline ......... .. 7

2 SEQUENCING OF SEGMENTED MULTILEAF COLLIMATORS .. .. 8

2.1 Methods ............. ........... 8
2.1.1 Discrete Profile ......... .. 8
2.1.2 Movement of Leaves . ... .. .. 8
2.1.3 Optimal Unidirectional Algorithm for one Pair of Leaves .. .. 11
2.1.4 Bi-directional Movement ... .. .. .. . 16
2.1.5 Algorithm Under Maximum Separation Constraint Condition 20
2.1.6 Algorithm Under Inter-Pair Minimum Separation Constraint .. 23
2.2 Conclusion ......... . .. 34

3 SEQUENCING OF DYNAMIC MULTILEAF COLLIMATORS .. .. .. 36

3.1 Methods ............. ........... 36
3.1.1 Movement of Leaves . ..... .. 36
3.1.2 Maximum Velocity Constraint ... ... .. .. 38
3.1.3 Optimal Unidirectional Algorithm for one Pair of Leaves .. .. 38
3.1.4 Minimum Separation Constraint .. .. 42
3.1.5 Bi-directional Movement ... .. .. .. . 46
3.1.6 Algorithm Under Maximum Separation Constraint Condition 51
3.1.7 Algorithm Under Interdigfitation Constraint .. .. .. .. .. 55
3.2 Conclusion ......... . .. 65

4 ELIMINATION OF TONGUE-AND-GROOVE UNDERDOSAGE .. .. .. 66

4.1 Algorithm with Interdigfitation and Tongue-and-Groove Constraints .. 67
4.1.1 Tongue-and-Groove Underdosage Effect ... .. .. 67
4.1.2 Algorithms ......... .. .. 68












4.1.3 Efficient Implementation of the Algorithms .. .. .. .. 79
4.2 Experimental Validation . ... ... .. 81
4.3 Comparison with Algorithm of Que et al. (2004)) .. .. .. .. 82
4.3.1 A!! 11, i of the Algorithm of Que et al. (2004) .. .. .. .. .. 83
4.3.2 Results ......... . .. 87
4.4 Conclusion ......... . .. 87

5 ALGORITHMS FOR SPLITTING LARGE FIELDS ... .. .. .. 90

5.1 Introduction ......... . .. .. 90
5.2 Field Splitting Without Featheringf .... .. . 90
5.2.1 Optimal Field Splitting for One Leaf Pair .. .. .. .. 90
5.2.2 Optimal Field Splitting for Multiple Leaf Pairs .. .. .. .. 97
5.3 Field Splitting with Feathering ..... ... .. 100
5.3.1 Splitting a Profile into Two .... ... .. 104
5.3.2 Splitting a Profile into Three .... ... . .. 105
5.3.3 Tongue-and-groove Effect and Interdigfitation .. .. .. .. .. 106
5.4 Results. ............ ............ 106
5.5 Conclusion . ...... . .. 109

6 CONCLUSION ............ ............ 110

REFERENCES ............. .............. 111

BIOGRAPHICAL SK(ETCH ......... .. .. 115


















LIST OF TABLES
Table page

4-1 Comparison of the number of segments ..... .... . 83

4-2 Number of MUs and segments ....... ... .. 88

4-3 Average number of MUs and segments ..... .... . 88

5-1 Total MUs for five clinical cases . .... .. 108


















LIST OF FIGURES
Figure page

1-1 A linear accelerator ......... . 2

12 A multileaf collimator ......... ... 3

13 Inter-pair minimum separation constraint .... .. 4

1-4 Cross section of leaves ......... ... 5

2-1 Geometry and coordinate system . ..... .. 9

2-2 Discretization of profile ......... ... 9

2-3 Leaf t! li.* br~!y during <\llC delivery ..... .... . 10

2-4 Obtaining a unidirectional plan . ..... .. 12

2-5 A profile and its plan ......... ... .. 13

2-6 Minimum separation constraint violation .... .. . 15

2-7 Bi-directional movement ......... .. .. 17

2-8 Bi-directional movement under minimum separation constraint .. .. .. 19

2-9 Bi-directional movement under maximum separation constraint .. .. .. 20

2-10 Obtaining a plan under maximum separation constraint .. .. .. .. .. 21

2-11 Maximum separation constraint violation .... .. .. 21

2-12 Obtaining a schedule ......... .. .. 24

2-13 Obtaining a schedule under the constraint .... .. .. 25

2-14 Eliminating a violation ......... .. .. 26

2-15 Eliminating a violation ......... .. .. 30

2-16 Intensity profiles of adjacent leaf pairs ..... .... . 31

2-17 Profiles violating inter-pair constraint ..... .. . 32

3-1 Leaf t! .1.* b r~!y during DMLC delivery ..... .... . 37

3-2 Obtaining a unidirectional plan . ..... .. 40

3-3 Obtaining a unidirectional plan with minimum separation constraint .. 44

3-4 Minimum separation constraint violation .... .. . 44











3-5 <\!LlC plan: feasible; DMLC plan: infeasible ... .. .. 47

3-6 Bi-directional movement ......... .. .. 48

3-7 Bi-directional movement under minimum separation constraint .. .. .. 50

3-8 Bi-directional movement under maximum separation constraint .. .. .. 52

3-9 Obtainingf a plan under maximum separation constraint .. .. .. .. .. 52

3-10 Maximum separation constraint violation .... .... .. 53

3-11 Obtainingf a schedule ......... .. .. 56

3-12 Obtainingf a schedule under the constraint .... .... .. 57

3-13 Eliminating a Typel violation ....... ... .. 58

3-14 Eliminating a Type2 violation ....... ... .. 59

4-1 Tongue-and-groove effect ......... .. .. 67

4-2 Counterexample ......... .. .. 70

4-3 Obtainingf a schedule under the tongue-and-groove constraint .. .. .. 71

4-4 Tongue-and-groove constraint violation: casel ... .. .. .. 72

4-5 Tongue-and-groove constraint violation: case2 ... .. .. .. 73

4-6 Obtainingf a schedule under both the constraints ... .. .. .. 77

4-7 Film measurement of the AP field . .... .. 82

4-8 Leaf positions ......... . .. .. 85

4-9 Worstcase example ......... .. .. 86

5-1 Splitting a profile (a) into two ....... ... .. 93

5-2 Tight upper bound for Lemma 32a . .. .. 95

5-3 Tight upper bound for Lemma 32b ...... .... . 95

5-4 Tight lower bound for Lemma 32c ...... .... . 96

5-5 Tight upper bound for Lemma 32c ...... .... . 97

5-6 Field matching problem ......... .. .. 102

5-7 Example of field splitting with featheringf ... . .. 103

5-8 Comparison of the field split line ...... .... .. 107

















Abstract of D~issertation P-resented to the Graduate"~ School--
of the Uj-,:1 11 of FI : in P~artial i ::11111::: : of the
P. i::' :.. for the Degree of Doctor o P':il

AL;GO iiili lilOR i -11 i = :IG i iii 1 '<1'COiii il'`C-1

By

SaullI KiAMbATH

AUrCusTi 2005

S. Sa: i .1 Sahni
' I.'..: Dllepartmencrt: Clomputer anid Ilnformnation Science and Elngineer~ing

In. I i .. radiiation t !E. ;-v.i-- fo~r cancer tIreatment., it, is diesirable t~o deliver high doses

of raldiat ion to the (tlarget, (umor, while permnitting a lown dosage on the surrl :::. : elh

tissues. In recent years, thel .-1 ... of intensity mnodulated radiation therapy (lil )

has made this 1-ll : be delivered by several t~echniques. 'i : delivery of

I' i l. ii with a mulileaf ... i : ( i C) :.. I the delivery of radliation from several

beamr orientation. ': -- intensity. ,- ..=1 for each beamn direction is desc~rib~ed as a. ~IvCI
sequenc31Ce, wh'ich' is: deve~lo~rped u1sing a leaf sequencingf algorithm. I:::i Itant considerations

in developing a leaf sequence fo~r a desired int~ensitly 1includte maximizing t~he monitor

unit (MU_) efficiency (equivalentlyl mninimnizing the beam--on time) and mninimnizing the total

trea~tmecnt time sub' to the leaf mnovemrent constraints of the Ii .C1 model. In this work,

we present a systematic study of the optimization of leaf sequencing algorithms and provide

rigorous mathematical proofs of optimizedl leaf sequence settings in terms of MUi rr: .

under most commron leaf mnovemrent constra~ints th-at include mninimnum and mnaximnum leaf
separationt leaif i. : :~ ~ :: :: lndl rlongue-alnd-g-~rr lroove. We also deve(-lopr aITlgoithms to sp~lit

large int~ensityi modulated fields intlo tw-io or three ;
















CHAPTER
INTRODUCTION

1.1 Problem Description

The objective of radiation therapy for cancer treatment is to deliver high doses of

radiation to the target tumor, while permitting a low dosage on the surrounding healthy

tissues. For example, for head and neck tumors, it is necessary for radiation to be delivered

so that the exposure of the spinal cord, optic nerve, salivary glands or other important

structures is niinintized. In recent years, this has been made possible due to the develop-

nient of confornial radiation therapy. In confornial therapy, treatment is delivered using a

set of radiation beams which are positioned such that the shape of the dose distribution

I .!!l .! .!." with the shape of the tumor. This is typically achieved by positioning beams

of varying shapes front different directions so that each beam approximately irradiates the

section of the tunior visible from its direction and avoids the organs at risk in the vicinity

of the tumor.

Intensity modulated radiation therapy (IMRT) is the state-of-the-art in confornial radi-

ation therapy. IMRT permits the iint, neityi of radiation beant to be varied across a treatment

area, thereby improving the dose conformity. Radiation is delivered using a medical linear

accelerator (Figure 1-1). A rotating gantry containing the accelerator structure can rotate

around the patient who is positioned on an adjustable treatment couch. Delivery of IMRT is

possible by several techniques. In conipensator-based IMRT, the beant is modulated with a

preshaped piece of material called the conipensator (modulator). The degree of modulation

of the beam varies depending on the thickness of the material through which the beant is

attenuated. The computer determines the shape of each modulator in order to deliver the

desired beam. This type of modulation requires the modulator to be fabricated and then

manually inserted into the tray mount of a linear accelerator. In toniotherapy-based IMRT,

the linear accelerator travels in multiple circles all the way around the gantry ring to deliver

the radiation treatment. The beant is colliniated to a narrow slit and the iintl, nityi of the










beant is modulated during the gantry movement around the patient. Care must be taken to

ensure that adjacent circular arcs do not overlap and thereby do not overdose tissues. This

type of delivery is referred to as serial toniotherapy. A modification of serial toniotherapy is

helical toniotherapy. In helical toniotherapy, the treatment couch moves linearly (continu-

ously) through the rotating accelerator gantry. So each time the accelerator contes around,

it directs the beam on a slightly different plane on the patient. In AILC-based IMRT the

accelerator structure is equipped with a computer controlled mechanical device called a

niultileaf colliniator (AILC, Figure 1-2) that shapes the radiation beam, so as to deliver

the radiation as prescribed by the treatment plan. The MLC may have up to 120 movable

leaves that can move along an axis perpendicular to the beam and can be arranged so as

to shield or expose parts of the a~natonly during treatment. The leaves are arranged in

pairs so that each leaf pair forms one row of the arrangement. The set of allowable MLC

leaf configurations may be restricted by leaf niovenient constraints that are manufacturer

and/or model dependent.























Figure 1-1: A linear accelerator (the figure is from http://www.lexnied .cont/-
niedicaL~services/IMRT .htni)


The first stage in the treatment planning process in IMRT is to obtain accurate three

dimensional anatomical information about the tunior and its surroundings. This is achieved































Figure 1-2: A multileaf collimator (the figure is from http://www.lexmed .com/-
medicaL~services/IMRT .htm)


using computed tomography (CT) and magnetic resonance (MR) imaging. An ideal dose

distribution would ensure perfect conformity to the target volume while completely sparing

all other tissues. However, such a distribution is impossible to realize in practice. Therefore,

minimum dose targets for tumors and tolerable doses for critical structures are prescribed

and an objective function that measures the quality of a plan is developed subject to these

dose based constraints. Next, a set of beam parameters (beam angles, profiles, weights)

that optimize this objective are determined using a computer program. This method is

called !!ni. !-n planning" since resultant dose distribution is first described and the best

beam parameters that deliver the distribution (approximately) are then solved for. It is

to be noted that inverse planning is a general concept and its implementation details vary

vastly among various systems. Following the inverse planning in MLC-based IMRT, the

delivery of radiation intensity profile for each beam direction is described as a MLC leaf

sequence, which is developed using a leaf sequencing algorithm. Important considerations

in developing a leaf sequence for a desired inltloneityi profile include maximizing the monitor

unit (MU) efficiency (equivalently minimizing the beam-on time) and minimizing the total

treatment time subject to the leaf movement constraints of the MLC model. Finally, when

the leaf sequences for all beam directions are determined, the treatment is performed from
























































L2
L3


R2
R3


various beam angles sequuentially using computer control. In this work, we develop optimized

leaf sequuencing algorithms for various MLC models.

1.2 MLC Models and Constraints

The purpose of the leaf sequuencing algorithm is to generate a sequence of leaf positions

and/or movements that faithfully reproduce the desired intensity map once the beam is
delivered, taking into consideration any hardware and dosimetric characteristics of the de-

livry-1-. I.The two most common methods of IMRT delivery with computer-controlled

MLCs are the segmental multileaf collimator (N\llC) and dynamic multileaf collimator

(DMLC). In WllC, the beam is switched off while the leaves are in motion. In other
words, the delivery is done using multiple static segments or leaf settings. This method is

also frequently referred to as the -I. pI and -I!s .. l or "stop and -I!s .. l method. In DMLC
the beam is on while the leaves are in motion. The beam is switched on at the start of

treatment and is switched off only at the end of treatment. The fundamental difference

between the leaf sequences of these two delivery methods is that the leaf sequence defines a

finite set of beam shapes for M:\!LC and ft 0 i r. 11. 4 of opposing pairs of leaves for DMLC.

In practical situations, there are some constraints on the movement of the leaves. The

minimum separation constraint requires that opposing pairs of leaves be separated by at

least some distance (Smin) at all times during beam delivery. In MLCs this constraint is

applied not only to opposing pairs of leaves, but also to opposing leaves of neighboring

pairs. For example, in Figure 1-3, L1l and R1, L2 and R2, L3 and R3, L1l and R2, L2 and

R1, L2 and R3, L3 and R2 are pairwise subject to the constraint. The case with Smin = 0

is called interdigitation constraint and is applicable to some MLC models. Wherever this

constraint applies, opposite adjacent leaves are not permitted to overlap.


Figure 1-3: Inter-pair minimum separation constraint










In most commercially available MLCs, there is a tongue-and-groove arrangement at

the interface between adjacent leaves. A cross section of two adjacent leaves is depicted in

Figure 1-4. The width of the tongue-and-groove region is 1. The area under this region gets

underdosed due to the mechanical arrangement, as it remains shielded if either the tongue

or the groove portion of a leaf shields it.

Radiation















Leaf
movement

Figure 1-4: Cross section of leaves


Maximum leaf spread for leaves on the same leaf bank is one more MLC limitation,

which necessitates a large field (intensity profile) to be split into two or more adjacent

abutting sub-fields. This is true for the Varian MLC (Varian Medical Systems, Palo Alto,

CA), which has a field size limitation of about 15 cm. The abutting sub-fields are then

delivered as separate treatment fields. This often results in longer delivery times, poor MU

efficiency, and field matching problems.

1.3 Prior Work

Optimization of the leaf sequencing algorithm has been the subject of numerous inves-

tigations (for example, Convery and Rosenbloom 1992, Bortfeld et al. 1994a, Dirkx et al.

1998, Ma et al. 1998, Xia and Verhey 1998, Siochi 1999, Langer et al. 2001, Luan et al.

2003, Chen et al. 2004). Treatment delivery with IMRT is not very efficient in terms of MU

efficiency, which is defined as the ratio of dose delivered at a point in the patient with an

IMRT field to the MU delivered for that field. Typical MU efficiencies of IMRT treatment










plans are 3 to 10 times lowver than those i : open/wedge field-based conventional treatments

plans. i i :..-e, total body dose due to increased leakage radciation reaching Ithe patient

in an iT i- i trea~tmernt is a; r concern (ollowill et al. : *., Intensit~y Modulateid F

ation ii ; py Collaboratiive Working C~roup :- 1). Lowl M/U efficiency of I i i. i' delivery

negatively impacts t~he room shielding design because t~he increased workload (Int~ensitly

Modulated T. : 1 il: pyll Collaborative Wo~rking Gr~oup : :1, Muti c al. -- --11). i~

M1U ( :i .. ;- (1 on both the degree of i : -;'".i modultion and the algorithm used

to convert the: intensity pattern into a. leaf :: for IE 7.T delivery. It is therefore im-

portant to design a leaf sequenceing algorithmrr thiat is; ini :. 1 for MIU iE^ .:. ..~~ to minimirrize

total body dose to the patie~nt. Foir dynamnic beamn delivery where dose rate is i--: :!- not

modulated, an algorithm that optimizes the M4U setting at a given dose rate also optimizes

the treatment time.

D~ynamnic~ :i i:: : :: algorithmns with the leaves in motion during radiation delivery

have been 1 ".. .1 (Convery and Ii..i I .. 1 ':; Spirou and ('.. 1 -),an lte

(van Santvoort and Hleit:- ; 1 *= .: D Iirkx et ali. 1998) to eimirinate the tongue-and-

groovec underdosag e etffects. Similar leaf sequlenci ng algorithmrrs have also beenr developed for

the segrmental mu~lt~ileaf i : ( :: = ) delivery methods (Borlitfeld et alj. 1994a., Bortfeli~d

at al. i;~ -i ,: M/a t al. i~~ ?( ia and '.' -1: 1-. i' 01 I c -- -, Engel : -- alinowsksiii:

L~i et al. ::: ). ... of these studies did not consider .. 1. movemecnt constraints.

Such? leaf sequencingf algorithms are i 1 :- i for certain I of MLC designs. Foir other

of MLC designs, notably the .. ..: : (Siemnens !.. =. 1 Systemrs, Inrc., 1lselin, NJ~)

MLIIC: design (D~as et al. : ) and 7 1 ( 1- O Cncologi y Systems~ Inc., NUorcross, GA)

MPLC design (Jordan aind Wliaiiitms 1' i ), other mechanical constraints need to be taken

into consideration when designing the leaf settings .. both dynamic and iC( delivery.

ii : minimnum- leaf .1 i 0 constraint, for example, was ::li incorporated into the

design of: ... (Cc;;- --- and We~bb 1 ). A general description and echara2cteristic s

of somer MLIC7 models can b~e found in Xia. a~nd Ve~r~rhey ( ::).










1.4 D~issertation Outline?

In this work, wie a syslemnatic stludyi the optimnizat~ion of sequ~encing

algorithms. Til dissertation is organized as follows. In ch-aptor 2, weC present loaf :

ing algorithms the: .' i i 0: beam delivery and provide: rigorous / of optimized leaf

sequence settings in terms of MU<~ ?- 'i :: under various : i movement constraints. Prac-

tical leaf mnovement constraints that are considered include thle mninimnu m leaf i..: Jion

constraint a~nd minimum int~er- !- separation constraint (leaf i-:- `.~:- -- constraint.).

qi uestion of whether bi-*1 i : .i leaf movemnen t : increase the M'U ii: :: i- when

ed with i.. .' .. : .. leaf movement 4 .1 is aliso addressed. In cha~pteir 3, we de-

velop : i sequencing algforithmns i : DM~LC beam d i---. A~lgorit~hms are presented to

sequence: leaves wczith maximum : separation constraint and Ithe leaf interdigitation c~on-

straint. In chlapte~r 4, we~ sttudy tongiue-and-gir oov:e ::- for SMC7. We provide bounds on

th~e maximnum extent to which tongfue-alnd-groove effetct can be li: .::. .1 and give neces-

sary; and sufficient? c~onditlions for a un~idirectional leaf sequetnce to attlain ti~he bound, Wei~

then present algorithms that generate i .i1 sequences that climrinate the tongue-and-gr~oove

effect and optionally satisfy the interdigitation constraint. --: also .. ... e our a~lgorithmlrs

to a. rec-entliy --:i:- i -1 leaf sequencing algorithmr th~at also ::::: t~ongue-and-groovee

ulnderdosage. Thl'ie : 11. of .1 j arge T .!.:. ti r mnodduatted fields into two or three

sub~fields is discussed in cha~pte~r 5. All our algorithms generation unidircectionali leaf mnovemennt

schedules and are i 1 to bec < ': 1:.1 in M/Us for :: .: l i i: :. .1 schedules.
















CHAPTER 2
SEQUENCING OF SEGMENTED MULTILEAF COLLIMATORS

In this chapter, we present a systematic study of the optimization of leaf sequenc-

ing algorithms for the M:\! LC beam delivery and provide rigorous proofs of optimized leaf

sequence settings in terms of MU efficiency under various leaf movement constraints. Prac-

tical leaf movement constraints that are considered include the minimum leaf separation

constraint and minimum inter-leaf separation constraint (leaf interdigitation constraint).

The question of whether bi-directional leaf movement will increase the MU efficiency when

compared with uni-directional leaf movement only is also addressed. We first introduce the

notation that will be used in the remainder of this work.

2.1 Methods

2.1.1 Discrete Profile

The geometry and coordinate --, -r. ill used in this study are shown in Figure 2-1.

We consider delivery of profiles that are piecewise continuous. Let I(z) be the desired

intensity profile. We first discretize the profile so that we obtain the values at sample

points zo,wi,z2, ...,zm. I(z) is assigned the value I(zi) for zi < z < zi 1, for each i.

Now, I(zi) is our desired inltlon-ityi profile. Figure 2-2 shows a piecewise continuous function

and the corresponding discretized profile. The discretized profile can be efficiently delivered

with the M:\!LC method. However, a WllC sequence can be transformed to a dynamic leaf

sequence by allowing both leaves to start at the same point and close together at the same

point, so that they sweep across the same spatial interval. We develop our theory for the

W11LlC delivery.

2.1.2 Movement of Leaves

In our I!! 11-, -R~ we assume that the leaves are initially at the left most position to

and that the leaves move unidirectionally from left to right. Figure 2-3 illustrates the leaf

ft i l, b r~!y during <:\!LC delivery. Let Il(zi) and Ir(zi) respectively denote the amount of

monitor units (MUs) delivered when the left and right leaves leave position zi. Consider

































I


.


Radiation
Beams


Radiation Source


Right Leaf


Left Leaf


'ff


x,


Figure 2-1: Geometry and coordinate --1 -r nl


xo x;


x, xo x,


Figure 2-2: Discretization of profile










the motion of the left leaf. The left leaf begins at to and remains here until Iz(zo) MUs

have been delivered. At this time the left leaf is moved to zl, where it remains until I;(zl)

MUs have been delivered. The left leaf then moves tO 23 where it remains until Ig(23) MUS

have been delivered. At this time, the left leaf is moved tO 26, where it remains until Ig(26)

MUs have been delivered. The final movement of the left leaf is to my, where it remains

until Iz(27) = Imaz MUs have been delivered. At this time the machine is turned off. The

total therapy time, TT(It, Ir), is the time needed to deliver Imax MUs. The right leaf starts

at z2; mOVeS tO 24 when Ir (z2) MUs have been delivered; moves to as when Ir (24) MUS

have been delivered and so on. Note that the machine is off when a leaf is in motion. We

make the following observations:






Ir3




II




I~~ I l
X I I ~ I ll$X X X






Figure 2-3: Leaf trajectory during 9llC delivery


1. All MUs that are delivered along a radiation beam along me before the left leaf passes

zi fall on it. The greater the a value, the later the leaf passes that position. Therefore

Iz(zi) is a non-decreasing function.

2. All MUs that are delivered along a radiation beam along me before the right leaf

passes zi are blocked by the leaf. The greater the a value, the later the leaf passes

that position. Therefore Ir (zi) is also a non-decreasing function.










From these observations we notice that the net amount of MUs delivered at a point is

given by I;(zi) I,(zi), which must be the same as the desired profile I(zi).

2.1.3 Optimal Unidirectional Algorithm for one Pair of Leaves

Unidirectional movement. When the movement of leaves is restricted to only one di-

rection, both the left and right leaves move along positive a direction, from left to right

(Figure 2-1). Once the desired intensity profile, I(zi) is known, our problem becomes that

of determining the individual "iil. 10; pr ol. / -~~I to be delivered by the left and right leaves,

I and I, such that



I(i) = II(i) I,(zi), O < i < m (2.1)

We refer to (I, 4) as the treatment plan (or simply plan) for I. Once we obtain the

plan, we will be able to determine the movement of both left and right leaves during the

therapy. For each i, the left leaf can be allowed to pass zi when the source has delivered

Ih(as) MUs. Also, we can allow the right leaf to pass zi when the source has delivered I,(as)

MUs. In this manner we obtain unidirectional leaf movement /pr oll.~ for a plan.

Algorithm. From Equation 2.1, we see that one way to determine It and I, from the

given target profile I is to begin with Ih(zo) = I(zo) and I,(zo) = 0; examine the remaining

zis from left to right; increase It whenever I increases; and increase I, whenever I decreases.

Once It and I, are determined the leaf movement profiles are obtained as explained in the

previous section. The resulting algorithm is shown in Figure 2-4. Figure 2-5 shows a profile

and the corresponding plan obtained using the algorithm.

Ma et al. (1998) shows that Algorithm SINGLEPAIR obtains plans that are optimal

in therapy time. Their proof relies on the results of Spirou and Chui (1994), Stein et al.

(1994) and Boyer and Strait (1997). We provide a much simpler proof below.

Theorem 1 Algorithm SINGLEPAIR obtains plans that are optimal in therapy time.

Proof: Let I(zi) be the desired profile. Let incl, inc2, ..., inck be the indices of the points

at which I(as) increases. So minel, zinc2, *inck are the points at which I(z) increases (i.e.,

I(zinci) > I(zinci-1)). Let ai = I(2inci) I(2inci-1)

Suppose that (IL, IR) is a plan for I(zi) (not necessarily that generated by Algorithm









Algorithm SINGLEPAIR

Iz(zo) = I(zo)
I,(zo) = 0
For j = 1 to m do

If (Ij)> I r(j_1) (j (j

Else

Iz(j) = Iz(j_1)
End for


Figure 2-4: Obtaining a unidirectional plan

SINGLEPAIR). From the unidirectional constraint, it follows that IL(2i) and IR(2 ) are

non-decreasing functions of z. Since I(as) = IL(as) IR(as) for all i we get

ai = (IL(2inci) In(2inci)) (IL(2inci-l) In(2inci-l))

= (IL(2inci) IL(2inci-l)) (In(2inci) In(2inci-l))

I L inci) L 2inci-1)*

Summing up Ai, we get

Li [I(2inci) I(2inci-)] < C i[IL(2inci) IL(2inci-)] = TT(IL, In)

Since the therapy time for the plan (It, I,) generated by Algorithm SINGLEPAIR is Cl,

[I(zinci) I(zinci-1)], it follows that TT(I I, ) is minimum.
Corollary 1 Let I(zi), O < i < m be a desired prll.~II Let Iz(zi), and I,(zi), O < i < m

be the left and right leaf tll.~II generated by Algorithm SINGLEPAIR. Iz(zi) and I,(as),

O < i < m define optimal therapy time unidirectional left and right leaf let..I;I. for I(zi),

S< i
Proof: Follows from Theorem 1 m

In the remainder of this paper, (Ig, I,) is the optimal treatment plan for the desired

profile I.

Properties of the optimal treatment plan. The following observations are made

about the optimal treatment plan (It, I,) generated using Algorithm SINGLEPAIR.

Lemma 1 At each zi at most one of the /; r oll. It and I, changes (increases).



























I1






rl


x, x


Figure 2-5: A profile and its plan










Lemma 2 Let (IL, IR) be any treatment plan for I.

(a) a(zi) = IL(2i) 1h(2i) = IR(2i) Ir(2i) > 0,0 < i < m.

(b) a(as) is a non-decreasing function.

Proof: (a) Since I(as) = IL(as) IR(a) = I (2i) I(zi),IL(2i) I (2) = IR(as) r(as).

Further, from Corollary 1, it follows that IL(2i) > 4(zi),0 O i < m. Therefore, A(zi) >

0, O
(b) We prove this by contradiction. Suppose that A(z,) > A(z, 1) for some n, O I

n < m. Consider the following three all encompassing cases.

Case 1: II(z,) = II(z,+l)

Now, IL(2n) = 1h(22) + A(an2) > 1h(z, 1) + A(z, +1) = IL(z, +1).

This is not possible because IL is a non-decreasing function.

Case 2: I,(z,) = I,(z,+l)

Now, IR(2n) = Ir(2n) + a(zn) > I,(z, 1) + a(2, 1) = IR(z, 1).

This contradicts the fact that IR is a non-decreasing function.

Case 3: II(z,) II(z,+l) and I,(z,) f &(zl)

From Lemma 1 it follows that this case cannot arise.

Therefore, A(zi) is a non-decreasing function. m

Theorem 2 If the optimal plan (It, ) violates the minimum separation constraint, then

there is no plan for I that does not violate the minimum separation constraint.

Proof: Suppose that (I, I,) violates the minimum separation constraint. Assume that the

first violation occurs when II MUs have been delivered. From the unidirectional movement

constraint, it follows that the left leaf has just been positioned at my (for some j, 0 < j < m)

at this time and that the right leaf is at zk, such that zk, zj is less than the permissible

minimum separation. Figure 2-6 illustrates the situation.

We prove the theorem by contradiction. Let (IL, IR) be a plan that does not violate

the minimum separation constraint. When j = 0, (II, ) has a violation at the initial

positioning to of the left leaf. Since the leaves move in only one direction, the violation

is when II = 0. When II = 0, the left leaf in (IL, IR) is also at to (because the left leaf

must begin at to in all plans; otherwise I(zo) = 0). For (IL, IR) not to have a violation

at II = 0, the right leaf must begin to the right of zk, I-, at some point p > zk, (nOte











Separation between
the jawsI













xo xj xk



Figure 2-6: Minimum separation constraint violation


that p may not be one of the zis). The MUs delivered at zk, by the plan (IL, IR) are

IL(2k) IR ~) = L k~) 1 k2) (Corollaryl). Also, I(z() k ~z) r kz) k ~z)

(Ir (zk) > 0). So (IL, IR) delivers more than I(zk) MUS at zk, and so is not a plan for I.

This contradicts the assumption on (IL, IR). Hence, j 0 .

Suppose that j > 0. Now, Iz(zj) > Iz(zj_l). Also, IL(zj) = Iz(zj) + A(zj) and

IL(2j_1) = Iz(2j_1) + A(2j_l). Since A(zj) > A(zj_1) (Lemma 2(b)), IL(zj) > IL(zj_i).

Therefore, the left leaf is positioned at my at some time during the on cycle of the plan

(IL, IR). Let the amount of MUs delivered when the left leaf arrives at zj in IL be I2. Let

the right leaf be at z = p at this time. Note that p may not be one of the zis. If p > zk, then

In(2k) 2 I. Also, from Lemma 2 we have IL(2k) = 1 k2) k (2) 1~~ k a(j-1)

I(zk) 2I 1I 1 k~z) 2I I, k) k ~z) 2I. Therefore, IL(2k) IR k) k (2). This

contradicts IL(2k) IR() k k() (since (IL, IR) is a plan for I). Therefore, j cannot

be > 0 either. So, there is no plan (IL, IR) that does not violate the minimum separation

constraint.

The separation between the leaves is determined by the difference in a values of the

leaves when the source has delivered a certain amount of MUs. The minimum separation

of the leaves is the minimum separation between the two profiles. We call this minimum










separation Sud-min. When the optimal plan obtained using Algorithm SINGLEPAIR is

delivered, the minimum separation is Sud--min(opt)*

Corollary 2 Let Sud--min(opt) be the minimum leaf separation in the plan (II, Ir). Let

Sud-min be the minimum leaf separation in any (not ii... ~,tale optimal) given unidi-

rectional plan. Sud-min < Sud-min(opt) *

2.1.4 Bi-directional Movement

In this section we study beam delivery when bi-directional movement of leaves is per-

mitted. We explore whether relaxing the unidirectional movement constraint helps improve

the efficiency of treatment plan.

Properties of bi-directional movement. For a given leaf (left or right) movement

profile we classify any 2-coordinate as follows. Draw a vertical line at z. If the line cuts the

leaf profile exactly once we will call z a unidirectional point of that leaf profile. If the line

cuts the profile more than once, z is a bi-directional point of that profile. A leaf movement

profile that has at least one bi-directional point is a bi-directional /pr oll.I All profiles that are

not bi-directional are unidirectional prll.~II Any profile can be partitioned into segments

such that each segment is a unidirectional profile. When the number of such partitions is

minimal, each partition is called a stage of the original profile. Thus unidirectional profiles

consist of exactly one stage while bi-directional profiles 1.h-- I-, have more than one stage.

In Figure 2-7, the leaf movement profile, BI, shows the position of the left leaf as

a function of the amount of MUs delivered by the source. The leaf starts from the left

edge and moves in both directions during the therapy. C'I, ..ly, BI is bi-directional. The

movement profile of this leaf consists of stages S1, S2 and S3. In stages S1 and S3 the

leaf moves from left to right while in stage S2 the leaf moves from right to left. zj is a

bi-directional point of BI. The amount of MUs delivered at my is Ll+L2. In stage S1, when

Li amount of MUs have been delivered, the leaf passes my. Now, no MU is delivered at my

till the leaf passes over my in S2. Further, L2~ MUs are delivered to my in stages S2 and S3-

Thus we have I(zjm) = L1 + L2a. Here, Lil = II, L2~ = 13 1 2* k is a unidirectional point

of BI. The MUs delivered at zk, are L3 = 14. Note that the inltloneityi profile It is different

from the leaf movement profile BI, unlike in the unidirectional case.













I Be UL






S L, S3









x, xk x



Figure 2-7: Bi-directional movement

Lemma 3 Let (II, Ir) be a plan delivered by the bi-directional leaf movement l; r oll.- pair

(BI, Br) (i.e., BI and Br are, ,. Ifr..1.: I:I. the left and right leaf movement /roll ~II )

(a) I is non-decreasing.

(b) Ir is non-decreasing.
Proof: (a)Whenever a point zi, O < i < m, is blocked by the the left leaf, the points

zo,zl,...,as_l are also blocked. It follows that Ih(zi) > Ih(zj), O < j < i < m.

(b)The proof is similar to (a)
From Lemma 3 we note that a bi-directional leaf movement profile B delivers a non-

decreasing intensity profile. This non-decreasing intensity profile can also be delivered

using a unidirectional leaf movement profile (Section 2.1.3). We will call this profile the

unidirectional leaf movement fol/.-II~ that corresponds to the bi-directional l; roll.- B and

we will denote it by U to emphasize that it is unidirectional. Thus every bi-directional

leaf movement profile has a corresponding unidirectional leaf profile that delivers the same

amount of MUs at each zi as does the bi-directional profile.










Theorem 3 The unidirectional treatment plan constructed by Algorithm SINGLEPAIR is

optimal in 'il.. ,~,Ite time even when bi-directional leaf movement is permitted.

Proof: Let BL and BR be bidirectional leaf movement profiles that deliver a desired

intensity profile I. Let IL and IR, respectively, be the intensity profiles for BL and BR.

From Lemma 3, we know that IL and In are non-decreasing. Also, IL(as) IR(as) =

I(zi), 1 < i < m. From the proof of Theorem 1, it follows that the therapy time for the

unidirectional plan (Ig, Ir) generated by Algorithm SINGLEPAIR is no more than that of

(IL, IR). m

Incorporating minimum separation constraint. Let Uz and Ur be unidirectional

leaf movement profiles that deliver the desired profile I(zi). Let BI and Br be a set of

bi-directional left and right leaf profiles such that Uz and Ur correspond to BI and Br re-

spectively, i.e., (BI, Br) delivers the same plan as (Ug, Ur). We call the minimum separation

of leaves in this bi-directional plan (BI, Br) Sbd-min*

Theorem 4 Sbd-min < Sud-min for a bi-directional leaf movement l; roll.- pair and its

corresponding unidirectional l;*<.-Gl.

Proof: Suppose that the minimum separation Sud-min occurs when Ims MUs are deliv-

ered. At this time, the left leaf arrives at my andl the~ bright leafI is poslitione atl Zk. ULet Bj

and Ul' respectively, be the profiles obtained when BI and Uz are shifted right by Sud-min.

Since Ul' is Uz shifted right by Sud-min and since the distance between Uz and Ur is Sud-min

when Ims MUs have been delivered, Ul' and Ur touch when Ims units have been delivered.

Therefore, the total MVVDUs delivered (UY' \Ur) at zk is zero. So the total MUs delivered by

(BJ, Br) at zk, is also zero (recall that Ul' and Ur, respectively, are corresponding profiles for
Bb and Br). This isn't possible if Br is 11. -- ; t the ih o ,(oreapli he situa-
tionof igur 2-, te M~ deliver b (B,~ B) atI Z alre (LI + L2)r (mL', + n L'2 + L3) 0)


Therefore Bf and Br must touch (or cross) at least once. So Sbd-min < Sud-min

Theorem 5 If the optimal unidirectional plan (II, Ir) violates the minimum separation con-

straint, then there is no bi-directional movement plan that does not violate the minimum

separation constraint.











B, U,


B
UI


I ,


1 %L~,L L,










Figure 2-8: Bi-directional movement under minimum separation constraint


Proof: Let BI and Br be bi-directional leaf movements that deliver the required profile.

Let the minimum separation between the leaves be Sbd-min. Let the corresponding unidi-

rectional leaf movements be UI and Ur and let Sud-min be the minimum separation between

Uz and Ur. Also, let Smin be the minimum allowable separation between the leaves. From

Corollary 2 and Theorem 4, we get Sbd-min < Sud-min < Sud-min(opt) < Smin

Incorporating maximum separation constraint. Let UI and Ur be unidirectional

leaf movement profiles that deliver the desired profile I. Let Sud-mat be the maximum

leaf separation using the profiles Ug and Ur and let Sud-maz(opt) be the maximum leaf

separation for the plan (It,Ir). Let BI and Br be a set of bi-directional left and right leaf

profiles such that UI and Ur correspond to BI and Br, respectively. Let Sbd-maz be the

maximum separation between the leaves when these bi-directional movement profiles are

used.

Theorem 6 Sbd-maz > Sud-mat for every bi-directional leaf movement I;I rol. and its cor-

responding unidirectional movement l; r oll.

Proof: Suppose that the maximum separation Sud-mat occurs when Ims MUs are deliv-

ered. At this time, the left leaf is positioned at my and the right leaf arrives at zk. Let BI'

and Ul' respectively, be the profiles obtained when BI and Uz are shifted right by Sud-max.










Since Uf is UI shifted right by Sud-mat and since the distance between UI and Ur is Sud-mat

when Ims MUs have been delivered, Uf and Ur touch when Ims units have been delivered.
Therefore, the total MUs delivered by (Ur, U') at Zk is zero. So the total M/lTs delivered~ by

(Br, Bf) at zk, is also zero (recall that Uf and Ur, respectively, are corresponding profiles for

Bf and Br). This isn't possible if Br is 11.h- I-, a to the left of Bf (for example, in the situation

of Figure 2-9, the MUs delivered by (Br, Bf) at sk, are (L': + L'2 + L'3 \- (L L2a) > 0).

Therefore Bf and Br must touch (or cross) at least once. So Sbd-maz > Sud-maz


U, B,
SB, UI U, B,









LL,






Figure 2-9: Bi-directional movement under maximum separation constraint


2.1.5 Algorithm Under Maximum Separation Constraint Condition

In this section we present an algorithm that generates an optimal treatment plan under

the maximum separation constraint. Recall that Algorithm SINGLEPAIR generates the

optimal plan without considering this constraint. We modify Algorithm SINGLEPAIR

so that all instances of violation of maximum separation (that may possibly exist) are

eliminated. We know that bi-directional leaf profiles do not help eliminate the constraint.

So we consider only unidirectional profiles.

Algorithm. The algorithm is described in Figure 2-10.

Theorem 7 Algorithm MiAXSEPARATION obtains plans that are optimal in therapy time,

under the maximum separation constraint.















Algorithm MAXSEPARATION
1. Apply Algorithm SINGLEPAIR to obtain the optimal plan (It, I).
2. Find the least value of inltlencity,) I,) such that the leaf separation in (Ig, I,) when II
MUs are delivered is > Smax, where Smax is the maximum allowed separation between
the leaves. If there is no such li, (II, 1,) is the optimal plan; end.
3. Let my and Zk, respectively, be the position of the left and right leaves at this time
(see Figure 2-11). Relocate the right leaf at a~ such that za zj Smax, when II
MUs are delivered. Let AI Il(zj) II I2 1I. Move the profile of I,, which
follows z(, up by AI along I direction. To maintain I(z) I(zs) I,(z) for every z,
move the profile of It, which follows z(, up by AI along I direction.
Goto Step 2.

Figure 2-10: Obtaining a plan under maximum separation constraint












A after __ __ __ __
I Sm modification~---L







I; Before


Figure 2-11: Maximum separation constraint violation










Proof: We use induction to prove the t~heoremn.

i ~statemnent wez prove, S(ub), is the ~

After Step 3 of the algor~ithmr is i i- -n times~, thelr resulting _! --lin,, I,,):

(a) It has no maximum separation violation when I < IgL(ul) M~s are delivered, where

I2(r?) is tIhe v-alue of I2 during t~he ut~h iteration of Algorithm 11 : = 'P: : = N`.

(b) For plans th-at satisfy (a), (/7,, In) is optimal in therapy time.

1. Consider ther base ca~se, a2 = 1.

Let (II, I) be the 1 :: generated by Algorithmn i GLEPAIR. i.i : 1 >,3 is appied

onice, the resulting i" (11:, l) meets the : :1: ...=.1 thiat there is no rnaximnurn

separation violation wheicn I < I2(1) MIl~s are delivered by th~e radiation sources. :

therapy timet increases by AI, i.e., TT(IIn, Ir) = TT(In: Ir) i aI.

Assume,, that~l threl:li is? anotherlll p~lan, (rl:y 17), which satisfies condition (a) of S(1) and

TT(II, 1:) < TT7'jli, Ir). We show- this assumption leads to a. c< ::i: 1: ndo

there is no such plan (It, 1~).

L~et my, and be a~s in Algorithmn I M/ A~RATIOCN-. We'c consider th~reec cases

for thie: 7 .r'.: 1.' >"c- bewe r"y)adIiz)

(a) If (xy) I~ln~x) 7(1)
Since there is no mnaximnumn :i :tion violation wh-en I < [,(1) Mljs are deliv-

eired, if,( ') > Itljxyg) 111zj) 1,('.Sne (') '(x ( ) -

rIii(') ri, ( '), we have It (x ) > Ill( '}. Wej now constructl a i : (Ii: n." i)
as follows:






I"(z (x) =r '1 >

(x) AI x: < 2 '


Ify~ ) Al2 Il~x l -It(x ) > It( .-1 > a a_1) .I- .i

no-ocesg So (.s /')' isC) a~ pla for- e~li~i IIix g).












This contradicts our knowledge that (II, Ir) is the optimal unconstrained plan.



This leads to a contradiction as in the previous case.



In this case, I',(zj) < Ill(zj) = Iz(zj). This violates Corollary 1. So this case

cannot arise.

Therefore S(1) is true.

2. Induction step

Assume S(u) is true. If there are no more maximum separation violations in the

resulting plan, (It,, Im,), then it is the optimal plan. If there are more violations, we

find the next violation. Apply Step 3 of the algorithm to get a new plan. Assume

that there is another plan, which costs less time than the plan generated by Algo-

rithm MAXSEPARATION. We consider three cases as in the base case and show by

contradiction that there is no such plan. Therefore S(n + 1) is true whenever S(u) is

true.

Since the number of iterations of Steps 2 and 3 of the algorithm is finite (at most one

iteration can occur when the left leaf is at me, O < i < m), all maximum separation

violations will eventually be eliminated.



Note that the minimum leaf separation of the plan constructed by Algorithm MAXSEP-

ARATION is min{Sud-min(opt), Sm,, }. From Theorem 7, it follows that Algorithm MAXSEP-

ARATION constructs an optimal plan that satisfies both the minimum and maximum sepa-

ration constraints provided that Sud-min(opt) > Smin. Note that when Sud--min(opt) < Smin,

there is no plan that satisfies the minimum separation constraint.

2.1.6 Algorithm Under Inter-Pair Minimum Separation Constraint

Introduction. We use a single pair of leaves to deliver inltlensityi profiles defined along

the axis of the pair of leaves. However, in a real application, we need to deliver intensity

profiles defined over a 2-D region. Each pair of leaves is controlled independently. If there

are no constraints on the leaf movements, we divide the desired profile into a set of parallel










profiles defined along the axes of the leaf pairs. Each leaf pair i then delivers the plan

for the corresponding inltloneityi profile lilt). The set of plans of all leaf pairs forms the

solution set. We refer to this set as the treatment schedule (or simply schedule). In this

section, we present leaf sequuencing algorithms for <\llC with and without constraints. The

constraints considered are (i) minimum separation constraint and (ii) tongue-and-groove

constraint and (optionally) interdigitation constraint.

We use the term intra-pair minimum separation constraint to refer to the constraint

imposed on an opposing pair of leaves and inter-pair minimum separation constraint to

refer to the constraint imposed on opposing leaves of neighboring pairs. Recall that, in

Section 2.1.3, we proved that for a single pair of leaves, if the optimal plan does not satisfy

the minimum separation constraint, then no plan satisfies the constraint. In this section we

present an algorithm to generate the optimal schedule for the desired profile defined over a

2-D region. We then modify the algorithm to generate schedules that satisfy the inter-pair

minimum separation constraint.

Optimal schedule without the minimum separation constraint. Assume we have n

pairs of leaves. For each pair, we have m sample points. The input is represented as a matrix

with n rows and m columns, where the ith row represents the desired intensity profile to

be delivered by the ith pair of leaves. We apply Algorithm SINGLEPAIR to determine the

optimal plan for each of the n leaf pairs. This method of generating schedules is described

in Algorithm MULTIPAIR (Figure 2-12).

Algorithm MULTIPAIR
For(i = 1;i n; i ++)
Apply Algorithm SINGLEPAIR to the ith pair of leaves to obtain plan (lil, Iir) that
delivers the inltloneityi profile lilt).
End For

Figure 2-12: Obtaining a schedule

Lemma 4 Algorithm M~ULTIPAIR generates schedules that are optimal in therapy time.

Proof: Treatment is completed when all leaf pairs (which are independent) deliver their

respective plans. The therapy time of the schedule generated by Algorithm MULTIPAIR

is maz {TT(Iu1, IIr), TT(I2, 2r,), TT ulI, Inr)}. From Theorem 1, it follows that this

therapy time is optimal.










Optimal algorithm with inter-pair minimum separation constraint. The schedule

generated by Algorithm MULTIPAIR may violate both the intra- and inter-pair minimum

separation constraints. If the schedule has no violations of these constraints, it is the desired

optimal schedule. If there is a violation of the intra-pair constraint, then it follows from

Theorem 2 that there is no schedule that is free of constraint violation. So, assume that only

the inter-pair constraint is violated. We eliminate all violations of the inter-pair constraint

starting from the left end, i.e., from to. To eliminate the violations, we modify those plans

of the schedule that cause the violations. We scan the schedule from to along the positive

a direction looking for the least z, at which is positioned a right leaf (say Ru) that violates

the inter-pair separation constraint. After rectifying the violation at z, with respect to

Ru we look for other violations. Since the process of eliminating a violation at me, may

at times, lead to new violations at my, my < z,, we need to retract a certain distance (we

will show that this distance is Smin) to the left, every time a modification is made to the

schedule. We now restart the scanning and modification process from the new position.

The process continues until no inter-pair violations exist. Algorithm MINSEPARATION

(Figure 2-13) outlines the procedure.

Algorithm MINSEPARATION
//assume no intra-pair violations exist
1. z = to
2. While (there is an inter-pair violation) do
3. Find the least me, 2, > 2, such that a right leaf is positioned at z, and this right leaf
has an inter-pair separation violation with one or both of its neighboring left leaves.
Let a be the least integer such that the right leaf Ru is positioned at z, and Ru has
an inter-pair separation violation. Let LL denote the left leaf (or one of the left leaves)
with which Ru has an inter-pair violation. Note that L E {u 1, a + 1}.
4. Modify the schedule to eliminate the violation between Ru and Lt.
5. If there is now an intra-pair separation violation between Rt and LL no feasible
schedule exists, terminate.
6. z = 2v Smin
7. End While

Figure 2-13: Obtaining a schedule under the constraint


Let M~ = ((III, 11r), (121 2r,), ul ar,)) be the schedule generated by Algorithm

MULTIPAIR for the desired intensity profile.

Let N(p) = ((III,, 11rp), (121p 2rp,), ulp arp)) be the schedule obtained after Step 4










of Algforithm MINSEPARATION is applied p times to the input schedule M~. Note that

M~ = N(0).


Figure 2-14: Eliminatingf a violation


To illustrate the modification process we use an example (see Figure 2-14). To make

things easier, we only show two neighboring pairs of leaves. Suppose that the (p + 1)th

violation occurs when the right leaf of pair n is positioned at z, and the left leaf of pair

t, L E {u 1, a + 1}, arrives at me, 2, 2, < Smin. Let at = 2, Smin. To remove this

inter-pair separation violation, we modify (Itlp, Itr). The other profiles of N(p) are not

modified. The new Illp *O*>. tl(p+1)) is aS defined below.


0o I a

n~ m 2


where AI = lurp v,) ftl( n~ 2 1 Itr(p+1)2 .= ,,! IL)(2 -t(2), where It(z)

is the target profile to be delivered by the leaf pair t. Since Itr(p 1) differs from Itrp for

z > z', = 2, -Smin there is a p. 11 .11 117r-, that N(p+1) has inter-pair separation violations for

right leaf positions z > z', = 2,-Smin. Since none of the other right leaf profiles are changed

from those of N(p) and since the change in Itl only delays the rightward movement of the

left leaf of pair t, no inter-pair violations are possible in N(p + 1) for z < z', = z, Smin.


Itlp 2
I~;!I )()= ma{Itlp(> fl(2 + r










One may also verify that since Islo and Isro are non-decreasing functions of z, so also are

Itlp and Itrp, P > 0.

Lemma 5 Let F = ((I' I' fi), (120 r' ',.,11 nD )) be any feasible schedule for the de-

sired in.~ 4/.I i. e., a schedule that does not violate the intra- or inter-pair minimum separation

constraints. Let S(p), be the following assertions.

(a) rlz (s > lilp (2), O < i < n, so < z < sm

(b) IL (z) > lirp (2), O < i < n, so < z < sm

S (p) is true for p > 0.
Proof: The proof is by induction on p.

1. Consider the base case, p = 0. From Corollary 1 and the fact that the plans

(lilo, firo), O < i < n, are generated using Algorithm SINGLEPAIR, it follows that

S(0) is true.

2. Assume S(p) is true. Suppose Algorithm MINSEPARATION finds a next violation

and modifies the schedule N(p) to N(p + 1). Suppose that the next violation occurs

when the right leaf of pair n is positioned at z, and the left leaf of pair t arrives at

me, 2, 2, < Smin (see Figure 2-14). Let z', = z, Smin. We modify pair t's plan

for z', < z < m,, to eliminate the violation. All other plans in the schedule remain

unaltered. Therefore, to establish S(p + 1) it suffices to prove that


I'l~) > tl~p1) ) n< 2 Im(2.2)



7 r (s > Itzp1 ( n< 2 < (2.3)

Weneed,, prove only one of hesetw re,,,lationships,, sinc I',() -I (s) = I,~,! I )(2) -

1tr(p+1) 02) three cases, that are exhaustive, and show that Equation 2.2 is true for each. This,

in turn, implies that S(p + 1) is true whenever S(p) is true and hence completes the

proof.

(a) No modification (relative to M~ = N(0)) has been made to pair t's plan for z > z'

prior to this. In this case, Itl,() = Itlo() = Itl(),a > '.









The situation is illustrated in Figure 2-14.

Since there is no niiinimun separation violation in F, the left leaf of pair t passes

2'( only after the right leaf of pair a passes ',., i.e.,


lil24) ir2,.)(2.4)

Since S(p) is true,

fir 2' ) > Zor; ( ) =L (a(,)(2.5)

Front Equations 2.4 and 2.5,


Itl\ ) I ;>j+1) (a (2.6)



tt u' =l \44; 2 l1\(a ), O <2 12' z(27


Sintila~rly,


In,,. I L) (2) = L ,, I L)\" (2;)+la )(p2) Itl;>+l) (a'(,, O o 2-

since Ist;,>(2 = stl(>) 2 > ,


f1(p+1) 2') = It1(2) a, 2' IC 2' 2,n (2.9)

Front Equations 2.8 and 2.9, we get

I, I1) 7) L~ /,, I )() + (Ist (2) + ar)


-(It(2'() + aI), 2'( < 2' < ak

= Icor I1) (2';) + rtl(2') rt,(2 ;), 2' I 2' I 2' (2.10)

Subtracting Equation 2.10 front Equation 2.7,


,ila') ltit;>+1) (' 77( ,It(2)4) + 1 fr (2))
-(11(24;) lIt/@{;), 2 ;< 2 2'm (.1










From Equations 2.6 and 2.11,



liz( (s -- Ill~u1, (p+1 1 1



From Lemma 2b,

II' (s) Itz (2) > I1(m' ) -- It (al),2 I' < I < m 2.3


From Equations 2.12 and 2.13, we get


Itl(s) > L;.,, I, g(), z', < m (2.14)


(b) Some prior modification has been made to pair t's plan for z > z's. There exists

a modification at z, such that Itlp2 fl wt(2 < ar <, I m, and there

is no a < 2, that satisfies this condition. Note that Itlp ua) < amount of MUs

delivered when profile Itlp Z) arriVeS at Zu (SinCe Itlp Z) is a DOn-decreasing func-

tion of 2) < lurp v,) (since there is a minimum separation violation when profile

Isrp(2) is at 2,). Therefore, Itlp(S a ftl n urp+ v,? 2)-I1() = fl n2 ~ *

So, 2( > z' .

In this case (see Figure 2-15),



I.,,,g1()= + ar )+AI z' < 2~,


Note that, in the example of Figure 2-15, a prior modification was made to pair

t's plan for z > sq. However, Itlp fl qt(2 w*r ,I ,

We get I,'1(2) > Itl(p+1)() 2 I zj < 2, fOr r88Sons similar to those in the

previous case. Also, lizs) > I. (s) = tlp() w1( < 2 < 2m, since S(p) is



(c) Some prior modification has been made to pair t's plan for z > z's. However,

Itlp fl nt(2 m*r ~I ,

In this case, Itl(p+1)(2 flt2 n case.



















tr p+1) -- - -
hhc ~I~

zx 4 x,~ x ,

Fiur 2-15: Elmntn ilto






(a)~~l I'z (s) Itlp 0






if Ilp nd Irp ave miimfum separ lmiation f violationtenoramntpn(IIs)ht



LExaml 1 Weilsraea ntac hr an inter-pair minimum separation violation dtce Sp INEP



obAtiONe using lgithmr sn MUTIAsIR. Tch.dl fo are dshonine F~~igue-7 ac f h

plans: ((Itz os),Isr the)) a d (Ist+1) ss)Iet+1) F ))) s feasible, ie., there isn intra-pair mn

miimum separation (Sin =io 7). Howee,p whe OrMINSEPARAIO is applied (fr imliit






consderlef paWeirls adte +1 inisolatin) it etect an inter-pair minimum separation vi-to





oaionm beprtwen I(t+1)z a Is owvr, when I~t1z rives at A = 6Nd is aposiioed ato si = 11.










To eliminate this violation, I~t+1)I is positioned at z = 4 (since 11 4 = 7 = Smin) and its

Ipo llI. is raised from z = 4. C .;-,, <;1:, ,1:1t~+1)r is also raised from z = 4 resulting in the

plan (I~t+1)11(2) ),I~t+1)rl(2)). This In..J.I.:; ,tl... t; results in an intra-pair violation for pair

t +1i, when I~t+1)I1 is at z = 1 and I~t+1)rl is at z = 4. From Lemma 6, there is no feasible

schedule.


La~ (x)


1 4 6


17 21


29 35


49 X


Figure 2-16: Inltlensityi profiles of adjacent leaf pairs


For N(p), p > 0 and every leaf pair j, 1 < j < n, define Ijlp -1) = Ijrp( l -1



Notice that App (i) giVeS the time (in monitor units) for which the left leaf of pair j stops

at position zi. Let Ajl,(zi) and Ajrp i) be zero for all me when j = 0 as well as when

j = + 1.

Lemma 7 For every j, 1 < j < n and every i, 1 < i < m,


nApp i> I mtz ajl0 i) a(j-1)rp zi + Smin), a(j+1)rp zi + Smin)}


(2.15)














ItlJ
I -- -- -- -- -- -



1..- - -
I__


(t+1)rl



0 1 4 6 9 11 13 17 21 29 35 47 49 x




Figure 2-17: Profiles violating inter-pair constraint


Proof: The proof is by induction on p. For the induction base, p = 0. Putting p = 0 into

the right side of Equation 2.15, we get


maz {Ajo (zi), A(j-l),o(2i + Smin), a(j+l)ro(2i + Smin)} > aj1o(2i) (2.16)


For the induction hypothesis, let q > 0 be any integer and assume that Equation 2.15

holds when p = q. In the induction step, we prove that the equation holds when p = q + 1.

Let t, u, and 2, be as in iteration p 1 of the while loop of algorithm MINSEPARATION.

Following this iteration, only atlp and atrp are different from Ac,;, _1) and atrp-1), respec-

tively. Furthermore, only Atlp(2w) and Atrp w,), where 2, = 2, Smin may be larger than

the corresponding values following iteration p 1. At all but at most one other a value

(where a may have decreased), Atlp and atrp are the same as the corresponding values

following iteration p- 1.

Since 2, is the right leaf position for the leftmost violation, the left leaf of pair t arrives

at 2, = 2, Smin after the right leaf of pair n arrives at z, = z, + Smin. Following the

modification made to L ;, 1), the left leaf of pair t leaves z, at the same time as the right leaf

of pair a leaves 2, + Smin. Therefore, Atlp 2w) Iur(p--1) w + Smin) = aurp 2w + Smin)

The induction step now follows from the induction hypothesis and the observation that

n E {t 1, t + 1}. m










Lemma 8 For every j, 1 < j < n and every i, 1 < i < m,


Ajrpi jp i /i) / -1)(2.17)

where ly (a ) = 0.

Proof: We examine N(p). The monitor units delivered by leaf pair j at zi are Ijl,(zi) -

Ijrp zi) and the units delivered at as_l are Ijlp i-1) Ijrp i-1). Therefore,


ly (i) Ijl i)- jr i)(2.18)



ly~a _1) Ijl i-1 jrp -1)(2.19)

Subtracting Equation 2.19 from Equation 2.18, we get


Isms -lyze1) = (jp i l -) r )- p i1

=ajlp i> ajrp 2i) (2.20)

The lemma follows from this equality.

Notice that once a right leaf a moves past m,, no separation violation with respect

to this leaf is possible. Therefore, z, (see algorithm MINSEPARATION) < zm. Hence,

njlp ~i) Ijl0 i), and Ajrp i) Ijr0 zi), sm Smin < zi < 2m, 1 < j < n. Starting
with these upper bounds, which are independent of p, on Ajrp i), 2m--Smin < 2i < 2m and

using Equations 2.15 and 2.17, we can compute an upper bound on the remaining Ajl,(zi)s

and Ajrp zi)S (frOm right to left). The remaining upper bounds are also independent of p.

Let the computed upper bound on App (i) be Ujl(as). It follows that the therapy time for

(rj,,, Ijrp) is at most Tmaz(j) = Co
at most Tmax = meazlign{Tmaz(j)}-

Theorem 8 The following are true of Algorithm M~INSEPARATION:

(a) The algorithm terminates.

(b) When the algorithm terminates in Step 5, there is no feasible schedule.

(c) Otherwise, the schedule generated is feasible and is optimal in 'I: r,ten~ time for uni-
directional schedules.










Proof: (a) As noted above, Lemmas 7 and 8 provide an upper bound, Tmax on the

therapy time of any schedule produced by algorithm MINSEPARATION. It is easy

to verify that

I.,, I L (2) > Iilp(2),0 O i < n, so < z < sm

lir(p+1)() >irp(2),0 O i < n, so < z < s
and that

I~., 1 1, (m',) > Idp u)6

Itr(p+1) (2',) > Itrp u~)
Notice that even though a a value (proof of Lemma 7) may decrease at an me, the

lilp and lirp ValueS never decrease at any zi as we go from one iteration of the while

loop of MINSEPARATION to the next. Since la increases by atleast one unit at

atleast one zi on each iteration, it follows that the while loop can be iterated at most

mnTmax times.

(b) Follows from Lemma 6.

(c) If termination does not occur in Step 5, then no minimum separation violations remain

and the final schedule is feasible. From Lemma 5, it follows that the final schedule is

optimal in therapy time for unidirectional schedules.



Corollary 3 When Smin = 0, Algorithm M~inseparation always generates an optimal fea-

sible schedule.

Proof: When Smin = 0, Algorithm Minseparation cannot terminate in Step 5 because

the Step 4 modification never causes the left leaf of a leaf pair to cross the right leaf of that

pair. The Corollary follows now from Theorem 8.
2.2 Conclusion

In conclusion, we presented mathematical formalisms and rigorous proofs of leaf se-

quencing algorithms for segmental multileaf collimation which maximize MU efficiency.

These leaf sequencing algorithms explicitly account for minimum leaf separation constraint

and leaf interdigfitation constraint. We have shown that our algorithms obtain all feasible











solutions that, are optimnal in t~rea~tment MT~s. Furthermore, our :: 1 shows that1 uni-

directional leaf move~ment is at least as : as bi-directiona~l movement. ii '..hese:

a~lgorithmirs are suited for commron use in ::.C1 barn delive~ry.
















CHAPTER 3
SEQUENCING OF DYNAMIC MULTILEAF COLLIMATORS

Delivery using DMLC is different from that using <\llC. The leaf positions change

with respect to time. In terms of the MLC controller it is the change in position with respect

to monitor units delivered that is important. The inputs required are the leaf positions at

various control points, the fractional number of monitor units to be delivered at each control

point, and the total number of monitor units to be delivered for that beam. In this chapter,

we present a systematic study of the optimization of leaf sequencing algorithms for the

dynamic beam delivery and provide rigorous proofs of optimized leaf sequence settings in

terms of MU efficiency under various leaf movement constraints. Practical leaf movement

constraints that are considered include the leaf interdigitation constraint. The question of

whether bi-directional leaf movement will increase the MU efficiency when compared with

unidirectional leaf movement only is also addressed.

3.1 Methods

3.1.1 Movement of Leaves

In our I!! 11-, i we will assume that I(zo) > 0 and I(zm) > 0 and that when the beam

delivery begins the leaves can be positioned ir-, -.-tu rie. We also assume that the leaves

can move with any velocity v, -vmax I v I vmax, where vmax is the maximum allowable

velocity of the leaves and that the leaf acceleration can be infinite. The consequences of

assuming infinite leaf acceleration are negligible. Figure 3-1 illustrates the leaf t l r .1. b!ry

during DMLC delivery. In the example, the leaves move from left to right. Let Il(zi) and

Ir (as), respectively, denote the amount of Monitor Units (MUs) delivered when the left and

right leaves leave position zi. Consider the motion of the left leaf. The left leaf begins at

to and remains here until Iz(zo) MUs have been delivered. At this time the left leaf leaves

to and is moved to 21, where it remains until I;(zl) MUs have been delivered. The left leaf

then moves tO 23 where it remains until Ig(23) MUs have been delivered. At this time, the

left leaf is moved to us, where it remains until I,(2s) MUs have been delivered. Then it










moves to where it remains until 41( -. ) MITs have been delivered. 'i : 1: : ; movement of

thle left leaf is to anlo. ... left leaf arrives at1 a to when Imax M~Ts have been delivered. At

this tirne the mnac~hine is turned off. Thel~ total i! timeo, TT ~ig, Ir), is theic tirne needed

to deliver Imax, M4~s. 'i .. rights leaf starts at. xo aind begins to moove :1 I .i .rtl t ece

22; leaVeS 22 when Ir (n2) M4Ts have been I i i leaveS 24 when Ir (n4) M4~s have been

delivered, and so on. Note that the machine is on th-roughout the treatment. All MIrs

that~ are delivered along a radiation bea~m along me, before Ithe left leaf n:: on it.

:::i- all M~ITs that are delivered alongf a radiation beam along x4 before the right, leaf

passes zi, ar~e blockedi by thie i So the net amount of MIUs dieliverecd at a 1= .1 is given

'.Il(x4) -- Irjmxi), wh-lich rust be th~e samne as the desired i T (z ).




I~x


I( r-- _




Ids Ix)


I~xI



Xo X, X2 X3 X4 X5 X6 X, X, X9 X ,o X

Figure 3 1: ILleaf t 1. i :-y diurinig DMLCI delivery

Th~eorem 9 The ." r;; are true: ,' pair %r that delivers a discreite:



(a) :' left~ -f' must reach z~o at some time.

(b) The leaf must reach at some? time.

(c) lef IJ !- rmust reach at some time.

(d) :1 .' i f mu~st reach .. at somecl timecl.

Proof: (a) Suppose: that, the: i leaf aliway2s stays to the right of mo: then xo does not,

receive any M/I~s: contradicting ou~r !: tha21t I(..) > 0).










(b) Similar to that of (a).

(c) If the left leaf doesn't reach 2m (i.e., it doesn't go to the right of sm-1), from (b), it

follows that the region between zm-1 and am receives a non-uniform distribution of MUs.

Hence the discrete profile can't be accurately delivered.

(d) Similar to that of (c). m

3.1.2 Maximum Velocity Constraint

As noted earlier, the velocity of leaves cannot exceed some maximum limit (say vme,)

in practice. This implies that the leaf profile cannot be horizontal at any point. From

Figure 2-3, observe that the time needed for a leaf to move from zi to ziay is > (asyl-

zi)/vmax. If # is the flux density of MUs from the source, the number of MUs delivered in

this time along a beam is > ##(zi41-zi)/vmax. So, 4(as 1)-4(as) > Os(as~ 1-zi)/vmax

# Az/vmax. The same is true for the right leaf profile I,.

3.1.3 Optimal Unidirectional Algorithm for one Pair of Leaves

Unidirectional movement. When the movement of leaves is restricted to only one di-

rection, both the left and right leaves move along the positive a direction, from left to right

(Figure 2-1). Once the desired intensity profile, I(zi) is known, our problem becomes that

of determining the individual "iil. 10; pr ol. / -~~I to be delivered by the left and right leaves,

It and I, such that



I(i) = r (i) I (zi), O < i < m (3.1)

Of course, It and I, are subject to the maximum velocity constraint. We refer to (I I, )

as the treatment plan (or simply plan) for I. Once we obtain the plan, we will be able to

determine the movement of both left and right leaves during the therapy. For each i, the

left leaf can be allowed to pass zi when the source has delivered Iz(zi) MUs. Also, we can

allow the right leaf to pass zi when the source has delivered I,(as) MUs. In this manner

we obtain unidirectional leaf movement /pr oll.~ for a plan. Some simple observations about

the leaf profiles are made below.

Theorem 10 In every unidirectional plan the leaves begin at to and end at zm.

Proof: Follows from Theorem 9 and the unidirectional constraint. m










Lemma 9 In the region between each pair of successive sample points, say zi and me 1,

both leaf lpr./.I maintain the same shape, i.e., one is I... ,. 10I a vertical translation of the

other.

Proof: As explained previously, the input profile is discretized to a square wave I. Since

the profile of I is horizontal between successive sample points and since it is equal to It Ir,

It and Ir must have the same shape. For example, if the left leaf moves at a constant

velocity v between points zi and me 1, so should the right leaf. m

Lemma 10 In an optimal plan, both leaves must move at vmax between every successive

pair of sample points they move across.

Proof: Suppose that in an optimal solution the leaves move between points zi and meil

at a possibly varying velocity v(z) < vmax. From Lemma 9, we know that both leaf profiles

are the same between zi and me 1. Setting v(z) = vmax results in new leaf profiles whose

difference remains the same as before (which is the desired profile I) and total therapy time

is lowered. This leads to a contradiction. m

Corollary 4 In an optimal plan, no leaf stops at an a that is not one of the zis.

Algorithm. From Equation 3.1, we see that one way to determine It and Ir from the

given target profile l is to begin from to; set Iz(zo) = I(zo) and Ir(zo) = 0; examine the

remaining zis to the right; increase It at zi whenever I increases and by the same amount

(in addition to the minimum increase imposed by the maximum velocity constraint); and

similarly increase Ir whenever I decreases. This can be done till we reach zm. So the

treatment begins with the leaves positioned at the leftmost sample point and ends with the

leaves positioned at the rightmost sample point. Once It and Ir are determined the leaf

movement profiles are obtained as explained earlier. Note that we move the leaves at the

maximum velocity vmax whenever they are to be moved. The resulting algorithm is shown

in Figure 3-2. Figure 2-3 shows a profile I and the corresponding plan (It,Ir) obtained

using Algorithm DMLC-SINGLEPAIR.

Ma et al. (1998) shows that Algorithm DMLC-SINGLEPAIR obtains plans that are

optimal in therapy time. Their proof relies on the results of Spirou and Chui (1994), Stein

et al. (1994) and Boyer and Strait (1997). We provide a simpler and direct proof below.










Algorithm DM4LC- .i -:i .ilZ IR,


Foir j = 1 to m do

ii (zj) It (xyl ) 1(xy) -- I jzj-) + 4,* Ax:/vmax,
I,(my) I,(y_1 *am/
Else

I,(my) -- I,(my_1) I$ y1 -, I*y + A/,,lt
E5nd for


Figure 3-2: ObtainingS a unidirectional plan


Theorem 11 .i t .: DMnLC'-SOVG'LEPAIR obtains planrs that arle optimail in :!

time.

Proof: Let. I(m,) be Ithe desired profile. Let 0 = .: < inrcl < ... < inack be: Ith

indices of the points a~t which 1(x:,) increases. So Xminco: wjinLCi:,.,Zick are thei ******i at

which I(x) increases (i.e., I(zi,,ce) > I(zi,,ci-1), assume that Ijx_1 =- 0)). L~et Ai

I( ) ( )? i > 0.

PEi:4 th~at (lL, Ty) is a _-1 for I(xri) (not thelr plan gernerated by Algorithm

DML '. \lLEPIR) '.=.I(xy) =- II( ) -- Injxi) for all i -we get



= (17 (: 1-I sni1)-(nxni 1))

1 (ici I(1) --~ n: + Ax/vaZ) -- (In(xLinc-i) -- In(xilnci--i) -- '" z/~nar

N-ote th~atf~r~om th~e maximum velocit~y constraint, I"Zll(:rii) IR(:rinc:i-1) > Ax/vmass,

i > 1. So ln(xinci) In(xinci-1) --: Ax/vmax o i > 1, andi Ai I II (xines) -- fr(xinci--1) --

Ax/vmazl!,,. Also, Ao = T(zo) -- T(2_1) = Tjzo) < II (Zo) -- IL(x_1), w~her3 1(x--1) =- 0.

Summingf up A~i, we: get

-~ .(ljinci) 1(xLilLCi-1j)] I Ei-[lL(2inc) -- 1L(xinc-i-1i) -- k.* i Az/vmax,~. Lect S1

o~i [1; (X~inc~i) -1;L sinLCi- 1)]. ii : ::, Si > E -0 [Ij~xines) I(wiLCj-j)] +k X;: (I Ax C/vmax.r Let

Sg' = E[I%-(xy) II,(1:j_1)], where the summation is carriedl out over indices j (0 < j < m)

suc~h that I(xy~) < I(xy~_l). Thel~re are a. total of m + 1 indiices of wh-lichl k + 1 do not satisfyi

thiis condiition. So there ar~e m1 -- k: indicees j at which I(zj) < I(xy-1). At Ceah of thelCse










j, IL(2j) > IL(2j_l) + # A2//vmax. Hence, S2 > (m k) # Az/vmax. Now, we get

Sl + S2 0~ [L zi> L 2i-1~ =0 C [(inci) I(inci-1)] + me A2/vmaz Finally,

TT(IL, IR) = I (2m) = IL(2m) IL(2_1) C= Eo IL(as) IL(2i_1)] > E =o[I(2inci) -

I(zinci-1)] + m* Az/vma = TT(II, Ir). Hence, the treatment plan (Ig, Ir) generated by

DMLC-SINGLEPAIR is optimal in therapy time.

Corollary 5 Let I(zi), O < i < m be a desired prll.~II Let Il(zi) and Ir(zi), O < i < m

be the left and right leaf tll.~II generated by Algorithm DM~LC-SINGLEPAIR. Iz(zi) and

Ir(zi), O < i < m define optimal 'it.. ,~,,ti time unidirectional left and right leaf let..I;I. for


Proof: Follows from Theorem 11 m

In the remainder of Section 3.1, (II, Ir) is the optimal treatment plan generated by

Algorithm DMLC-SINGLEPAIR for the desired profile I.

Properties of the optimal treatment plan. The following observations are made

about the optimal treatment plan (It, Ir) generated using Algorithm DMLC-SINGLEPAIR.

Lemma 11 At most one of the leaves stops at each me.

Lemma 12 Let (IL, IR) be any treatment plan for I.

(a) a(zi) = IL(2i) 1h(2i) = IR(2i) Ir(2i) > 0,0 < i < m.

(b) a(as) is a non-decreasing function.

Proof: (a) SincelI(as) = IL(as) IR(a) = I (as) -Ir (z), IL (as) I () = IR(as) -Ir (zi).

Further, from Corollary 5, it follows that IL(2i) > 4(zi),0 O i < m. Therefore, A(zi) >

0, O
(b) We prove this by contradiction. Suppose that A(z,) > A(z, 1) for some n, O I
n < m. Consider the following three all encompassing cases.

Case 1: Iz(z, 1) = I(za) + # Az/vmax (left leaf does not stop at an 1)

Now, IL(2n) = 1h(22) + A(an1) > 1h(z, 21) #. A2/vmaz + A(2n 1) = IL(2n 1) #. *


This is not possible because IL(2n 1) > IL(2n) + # s A2/vmaz from the maximum velocity

constraint.

Case 2: Ir(as 1) = Ir(z,) + # Az/vmax (right leaf does not stop at an 1)

Now, IR(2n) = Ir (2n) + A(212) > Ir (2n1) '" Az/vmax + A(z, 1) = IR(z, 1) # s












This is not possible because IR(z, 1) > IR(2n) + # A2/vmaz from the maximum velocity

constraint.

Case 3: Il(z,+ 1) I(z,) + # Az/vmax and Ir(z,+ 1) Ir(z,) + # Az/vmax (both

leaves stop at an 1)

From Lemma 11 it follows that this case cannot arise.

Therefore, A(zi) is a non-decreasing function. m

Corollary 6 Let Alzia) (Ar(zi)) and AL(as) (An(as)), ,. ,7. .: Ho lo. denote the amount of

time for which the left (right) leaf stops at zi in plans (I I,r) and (IL, IR). Then

(a) AL(as) > Az(ws).

(b) An(as) > Ar (as).

Proof: (a) Suppose that AL(as) < Az(zi). We have the following two cases:

Case 1: Both leaves move at the maximum velocity between as_l and me in (IL, IR).

We get a(z,) < a(z,_l) contradicting Lemma 12(b).

Case 2: In (IL, IR), the leaves do not travel uniformly at the maximum velocity between

as_l and me. In this case, transform plan (IL, IR) to a plan (I I'9) as follows. Between as_l

and me move the leaves at the maximum velocity. The leaves now arrive at my earlier than

they did in (IL, IR) by an amount 6i. Propagate this difference to the right from zi so that

I (2j) = IL(2j)--6i and I'9(2j) = IR(2j)--bi, j > i. Note that this transformation preserves
the As, i.e., A',(2j) = AL(2j). Also, the resulting leafprofile, rI and I',silfomapn

for I. Let a'(z,) = I (zj) Il(zj) = I' (zj) Ir(zj). Since A',(as) = AL(as) < Aza) and

since the leaves move at maximum velocity from zi_l to me in (It, Ir) and (If, I'9), we have

a'(z,) < a'(z,_l) contradicting Lemma 12(b).

(b) Similar to proof of (a).

3.1.4 Minimum Separation Constraint

Some MLCs have a minimum separation constraint that requires the left and right

leaves to maintain a minimum separation Smin at all times during the treatment. Notice

thlat in1 the planl generated by AIU~lgrithm DMLCU~-SING~LEPAIR, the two leaves start and

end at the same point. So they are in contact at to and zm. When smin > 0, the minimum

separation constraint is violated at to and zm. In order to overcome this problem we modify










Algorithm DMLC-SINGLEPAIR to guarantee minimum separation between the leaves in

the vicinity of the end points (zo and 2m). In particular, we allow the left leaf to be

initially positioned at a point zo/ = to smin and the right leaf to be finally positioned

at am/ = 2m + Smin. The only changes made relative to Algorithm DMLC-SINGLEPAIR

are for the movement of the left leaf from zo/ to to and for the right leaf from zm to sm*-

We define the movement of the left leaf from zo/ to to (and as ;- en?~rowil: definition for the

right leaf from zm to m,/) to be such that it maintains a distance of exactly Smin from

the right leaf at all times. Once the left leaf reaches to it follows the trajectory as before.

While this modification results in the exact profile being delivered between to and am it

also results in some undesirable exposure to the intervals (zo/,, o) and (zm, s,/). In the

remainder of this section we will consider an exposure of this kind permissible, provided

the exact profile is delivered between to and zm. Note that for most accelerators (Varian,

Elekta) undesirable exposure to the intervals (zo/,, o) and (mm, sm,) can be avoided by

positioning the back-up jaws at to and am respectively. However, a difficulty arises when

the number of monitor units delivered at the time the left leaf reaches to in this new plan

(call it (If, II\ )) i greter than 7./o). This would prevent us, fromusing theold plan from

to to m,, since the leaf cannot pass to before it arrives there. Observe however, that if

the left leaf were to arrive at to any earlier, it would be too close to the right leaf. In the

discussion that follows we show that in this and other cases where the original plan violates

the constraint, there are no feasible solutions that deliver exactly the desired profile between

to and m,, while permitting exposure outside this region. The modified algorithm, which

we~ calll DMVLC~-MVINS I~V-,,1 ~n nNGanLEPAIR, is described in Figure 3-3. Note that the therapy time

of the plan produced by DMLC-MINSINGLEPAIR is the same as that of the plan produced

by DMLC-SINGLEPAIR. Therefore, the modified plan is optimal in therapy time.
Theorem 12 (a) Smin O/Umax` > h-\o) or' (b I the plan (If ,' If ) enerted by DMLC-

M~INSINGLEPAIR violates the minimum separation constraint, then there is no plan for I

that does not violate the minimum separation constraint.

Proof: Suppose~ that (I ,/ I ) violates the minimum separation constraint. Assume that the

first violation occurs when II MUs have been delivered. Since there is no violation when less

than II MUs are delivered and since the leaves are either stationary or move at the maximum







44

Algorithm DMLC-MINSINGLEPAIR

1. Apply Algorithm DMLC-SINGLEPAIR
2. Modify the profile of the left leaf from zo/ to to and the right leaf from zm to m,/ to
maintain a minimum interleaf distance of Smin. Call this profile (I,', I').
3. If the number of MUs delivered when the left leaf arrives at to is greater than Iz(zo)
there is no feasible solution. End.
4. Else output (I,', I'). There is a feasible solution only if (I', I') is feasible.

Figure 3-3: Obtaining a unidirectional plan with minimum separation constraint

velocity, at the time of the violation, it must be the case that the right leaf is stationary

at a sample point (say zk) and the left leaf is moving. The violation occurs when the left

leaf passes 2' = zk, Smin. Since the left leaf is moving, II = I,'(z') < I'(zk). Figure 3-4
illustrates the situation. Spoetathrisnterln(I!" htde o ilt h

minimum separation constraint. Let If"(m') = It'(z')+A(z') and let I"(zk) k k)*~+~z)

From Lemma 12(a), A(z'), A(zk) > 0 and from Lemma 12(b), A(z') < A(zk). Here, we

have made use of the fact that in the statement of Lemma 12, we can replace the plan

(II, 1r) with the plan (If, I'). Now, I"(zk) kf(' I(~ kz) 1I( az)

(I'(zk) 1 ~z)) k)a~, a(z')). Since I'(zk) > I'(z') and A(zk) > a(z'), We get

I"(Zk) l "(2'). Therefore there is a minimum separation violation in (I"', I") when the
the left leaf passes 2'.






Separation between
the leaves



I-





xo xo x' xk Xm Xm,, X

Figure 3-4: Minimum separation constraint violation












The separation between the leaves is determined by the difference in a values of the

leaves when the source has delivered a certain amount of MUs. The minimum separation of

the leaves is the minimum separation between the two profiles. We call this minimum sep-

aration Sud-min. When the optimal plan obtained using Algorithm DMLC-SINGLEPAIR

is delivered, the minimum separation is Sud--min(opt) *

Corollary 7 Let Sud-min(opt) be the minimum leaf separation in the plan (I', I'). Let

Sud-min be the minimum leaf separation in any (not ii... whil~ optimal) given unidirec-

tional plan. Sud-min < Sud-min(opt) *

Theorem 13 If Algorithm DM~LC-M~INSINGLEPAIR terminates in Step 3, then there is

no plan for I that does not violate the minimum separation constraint.

Pro:Let (I"', I" ) be a feasible plan (i.e., a plan that delivers I and satisfies the minimum

separationl cvlonstaint). ULet (II I'.) be. Ilthe pllanI of Srtep 2i of DMVLC~-MVINS ING~LEPA I. FromII

Corollary 5, it follows that I"(zo +Smin) > I:((o +Smin)+I"(zo), where Ir(z) is the number

of MUs delivered when the right leaf reaches n (note that Ir (z) is the number of MUs



If"(zo) > If'(zo) > I('(o + Smin). Also, because DMLC-MINSINGLEPAIR terminates in


So, (If", I") does not deliver the proper dose at to and so cannot be feasible. m

Comparison with SMLC. K~amath et al. (2003) discuss the optimal therapy time

algorithm for M \! LC. Their algorithm generates an optimal plan that satisfies the minimum

separation constraint, whenever one exists. We prove the existence of profiles for which

there are feasible plans using 9llC, but no feasible plans using DMLC.

Lemma 13 Let the minimum separation between the leaves in the optimal SM~LC plan

be Ss-min. Let the minimum leaf separation in the plan generated by algorithm DM~LC-

M~INSINGLEPAIR be Sd-min. Then Sd-min < Ss-min

Proof: Consider the delivery of a profile I by the optimal <\llC plan of K~amath et
al. (2003). Call this plan (If, If) Let IIl- (s b the number of M"Ts de~livered~ when the

left leaf arrives at a using the plan IfS. I S(z), I,'(z), and I'(z) are defined similarly. Let

fl(2k) = 1~s k) ls k~) = 1 kz) 1 kz). r k2) is defined similarly. Note that fl(zk) > 0










iff the left leaf stops at zk, and Fl(2k) gives the amount of MUs delivered while the left leaf

is stopped at zk. Let me and my, j > i, be such that Ss-min = zj zi and If (zi) < I, (zj).

Such an zi and my must exist by definition of Ss-min.

It is easy to see that I,"(zj) I, (zi) ,= E Fr(2k). From this and lf (as) < I, (zj), we

get



Er (k I i) i)(3.2)
k=i+1

Since I(as) = If (as) I, (as) = If (as) + El(z ) I, (as) = It'(zi) + El(zi) I'(as),



I'r\ I ( -(Ia I'(a )) (3.3)

Also, we see that I' (x ) = I' (ws\I'" ~ ,z) + (j F (Zk azlUmaz Izi) 1Is zi) -

I'(mi))+] Fr(23) + (j i) # A2/vmaz (from (3.3)) > II (mi) + (J i) -Y # us Azmaz

(from (3.2)). So, Sd-min < my 2i = Ss-min

The following result immediately follows and can be easily verified.

Corollary 8 All l; r oll. that have feasible plans using DM~LC have feasible plans using

SM~LC. There exist foll.~II for which there are feasible plans using SM~LC, but no feasible

plans using DM~LC.

Figure 3-5 shows two plans for an in-tens-ityi profile. The TU L IC plan for the profile is fea-

sible. The corresponding DMLC plan obtained using Algorithm DMLC-MINSINGLEPAIR

is infeasible.

3.1.5 Bi-directional Movement

In this section we study beam delivery when bi-directional movement of leaves is per-

mitted. We explore whether relaxing the unidirectional movement constraint helps improve

the efficiency of treatment plan.

Properties of bi-directional movement. For a given leaf (left or right) movement

profile we classify any 2-coordinate as follows. Draw a vertical line at z. If the line cuts the

leaf profile exactly once we will call z a unidirectional point of that leaf profile. If the line

cuts the profile more than once, z is a bi-directional point of that profile. A leaf movement

profile that has at least one bi-directional point is a bi-directional /pr oll.I All profiles that are










































Constraint violation



















XO XO

Figure 3-5: C ( plan: feasible; DMLC: i i.









not bi-directional are unidirectional prll.~II Any profile can be partitioned into segments
such that each segment is a unidirectional profile. When the number of such partitions is

minimal, each partition is called a stage of the original profile. Thus unidirectional profiles

consist of exactly one stage while bi-directional profiles 11.h- I-, have more than one stage.

In Figure 3-6, the bi-directional leaf movement profile, BL, shows the position of the

left leaf as a function of the amount of MUs delivered by the source. The movement profile

of this leaf consists of stages S1, S2 and S3. In stages S1 and S3 the leaf moves from left

to right while in stage S2 the leaf moves from right to left. zj is a bi-directional point of

BL. Let IL be the inltlencityi profile corresponding to the leaf movement profile BL. IL(2)

gives the number of MUs delivered at a using movement profile BL. The amount of MUs

delivered at my is Li + L2~. In stage S1, when II amount of MUs have been delivered, the

leaf passes my. Now, no MU is delivered at my till the leaf passes over my in S2. Further,

I3 2~ MUs are delivered to my in stages S2 and S3. Thus we have Iz(zj) = Li + L2, where

Li = II and L2 = 13 1 2- k is a unidirectional point of BL. The MUs delivered at zk, are

L3S = 14. Note that the intensity profile IL is different from the leaf movement profile BL,

unlike in the unidirectional case.



I,



I4 S3

SL,




I,~ ------




Figure 3-6: Bi-directional movement


Lemma 14 Let IL and In be the 1. /10 pr l~l. l -~II corresponding to the bi-directional leaf

movement l],.-61.- pair (BL, BR) (i~e., BL and BR are, e.I .1/.:: I:. the left and right leaf










movement /pr oll.~ ). Let I(zi) = IL(2i) IR(2i), O < i < m, be the il. i i~ /10~ pr;I l delivered

by (BL, BR). Then

(a) IL(ziy1) > IL(2i) + # A2//vmax.

(b) IR(as 1) > IR(as) + # Az//vmax.

Proof: (a) Between the time, 1, the left leaf moves rightward from zi for the last time

(since the left leaf ends at m,, such a last right move must occur) and the least time t2,

t2 1 t, that the left leaf reaches zi 1 (again, since the left leaf ends at m,, such a 2

exists), sail receives at least # Az/vmax MUs that are not delivered to me. At all other

times during the therapy, whenever the left leaf doesn't cover me, it also doesn't cover me 1.

Hence, outside the time interval [tl, t2], the number of MUs delivered to meil is atleast as

many as delivered to me. Therefore, for the entire therapy, IL(2i+1) > IL(2i)+ *Az/vmax.

(b) The proof is similar to that of (a). m

From Lemma 14 we note that every bi-directional leaf movement profile (BL, BR)

delivers an intensity profile (IL, IR) that satisfies the maximum velocity constraint. Hence,

(IL, IR) is deliverable using a unidirectional leaf movement profile (Section 3.1.3). We will

call this profile the unidirectional leaf movement /pr oll.- that corresponds to the bi-directional

pil.Thus every bi-directional leaf movement profile has a corresponding unidirectional

leaf profile that delivers the same amount of MUs at each zi as does the bi-directional

profile.

Theorem 14 The unidirectional treatment plan constructed by Algorithm DM~LC SIN-

GLEPAIR is optimal in 'it.. ,~,I;*0 time even when bi-directional leaf movement is permitted.

Proof: Let BL and BR be bidirectional leaf movement profiles that deliver a desired

intensity profile I. Let IL and IR, respectively, be the corresponding inltlensityi profiles for

BL and BR. From Lemma 14, we know that IL and In are deliverable by unidirectional leaf

movement profiles. Also, IL(as) -IR(as) = I(as), 1 < i < m. From the proof of Theorem 11,

it follows that the therapy time for the unidirectional plan (Ig, Ir) generated by Algorithm

DMLC-SINGLEPAIR is no more than that of (IL, IR). m

Incorporating minimum separation constraint. Let Uz and Ur be unidirectional

leaf movement profiles that deliver the desired profile I(zi). Let BI and Br be a set of










bi-directional left and right leaf profiles such that Uz and Ur correspond to BI and Br re-

spectively, i.e., (BI, Br) delivers the same plan as (Ug, Ur). We call the minimum separation

of leaves in this bi-directional plan (BI, Br) Sbd-min. Sud-min is the minimum separation

of leaves in (UI, Ur).

Theorem 15 Sbd-min < Sud-min for every bi-directional leaf movement I;I rol. pair (BI, Br)

and its corresponding unidirectional l *<. -Gl. (UI Ur) -

Proof: Suppose that the minimum separation Sud-min occurs when Ims MUs are deliv-

ered. At this time, the left leaf arrives at my andl thel bright IClea is poslitione atl Zk. ULet BJ

and Uf respectively, be the profiles obtained when BI and Uz are shifted right by Sud-min.

Since Uf is Uz shifted right by Sud-min and since the distance between Uz and Ur is Sud-min

when Ims MUs have been delivered, Uf and Ur touch when Ims units have been delivered.

Therefre, th totalrC YV M~ILUsdlvrdb (Uf, \Ur) at zk is zero. So the total MUs delivered by

(Bj, Br) at zk, is also zero (recall that UI and Ur, respectively, are corresponding profiles
for Bb and Br). This isn't possible if Br is 11. --; c- t the right of BJ(o eapeih

situation of Figure 3-7, the MUs delivered by (Bf, Br) at zk, are (L1 + L2a)- (L', + L2) > 0).

Therefore Bf and Br must touch (or cross) at least once. So Sbd-min < Sud-min


I, B n ; u


X1 Xk X


Figure 3-7: Bi-directional movement under minimum separation constraint










Theorem 16 If the optimal unidirectional plan (If I') violates the minimum separation

constraint, then there is no bi-directional movement plan that does not violate the minimum

separation constraint.

Proof: Let BI and Br be bi-directional leaf movements that deliver the required profile.

Let the minimum separation between the leaves be Sbd-min. Let the corresponding unidi-

rectional leaf movements be Uz and Ur and let Sud-min be the minimum separation between

Uz and Ur. Also, let Smin be the minimum allowable separation between the leaves. From

Corollary 7 and Theorem 15, we get Sbd-min < Sud-min < Sud-min(opt) < Smin

Incorporating maximum separation constraint. Let UI and Ur be unidirectional

leaf movement profiles that deliver the desired profile I. Let Sud-mat be the maximum

leaf separation using the profiles Uz and Ur and let Sud-maz(opt) be the maximum leaf

separation for the plan (Ig, I) generated by Algorithm DMLC-SINGLEPAIR. Let BI and

Br be a set of bi-directional left and right leaf profiles such that UI and Ur correspond to

BI and Br, respectively. Let Sbd-maz be the maximum separation between the leaves when

these bi-directional movement profiles are used.

Theorem 17 Sbd-maz > Sud-mat for every bi-directional leaf movement l; roll.- and its

corresponding unidirectional movement p ll.~II

Proof: Suppose that the maximum separation Sud-mat occurs when Ims MUs are deliv-

ered. At this time, the left leaf is positioned at my and the right leaf arrives at zk. Let BI'

and Uf respectively, be the profiles obtained when BI and Uz are shifted right by Sud-max.

Since Uf is Uz shifted right by Sud-mat and since the distance between Uz and Ur is Sud-mat

when Ims MUs have been delivered, Uf and Ur touch when Ims units have been delivered.

Therefore, the total MUs delivered by (Ur, Uf) at zk, is zero. So the total MUs delivered by

(Br, BJ) at Z is also zero (recall that Uf and Ur, respectively, are corresponding profiles

for Bf and Br). This isn't possible if Br is 1.h-- .-, to the left of Bf (for example, in the
situation of Figure 3-8, the- M"Ts delivered~ by, (Br Bj ts ae( L'2 L L2a) > 0).

Therefore Bf and Br must touch (or cross) at least once. So Sbd-maz > Sud-maz

3.1.6 Algorithm Under Maximum Separation Constraint Condition

In this section we present an algorithm that generates an optimal treatment plan

under the maximum separation constraint. Recall that Algorithm DMLC-SINGLEPAIR












U, B, B B; Ur U;













X1 4 x

Figure 3-8: Bi-directional movement under maximum separation constraint


generates the optimal plan without considering this constraint. We modify Algorithm

DMLC-SINGLEPAIR so that all instances of violation of maximum separation (that may

possibly exist) are eliminated. We know (Theorem 17) that bi-directional leaf profiles do

not help eliminate the constraint. So we consider only unidirectional profiles.

Algorithm. The algorithm is described in Figure 3-9.

Algorithm DMLC-MAXSEPARATION
1. Apply Algorithm DMLC-SINGLEPAIR to obtain the optimal plan (It,Ir).
2. Find the least value of inltlencity,) I,) such that the leaf separation in (Ig,Ir) when II
MUs are delivered is Smax, where Smax is the maximum allowed separation between
the leaves and the leaf separation when II + t MUs are delivered is > Smax, for some
positive constant t. If there is no such li, (II, 1r) is the optimal plan; end.
3. From Lemma 10 it follows that when II MUs are delivered, the left leaf is stopped at
some my. Let 2' be the position of the right leaf at this time (see Figure 3-10). Note
that 2' may not be one of the sample points zi, j < i < m. Let AI Il(zj) II
I2 1I. Move the profile of Ir, which follows z', up by AI along I direction. To
maintain I(z) = I;(z) Ir(z) for every 2, move the profile of It, which follows z', up
by aI along I direction.
Goto Step 2.

Figure 3-9: Obtaining a plan under maximum separation constraint


Theorem 18 Algorithm DM~LC-MiAXSEPARA TION obtains plans that are optimal in ther-


apy time, under the maximum separation constraint.







53



Ife oifcto





After modificatio









x1 xX

F o:: 3 10: Maximnum separation constraint violation


Proof: We~ usse induction to i :..- the theoremn.

'i statements wre prove, S(nz), is the I

.i:19i (iii) of the algorithm- is :_ i iT n times, the resulting plan, (la,, In): ii

(a.) It has no maximum separation violation when I < Ig(ub) Mijs are delivered, where

[2(8) is the V8,1UO Of I2 during theic nith iteration of Algorithmr DMLIIC N= --. =A

RATO111 >Nr.

(b) F~or plans tha~t sa~tisfyi (a2), (Ita, Im.7) is o~trimai'l in '" i time.

1. C~onsider the base case, a =- 1.

L~et (11, .: ) be the plan generated 1 Algorithm DI) C- i::i ..' l i vi After Step (iii)

is ii-=--' once, the resulting i : (Il, 1ry) meets the requirement that there is no

ma~ximumn separation violation when I < I,(1) MITs are delivered by the radiation

source. T'~ therapy time increases '.AieTTTi r)=Ti l r l

A~ssumeo that there is another plan, (I,', 11): which saitlisfies c :.': :(a) of S(1) aind

TT(i(, I1) < TT(In, 1,r3). We't showi this assumption leads to a at : `; : = -- and so

there is no such plan ((Ill I1).

L~et xyg and z' b-e a~s in Algorithm Dn 1) CI ': i i'li iON). ':'. consider three

cases for the relationshhip between Ill(xy) a~nd sI;(xy)>.












Since there is no maximum separation violation when I < I2(1) MUs are de-



h1(2') 1,1(2'), we have If1(2') > 1hi(2'). We now construct a plan (I'(, I",) as
follows :




If,(2) aI z 2 z


I",(s)= &(2) 0 I z < 2'


CI.~ ~ 7/ / yI(s "i\s = s,\ 0 < x < 2m. Also, If( is non-decreasing and
satisfies the maxi;mum ve~loncity cnstraint~ (I/ m)= f m)- I>h/m)-A

1h(2') > 1h(2' Az) + # A2/vmaz = If((m' Az) + `f sP Az/ vmaz). Smlrly

is non-decreasing and satisfies the maximum velocity constraint. So (If:, I",) is

a plan for I(zi).

Alo TT'( (', = T( 7 I
This contradicts our knowledge that (I, I,) is the optimal unconstrained plan.



This leads to a contradiction as in the previous case.



In this case, If,(zj) < hIl(zj) = Ih(zj). This violates Corollary 5. So this case

cannot arise.

Therefore S(1) is true.

2. Induction step

Assume S(u) is true. If there are no more maximum separation violations in the

resulting plan, (4,, Im,), then it is the optimal plan. If there are more violations, we

find the next violation. Apply Step (iii) of the algorithm to get a new plan. Assume

that there is another plan, which costs less time than the plan generated by Algorithm

DMLC-MAXSEPARATION. We consider three cases as in the base case and show by










contradiction that there is no such plan. Therefore S(n + 1) is true whenever S(u) is

true.

Since the number of iterations of Steps (ii) and (iii) of the algorithm is finite (for each

iteration, the left leaf must be stationary at my and there can be at most one iteration

for each such my), all maximum separation violations will eventually be eliminated.



When the minimum separation constraint is also applicable, we can use Algorithm

DMLC-MINSINGLEPAIR in place of Algorithm DMLC-SINGLEPAIR in Step (i) of Algo-

rithm DMLC-MAXSEPARATION. Note that in this case the minimum leaf separation of

the plan constructed by Algorithm DMLC-MAXSEPARATION is min{Sud--min(opt), Smax }.
From Theorem 18, it follows that Algorithm DMLC-MAXSEPARATION constructs an op-

timal plan that satisfies both the minimum and maximum separation constraints provided

that Sud-min(opt) > Smin. Note that when Sud--min(opt) < Smin, there is no plan that

satisfies the minimum separation constraint.

3.1.7 Algorithm Under Interdigitation Constraint

Introduction. The inter-pair minimum separation constraint with Smin = 0 is of special

interest and is referred to as the interdigitation constraint. Recall that, in Section 3.1.3, we

proved that for a single pair of leaves, if the optimal plan does not satisfy the minimum

separation constraint, then no plan satisfies the constraint. In this section we present

an algorithm to generate the optimal schedule for the desired profile defined over a 2-D

region. We then modify the algorithm to generate schedules that satisfy the interdigitation

constraint. Note that in our discussion on single pair of leaves (Section 3.1.1), we assumed

that I(zo) > 0 and that I(zm) > 0. However, with multiple leaf pairs, the first and last

sample points with non-zero intensity levels could be different for different pairs. Hence we

will no longer make this assumption.

Optimal schedule without the interdigitation constraint. For sequencing of mul-

tiple leaf pairs, we apply Algorithm DMLC-SINGLEPAIR to determine the optimal plan

for each of the n leaf pairs. This method of generating schedules is described in Algorithm

DMLC-MULTIPAIR (Figure 3-11). Note that since to, am are not necessarily non-zero for

any row, we replace to by zl and am by z, in Algorithm DMLC-SINGLEPAIR for each row,










where 2z and my, respectively, denote the first and last non-zero sample points of that row.

Also, for rows that contain only zeroes, the plan simply places the corresponding leaves at

the rightmost point in the field (call it zm+l).

Algorithm DMLC-MULTIPAIR
For(i = 1;i n; i ++)
Apply Algorithm DMLC-SINGLEPAIR to the ith pair of leaves to obtain plan (lil, lir)
that delivers the intensity profile lilt).
End For

Figure 3-11: Obtaining a schedule


Lemma 15 Algorithm DM~LC-MULTIPAIR generates schedules that are optimal in 'I: ,**?/*

time.

Proof: Treatment is completed when all leaf pairs (which are independent) deliver their

respective plans. The therapy time of the schedule generated by Algorithm DMLC MUL-

TIPAIR is maz {TT(I11, IIr), TT(I2, 2r,), TT ulI, Inr)}. From Theorem 11, it follows

that this therapy time is optimal.

Optimal algorithm with interdigitation constraint. The schedule generated by Al-

gorithm DMLC-MULTIPAIR may violate the interdigitation constraint. Note that no intra-

pair constraint violations can occur for Smin = 0. So the interdigitation constraint is essen-

tially an inter-pair constraint. If the schedule has no interdigitation constraint violations,

it is the desired optimal schedule. If there are violations in the schedule, we eliminate

all violations of the interdigitation constraint starting from the left end, i.e., from to. To

eliminate the violations, we modify those plans of the schedule that cause the violations.

We scan the schedule from to along the positive a direction looking for the least z, at

which is positioned a right leaf (say R,) that violates the inter-pair separation constraint.

After rectifying the violation at z, with respect to R, we look for other violations. Since

the process of eliminating a violation at me, may at times, lead to new violations involving

right leaves positioned at me, we need to search afresh from z, every time a modification

is made to the schedule. We now continue the scanning and modification process until

no interdigitation violations exist. Algorithm DMLC-INTERDIGITATION (Figure 3-12)

outlines the procedure.










Algorithm DMLC-INTERDIGITATION
1. z = zo
2. While (there is an interdigitation violation) do
3. Find the least me, 2, > 2, such that a right leaf is positioned at z, and this right
leaf has an interdigfitation violation with one or both of its neighboring left leaves.
Let a be the least integer such that the right leaf R, is positioned at z, and R,
has an interdigfitation violation. Let Le denote the left leaf with which R, has an
interdigitation violation. Note that L E {u 1, a + 1}. In case R, has violations with
two adjacent left leaves, we let t = 1.
4. Modify the schedule to eliminate the violation between R, and Lt.
5. z = 2v
6. End While

Figure 3-12: Obtaining a schedule under the constraint


Let M~ = ((III, 11r), (121 2r,), ul ar,)) be the schedule generated by Algorithm

DMLC-MULTIPAIR for the desired intensity profile.

Let N(p) = ((Illp IIrp), 21p 2rp),.( I,, up r)) be the schedule obtained after Step 4 of
A~lgorithm~nr DMC-NTR~Dr/ IGITA`\TIO is applied p times to the input schedule M~. Note

that M~ = N(0).

To illustrate the modification process we use examples. There are two types of viola-

tions that may occur. Call them Typel and Type2 violations and call the corresponding

modifications Typel and Type2 modifications. To make things easier, we only show two

neighboring pairs of leaves. Suppose that the (p + 1)th violation occurs between the right

leaf of pair u, which is positioned at me, and the left leaf of pair t, L E {u 1, a + 1}.

In a Typel violation, the left leaf of pair t starts its sweep at a point zStart(t, p) > z,

(see Figure 3-13). To remove this interdigitation violation, modify (Itlp, trp) tO (tl(p+1),

1tr(p+1)) aS follows. We let the leaves of pair t start at z, and move them at the maximum ve-
locity vmax towards the right, till they reach zStart(t, p). Let the number of MUs delivered

when they reach zStart(t, p) be II. Raise the profiles Itlp() and Itrp(z), a > zStart(t, p),

by an amount II = s (zStart(t, p) z,)/vmax. We get,




Itlp~z 1 xI '> zStart(t, p)

1tr(p+1)(2 )2 I t(2), where It(z) is the target profile to be delivered by the leaf

pair t.















Iul













x, xStart(t,p) x

Figure 3-13: Eliminating a Typel violation


A Type2 violation occurs when the left leaf of pair t, which starts its sweep from z < ze,

passes 2, before the right leaf of pair a passes z, (Figure 3-14). In this case, I ;,!, I) is as
defined below.




where AI = lurp v,) lp v(2) = 13 12. Once again, Itr(p+1)(> 1tcl(p+1) Itz

where It(z) is the target profile to be delivered by the leaf pair t.

In both Typel and Type2 modifications, the other profiles of N(p) are not modified.

Since Itr(p+1) differs from Itrp for z > 2, there is a possibility that N(p + 1) has inter-

pair separation violations for right leaf positions z > z,. Since none of the other right leaf

profiles are changed from those of N(p) and since the change in Itl only delays the rightward

movement of the left leaf of pair t, no interdigitation violations are possible in N(p + 1)

for z < 2,. One may also verify that since Itlo and br~o are feasible plans that satisfy the

maximum velocity constrains, so also are y,~ and Itrp, P > 0.

Lemma 16 Ijrp(zStart(j, p)) = 0, 1 < j < m, p > 0.

Proof: The proof is by induction on p. Let T(p) be the following statement: Ijrp

(zStart(j, p)) = 0.


















Itl











x, x

Figure 3 14l: li:: ..: oaT'l` i.. violation (close parallel dottedi and solid linre segmeonts
c they have been drawn writh a small separation to enhance readabiiliy)


*t For the base case, p- = 0. (lIyol, lyr) is th~e 1 :: generatled by A~lgorit~hm DMILC-

SlnG jINGLEPAR and it satisfies thle statedi proper

Assume th-at T(.. .) is true. Fo~r theic (. +1)th violation, we h~ave th~e followingi two cases:

-- The (-- i 1)t~h violation is a Ti 1 violation.

A TL.- mnodification is ~;;'i-i .1 Such a ...... : .= .. results in chang-

ing thel start position of thel leaves of pair t (as defined in Algorithm DMLZC:-

II i 1i)IC' i i. ; i ON)) to x:,: whichh becomes xStart(t, p) + 1)) and Isr (+1)(x,) =

0. F~or j 7' to Ir(p+)(x:Start(j ,p + 1)) = lyr,,(xStart(j? p)) = 0 by induction



-- The: (p 1 ) th violation is a T : violation.

AZT i modification is is i Let I be: as in Alg~orithm DM4LC IN-TER-

DI>( ~i'I i iSN. Suppose that zSta~tjt, p) < Sincei a T:`. mnodification does

not alter th-e plan for x < x.,,, Itr.jp+,(x:Start(t, p +1)) =- Itrp(x:Start(t, pu)) = 0).

If zStart(t, p) = n:,:, it must be Ithe case: that xStart(u, p) = xStart(t, p) (as

otherwi~W se there is aT'l i violation at zStarstju,p) < x,i). So th~e right 1 I of

pair n is not ri.. at Hience, there is no interdiigitation violations at z,.










So, the case zStart(t, p) = 2, cannot arise. For j t, the plan is unchanged.

So, Ijr(p+1)(zStart(j, p+1)) = Ifrp(zStart(j, p)) = 0 by induction hypothesis.


Corollary 9 A Lt;*~ 1' violation in which Itlp v,) = 0 cannot occur.

Proof: From the proof of Lemma 16, it follows that whenever there is a Type2 violation,

zStart(t, p) < 2,. Hence, Itlp v,) > 0. m

Lemma 17 In case of a Typel violation, (Itl, Itrp) iS the Same aS (Itl, Itr0)-

Proof: Let p be such that there is a Typel violation. Let t, u and v be as in Algo-

rithm DMLC-INTERDIGITATION. If (Itlp, Itr) is different from (Itlo, Itro), leaf pair t was

modified in an earlier iteration (say iteration q < p) of the while loop of Algorithm DMLC-

INTERDIGITATION. Let v(q) be the v value in iteration q. If iteration q was a Typel vio-

lation, then zStart(t, p) < zStart(t, q + 1) = 2,(,) < 2,. So, iteration p cannot be a Typel

violation. If iteration q was a Type2 violation, zStart(t, p) < zStart(t, q) < z,(,) < z,.

Again, iteration p cannot be a Typel violation. Hence, there is no prior iteration q, q < p,

when the profiles (la, Itr) were modified. m

Lemma 18 (a) A Typel n,~~.:.lI;. al... r; eliminates a Typel violation.

(b) A 07.1~ 1' n,~~.:.I; .al...r; eliminates a 07.1~ 1' violation.

Proof: (a) From Lemma 16, lurp v,) = 0. By changing the start position of leaf pair t to

me, we eliminate this violation.

(b) Follows from the construction of (I,~, 1, Itr(p+1))*
Note that Itlp(2) and Itrp(z) are defined only for z '> zStart(t,p). In the sequel, we

adopt the convention that z > Iip(2 (X> rp(2)) is true whenever Alpl( rp ~z>> is
undefined, irrespective of whether z is defined or not.

Lemma 19 Let F = ((I' I r), 11 I' r),. 0 )) be any feasible schedule for the de-

sired l; roll.~. i.e., a schedule that does not violate the interdigitation constraint. Let S(p),

be the following assertions.

(a) IL1(2) > Iilp(2), 0 < i < n, to < z < am


S(p) is true for p > 0.
Proof: The proof is by induction on p.










1. Consider the base case, p = 0. From Corollary 5 and the fact that the plans

(lio, firo), O < i < n, are generated using Algorithm DMLC-SINGLEPAIR, it fol-
lows that S(0) is true.

2. Assume S(p) is true. Suppose Algorithm DMLC-INTERDIGITATION finds a next

violation and modifies the schedule N(p) to N(p +1). Suppose that the next violation

occurs between the right leaf of pair u, positioned at me, and the left leaf of pair t.

We modify pair t's plan for z > z,, to eliminate the violation. All other plans in the

schedule remain unaltered. Therefore, to establish S(p + 1) it suffices to prove that

II'ls >~M Itp1 T M v (3.4)




1sr(2) > Itr(p+1)() v 2 (3.5)

We need prove only one of these two relationships since I',(z) Ir(z) = I,~, l)(z) -

Itr(p+1)(),z 0 < 2 z,. Note that
the (p + 1)th violation may either be a Typel or Type2 violation. We show that

Equation 3.4 is true in both cases. This, in turn, implies that S(p+1) is true whenever

S(p) is true and hence completes the proof. Note that in (Itl(p+1), >tr(p+1)), the leaves
move at maximum speed between adjacent sample points. So, it is sufficient to show

Equation 3.4 for sample points > z,.

(a) The (p + 1)th violation is a Typel violation.
From S(p) it follows that lI,(z,) > lurp v,). So, the right leaf of pair a leaves z,

no earlier in I'r than it does in lurp. From this and the fact that F satisfies the

interdigitation constraint, we conclude that leaf pair t cannot begin its sweep at

the right of 2,. This observation together with the fact that in (I,,. 1), Itr(p+1))

the leaves move at the maximum velocity from z, to z' = zStart(t, p) implies

that I' ~') > Itl(p+1)(2') and I' (5') > Itr(pi+1)(Z'), where I denotes an arrival

time. Now, from Lemma 16, we get IAr(z') > I$r(z') > Itr(p+1)z) trI,(p+1) (z)*

So I'z (2') = Ir (z') + It(2') > Itr(p+1)2) + It2/ = .! I L) (2'). From this and

the fact that the left leaf of pair t moves at the maximum velocity between










2, and 2', it follows that Equation 3.4 holds for all a between z, and z'. To

prove that Equation 3.4 holds for all sample points to the right of z' (and so
holds for all a between to and zm), consider a sample point me, that is to the

right of 2'. Let aI' = I'l(z') I,~,. I)(z') > 0 and let II be as in Algorithm
DMLC-INTERDIGITATION. Define a new plan (I"~, I"~) for leaf pair t as below


I"(s = unde fined a < 2'
I'z< (s I =x>z


I"(s = unde fined a < 2'


Note that I'~((') = I'l(2') AI' II = I,,! I )(2') II = Itlp(2') > 0. Sim-

ilarly, I" (z') > 0. Hence (I'(, I") is a plan for It. Also, Iff(as) = liz(2,)-



Itlp w,) = 102w,) (Lemma 17). This contradicts Corollary 5. Hence, Ifl(as,) >


(b) The (p + 1)th violation is a Type2 violation.
The situation is illustrated in Figure 3-14. Since F satisfies the interdigfitation

constraint, the left leaf of pair t does not pass 2, before the right leaf of pair a

passes 2,. So,

I'z (me) > lu (2v) (3.6)

From S(p) and the definition of a Type2 modification, we get,


~r (2v) > lurp zv) =tl(p+1) 2v) (3.7)

Equations 3.6 and 3.7 yield


I'z me)- It~p+) v)> 0(3.8)

Lemma 12b implies,


I1'l() -Irtl() > Itll(2v) Itl (2v),a > z,


(3.9)










(Lemma 12b yields Equation 3.9 only for z > z, and z is a sample point. From

this and the fact that the left leaf moves at maximum velocity in Itl between

adjacent sample points, we get Equation 3.9 for all z, a > z,.)

From Equation 3.9, we get


I1() -I -., 1)(s > 1(m ) -Itz(m ) +Itz(s)- I-,, I1) s),a > z, (3.10)

From the definitions of Typel and Type2 modifications and the working of Al-

gorithm DMLC-INTERDIGITATION, it follows that


I -,., I I) () Itl () = Itl(p+1) 2v) ftl (v), v 1 (3.11)

From Equations 3.10, 3.11 and 3.8, we get


I' z(2) I -,., I L) (2) > I1~(2,) I -,.! I L) (2,) > 0, a > 2, (3.12)

Therefore,

IL () >I .,I t)(s) m z,(3.13)



Lemma 20 For the execution of Algorithm DM~LC-INTERDIGITA TION

(a) O(u) Typel violations can occur.

(b) O(n2m) 71/** .' viOatliOnS CGH OCCUT.

(c) Let Tmax be the optimal 'I: t -ten time for the input matrix. The time complexity is

O(mn + u s min~nm, Tmax})).

Proof: (a) It follows from Lemma 17 that each leaf pair can be involved in at most one

Typel violation as pair t, i.e, the pair whose profile is modified. Hence, the number of

Typel violations is I n.

(b) We first obtain a bound on the number of Type2 violations at a fixed z,. Let n, t be as

in AIU~lgrithml DMVLC-INTERDIGUI[TATION. Note that n is chosen to be the least possible in-

dex. Let ui be the value of a in the ith iteration of Algorithm DMLC-INTERDIGITATION

at 2,. 14 is defined similarly. Let Umax = mazjgi~uj}. If ti = ui 1, it is possible that

ni+1 = ti = ui 1 and tis = ui 2. Note that in this case, til f ui = uity + 1. Next,










it is possible that ui+2 = ui 2 and ti+2 = ui-3 (again ti+2 f i 1 = ui+2 1 ). In

general, one may verify that ti = ui + 1 is possible only if ofax = Ui. If ti = Ui + 1, then

ui 1 > ti = ui + 1, since the violation between ui and ti has been eliminated and no profiles

with an index less than ti have been changed during iteration i at z,. It is also easy to

verify that ti = 1, as = 2 -> ui > u$"m, age > u$"m. From this and tis E {ui+1, as -1} it
follows that urnex--- > upm. W knowi,,, that afax > 1. It follows that sy"" > 2, afax" > 3,

lly""i '> 4 and in general, a x-1l)/2)+1 > i + 1. Clearly, for the last violation (say jth) at
me, Urnam < n and for this to be true, j = O(n2). So the number of Type2 violations at z,

is O(n2). Since 2, has to be a sample point, there are m possible choices for it. Hence, the

total number of Type2 violations is O(n2m).

(c) Since the input matrix contains only integer intensity values, each violation modification

raises the profile for one pair of leaves by at least one unit. Hence, if Taw is the optimal

therapy time, no profile can be raised more than T,,, times. Therefore, the total number of

violations that Algorithm DMLC-INTERDIGITATION needs to repair is at most sham,.

Combining this bound with those of (a) and (b), we get O(min~n2m, nTrum}) as a bound

onI thel totarl nIumber~ of violllations repaired by A1~~lgrithml DMLC-INTERDIGUIUTATION. By

proper choice of data structures and programming methods it is possible to implement Al-
gorithm~nr DMLCINTERDr/ IGITA`\TIO so, as to ru in O(mn + u s min {nm, T,, }) time.



Note that Lemma 20 provides two upper bounds of on the complexity of Algorithm
DMLC-N TE~mrR DIG~rr I\TARTION Or2m) and O(n maz {m, T,,, }). In most practical situa-

tions, T,,, < nm and so O(n maz {m, T,,, }) can be considered a tighter bound.

Theorem 19 The following are true of Algorithm DM~LC-INTERDIGITA TION:

(a) The algorithm terminates.

(b) The schedule generated is feasible and is optimal in therapy time for unidirectional
schedules.

Proof: (a) Lemma 20 provides a polynomial upper bound (O(n2m)) on the complexity

of A~,,:lgorithm r DMCITERT ~~Dr~IGI~TATION The result follows from this.










(b) When the algorithm terminates, no interdigitation violations remain and the final

schedule is feasible. From Lenina 19, it follows that the final schedule is optimal in

therapy time for unidirectional schedules.



3.2 Conclusion

We have presented niathentatical fornialisnis and rigorous proofs of leaf sequencing

algorithms for dynamic niultileaf colliniation that niaxintize MU efficiency. These leaf se-

quencing algorithms explicitly account for leaf interdigitation constraint. We have shown

that our algorithms obtain feasible solutions that are optimal in treatment MUs. Fur-

therniore, our .!!l- .h~ shows that unidirectional leaf niovenient is at least as efficient as

bi-directional movement. Thus these algorithms are well suited for coninon use in DAILC

beant delivery.

















C il T"ER 4
E M IN -.iN O nr F TO~NGUE ,-AND,-GROV UNDERDOSAGE,:;,,,,,,,:

Deiiveired of li i' with i i0C in the i. .. -shloot mode uses mnultiplei static i i0C

segments to achieve intensity modulation. i i sides of each leaf of a. MLC' have a pr, :

tonigue or a i i on one side thiat fits into a similar groove of thie ..11 : i leaf. 'i1. results

in :: --= radiologiical path lenith~s across different parts of the leaves. G~alvin e~t al.

(1~~I :` i'. :4 that the diffetrent radiologicall path lengths mnanifest, themselves a~s

v doses in a, plane perpendicular to the leaf motion. 'i i low- dose region between twvo

ad1 : leaves was I. :i: as th-e tongue-and-gr~ ooveo ii : In an IMI-1'1 treatment using

an MILC? the tongue-a~nd-groov-e effetct occurs wrhen the tongue, or Ithe groove or both for

the rnost timer during treatments deivery cove~r th~e ove~rla~pping region be~twee~c n twLo :P r e~nt

Sof leaves. As io-inted out investigators, the tonguec-andi-gr oovec arrangement

always resullt~s in underdosagfes of as mulch as 10- inl the treatmnentl :-- in both static
and dynamnic multileaf 1' ::: .: (Galvin:- at al. i~ -'-- -" Gav at al -,(1::o l

::' M ohan =- : '\: ; .., t al. :: Sykes and '': i I :- -).

Severali recent publications (van Sanovroort and HT 1::: : 1=- :' We~ bb et al. 1: --., Con-

very a~nd WelTbb 1 : D~irkxi et al1. 1 --. ; Xia? and Verhey 1 = ;) have: shown that the tongue-

and-groovei :: can be significantly reduced by synchr~onizatio n of th~e leaves. However,

thle cost of leaf ~synchir'onizationi is :: :: -11 an icrierase in thle total number of sub i

and monitor units. van Santvoort a~nd i i : (: ::) ] a.an algorithm to elimrinate

tongue-and-groove effects for D>MLC treatment Although 11 note that their al-

gorithm increases Ithe number of monitor units, they do not examine the optimality or

subopt~imality of the plans bi obta~in. We recently published a. paper (K~armath et al1.

.:: ) that gave rnathernatical .. .1 : : and rigorous proofs of leaf sequencing algorithmns

for se~grne~ntal rnuitileaf collimration, wh-ich rnaxirnize MUj ~ : ~ ~- We' 1 that our leaf

sequencing algorithms thatl explicitly account for minimum leaf separation obtain feasible

unidirectional solutions that are optimal. ':' now- extend that wTLork to develop a~lgorithmlrs










that explicitly account for leaf interdigitation and the tongue-and-groove effect and are

optimal in MU efficiency for unidirectional schedules. We show also that the algorithm of
van Santvoort and Heijmen (1996) obtains optimal dynamic multileaf collimation treatment

schedules.

4.1 Algorithm with Interdigitation and Tongue-and-Groove Constraints

4.1.1 Tongue-and-Groove Underdosage Effect

Figure 4-1 shows a '.. I!n-- -, 1 view of the region to be treated by two adjacent leaf pairs,

t and t+1. Consider the shaded rectangular areas At(zi) and At+l(zi) that require exactly

It(zi) and It+l(zi) MUs to be delivered, respectively. The tongue-and-groove overlap area

between the two leaf pairs over the sample point zi, At~t+l(zi), is colored black. Let the

amount of MUs delivered in At,t+l(zi) be It,t+1(as). Ignoring leaf transmission, the following

lemma is a consequence of the fact that At~t+l(z ) is exposed only when both At(zi) and

At+l(zi) are exposed.

Xi-1 Xi Xi+1

It ~ At
It,t+1 :::: ::::m : 11: ::::II At, t+1
It+1i i At+1

Figure 4-1: Tongue-and-groove effect

Lemma 21 It,t+1(i) < min{It(zi), It+1(zi)}, O < i < m, 1 5 t < n.

Schedules in which It,t+1(as) = min{It(zi), It+1(as)} are said to be free of tongue-and-

groove underdosage effects.

Unless treatment schedules are carefully designed, it is possible that It,t+1(zi) <<

min{It(zi), It+1(zi)} for some i and t. For example, in a schedule in which Isr(zi) = 30,

Itz (zi) = 50, I~t+1)r (zi) = 50 and Irt+1)z (as) = 60, we have It,t+1(2i) = Itz (2i) I~t+1)r (2i) =
50 50 = 0. Note that in this case, min{It(zi), It+1(zi)} = I(t+1)l(zi) Itl (zi) = 60 50 =

10. It is clear from this example that It~t+1(zi) could be 0 even when min{It(zi), It+1(2i)}

is arbitrarily large.










4.1.21 Algoritihms

K~ama~th etl al. ( ) an algorithm that greneratles a schedlelt t~hat satisfies

inter-pair mninimnum separation constraint. T1: schedule is optimal in therapy time. Hlow-

ever, it does not account for the longue-a~nd-groove~ : In this section, wre present two

algorithms. Algorithm TON\( : i1 ANDC i :OOVE generates minimum therapy time -:

rectional schiedules that are free of tongue-and-groovev underdiosage and : .. 'e used for

MiL(' s that do not have a inllterdigitat2(ion cionstlraint,. Algorithm T'ON(': : : \;Dc : =)OVE-

ID generates minimum i"..: .1 y time ::. `:: : :: 1 schedules that are: free of (ongue-alnd-

groovie underdosa~ge 1.. simultaneously sa~tisfying thle interdigitation constraint and is for

MLc th~at have an interdigitatlion constraint,.

following lemma provides a necessary and suficiient condition for a

se-chedule to be free of tongiue-and-gir oov:e underdosa~ge ef~ects.

Lemmua 22 A1 ubid'ilr~etionbal scheduled is f~ree~ of '. . -; .7 ..: :!



(a) I?(xi) =- 0 or I?+l(xyi) =- 0, or

(b) < I(t+1: ) (+4(x)<. or


0
Proof: It is easy to see th-at slchedule that satisfie~s theic above conditions is freec of

(ongfue-alnd-groove underdosage: effetcts. So w~hat remains is fo~r us to show that ( sched-

ule that is free of tongue-and-groov-e underdosage : thle above conditions.

Consider :i such schedule. If condition (a) is :i at : i and t, the proof is

So assume i and t such that Ii(x4) 0 ( and ~fIt(x4) 0- exist. We need to

showr that eitheicr (b) or (c) is true for th-is valued of i and t. -- th-e sch-edule is free of

tonigue-and-groove effects,


It,>1(:i) mi {I(x4) It1(x) }> 0(4.1)


the unidirectional constraint, it t i that A?;t+i(x4)i : gets exposed when both

right leaves pass rs, and it remains exposed till the first of th~e : leaves 2,. Further,

if a left leaf i ; a neighiborinig right leaf xi ,.+ 1( ) is niot exposed at









all. So,

I,,t+1(ws) = max {, 0, I~tt+1)(2i) I~t,t+1)r (a) } (4.2)

where I~t,t+1)r (zi ) = max {Itr(i), I~t+1)r (zi)) and I~,t,t+l)lz () = min {It,(i), (,+l), (zi) }.
From 4.1 and 4.2, it follows that


It~t1(w) = ~t~+1)z(ws It~t1)r as)(4.3)

Consider the case It(zi) > It+1(zi). Suppose that Isr(zi) > I~t+1)r(as). It follows that

I~t,t+1)r (2i) = Isr (2i) and I~,t,t+l)lz () = I~t+l)lz (a). Now from 4.3, we get


It,t+1(2i) = I~t+1)z (2i) Isr (2i)

< Irt+1)z (ws) I~t+1)r (2i)
= It+1(2i)

< It(i)

So It~,t+(zi) < min{It(zi), It+1(a))}, which contradicts 4.1. So


Isr (2i) < I~t+1)r (as) (4.4)

Now, suppose that Itl(i) < I~t+1)l(as). From It(zi) > It+1(zi), it follows that I~t,t+1)l(zi) =

rtz(,) and I~t,t+l)r(zi) = I~t+l)r(zi). Hence, from 4.3, we get

It,t+1(2i) = Itz (2i) I~t+1)r (2i)

< Irt+1)z (ws) I~t+1)r (2i)
= It+1(2i)



So It~,t+(zi) < min{It(zi), It+1(a))}, which contradicts 4.1. So


Itz (i) > I~t+1)z (as) (4.5)

From 4.4 and 4.5, we can conclude that when It(zi) > It+1(zi), (b) is true. Similarly one

can show that when It+1(zi) > It(zi), (c) is true. m

Lemma 22 is equivalent to saying that the time period for which a pair of leaves (say

pair t) exposes the region At~t+l(zi) is completely contained by the time period for which










pair t + 1 exposes region At~t+l(zi), or vice versa, whenever It(zi) 0 and It+l(zi) 0 .

Note that if either It(zi) or It+1(zi) is zero the containment is not necessary. We will refer

to the necessary and sufficient condition of Lemma 22 as the tongue-and-groove constraint

condition. Schedules that satisfy this condition will be said to satisfy the tongue-and-

groove constraint. van Santvoort and Heijmen (1996) present an algorithm that generates

schedules that satisfy the tongue-and-groove constraint for DMLC.

Xia and Verhey (1998) claim that every schedule that violates the interdigitation con-

straint also violates the tongue-and-groove constraint. We demonstrate with a counterex-

ample that this is not necessarily the case. The intensity matrix shown in Figure 4-2(a) can

be exposed in a single segment as shown in Figure 4-2(b). The segment is free of tongue-

and-groove constraint violations, while it clearly violates the interdigitation constraint.


0 0 50 50




50 0 0 50 0 0

(a) (b)

Figure 4-2: Counterexample. The inltlencityi matrix shown in (a) can be treated using
a single segment with 50 MUs as shown in (b). Areas shaded dark are covered by left
leaves and those shaded light are covered by right leaves. Areas not shaded are exposed.
Interdigfitation constraint violation occurs though there is no tongue-and-groove violation.


Elimination of tongue-and-groove effect. Note that the schedule generated by Al-

gorithm MULTIPAIR may violate the tongue-and-groove constraint. If the schedule has

no tongue-and-groove constraint violations, it is the desired optimal schedule. If there are

violations in the schedule, we eliminate all violations of the tongue-and-groove constraint

starting from the left end, i.e., from to. To eliminate the violations, we modify those plans

of the schedule that cause the violations. We scan the schedule from to along the positive

a direction looking for the least z, at which there exist leaf pairs n, t, L E {u 1, a + 1},

that violate the constraint at me. After rectifying the violation at z, we look for other

violations. Since the process of eliminating a violation at e,, may at times, lead to new










violations at e,, we need to search afresh from z, every time a modification is made to

the schedule. However, we will prove a bound of O(u) on the number of violations that

can occur at me. After eliminating all violations at a particular sample point, e,, we move

to the next point, i.e., we increment w and look for possible violations at the new point.

We continue the scanning and modification process until no tongue-and-groove constraint
vioaton e ist A,,:lgorithml~ TONGUEA DGOOVE (Figure 4-3) outlines the procedure.

Algorithm TONGUEANDGROOVE
1. z = zo
2. While (there is a tongue-and-groove violation) do
3. Find the least me, 2, > 2, such that there exist leaf pairs n, a + 1, that violate the
tongue-and-groove constraint at me.
4. Modify the schedule to eliminate the violation between leaf pairs n and n + 1.
5. z = 2w
6. End While

Figure 4-3: Obtaining a schedule under the tongue-and-groove constraint


Let M~ = ((III, 11r), (121 2r,), ul ar,)) be the schedule generated by Algorithm

MULTIPAIR for the desired intensity profile.

Let N(p) = ((Illp, IIrp), 21p 2rp),.( I,, ** lpar)) be the schedule obtained after Step 4 of

Algorithm TONGUEANDGROOVE is applied p times to the input schedule M~. Note that

M = N(0).

To illustrate the modification process we use examples. To make things easier, we only

show two neighboring pairs of leaves. Suppose that the (p + 1)th violation occurs between

the leaves of pair n and pair t = u + 1 at me. Note that Itlp(, w ulp 2w), aS Otherwise,

either (b) or (c) of Lemma 22 is true. In case Itlp w,) >ulp(2w,), swap n and t. Now,

we have Itlp zw) < ulp w;). 11 the Sequel, we refer to these n and t values as the n and I

of Algorithm TONGUEANDGROOVE. From Lemma 22 and the fact that a violation has

occurred, it follows that Itrp w)
violation, we modify (Itlp, Itr). The other profiles of N(p) are not modified.










The new plan for pair t, (I,~, Itr1,(p+1)) is aS defined below.

If lulp 2w) ftlp 2w) Iurp 2w) -trp 2w), then


Itl~p1) fl 0 w(4.6)


where AI = lulp2, w ftlp(> Itr(p+1)() I L( t( ), where It(z) is the target

profile to be delivered by the leaf pair t.

Otherwise,

Itr~+1)trp(2 0 w (4.7),


where AI' = lurp w, Itrp 2w)* Itl(p+1)(2 =tr(p 1)() + t(2), where It(z) is the target

profile to be delivered by the leaf pair t.

The former case is illustrated in Figure 4-4 and the latter is illustrated in Figure 4-5. Note

that our strategy for plan modification is similar to that used by van Santvoort and Heijmen

(1996) to eliminate a tongue-and-groove violation for dynamic multileaf collimator plans.


1~(p+1)


Ir(p+1)


Itrp


Xw

Figure 4-4: Tongfue-and-groove constraint violation: casel

















Ilp




Tr(p+1)




t~rp

Xw X

Figure 4-5: Tongue-and-groove constraint violation: case2 (close parallel dotted and solid
line segments overlap, they have been drawn with a small separation to enhance readability)


Since (Itl(p+1), I r(p+1)) differs from (Ilip, Itrp) for z > 2, there is a possibility that

N(p + 1) is involved in tongue-and-groove violations for z > z,. Since none of the other

leaf profiles are changed from those of N(p) no tongue-and-groove constraint violations

are possible in N(p + 1) for z < z,. One may also verify that since Itlo and btro are

non-decreasing functions of z, so also are Atlp and Itrp, P > 0.

ULemmla 23) LetI F = (()I~ofr)) (20I Ibr).)(ll "' 0r)) be any unidirectional schedule for

the desired /poll.-~ that -il. 6. -~ the tongue-and-groove constraint. Let S(p), be the following
assertions.

(a) IL1(2) > Iilp(2), O < i < n, to < z < am

(b) lI,(z) > lirp(2), O < i < n, so < z < sm

S(p) is true for p > 0.
Proof: The proof is by induction on p.

1. Consider the base case, p = 0. From Corollary 1 and the fact that the plans

(lio, liro), O < i < n, are generated using Algorithm SINGLEPAIR, it follows that

S(0) is true.










2. Assume S(p) is true. Suppose Algorithm TONGUEANDGROOVE finds a next vio-

lation and modifies the schedule N(p) to N(p + 1). Suppose that the next violation

occurs between leaf pairs n and t, te { u 1, a + 1}. Hence, Itlp it>) < ulp it,). We

modify pair t's plan for z > at,, to eliminate the violation. All other plans in the

schedule remain unaltered. Therefore, to establish S(p + 1) it suffices to prove that

II I~s) I Tl IMp+1) t (4.8)




1~,(2) > Itr(p+1)() it, (4.9)
Weneed, prove only one of these two relationships;,, sinc I'(s) -r(2) = I,~, 1)(2) -

Itr~~~p+1)~I 0 m *O )tr(p+1) (z)). We now

consider pair t's plan for z > z, and show that Equation 4.8 is l.h-- .-, true. This,

in turn, implies that S(p + 1) is true whenever S(p) is true and hence completes the

proof.

Suppose that lulp(, It lp(w It tirp it) Itrp it,). Then, lulp it) tIrpz to

Itlp it>) Itrp to *O ise. It>z) I Itz,). Clearly, in a schedule F, which is free of
tongue-and-groove violation between pairs n and I at at,, only the- ordering- li(' ,

ler(zwi) < I'sz(2,) < I'l(2z,) is possible (refer Lemma 22) in this scenario (the ex-

ception being when Iz,(zz,) = It(z,,), in which case all the quantities in the ordering

are equal). From this ordering, I'l(zz,) > I'st(zz,). From the induction hypothesis,

I'l(2z,) > lulp2, it isl(p+1) i,). From Equation 4.6, Itl(p+1)z) it> telp(, i

litl(p+1) it,). Hence, I'l(2z,) > Itl(p+1) it,) when litlp it) ftlp it>) tI rp(w t
Itrp i,). A ;-, my,?! ril: argument can be presented to show that I'l~z, >- Itp1 i

when litlp it) ftlp it)>) tirp it) Itrp it,). So I'l(2,,) > I~.,, I )(z,).

It remains to be proved that I'l(as) > I,, I t)(zi), w < i I m. Suppose for a

contractionon that 3,y > w., I'lz,) < I.,;, I L)(2v). Let aI"l = I'l(2z,) Itlo(zz,). Note

that I'z(4,,) > I,,! ,)(4,,) and so AI" > 1t(p+1)2) -It>) 10 f>) = ..,! I ,) v,)- 10 v)>

(from the working of Algorithm TONGUEANDGRO OVE). DefineI aC new pla (ICCI ft I"~)
as follows:












I'iws = Itlo(zi) i< w
I'l(as) aI"l w < i < m


I" ws = Isro(i) i< w


Note that I'~((a) = I'l(2,) AI" = Itlo(z,) > Itlo(z,_l) = If((z,_l). Similarly,
I"(a ) > I"(as_ ) So (I'(,If) is a plan for t. Also, I'~(z,) = I' /-,) AI" I

liz(2v) I,~,, 9(2,) +Itlo(z,) (since AI" > Itl(p+1)2) v 10 vz) as explained above).
From this and our assumption that liz(2,) < Itl(p+1) v,), it follows that I'((z,) <

Ito(z,). Since plan (Itlo, Isro) was generated using Algorithm SINGLEPAIR, I'((z,) <

Itlo(z,) violates Corollary 1. So our assumption was wrong and hence Equation 4.8
is 1.h-- -,. true.



Elimination of tongue-and-groove effect and interdigitation. As we have pointed

out, the elimination of tongue-and-groove constraint violations does not guarantee elimina-

tion of interdigitation constraint violations. Therefore the schedule generated by Algorithm

TONGUEANDGROOVE may not be free of interdigitation violations. The algorithm we

propose for obtaining schedules that simultaneously satisfy both constraints, Algorithm
TO>NGUIEANDG ROOVE-ID,: is -'-' similar-- toAloitmTONGUEAND GROOVE. The only

difference between the two algorithms lies in the definition of the constraint condition. To

be precise we make the following definition.

Definition 1 A unidirectional schedule is said to satisfy the tongue-and-groove-id con-

straint if




for 0 < i < m, 1 5 t < n.

The only difference between this constraint and the tongue-and-groove constraint is

that this constraint enforces condition (a) or (b) above to be true at all sample points zi

including those at which It(zi) = 0 and/or It+l(zi) = 0.










Lemma 24 A schedule -,t. G. the tongue-and-groove-id constraint iffit -,t. G. the tongue-

and-groove constraint and the interdigitation constraint.

Proof: It is obvious that the tongue-and-groove-id constraint subsumes the tongfue-and-

groove constraint. If a schedule has a violation of the interdigfitation constraint, 3i, t,

I(t+1)1(zi) < Isr(zi) or Itlziw) < I~t+1)r(as). From Definition 1, it follows that schedules
that satisfy the tongue-and-groove-id constraint do not violate the interdigitation constraint.

Therefore a schedule that satisfies the tongue-and-groove-id constraint satisfies the tongfue-

and-groove constraint and the interdigfitation constraint.

For the other direction of the proof, consider a schedule O that satisfies the tongue-and-

groove constraint and the interdigfitation constraint. From the fact that O satisfies the

tongue-and-groove constraint and from Lemma 22 and Definition 1, it only remains to be

proved that for schedule O,

(a) Isr (w) < I~t+1)r (2) < I~t+1) (2i) < Itz (2i), or


whenever It(zi) = 0 or It+1(zi) = 0, O < i < m, 1 < t < n.

When It(zi) = 0,

Itz (i) = Isr (a) (4.10)

Since O satisfies the interdigfitation constraint,


Isr (2i) < I~t+1)1(as) (4.11)

and

I~t+)r (i)
From Equations 4.10, 4.11 and 4.12, we get I~t+1)r(zi) I Isr(zi) = rtl(zi) I I(t+1)l(zi). So

(b) is true whenever It(zi) = 0. Similarly, (a) is true whenever It+1(zi) = 0. Therefore, O

satisfies the tongue-and-groove-id constraint. m

Algorithm TONGUEANDGROOVE-ID finds violations of the tong ue-and-groove-id

constraint from left to right in exnactly ~lthe samelt mannerllt~ in whIich AI~lgrithm TONGUE

ANDGROOVE detects tongue-and-groove violations. Also, the violations are eliminated as

before, i.e., as prescribed by Equations 4.6 and 4.7 and illustrated in Figures 4-4 and 4-5,










respectively. Algorithm TONGUEANDGROOVE-ID is shown in Figure 4-6. All notation

used in the algorithm and the related discussion in the remainder of Section 4.1.2 is also

the same as that used in Section 4.1.2 and corresponds directly to the usage in Algorithm

TONGUEANDGROOVE.

Algorithm TONGUEANDGROOVE-ID
1. z = zo
2. While (there is a tongue-and-groove-id violation) do
3. Find the least me, 2, > 2, such that there exist leaf pairs n, a + 1, that violate the
tongue-and-groove-id constraint at me.
4. Modify the schedule to eliminate the violation between leaf pairs n and n + 1.
5. z = 2w
6. End While

Figure 4-6: Obtaining a schedule under both the constraints


Lemma 25 Let F = ((I(1 r'), (1fe I' ), >, (Inc )) be any unidirectional schedule for

the desired prll.-II~ that -,t. 6. -~ the tongue-and-groove-id constraint. Let S(p), be the fol-

lowing assertions.

(a) IL1(2) > Iilp(2), 0 < i < n, to < z < am

(b) lI,(z) > lirp(2), O < i < n, so < z < sm

S(p) is true for p > 0.

Proof: The proof is by induction on p.

1. Consider the base case, p = 0. From Corollary 1 and the fact that the plans

(lio, br~o), O < i < n, are generated using Algorithm SINGLEPAIR, it follows that

S(0) is true.

2. Assume S(p) is true. Suppose Algorithm TONGUEANDGROOVE-ID finds a next

violation and modifies the schedule N(p) to N(p +1). Suppose that the next violation

occurs between leaf pairs n and t, te { u 1, a + 1}. As in the proof of Lemma 23,

we only need prove either Equation 4.8 or Equation 4.9 to complete this proof. We

complete the proof for the following three cases that are exhaustive.

case 1: It(as) 0 and I,(as) 0 .

The remainder of the proof for this case is the same as that of Lemma 23.

case 2: It(as) = 0.

In this case, Itlp 2w) = Itrp w). Since lulp2, w Irp w), we have lulp ,)-










Itlp w,) > urp 2w) Itrp2w,). The modification prescribed by Equation 4.7 is

applicable. Note that if lurp w,) Itrp zw) = Iulp w,) ftlp2w,), Equation 4.6
is the same as Equation 4.7. In particular,


Itr(p+1) 2w) = Itrp 2w> +urp 2w) Itrp 2w) = urp2w,) (4.13)

Since It(m,,) = 0,

Itr~+1) IL) (s,)(4.14)

From Equations 4.13 and 4.14,


Isrp(w f Il(p+1) w,) (4.15)

Since F satisfies the interdigitation constraint, the left leaf of pair t does not

pass me, before the right leaf of pair a passes me,. So,


I1(a,) >lur as,)(4.16)

From S(p) and Equation 4.15, we get,


for as, > lrp ) ; L)w,)(4.17)

Equations 4.16 and 4.17 yield


I1(as,) Itl(p+1)2w,) > 0 (4.18)

Lemma 2b implies,


I1l() Itl() > I~1(as,) It1(as,), a > 2, (4.19)

Subtracting L a, l) (z) from Equation 4.19, and rearranging terms we get


I1(s) Itl(p+1)(2 I \w> 1 (w) f Il f2 -Il(p+1) n, (4.20)

From Equations 4.6 and 4.7 and the working of Algorithm TONGUEAND-

GROOVE ID, it follows that


rtlcp+1) fl t12 ;.! I L)2w> 1 I (w), > ,( (4.21)










From Equations 4.20, 4.21 and 4.18, we get


Izl() Itlg+)(2) '> Itllas) -I-. I ;,!,)(as) > 0, a > 2, (4.22)

Therefore,

IlI(2) > I -,.!, 1 )(2), 2 > 2, (4.23)

case 3: I,(z,) = 0.

The proof is similar to that of case 2.


4.1.3 Efficient Implementation of the Algorithms

In the remainder of this section we will use 'algorithm' to mean Algorithm TONGUE-

ANDGROOVE or Algorithm TONGUEANDGROOVE-ID and 'violation' to mean tongue-

and-groove constraint violation or tongue-and-groove-id constraint violation (depending on

which algorithm is considered) unless explicitly mentioned.

The execution of the algorithm starts with schedule M~ at z = zo and sweeps to the

right, eliminating violations from the schedule along the way. The modifications applied to

eliminate a violation at e,, prescribed by Equations 4.6 and 4.7, modify one of the violating

profiles for z > z,. From the unidirectional nature of the sweep of the algorithm, it is clear

that the modification of the profile for z > z, can have no consequence on violations that

may occur at the point me. Therefore it suffices to modify the profile only at z, at the

time the violation at 2, is detected. The modification can be propagated to the right as

the algorithm sweeps. This can be done by using an (n x m) matrix A that keeps track of

the amount by which the profiles have been raised. A(j, k) denotes the cumulative amount

by which the jth leaf pair profiles have been raised at sample point zk, from the schedule M

generated using Algorithm MULTIPAIR. When the algorithm has eliminated all violations

at each e,, it moves to 2,+ to look for possible violations. It first sets the (w+1)th column

of the modification matrix equal to the wth column to reflect rightward propagation of the

modifications. It then looks for and eliminates violations at z,+l and so on.










The process of detecting the violations at z, merits further investigation. We show

that if one carefully selects the order in which violations are detected and eliminated, the

number of violations at each e,, O < w < m will be O(u).

Lemma 26 The algorithm can be implemented such that O(u) violations occur at each e,,

O < w
Proof: The bound is achieved using a two pass scheme at me. In pass one we check

adjacent leaf pairs (1, 2), (2, 3), .. ., (n 1, n), in that order, for possible violations at me. In

pass two, we check for violations in the reverse order, i.e., (n 1, n), (n 2, a 1), .. ., (1, 2).

So each set of adjacent pairs (i, i + 1), 1 < i < n is checked exactly twice for possible

violations. It is easy to see that if a violation is detected in pass one, either the profile of

leaf pair i or that of leaf pair i + 1 may be modified (raised) to eliminate the violation.

However, in pass two only the profile of pair i may be modified. This is because the profile

of pair i is not modified between the two times it is checked for violations with pair i + 1.

The profile of pair i + 1, on the other hand, could have been modified between these times

as a result of violations with pair i + 2. Therefore in pass two, only i can be a candidate

for t (where t is as explained in the algorithm) when pairs (i, i + 1) are examined. From

this it also follows that when pairs (i 1, i) are subsequently examined in pass two, the

profile of pair i will not be modified. Since there is no violation between adjacent pairs

(1,2), (2, 3),...,(i, i +1) at that time and none of these pairs is ever examined again, it

follows that at the end of pass two there can be no violations between pairs (i, i + 1),

1< i
Lemma 27 For the execution of the algorithm, the time complexity is O(nm).

Proof: Follows from Lemma 26 and the fact that there are m sample points.

Theorem 20 (a) Algorithms TONGUEANDGROOOVE and TONGUEANDGROOVE-

ID terminate.

(b) The schedule generated by Algorithm TONGUEANDGROOVE is free of tongue-and-

groove constraint violations and is optimal in fl... tuptI~ time for unidirectional schedules.

(c) The schedule generated by Algorithm TONGUEANDGROOVE-ID is free of interdig-

itation and tongue-and-groove constraint violations and is optimal in fl... tuptI~ time for

unidirectional schedules.










Proof: (a) Lemma 27 provides a polynomial upper bound (O(n am)) on the complexity

of Algorithms TONGUEANDGROOVE and TONGUEANDGROOVE-ID. The result

follows from this.

(b) When Algorithm TONGUEANDGROOVE terminates, no tongue-and-groove viola-

tions remain. From this and Lemma 23, it follows that the schedule generated by
A~,,c~ ~Tlgorth TONGUEADGOOVE is optimal in therapy time for unidirectional

schedules free of tongue-and-groove violations.

(c) When Algorithm TONGUEANDGROOVE-ID terminates, no tongue-and-groove-id

violations remain and from Lemma 24 the final schedule satisfies the tongfue-and-

groove and interdigitation constraints. From this and Lemma 25, it follows that the

schedule generated by Algorithm TONGUEANDGROOVE-ID is optimal in therapy

time for unidirectional schedules free of both types of violations.



Theorem 21 The schedule generated by the algorithm of van Santvoort and Heij~men (1996)

is free of interdigitation and tongue-and-groove constraint violations and is optimal in ther-

apy time for unidirectional DM~LC schedules with this /****/** '///

Proof: Similar to that of Theorem 20(c). m

4.2 Experimental Validation

The algorithms were validated on a Varian 2100 C/D with 120-leaf MLC (Varian

Medical Systems, Palo Alto, CA). The inltloneityi maps of a 7-field head and neck plan

from a commercial inverse treatment planning --, -r. inl (CORVUS 5.0, NOMOS Corporation,

Cranberry, PA) were sequenced using Algorithm MULTIPAIR, which optimizes the MU

efficiency, and Algorithm TONGUEANDGROOVE-ID, which eliminates the tongue-and-

groove effect and interdigitation. The intensity maps have a bixel size of 1 cm x 1 cm

and a -'II' inltloneity step. Figure 4-7 shows the film measurement of the fluence maps

of the AP field. The tongue-and-groove effect is readily seen in Figure 4-7(a), while it
is completely eliminated in Figure 4 /-7(b):- using '-- Algorithm TONGUEADGOOVE-ID.

Table 4-1 compares the number of segments and the MU efficiencies of all three algorithms.

The MU efficiency is defined as the ratio of the maximum fluence of intensity modulated

field per MU to the fluence of an open field per MU. Compared to the leaf sequences with no










constraints, the consideration of tongue-and-groove correction increased both the number of

segments and MUs, with an average increase of 21% and 19l' i respectively, for the 7 inltloneityi

maps considered here. With the additional elimination of interdigitation, the increases were

-''.' and 2!1' .,i respectively. Examination of all the sub fields of the leaf sequences generated

with Algorithm TONGUEANDGROOVE-ID verified that no interdigfitation constraint has

been violated.
















(a) (b~)

Figure 4-7: Film measurement of the AP field (field ID 1 in Table 1) of a seven-field head and
neck plan. The optimized leaf sequences were generated without (Algorithm MULTIPAIR,

(b)).


4.3 Comparison with Algorithm of Que et al. (2004))

Recently a new algorithm to eliminate tongue-and-groove effects in step and shoot de-

livery has been proposed (Que et al. 2004). The algorithm of Que et al. (2004) is designed to

eliminate tongue-and-groove effect. Although this algorithm eliminates tongue-and-groove

effect on all 1000 random matrices tried in Que et al. (2004), no proof that the algorithm

eliminates tongue-and-groove effect on all possible matrices has been provided. Further, it

is not known whether or not the algorithm of Que et al. (2004) minimizes therapy time.

We .., 1-,... the algorithm of Que et al. (2004) and show that it is 1.h- .-,. successful in

eliminating tongue-and-groove effect; the generated leaf sequence is also free of interdigfi-

tation. We also perform a theoretical and experimental comparison of this algorithm with

our algorithms (K~amath et al. (2004)).










same leaf sequence as that obtained using the 'slidingf window' method of Bortfeld et al.

(1994b3).

Note that the discrete intensity profile that needs to be delivered, I, is output from

the optimizer. Let n be the number of leaf pairs and m be the number of sample points for

each leaf pair (i.e., for each row of the profile). We denote the rows of I by Il, I2, In. Let

It(zi) denote the number of MUs that need to be delivered at sample point i (ith column)

of leaf pair t (tth row).

Lemma 28 The algorithm of Que et al. (I',ir),) generates unidirectional schedules.

Proof: During each iteration, the next segment generated using the 'sliding window'

method is such that the left leaves are positioned at the leftmost non-zero sample point

(i.e., the least i such that It(zi) > It(zi_l), where It(z_l) = 0) for each row t in the residual

matrix I. The right leaves are positioned at the first columns of the respective rows where

there is a decrease in inltlensityi profile (i.e., the least j for which It(zj) < It(zj_1)). For

example, for the single row intensity profile of Figure 4-8, the left leaf will be positioned at

z2 and the right leaf will be positioned at 26. The algorithm of Que et al. (2004) repositions

all right leaves to the position of the leftmost right leaf thus obtained. During the delivery

of this segment, the inltloneityi values in the matrix in the exposed areas decreases, while the

other values remain unaltered. Therefore in the new residual matrix, the leftmost non-zero

points of rows either remain at the same positions as in the residual matrix of the previous

iteration (or the original intensity matrix for the first iteration) or they move to the right.

So the left leaves cannot move to the left between successive segments. It is also easy to

verify that there can be no index k such that in the updated residual inltloneityi matrix zrk

is to the left of the column of right leaf positions in the segment and It(zk) < It k-1>

for some row. It follows that the right leaves cannot move to the left either. So the leaf

movements are unidirectional and from left to right. m

Definition 1 and Lemma 24 are from K~amath et al. (2004) and are used in the proof

of Theorem 22.

Theorem 22 The algorithm of Que et al. (I',ir),) generates schedules free of tongue-and-

groove effect and interdigitation.





















XO X1 X2 X3 X4 X5 X6 X7 X8 X

Figure 4-8: Leaf positions: The left leaf will be positioned at z2, i.e., it will shield to and
zl. The right leaf will be positioned at 26 and will shield me, i > 6.


Proof: Let I l(zi) and Ir (zi), respectively, be the number of MUs delivered when the left

and right leaves of pair t pass zi in the schedule generated by the algorithm of Que et al.

(2004). In the schedule generated, all right leaves pass point zi, O < i < m (during their left

to right movement) after exactly the same number of monitor units (MUs) are delivered.

So Ir (zi) = I t+,)r (zi), O < i < m, 1 < t < n. From this equality, Lemma 28, Definition 1,
and Lemma 24, it follows that the schedule generated by the algorithm of Que et al. (2004)

is free of tongue-and-groove effect and interdigfitation. m

Theorem 23 Let Teng-id be the optimal 'il.. ,~,,;*0 time unidirectional leaf sequence that deliv-

ers an "iil. i10 il ol l.-~II I, while eliminating the tongue-and-groove effect and interdigitation.

The 'it.. r,tya~ time for the schedule generated by the algorithm of Que et al. (.',ie),) is at most

usTeng-id, where n is the number of involved leaf pairs. Further, usTeng-id is a tight bound,

i.e., there exist prll.~II I for which the schedule generated by the algorithm of Que et al.

(.',ir),) requires a therapy time of n*Teng-id-

Proof: Let ar (zi) denote the amount of therapy time for which the right leaf of leaf

pair j stops at zi in the schedule obtained for I using Algorithm MULTIPAIR (K~a-

math et al. 2003). The therapy time for the plan of leaf pair j is the sum of times
for which its right leaf stops at all sample points, which is p m ~ s. h hrp

time of the entire schedule, T, is the maximum of the therapy times of all leaf pairs, i.e.,

T = maxj( {E o Ajr(zi)}. Clearly, Ttng_44 > T. In the schedule generated by the algo-

rithm of Que et al. (2004), all the right leaves stop at each zi for the same amount of










time, say A (zi), which is equal to the maximum of the times for which a right leaf stops

at zi in the schedule generated by Algorithm MULTIPAIR, i.e., A (as) = maxy {Ajr(zi)}.

The therapy time for the schedule generated by the algorithm of Que et al. (2004) is

therefore T/_4= A(a)= o maxy {Ajr(z))}. Since each Ajr(zi), O < i < m,

1 < j < n can contribute a term to this expression for T/ng at most once, Ting-id

L=1 io nir(i) < u s maxjy { o nir(zi)} = n *T I n Ttng-id. Note that the al-
gorithm of K~amath et al. (2004) generates schedules that are optimal in therapy time for

unidirectional schedules. Hence the algorithm of Que et al. (2004) may generate sched-

ules requiring up to n times the therapy time required by the schedules generated by the

algorithm of K~amath et al. (2004).

The above .!! .11- i assumes that leaf pairs are allowed to close within the field as

defined by the collimator jaws. This is true for certain designs of MLCs. For MLCs with

rounded leaf-end design, significant radiation transmission through the closed leaf pairs

requires them to be moved under the collimator jaws. In this case, both the algorithms of

K~amath et al. (2004) and Que et al. (2004) violate the interdigitation constraint, and only

tongue and groove effect is eliminated.

Figure 4-9 shows an intensity map with 4 rows for which the algorithm of Que et

al. (2004) requires 4 20 = 80 MUs. The map can be delivered using 20 MUs without

violating the tongue-and-groove constraint and interdigfitation constraint using Algorithm

TONGUEANDGROOVE-ID (K~amath et al. 2004). The example can be generalized for n

rows. m


20 0 0 0

0 20 0 0

0 0 20 0

0 0 0 20


Figure 4-9: Worstcase example. This intensity map can be delivered using 20 MUs using
Algorithm TONGUEANDGROOVE-ID (K~amath et al. 2004). The algorithm of Que et al.
(2004) delivers this map using 80 MUs.










4.3.2 Results

We implemented Algorithms TONGUEANDGROOVE and TONGUEANDGROOVE-

ID (K~amath et al. 2004) and the algorithm of Que et al. (2004). For performance compar-

ison, we used two separate data sets. The first set consisted of three clinical IMRT plans

with 7, 5 and 7 beams, respectively. The first two plans had a -'II' fluence step and last plan

had a 10% fluence step. Table 4.3.2 gives the total MUs and number of segments required

for each of the 19 beams in the 3 clinical plans. On our clinical data set, the algorithm of

Que et al. (2004) generated schedules with 2-4 times as many MUs and segments as did

the algorithms of K~amath et al. (2004). The second data set consisted of 100,000 randomly

generated 15 x 15 matrices. The intensity values in these matrices were random integers

from 0 to 10. The average MUs and segments for schedules generated using the three

algorithms for this set and their respective standard deviations are shown in Table 4.3.2.

On this set, the algorithm of Que et al. (2004) generated schedules with about 2.5 times

as many MUs and segments as did the algorithms of K~amath et al. (2004). Note that in

both cases the number of MUs and segments in the schedules generated using Algorithm
~~T l~ TONGEA DGROOVE-ID (K~amath et al. 2004) are only slightly greater than in those

generated using Algorithm TONGUEANDGROOVE (K~amath et al. 2004).
4.4 Conclusion

We have described mathematical formalism and rigorous proofs of leaf sequencing algo-

rithms for segmental multileaf collimation, which maximize MU efficiency while completely

eliminating the tongue-and-groove underdosagfe. Even though it has been shown that for

a multiple field IMRT plan (> 5), the tongue-and-groove effect on the IMRT dose distri-

bution is clinically insignificant (Deng et al. 2001) due to the smearing effect of individual

fields, yet it still can be problematic for a small number of fields and for the patient setup

with minimal uncer' lnir -,. Compared to the unconstrained leaf sequencing algorithms, the

presented methods yield leaf sequences, which decreases the MU efficiency a little. But they

completely overcome tongue-and-groove underdosages. One of the methods also eliminates

leaf interdigitation. Most importantly, mathematical proofs show that these algorithms

are optimal in MU efficiency for unidirectional schedules. We have also proved that the

algorithm of Que et al. (2004) generates schedules that are free of the tongue-and-groove













Table 4-2: Number of MUs and segments generated for 19 clinical intensity modu-
lated beams from 3 IMRT plans using algorithms A (Algorithm of Que et al. 2004),
B (Algorithm TONGUEANDGROOVE (K~amath et al. 2004)) and C (Algorithm
TONGUEANDGROOVE-ID (K~amath et al. 2004)). Beams 1-12 have a -'II' fluence step,
while beams 13-19 have a 10% fluence step.

Beam number A BC
MUs Segments MUs Segments MUs Segments
1 780 38 280 14 280 14
2 520 26 200 10 220 11
3 760 33 300 15 320 16
4 840 40 420 21 420 21
5 740 32 280 14 280 14
6 780 34 260 13 280 14
7 640 29 260 11 280 11
8 1500 74 380 19 420 21
9 860 43 240 12 240 12
10 1 500 67 420 20 420 20
11 1660 78 420 21 440 22
12 840 39 280 14 280 14
13 880 78 280 25 280 24
14 1 080 102 300 30 340 33
15 1070 90 310 27 320 26
16 1000 90 340 31 390 36
17 890 71 340 29 340 28
18 990 75 310 29 310 30
19 1 010 84 330 30 330 30


Table 4-3: Average number of MUs and segments generated over a set of 100,000
random 15 x 15 matrices using algorithms A (Algorithm of Que et al. 2004),
B(Algorithm ~~~T'';"TONGUEAD GROOVEi (K~amath et al. 2004)) and C (Algorithm
TONGUEANDGROOVE-ID (K~amath et al. 2004)). The respective standard deviations
are also shown. The intensity values in the matrices were randomly generated integers from
Sto 10.

A B C:
MUs Segments MUs Segments MUs Segments
Average 114.3 111.6 47.5 45.7 48.2 46.4
Standard Deviation 6i.1 6.0 3.4 3.0 3.5 3.0










effct, and i::- : l :: Our analysis shows that th~e algorithmn of i`'-- et, al. generates

schedules that miyv require: up to a1 times the therapy time required by tha~t fo~r an optimal

I seqcuenc~e free of tongue-and-gr oovec :: : and inter~digita~tion, wh-ere n is th-e number of

involved leaf palirs. In experiments witlh < .' and ralndomnly generatled data sets we~t find

that, the alg~rorit~hmn <. (`'.- et alj. ( ::: = ) gen31erartet s schedules tlhat require 2 t~o 4 times tihe

therapy time required the schedules generated our algorithms.

















C il T"ER 5
ALGR YI /0-u i FOR SPLITTING LARGE FIELDS'""'T"

5.1 Introduction

1 of abutting sub-feieds thatl result fromn the ; 1: of a large (

results in longer delivery times, poor ii !I ..., and field matching problems. D~ogan

at al. ( : ) point out that th-e uncertainties in loaf and carriage positions cause errors in

thle delivered dlose (hot or cold spots) along Ithe match line of the a~butting i

observed dose~l -7 of up to 11= aiongi th~e field split "v--- whn theo split line crossed

through the center of thle target for all thie fields. 'ii.. problems of doshrnetric perturbationi

along t~he fieldi split line has been addressed in several recent "!: i: 1 -. (W,~u et al. :::

Hiong at al. : -- Dogan at al. .T solutions included automatic feath-ering of i ir'

fields 1 mnodif -- thei split line position foreach ...i position (11 i ;, et al. :: D ogan

et al. : -- ) or by d~ : : :- ,~ changing radiation intensity in the overlap region of th~e split

fields. NJone of thle field splitting techiniques~ reported in thie literature has addressed the

issue of trecatmnent delivery a~nd MU; :I7- --. We believe th~at it is important to

address this issue.

O~ur optimnal i splitting algorithms with and without :1 .., 1 be integrated

into our previously developed i .1~ sequencing algorithmlrs to < :: i account for interdig-

itation a~nd tonguei-and-gr oovec :i: of some multilea~f ii We provide rigorous

ma2them atlical proofs t~hat t~he i : schemes for 1 1 !; :- are optimal in MUT

clency. 7; i..:T: .:.i 1 results showr thiat our (. .1: .' field i. I:: algor~ithmr -withiout feathi-

ering re~duices total M~is '-` up to --- on i-- 1 cases a~nd up to on synthetic cases

compa--red-- to ~ a comme-----rcia planning i. .. that, also splits fields without featheringS.

5.2 F~ield Splitting W\~ithout Ferathering

5.2.1 Optirnal Field Splitting for One Leaf Pair

In this section we deviate slightly from our -: :- : notation and assume that1 the: sample

points ar~e 2i, 2, .., 4, rather than : 2i, .., 42. All other notation remains unc~hanged..