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Optimization Models for Radiation Therapy

Permanent Link: http://ufdc.ufl.edu/UFE0025021/00001

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Title: Optimization Models for Radiation Therapy Treatment Planning and Patient Scheduling
Physical Description: 1 online resource (114 p.)
Language: english
Creator: Men, Chunhua
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: dao, fmo, imrt, optimization, patient, radiation, stochastic
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: My research addressed optimization models for radiation therapy treatment planning and patient scheduling. In intensity modulated radiation therapy (IMRT) treatment planning problems, I use direct aperture optimization (DAO) that explicitly formulates the fluence map optimization (FMO) problem as a convex optimization problem in terms of all multileaf collimator (MLC) deliverable apertures and their associated intensities and solve it using column generation method. In addition, the interfraction motion has been incorporated to the stochastic-programming based FMO and DAO models. Optimization models for patient scheduling problems in proton therapy delivery have also been studied in this research.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Chunhua Men.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Romeijn, Hilbrand E.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0025021:00001

Permanent Link: http://ufdc.ufl.edu/UFE0025021/00001

Material Information

Title: Optimization Models for Radiation Therapy Treatment Planning and Patient Scheduling
Physical Description: 1 online resource (114 p.)
Language: english
Creator: Men, Chunhua
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: dao, fmo, imrt, optimization, patient, radiation, stochastic
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: My research addressed optimization models for radiation therapy treatment planning and patient scheduling. In intensity modulated radiation therapy (IMRT) treatment planning problems, I use direct aperture optimization (DAO) that explicitly formulates the fluence map optimization (FMO) problem as a convex optimization problem in terms of all multileaf collimator (MLC) deliverable apertures and their associated intensities and solve it using column generation method. In addition, the interfraction motion has been incorporated to the stochastic-programming based FMO and DAO models. Optimization models for patient scheduling problems in proton therapy delivery have also been studied in this research.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Chunhua Men.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Romeijn, Hilbrand E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0025021:00001


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Iwouldliketothankallofthosepeoplewhohelpedmakethisdissertationpossible.First,Iwishtothankmyadvisor,Dr.H.EdwinRomeijnforallhisguidance,encouragement,support,andpatience.Also,IwouldliketothankmycommitteemembersDr.JamesF.Dempsey,Dr.JosephP.Geunes,Dr.StanislavUryasev,andDr.FazilT.Najafortheirveryhelpfulinsights,commentsandsuggestions.Additionally,IwouldliketoacknowledgeZ.CanerTasknandEhsanSalariwhoprovidedtechnicalsupportandassistancewithmyprojects. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 12 CHAPTER 1INTRODUCTION .................................. 13 2ANEXACTAPPROACHTODIRECTAPERTUREOPTIMIZATIONINIMRTTREATMENTPLANNING ............................. 18 2.1Introduction ................................... 18 2.2DirectApertureOptimization ......................... 20 2.3ColumnGenerationAlgorithm ......................... 22 2.3.1Introduction ............................... 22 2.3.2DerivationofthePricingProblem ................... 23 2.3.3SolvingthePricingProblem ...................... 25 2.4Results ...................................... 29 2.4.1ClinicalProblemInstances ....................... 29 2.4.2Dose-VolumeHistogram(DVH)Criteria ............... 32 2.4.3StoppingRules ............................. 33 2.4.4Results .................................. 34 2.4.4.1Deliveryeciency ...................... 36 2.4.4.2Transmissioneects ..................... 37 2.5ConcludingRemarks .............................. 39 3NEWMODELSFORINCORPORATINGINTERFRACTIONMOTIONINFLUENCEMAPOPTIMIZATION ......................... 46 3.1Introduction ................................... 46 3.2Beamlet-basedStochasticOptimizationModels ............... 48 3.2.1Beamlet-basedDeterministicModel .................. 48 3.2.2FirstNewStochasticOptimizationModel ............... 49 3.2.3SecondNewStochasticOptimizationModel ............. 50 3.2.4Results .................................. 52 3.2.4.1Clinicalprobleminstances .................. 52 3.2.4.2Results ............................ 53 3.3Aperture-basedStochasticOptimizationModels ............... 65 5

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........... 65 3.3.2DerivationofthePricingProblem ................... 66 3.3.3Results .................................. 68 3.4ConcludingRemarks .............................. 70 4OPTIMIZATIONMODELSFORPATIENTSCHEDULINGPROBLEMSINPROTONTHERAPYDELIVERY ......................... 73 4.1Introduction ................................... 73 4.2BuildingtheStrategicModels ......................... 75 4.2.1TheObjectiveFunction ......................... 75 4.2.2TheConstraints ............................. 76 4.2.2.1Thecapacity ......................... 76 4.2.2.2Thepatientmix ........................ 77 4.2.2.3Newpatients'treatmentstarting-time ........... 77 4.2.2.4Anesthesiacasesandtwice-a-dayfractions ......... 77 4.2.2.5Gantryspecializationandgantryswitching ......... 78 4.2.3TheOptimizationModel ........................ 79 4.2.4TheModelwithPenaltyFunction ................... 81 4.2.5Results .................................. 82 4.2.5.1Inputdata ........................... 82 4.2.5.2Resultsforthebasicscenario ................ 83 4.2.5.3Sensitivityanalysis ...................... 83 4.2.6ConcludingRemarks .......................... 88 4.3AHeuristicApproachtoOn-linePatient(Re-)scheduling .......... 95 4.3.1ProblemDenition ........................... 95 4.3.2ModelDescription ........................... 96 4.3.2.1Sequenceandtimingdecomposition ............. 96 4.3.2.2Unionsequence ........................ 97 4.3.2.3Restrictedanddesiredtimewindows ............ 98 4.3.2.4Evaluationcriteriafortheunionsequence ......... 98 4.3.3SolutionApproach ............................ 99 4.3.3.1Timingoptimizerforasingleday .............. 99 4.3.3.2Snoutchangeindex(SCI) .................. 102 4.3.3.3Patientdissatisfactionindex(PDI) ............. 102 4.3.3.4Sequenceoptimizer ...................... 103 4.3.3.5Fittinganewpatient ..................... 106 4.3.3.6Numericalexample ...................... 107 4.3.4ConcludingRemarks .......................... 108 REFERENCES ....................................... 109 BIOGRAPHICALSKETCH ................................ 115 6

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Table page 2-1Modeldimensions. .................................. 30 2-2Numberofapertureswithouttransmissioneects. ................. 41 2-3Beam-ontimewithouttransmissioneects. ..................... 41 2-4Aperture-basedFMO:numberofapertureswithtransmissioneects. ...... 42 2-5Aperture-basedFMO:beam-ontimewithtransmissioneects. .......... 42 2-6Aperture-basedFMO:DVHcriteriaunderC1withouttransmissioneects. ... 42 2-7Aperture-basedFMO:DVHcriteriaunderC1withtransmissioneectsaddedafteroptimization. .................................. 43 2-8Aperture-basedFMO:DVHcriteriaunderC1withtransmissioneects. ..... 43 2-9Beamlet-basedFMO:DVHcriteriaunderC1withouttransmissioneects. ... 43 2-10Beamlet-basedFMO:DVHcriteriaunderC1withtransmissioneects. ..... 44 3-1Modeldimensions. .................................. 52 3-2RadiationTherapyOncologyGroup(RTOGP0126)criteriaforprostatecancer. 53 3-3Hotspots(%)forthreemodels. ........................... 59 3-4DVHcriteriaforcriticalstructures(%). ...................... 63 3-5CPUrunningtime(seconds)forthreemodels(%). ................ 63 3-6C1:Numberofaperturesandbeam-ontime. .................... 71 3-7C2:Numberofaperturesandbeam-ontime. .................... 71 3-8C3:Numberofaperturesandbeam-ontime. .................... 71 3-9C4:Numberofaperturesandbeam-ontime. .................... 71 4-1PatientsClassication. ............................... 89 4-2Theresultsforthebasicscenariomodel. ...................... 89 4-3GantryutilizationperdayusingIPmodel(%). .................. 89 4-4Thedenitionsofscenarios. ............................. 90 7

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.................................. 91 4-6Patientmixscenarios(%). .............................. 92 4-7Scenarios17{19:Gantryavailabletime(minutes). ................. 92 4-8Scenario17{19:Gantryutilization(%). ....................... 92 4-9Scenario20:Revenueperfraction. ......................... 93 4-10ResultsforScenario21. ............................... 93 4-11Scenario22:Patientmixfordierentanesthesiaavailabilities. .......... 93 4-12Scenario22:Otherstatisticsfordierentanesthesiaavailabilities. ........ 93 4-13Scenario23:Patientmixfordierentanesthesiaavailabilities. .......... 93 4-14Scenario23:Otherstatisticsfordierentanesthesiaavailabilities. ........ 93 4-15Scenario24:Patientmixfordierentanesthesiaavailabilities. .......... 94 4-16Scenario24:Otherstatisticsfordierentanesthesiaavailabilities. ........ 94 4-17Scenario25:Patientmixfordierentanesthesiaavailabilities. .......... 94 4-18Scenario25:Otherstatisticsfordierentanesthesiaavailabilities. ........ 94 4-19Scenario26:Patientmixfordierentanesthesiaavailabilities. .......... 94 4-20Scenario26:Otherstatisticsfordierentanesthesiaavailabilities. ........ 95 4-21Resultsforschedulinganewpatientondierentgantries. ............. 107 8

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Figure page 1-1(a)TargetsandcriticalstructuresdelineatedonasliceofaCTscan;(b)Radiationbeamspassingthroughapatient. .......................... 14 1-2(a)Amultileafcollimatorsystem1;(b)theprojectionofanapertureontoapatient. ........................................ 15 2-1AtypicalCTsliceillustratingtargetandcriticalstructuredeliniation.Inparticular,thetargetsPTV1andPTV2areshown,aswellastherightparotidgland(RPG),thespinalcord(SC),andnormaltissue(Tissue). ................. 31 2-2DVHsoftheoptimaltreatmentplanobtainedforCase5withC1apertureconstraintsandtheConvergencestoppingrule. ......................... 35 2-3Isodosecurvesfor73.8Gy,54Gy,and30GyonatypicalCTslice,correspondingtotheoptimaltreatmentplanobtainedforCase5withC1apertureconstraintsandtheConvergencestoppingrule. ......................... 35 2-4Case5:Therelativevolumeof(a)PTV1and(b)PTV2thatreceivesinexcessofitsprescriptiondose. ................................ 44 2-5Therelativevolumeof(a)LSGand(b)RSGthatreceivesinexcessof30Gy. 45 3-1TheprobabilitythataspeciedpercentageofthetargetreceivedtheprescriptiondoseforCase1(rststochasticmodel). ....................... 55 3-2TheprobabilitythataspeciedpercentageofthetargetreceivedtheprescriptiondoseforCase1(secondstochasticmodel). ..................... 56 3-3TheprobabilitythataspeciedpercentageofthetargetreceivedtheprescriptiondoseforCase1. .................................... 58 3-4TheprobabilitythataspeciedpercentageofthetargetreceivedtheprescriptiondoseforCase2. .................................... 58 3-5TheprobabilitythataspeciedpercentageofthetargetreceivedtheprescriptiondoseforCase3. .................................... 58 3-6TheprobabilitythataspeciedpercentageofthetargetreceivedtheprescriptiondoseforCase4. .................................... 59 3-7TheprobabilitythataspeciedpercentageofthetargetreceivedtheprescriptiondoseforCase5. .................................... 59 3-8DVHsforcriticalstructures(rststochasticmodelvs.traditionalmodel). ... 60 9

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.. 61 3-10DVHsforCase1inScenario135(rstmodelvs.traditionalmodel). ...... 62 3-11DVHsforCase1inScenario135(secondmodelvs.traditionalmodel). ..... 62 3-12PerceivedandactualDVHsforcriticalstructuresforCase1andCase2 ..... 63 3-13PerceivedandactualDVHsfortarget. ....................... 64 3-14Targetdosecoveragefortwomodelswithtwostoppingrules. ........... 72 10

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Myresearchaddressedoptimizationmodelsforradiationtherapytreatmentplanningandpatientscheduling.Inintensitymodulatedradiationtherapy(IMRT)treatmentplanningproblems,Iusedirectapertureoptimization(DAO)thatexplicitlyformulatestheuencemapoptimization(FMO)problemasaconvexoptimizationproblemintermsofallmultileafcollimator(MLC)deliverableaperturesandtheirassociatedintensitiesandsolveitusingcolumngenerationmethod.Inaddition,theinterfractionmotionhasbeenincorporatedtothestochastic-programmingbasedFMOandDAOmodels.Optimizationmodelsforpatientschedulingproblemsinprotontherapydeliveryhavealsobeenstudiedinthisresearch. 11

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Everyyear,approximately300,000peopleintheU.S.thatarenewlydiagnosedwithcancermaybenetfromradiationtherapy(AmericanCancerSociety[ 2 ]).Withthistreatmentmodality,externalbeamsofradiationpassthroughapatientwiththeaimofkillingcancerouscellsandtherebycuringthepatient.However,radiationkillsbothcancerousandnormalcells.Manypatientswhoareinitiallyconsideredcurabledieoftheirdiseaseandothersmaysuerunintendedsideeectsfromradiationtherapy.Themajorreasonisthatradiationtherapytreatmentplansoftendelivertoolittleradiationdosetothecancerouscells,toomuchradiationdosetohealthyorgans,orboth.Thegoalofradiationtherapytreatmentplanningisthereforetodesignatreatmentplanthatdeliversaprescribeddosetoregionsinthepatientthatcontain(oraresuspectedtocontain)cancerouscells(oftencalledtargets),whilesparingnearbyfunctionalorgans(oftencalledcriticalstructures).Figure 1-1 (a)showsasliceofaCTscanonwhichseveraltargets(PTV1andPTV2)andcriticalstructures(spinalcord(SC),rightparotidgland(RPG)),andtissuearedelineated.Typically,thereareseveralclinicaltargetsthatwewishtoirradiateandseveralcriticalstructuresthatwewishtospare. Itmaybepossibletoeradicatethediseasewithasinglebeamofradiation.However,suchatreatmentmaysignicantlydamagenormalcellsincriticalstructureslocatedalongthepathofthebeam.Hence,multiplebeams(usually3{9)areused,whoseintersectionprovidesahighdose,whereasregionscoveredbyasinglebeamoronlyafewbeamsreceivemuchlowerradiationdoses(seeFigure 1-1 (b)).Inparticular,patientsreceivingradiationtherapyaretypicallytreatedonaclinicalradiation-deliverydevicecalledalinearaccelerator,whichcanrotatearoundthepatient.Conformalradiationtherapyseekstoconformthegeometricshapeofthedeliveredradiationdoseascloselyaspossibletothatoftheintendedtargets.Inconventionalconformalradiationtherapy,thisusuallymeans 12

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(b) Figure1-1. (a)TargetsandcriticalstructuresdelineatedonasliceofaCTscan;(b)Radiationbeamspassingthroughapatient. thatfromeachbeamdirectionwedeliverasinglebeamwithuniformintensitylevelwhoseshapeconformstothebeam'seyeviewofthetargetsinthepatientasseenfromthatbeam.Sincethegeometryofthetargetsandcriticalstructureisdierentineachpatient,customizedapertureshavetobemanufacturedforeachpatient.ArelativelyrecentradiationtherapytreatmentdeliverytechniqueisIntensityModulatedRadiationTherapy(IMRT).Withthistechnique,whichhasbeenemployedinclinicssince1994,thelinearacceleratorisequippedwithaso-calledmultileafcollimator(MLC)system.Thissystemisabletoblockdierentpartsofthebeam,sothatitcandynamicallyformalargenumberofdierentapertures(seeFigure 1-2 ).IMRTthusallowsforthedeliveryofatreatmentplanthatusesamuchlargernumberofdierentaperturesthanconventionalconformalradiationtherapy,andthusthecreationofverycomplexnonuniformdosedistributionsthatdeliversucientlyhighradiationdosestotargetswhilelimitingtheradiationdose 13

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(b) Figure1-2. (a)Amultileafcollimatorsystem1;(b)theprojectionofanapertureontoapatient. deliveredtohealthytissues.TheadventofIMRThasdramaticallyimprovedtreatmenteciencyandquality. Traditionally,IMRTtreatmentplansaredevelopedusingatwo-stageprocess.Inparticular,wemodeleachbeamasacollectionofhundredsofsmallbeamlets(orbixels),andconsidertheintensitiesofeachofthesebeamletstobecontrollableonanindividualbasis.Theproblemofndinganoptimalintensityprole(alsocalleduencemap)foreachbeamiscalledthe(beamlet-based)uencemapoptimization(FMO)problem.Thegoalofthisproblemistodevelopatreatmentplanthatsatisesand/oroptimizesseveralclinicaltreatmentplanevaluationcriteria.However,thisuencemapmustthenbefollowedbyaleaf-sequencingstageinwhichtheuencemapsaredecomposedintoamanageablenumberofaperturesthataredeliverableusingamultileafcollimator(MLC)system.Theobjectiveofthissecondstageproblemistoaccuratelyreproducetheidealuencemapwhilelimitingthetotaltreatmenttime.Moreformally,inthissecondstageitisdesirabletolimitboththetotaltimethatradiationisdelivered,i.e.,thetotalbeam-on-time,andthetotalnumberofaperturesused.InChapter 2 ,weconsidertheproblemofIMRTtreatmentplanningusinganapproachwhichintegrates 14

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Ingeneral,theIMRTtreatmentplansaredeliveredduring5-7weeksandadailytreatmentiscalledafraction.Duringacourseoffractionatedradiationtherapyandbetweenthefractionstheorgansofthepatients'bodymaymove.Organmovementsaredividedintotwogeneralcategories:interfractionmotionandintrafractionmotion([ 36 ]).Interfractionorganmotionconsistsofday-to-daychanges,suchasset-uperrors,tumorshrinkage,weightlost,etc.Intrafractionmotionreferstotheinternalorganmotionoccurringduringtheactualradiationtreatmentduetobreathing,swallowing,etc.Traditionally,theIMRTtreatmentsystemsarebasedonastaticpatientmodel,whichreliesonasinglestaticplanningcomputedtomography(CT)imagefortreatmentplanningandevaluation.Toaccountfororganmotion,theconventionalmethodistoaddamarginaroundtheclinicaltumorvolume(CTV)togetaplanningtargetvolume(PTV)([ 26 27 ]).Insteadofusingamargin,inChapter 3 ,weintroducenewstochastic-programmingbasedmodelsforincorporatinginterfractionmotioninFMO. Finally,achallengeofradiationtherapyhasbeentoecientlydeliverhighqualitytreatmentusinglimitedandexpensiveresources.Eectiveschedulingsystemshavetheconictinggoalsofmaximizingtheutilizationofresourcesandthenumberofpatients,andminimizingthepatientwaitingtime.Thetreatmentprovidersareunderpressuretoreducethecostandimprovetreatmentquality.Inrecentyears,patientschedulingisgraduallybecominganessentialcomponentinmedicalcare.However,littleattentionhasfocusedonpatientschedulinginradiationtherapy.InChapter 4 ,wedevelopoptimization 15

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64 ].Morerecently,Leeetal.[ 40 41 ]studiedmixed-integerprogrammingapproaches,Romeijnetal.[ 61 ]proposednewconvexprogrammingmodels,andHamacherandKufer[ 24 ]andKuferetal.[ 37 ]consideredamulti-criteriaapproachtotheproblem.Theproblemofleaf-sequencingwhileminimizingtotalbeam-on-timeisveryecientlysolvableingeneral.WereferinparticulartoAhujaandHamacher[ 1 ],Bortfeld[ 11 ],Kamathetal.[ 31 ],andSiochi[ 66 ];inaddition,Baataretal.[ 5 ],Bolandetal.[ 10 ],Kamathetal.[ 32 33 34 35 ],Lenzen[ 42 ],andSiochi[ 66 ]studytheproblemunderadditionalMLChardwareconstraints,whileKalinowski[ 30 ]studiesthebenetsofallowingrotationoftheMLChead.TheproblemofdecomposingauencemapintotheminimumnumberofapertureshasbeenshowntobestronglyNP-hard(seeBaataretal.[ 5 ]),leadingtothedevelopmentofalargenumberofheuristicsforsolvingthisproblem.NotableexamplesaretheheuristicsproposedbyBaataretal.[ 5 ](whoalsoidentifysomepolynomiallysolvablespecialcases),DaiandZhu[ 20 ],Que[ 54 ],Siochi[ 66 ],andXiaandVerhey[ 70 ].Inaddition,Engel[ 23 ],Kalinowski[ 29 ],andLimandChoi[ 43 ]developedheuristicstominimizethenumberofapertureswhileconstrainingthetotalbeam-on-timetobeminimal.Langeretal.[ 39 ]developedamixed-integerprogrammingformulationoftheproblem,whileKalinowski[ 28 ]proposedanexactdynamicprogrammingapproach. 17

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68 ]proposeanewexactoptimizationapproachtotheproblemofminimizingtotaltreatmenttime. Amajordrawbackofthedecouplingofthetreatmentplanningproblemintoabeamlet-basedFMOproblemandaMLCleafsequencingproblemisthatthereisapotentiallossoftreatmentquality.Thishasledtothedevelopmentofapproachesthatintegratethebeamlet-basedFMOandleaf-sequencingproblemsintoasingleoptimizationmodel,whichareusuallyreferredtoasdirectapertureoptimizationapproachestoFMO.Inthisapproach,weexplicitlysolveforasetofaperturesandcorrespondingintensitiesinasingleaperture-basedFMOproblem.Forexamplesofintegratedapproachestouencemapoptimization,sometimesalsocalledaperturemodulationoraperture-baseduencemapoptimization,werefertoPreciado-Waltersetal.[ 52 ],Shepardetal.[ 63 ],Siebersetal.[ 65 ],Bednarzetal.[ 8 ]andRomeijnetal.[ 60 ].Thewaythedosedistributionreceivedbythepatientismodeledinabeamlet-basedFMOmodelisnecessarilyanapproximationsincethisdistributiondependsnotonlyontheintensityprolebutalsoontheactualaperturesusedtodeliverthisprole.Thecurrentliteratureonaperturemodulationhas,however,hasnotyetexploitedtheabilityofaperturemodulationtotakeintoaccountsucheects.Inparticular,whiletheleavesintheMLCsystemdoblockmostoftheradiationbeam,thereissomesmallbutnotinsignicantamountofdose(ontheorderof1.5{2%,seeArneldetal.[ 4 ])thatistransmittedthroughtheleavesintheMLCsystem.Finally,whileseveralaperture-basedFMOapproachesattempttolimittotaltreatmenttimebylimitingthenumberofaperturesused,thesemodelsdonotexplicitlyincorporatetotalbeam-on-timeasameasureoftreatmentplaneciency. Inthischapter,weextendtheapproachdevelopedbyRomeijnetal.[ 60 ]by(i)allowingfortheincorporationofmoregeneraltreatmentplanevaluationcriteriaandtreatmentplanconstraints;(ii)accountingfortransmissioneects.Inaddition,wealso 18

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Thedosedistributioninapatientisevaluatedonadiscretizationofthe3-dimensionalgeometryofthepatient,obtainedviaaCTscan,intoanumberofvoxels.WedenotethesetofallvoxelsbyV,andassociateadecisionvariablezjwitheachvoxelj2Vthatindicatesthedosereceivedbythatvoxel.Thevectorofvoxeldosescanbeexpressedasalinearfunctionoftheintensitiesoftheaperturesthroughtheso-calleddosedepositioncoecientsDkj,thedosereceivedbyvoxelj2Vfromaperturek2Katunitintensity. 19

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62 ]).Examplesofsuchcriteriaaretumorcontrolprobability(TCP),normaltissuecomplicationprobability(NTCP),equivalentuniformdose(EUD),conditionalvalue-at-risk(CVaR),voxel-basedpenaltyfunctions,etc.(see,e.g.,Niemierko[ 49 ]and[ 50 ],LuandChin[ 45 ],KutcherandBurman[ 38 ],RockafellarandUryasev[ 58 ]). Ouraperture-basedFMOmodelcannowbeformulatedasfollows:minimizeX`2LG`(z) subjectto(A) Herez2RjVjandy2RjKjarethevectorscontainingthevoxeldosesandapertureintensities,respectively.Manyotheraperture-basedFMOmodelsthathavebeenproposedintheliteratureareheuristicsthatarebasedondeterministicorstochasticsearch,suchassimulatedannealing,forwhichitoftencannotbeguaranteedthatalldeliverableapertures 20

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Traditionalbeamlet-basedFMOmodelsaswellasallaperture-basedFMOmodelstodatehaveassumedthatthedosedepositioncoecientscanbewrittenas whereDijisthedosereceivedbyvoxeljfrombixeliatunitintensity.However,thisdenitionignoresanytransmissionandscattereectsthatareduetotheshapeoftheaperturesused.Bothoftheseeectscannotbemodeledinabeamlet-basedFMOmodel.Wewillexplicitlyincorporatethetransmissioneect.Inparticular,theexpressionforthedosedepositioncoecientsgivenin( 2{3 )assumesthatanybixelthatisblockedinanaperturedoesnottransmitanyradiation.Ifwedenotethefractionofdosethatistransmittedby2[0;1],weobtainthefollowingexpressionforthedosedepositioncoecients: 2{3 )correspondstothespecialcasewhere=0. 2.3.1Introduction 21

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(i) identiesoneormorepromisingaperturesthatwillimprovethecurrentsolutionwhenaddedtothecollectionofconsideredapertures;or (ii) concludesthatnosuchaperturesexist,andthereforethecurrentsolutionisoptimal. Incase(i),weaddtheidentiedaperturesto^K,re-optimizethenewaperture-basedFMOproblem,andrepeattheprocedure.Intuitively,thepricingproblemidentiesthoseaperturesforwhichtheimprovementoftheobjectivefunctionperunitintensityislargest(andthereforeshowpromiseforsignicantlyimprovingthetreatmentplan).Theverynatureofourapproachthusallowsustostudytheeectofaddingaperturesonthequalityofthetreatmentplan,therebyenablingasoundtrade-obetweenthenumberofaperturesandtreatmentplanquality. 2{1 )and( 2{2 )byj(j2V)andk(k2K).TheKarush-Kuhn-Tucker(KKT)optimalityconditions(see,e.g.,Bazaraaetal.[ 7 ])for(A),whicharenecessaryandsucientforoptimalitybecauseofthe 22

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Wecannowjustifytheintuitionbehindthepricingproblemandthecolumngenerationalgorithmthatwasprovidedearlier:realizingthatPj2VDij^jmeasurestheper-unitchangeinobjectivefunctionvalueiftheintensityofbeamletiisincreased,itfollowsthatthepricingproblemforagivenbeamidentiestheaperturewiththepropertythattherateofimprovementinobjectivefunctionvalue,astheintensityoftheapertureisincreased,islargestamongalldeliverableapertures.Furthermore,thisapertureisaddedtothemodelonlyifincreasingtheintensityofthatapertureactuallycorrespondstoanimprovementinobjectivefunctionvalue. C1. 24

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C3. C4. NotethatconstraintC4correspondstotheuseofconventionaljawsonly.Recently,theviabilityofthisdeliverytechniquehasbeenshownbyKimetal.[ 71 ]fortreatingprostatecancerandbyEarletal.[ 21 ]fortreatingpancreas,breast,andprostatecancer. Romeijnetal.[ 60 ]providepolynomial-timealgorithmsforsolvingthepricingproblemcorrespondingtoC1{C3.Inparticular,supposethateachbeamisdiscretizedintoanmnmatrixofbixels.TheythenshowthatthepricingproblemforaparticularbeamcanbesolvedinO(mn)timeforC1andinO(mn4)timeforC2andC3.Forcompletenesssake,wewillbrieydescribethesealgorithmsbelow.Next,wewilldevelopanecientalgorithmforsolvingthepricingproblemunderC4. Itiseasytoseethat,underC1,thepricingproblemdecomposesbybixelrow,i.e.,wemayndtheoptimalleafsettingsforeachrowindividuallyandthenformtheoptimalaperturebysimplycombiningtheseleafsettings.Wearethusinterestedinnding,foreachbixelrow,aconsecutivesetofbixelsforwhichthesumoftheircoecientsintheobjectivefunctionofthepricingproblemisminimal.Wecanndsuchasetofbixelsbymakingasinglepass,fromlefttoright,throughthenbixelsinagivenrowandbeam.Indoingso,weshouldkeepingtrackof(i)thesumofthecoecientsforallbixelsconsideredsofar,and(ii)themaximumvalueofthesesumsencounteredsofar.Nownotethat,atanypointinthisprocedure,thedierencebetweenthesetwoisacandidateforthebest 25

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6 ]andBentley[ 9 ].) ThealgorithmforidentifyingtheoptimalaperturetoaddunderC2andC3issomewhatmorecomplicated.Forthesetwosituations,weformulatethepricingproblemastheshortestpathprobleminanappropriatelydenednetwork.Inparticular,wedeneanodecorrespondingtoeachpotentialleafsettingineachbixelrow,i.e.,(r;c1;c2)forr=1;:::;mandc1;c2=1;:::;nwithc1
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Clearly,if,foreachbeamb2B,thebestsolutionfoundhasanobjectivefunctionvaluethatexceedsthecorrespondingthresholdvalue( 2{4 )derivedinSection 2.3.2 ,no 27

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2.4.1ClinicalProblemInstances Foralltencases,wedesignedplansusingveequispaced60Co-beams.Notethatthisdoesnotaecttheoptimizationalgorithm,whichis,withoutmodication,applicabletohigh-energyX-raybeamsaswell(forexample,seeRomeijnetal.[ 60 ]forresultswith6MVphotonbeams).Thevebeamsareevenlydistributedaroundthepatientwithangles0;72;144;216;288,respectively.Thenominalsizeofeachbeamis4040cm2.Thebeamsarediscretizedintobixelsofsize11cm2,yieldingontheorderof1,600bixels.However,wereducethesetofbeamletsthatactuallyneedtobeconsideredintheoptimizationbyusingthefactthattheactualvolumetobetreatedisusuallysignicantlysmaller.Thatis,foreachbeam,weidentifya\mask"consistingofonlythosebixelsthatcanhelptreatthetargets,i.e.,weidentifythebixelsforwhichthedosedepositioncoecientDijassociatedwithatleastonetargetvoxelisnonzero.WethenextendthemasktoarectangleofminimumsizetoensurethatalldeliverableaperturesfromC1{C4thatcanhelptreatthetargetsareconsidered.Forallcaseswegenerateavoxelgridwithvoxelsofsize444mm3.Todecreasethesizeofproblems,weuseavoxelsize 28

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2-1 showstheproblemdimensionsforthetencases. Table2-1. Modeldimensions. case#structurestotal#voxels#voxelsinthemodels#bixels 11485,01717,576990 213104,29824,3001,637 38189,23436,4141,658 411195,11338,6802,006 51286,25515,9851,113 61358,63613,878765 710102,26221,3861,247 81084,36918,6611,149 91071,83714,664938 1012148,29440,2022,183 Eachcasecontainedtwotargets,whicharereferredtoasPlanningTargetVolume1(PTV1)andPlanningTargetVolume2(PTV2).PTV1consiststheGrossTumorVolume(GTV,whichreferstothebestclinicalestimationoftheexactareasoftheprimarytumorvolume)expandedtoaccountforbothsub-clinicaldiseaseaswellasdailysetuperrorsandinternalorganmotion;PTV2isalargertargetthatalsocontainshigh-risknodalregions,againexpandedtoaccountforsub-clinicaldiseaseandsetuperrorsandorganmotion.PTV1andPTV2haveprescriptiondosesof73.8Gyand54Gy,respectively.Figure 2-1 showsanexampleoftargetdeliniation. OurFMOmodelemployedtreatmentplanevaluationcriteriathatarequadraticone-sidedvoxel-basedpenalties.Inparticular,denotingthesetoftargetsbyTthesetofcriticalstructuresbyC,thesetofallstructuresbyS=T[C,andthesetofvoxelsin 29

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AtypicalCTsliceillustratingtargetandcriticalstructuredeliniation.Inparticular,thetargetsPTV1andPTV2areshown,aswellastherightparotidgland(RPG),thespinalcord(SC),andnormaltissue(Tissue). structures2SbyVs,weusethefollowingtreatmentplanevaluationcriteria: (Clearly,thismeansthatthesetoftreatmentplanevaluationcriteriacanbeexpressedasL=fs:s2Tg[fs+:s2Sg.)Thecoecientssands(s2S)arenonnegativeweightsassociatedwiththeclinicaltreatmentplanevaluationcriteria.Criteria( 2{5 )penalizeunderdosingbelowtheunderdosingthresholdTsinalltargetss2T,whilecriteria( 2{6 )penalizeoverdosingabovetheoverdosingthresholdT+sinallstructuress2S.Wechoosethismodelbasedonthefactthatit,inourexperience,canbesolvedveryecientlyandyieldshigh-qualitytreatmentplans.However,recallthatouralgorithmcaneasilybeappliedtomodelsthatincludeotherconvextreatmentplanevaluationcriteria,suchasvoxel-basedpenaltyfunctionswithhigherpowers,orEUD.Theresulting 30

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55 ],[ 56 ]): { Atleast99%shouldreceive93%oftheprescribeddose(0:9373:8Gy) { Atleast95%shouldreceivetheprescribeddose(73.8Gy) { Nomorethan10%shouldbeoverdosedbymorethan10%ofprescribeddose(1:173:8Gy) 31

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Nomorethan1%ofPTV1shouldbeoverdosedbymorethan20%ofprescribeddose(1:273:8Gy) { Atleast99%shouldreceive93%oftheprescribeddose(0:9354Gy) { Atleast95%shouldreceivetheprescribeddose(54Gy) { Nomorethan50%ofeachglandshouldreceivemorethan30Gy { Tissueshouldreceivelessthan60Gy { Spinalcordshouldreceivelessthan45Gy { Mandibleshouldreceivelessthan70Gy { Brainstemshouldreceivelessthan54Gy { Eyeshouldreceivelessthan45Gy { Opticnerveshouldreceivelessthan50Gy { Opticchiasmshouldreceivelessthan55Gy 32

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Ineithercase,wesaythatthealgorithmhasconvergedifthestoppingrulehasbeensatisedforallconvergence(resp.clinical)criteria.Moreover,wereportthesolutionobtainedintherstiterationofthelastsequenceof5iterations. 1 .First,weevaluatetheabilityofourapproachtoecientlyndhigh-qualitytreatmentplanswithalimitednumberofapertures,aswellastheeectofMLCdeliverabilityconstraintsontherequirednumberofaperturesandbeam-on-time.Next,weevaluatetheimportanceofexplicitlyincorporatingtransmissioneectsintotheoptimizationmodel.However,beforewedoso,inFigures 2-2 and 2-3 weillustratethebehaviorofourFMOmodelbyshowingtheDVHsandisodosecurvessuperimposedonatypicalCTslice,bothcorrespondingtoanoptimaltreatmentplanfoundforCase5.The 33

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Figure2-2. DVHsoftheoptimaltreatmentplanobtainedforCase5withC1apertureconstraintsandtheConvergencestoppingrule. Figure2-3. Isodosecurvesfor73.8Gy,54Gy,and30GyonatypicalCTslice,correspondingtotheoptimaltreatmentplanobtainedforCase5withC1apertureconstraintsandtheConvergencestoppingrule. 34

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2-2 and 2-3 showthenumberofaperturesandbeam-on-time(inminutes)forthetencasesobtainedwithouraperture-basedapproach.WeusedthetwostoppingrulesdescribedinSection 2.4.3 .Fortheseexperiments,wehavenotincorporatedanytransmissioneects.Forcomparisonpurposes,thetablesalsoshowtheresultsoftraditionalbeamlet-basedFMOwheretheoptimaluencemapswererstdiscretizedtointegermultiplesof5%ofthemaximumbeamletintensityineachbeam,andsubsequentlydecomposedintoapertureswiththeobjectivefunctionofminimizingthebeam-ontime.WeusedthealgorithmsbyKamathetal.[ 31 32 ]tominimizethebeam-on-timeunderC1andC2,respectively.Furthermore,weappliedamodicationoftheapproachbyBolandetal.[ 10 ]tominimizethebeam-on-timeunderC3,whileweusedalinearprogrammingmodelforthecaseofC4.(Notethatthereare,ingeneral,manydecompositionsthatattaintheminimumbeam-on-time;thenumberofaperturesthatisgivenisforaparticularsolutionthattheso-calledleaf-sequencingalgorithmfound,butthatnumberisnotminimizedexplicitly.) Ourmainconclusionsare: 35

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Theamountoftimerequired,onaverage,byouroptimizationalgorithmtondtheaperturebasedsolutionsrangesfromabout1{3minutesofCPUtimeforthecaseofconsecutivenessconstraintsonly.Therequiredtimeincreasestoupto4minuteswheninterdigitationandconnectednessareimposed,whileitfurtherincreasestouptoabout12minuteswhenonlyrectangularaperturesareallowed.Thisisincomparisonwithanaverageofalittleover1minuteofCPUtimerequiredforthetraditionaltwo-phaseapproach.NotethatthesetimesdonotincludethetimerequiredtoreadtheDICOMdataandcomputethedosedepositionconstraints,whichtookabout10{25minutesofCPUtimedependingonthesizeofthecase.However,notethatthesetasksonlyneedtobeperformedonceforeachpatient. Toillustratehowourapproachmaybeusedtomakeatrade-obetweentreatmentplanqualityanddeliveryeciency,Figures 2-4 { 2-5 showthebehavioroftargetcoverageandsubmandibularglandsparingasafunctionofthenumberofaperturesusedforC1{C4foratypicalexample,Case5. 2-4 and 2-5 showthenumberofaperturesandbeam-on-timewithincorporationoftransmissioneects.Wehaveused=1:7%asthetransmissionrate(seeArneldetal.[ 4 ]).AcomparisonwithTables 2-2 and 2-3 revealsthattheincorporationoftransmissioneectshasverylittleeectonthetreatmentdeliveryeciency. 36

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2-6 { 2-8 showtheresultsoftheaperture-basedFMOproblemunderC1withoutandwithtransmissioneectsusingtheconvergencestoppingrule.Inparticular,thetablesshowvaluesoftheDVHcriteriaforthetargetsandmaincriticalstructuresforalltencases.Forexample,thedatainthecolumnlabeledPTV2@93%representsthefractionofvolume(in%)ofPTV2receiving0:9354Gy.Thelabelsoftheothercolumnsfollowasimilarformat,wheretheacronymscorrespondtotheleftparotidgland(LPG),rightparotidgland(RPG),leftsubmandibulargland(LSG),andrightsubmandibulargland(RSG).ComparingTables 2-6 and 2-8 suggeststhattreatmentplansofverysimilarqualitycanbefoundwithandwithoutincorporationoftransmissioneects.However,theresultsofTable 2-6 neitherincorporatetransmissionintheoptimizationproblemnoraccountfortransmissioninthepresentationoftheactualresults,sothatthetreatmentplanqualityinthattableisaperceivedratherthananactualone.Table 2-7 showstheactualqualityofthetreatmentplanthatwasobtainedwhensolvinganoptimizationmodelthatdoesnottaketransmissioneectsintoaccount,i.e.,fortheresultsinTable 2-7 transmissioneectswereaddedaposterioritotheplan. Finally,weanalyzedtheresultsofusingabeamlet-basedFMOapproachfollowedbyaleafsequencingphase,underC1.Table 2-9 showstheperceivedqualityoftheobtainedtreatmentplan(inwhichtransmissioneectsareignored),whileTable 2-10 showstheactualqualityofthetreatmentplan(inwhichtransmissioneectsareaddedtothenaltreatmentplan). Fromthelattertwotables,itisclearthatusingabeamlet-basedFMOoptimizationapproachmayseverelyunderestimatetargethotspots(overdosing)andeectsoncriticalstructures.Thedirectapertureoptimizationapproachwithtransmissioneectsincorporated,however,providesahigh-qualitytreatmentplanwith,onaverage,acomparablenumberofaperturesandbeam-on-time.TakingCase5asanexample,a 37

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38

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39

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Numberofapertureswithouttransmissioneects. Aperture-basedBeamlet-based caseClinicalConvergence C1C2C3C4C1C2C3C4C1C2C3C4 117292259574851142170189329334 2354533115474733132213230390650 31919203335313339212252384592 42324245129292452262293449636 53219264239376455185227348378 61829425254659383183211285321 71818193249223055196219357445 85072861377710086137173198327394 91920233830302766181203312384 101719194724273459248295418611 Average24.829.431.460.644.143.647.582.0202.3231.7359.9474.5 Table2-3. Beam-ontimewithouttransmissioneects. Aperture-basedBeamlet-based caseClinicalConvergence C1C2C3C4C1C2C3C4C1C2C3C4 12.562.902.706.593.983.523.5612.437.438.2314.5114.67 22.983.092.779.143.213.122.779.848.469.1615.6526.00 32.922.722.704.663.303.083.064.988.079.4314.4822.86 42.732.682.656.262.892.822.656.3010.3511.6818.8527.19 53.242.602.785.053.493.093.896.138.9410.9517.5219.45 62.562.723.265.003.543.694.066.916.817.8110.7012.11 72.562.522.493.983.302.622.825.777.988.7914.1118.00 84.044.424.6812.124.484.924.6812.127.318.3513.9216.81 92.652.562.694.423.122.962.826.297.097.7012.5117.18 102.722.762.706.862.972.973.257.9910.7112.8118.3326.79 Average2.892.892.946.413.423.283.367.878.319.5015.0520.10 40

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Aperture-basedFMO:numberofapertureswithtransmissioneects. ClinicalConvergence caseC1C2C3C4C1C2C3C4 1262529874432102106 232313879356259140 31718163637232351 42832203928322963 52324246860683168 62431274979764355 72417222441413068 8687466138707466138 91819202356685581 101621223327363451 Average27.629.228.457.647.751.247.282.1 Table2-5. Aperture-basedFMO:beam-ontimewithtransmissioneects. ClinicalConvergence caseC1C2C3C4C1C2C3C4 12.842.802.897.173.442.974.047.65 22.842.682.786.622.923.253.188.91 32.702.652.584.703.242.742.765.55 42.782.812.535.052.782.812.726.10 52.722.642.656.833.693.812.966.83 62.652.822.604.653.693.652.944.70 72.652.462.533.423.083.022.725.25 84.044.093.788.234.064.093.788.23 92.562.502.523.363.683.763.566.47 102.612.782.904.852.963.223.166.11 Average2.842.822.785.493.363.333.186.58 Table2-6. Aperture-basedFMO:DVHcriteriaunderC1withouttransmissioneects. 199.498.7100.099.04.50.021.818.533.1100.0 299.198.999.797.99.90.049.3100.0100.0100.0 399.899.6100.099.32.70.020.418.3n/an/a 498.898.0100.0100.02.70.010.34.154.117.7 598.697.7100.0100.08.10.042.20.044.434.6 699.599.0100.098.66.70.026.236.2n/an/a 799.198.6100.099.22.40.00.546.724.6100.0 899.298.9100.098.69.90.028.64.9100.052.3 999.298.1100.099.62.10.05.939.836.4100.0 10100.0100.0100.0100.04.30.00.225.47.7100.0

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Aperture-basedFMO:DVHcriteriaunderC1withtransmissioneectsaddedafteroptimization. 199.699.2100.099.919.50.023.821.438.6100.0 299.399.099.999.034.00.051.0100.0100.0100.0 399.899.7100.0100.024.30.021.819.1n/an/a 498.898.1100.0100.020.90.010.54.155.717.7 598.898.1100.0100.030.60.043.00.046.839.5 699.699.2100.099.225.70.028.238.4n/an/a 799.298.8100.099.815.00.00.748.229.8100.0 899.399.0100.0100.041.60.029.55.5100.058.1 999.598.8100.099.918.90.07.246.140.2100.0 10100.0100.0100.0100.011.20.00.225.49.1100.0 Aperture-basedFMO:DVHcriteriaunderC1withtransmissioneects. 199.398.499.796.94.90.022.920.233.1100.0 299.098.699.997.56.70.044.6100.0100.0100.0 399.799.6100.099.31.30.020.018.1n/an/a 498.497.5100.0100.02.40.09.51.453.015.6 598.597.7100.099.43.10.038.90.042.733.3 699.398.9100.097.44.00.024.835.7n/an/a 799.098.4100.099.10.50.00.547.517.5100.0 898.998.6100.097.78.80.026.93.9100.048.9 999.198.3100.099.21.20.02.635.133.6100.0 10100.0100.0100.0100.08.60.00.223.11.4100.0 Beamlet-basedFMO:DVHcriteriaunderC1withouttransmissioneects. 199.499.099.998.35.00.019.417.532.3100.0 299.298.899.998.73.80.043.096.7100.0100.0 399.899.7100.099.72.60.019.317.8n/an/a 498.898.3100.0100.01.10.09.01.653.016.7 598.898.1100.0100.03.10.037.70.039.532.1 699.998.9100.098.38.40.024.234.3n/an/a 799.298.8100.099.11.80.00.346.514.9100.0 899.298.9100.099.314.10.028.34.4100.051.7 999.398.6100.098.60.50.02.134.633.6100.0 10100.0100.0100.0100.03.70.00.222.80.0100.0

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(b) Figure2-4. Case5:Therelativevolumeof(a)PTV1and(b)PTV2thatreceivesinexcessofitsprescriptiondose. Table2-10. Beamlet-basedFMO:DVHcriteriaunderC1withtransmissioneects. 199.999.8100.0100.069.11.332.527.255.1100.0 299.699.5100.099.996.50.459.199.3100.0100.0 399.999.9100.0100.094.30.125.821.0n/an/a 499.799.5100.0100.0100.06.817.710.474.640.3 599.699.3100.0100.083.11.348.02.156.556.8 699.999.4100.0100.075.80.033.641.7n/an/a 799.799.5100.0100.096.71.84.459.342.1100.0 899.799.5100.0100.086.91.440.218.5100.070.69 999.899.7100.0100.072.70.09.454.551.4100.0 10100.0100.0100.0100.0100.070.64.741.716.1100.0

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(b) Figure2-5. Therelativevolumeof(a)LSGand(b)RSGthatreceivesinexcessof30Gy. 44

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3 ],Stroometal.[ 67 ],Herketal.[ 25 ]andParkeretal.[ 51 ])derivedexpressionsforthesizeofmargin.However,thesemargin-basedmethodshavesometheoreticalandpracticalproblems:theprescribeddoseisappliedtotheentirePTV(ratherthantheCTVonly),andthemarginmaynotadequatelymodelthechangesindosedistributionsduetotherandomdeviationsoforganmotion:theCTVmayonlyhaveaverysmallchancetoreachsomeoftheedgesofthishighdoseareabutinotherdirectionsitmayextendbeyondthisarea.Therefore,theCTVmaynotreceiveadequatedosewhileorgansclosetotheCTV(criticalstructuresororgansat-risk)maynotbesparedduetooverdosing;moreover,thedosedistributionofPTVcannotbeusedtoaccuratelyevaluatethetreatmentplanquality;nally,theplanningdoseisdeliveredoveranumberoffractionssothedosereceivedbyPTVisanaccumulateddose.Hence,theactualdosereceivedineachfractionmaysignicantlydierfromtheplannedoneandthetotaldosemaybebadlyestimatedaswell. Becauseoftheseproblemsofmargin-basedmethods,attentionhasrecentlyshiftedtothedevelopmentofalternativemethods.Themostcommonmethodsaretoprocessthe 45

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46 ]andCarterandBeckham[ 47 ]).Thisdose-convolutionmethodassumesshiftinvarianceofthedosedistribution.However,internalinhomogeneitiesandsurfacecurvaturemayleadtoviolationsofthisassumption(Craigetal.[ 18 ]);Inaddition,thismethodassumesthatthepatientistreatedwithaninnitenumberoffractions,eachdeliveringaninnitesimallysmalldose.Theeectofnitefractionationappearstohaveagreaterimpactonthedosedistributionthanplanevaluationparameters(Craigetal.[ 19 ]).Recently,Luetal.[ 44 ]showedthatthemotioneectscanbeaccountedforbymodifyingtheuencemaps.Aftersuchmodication,dosecalculationisthesameasthosebasedonastaticplanningimage.However,thismethodisonlysuitableforthecaseswhenthepatientmotionissmall(Luetal.[ 44 ]).Chanetal.[ 15 ]introducedarobustmethodologyfordealingwithIMRToptimizationproblemsunderintrafractionaluncertaintyinducedbybreathingmotion.Theyusedtheideaofamotionprobabilitymassfunction(PMF)alongwithanassociatedsetdescribingtheuncertaintyofthisPMFastheirmodelofdatauncertainty. InChapter 2 ,weintroducedanaperture-baseddeterministicFMOmodelwhichisactuallyamargin-basedmodel.Inthischapter,wewillrstfocusonbeamlet-basedstochasticmodelsandthenmovetoaperture-basedstochasticmodelsandbothstochasticmodelsaccountforinterfractionmotion.Wewillnotconsiderintrafractionmotionwhichneedtobeanalyzedseparately.Ourgoalsinthischapterareto 46

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3.2.1Beamlet-basedDeterministicModel 2 .Beamlet-basedFMOistondtheoptimalintensity(uencemap)foreachbeamletandwewilldenotetheintensityofbeamleti2Nbyxi,thenthevectorofvoxeldosescanbeexpressedasalinearfunctionoftheintensitiesofthebeamletsthroughdosedepositioncoecientsDij.Thenthebeamlet-baseddeterministicmodelcanbewrittenasminimizeX`2LG`(z)subjecttozj=Xi2NDijxiforj2Vxi0fori2N: 2{5 and 2{6 ).IfwedenotethesetoftargetvoxelsbyVTandthesetofcriticalstructurevoxelsbyVC(VT[VC=V),thenwecanrewritethecriteriaas( 2{5 )and( 2{6 )asFj(zj)=s

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48

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OurstochasticFMOmodelisthenformulatedasfollows:minimize1 49

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50

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3.2.4.1Clinicalprobleminstances 2 fordetailinformationaboutthesizesofbixesandvoxels,howtoimportdata,etc.Forallvecases,wedesignplansusingnineequispaced60Co-beams.Theninebeamsareevenlydistributedaroundthepatientwithangles0,40,80,120,160,200,240,280,320,respectively.Thereareonetarget(prostate)withprescriptiondose73.8Gyandthreecriticalstructures:bladder,rectumandfemoralhead.Table 3-1 showstheproblemdimensionsforthevecases. Table3-1. Modeldimensions. Casetotal#voxels#voxelsinthemodels#bixels 1205,91135,9882,055 2131,21622,3302,253 3226,32438,7312,653 4162,62829,8282,413 5262,18440,6362,736 Sinceourmodelcouldaccountfortheinterfractionmotion,thetargetintheoptimizationmodelistheCTVwiththeintrafractionmotionmargin.Empirically,thestandarddeviationofset-uperrorsineachdirectionhasbeenfoundtobeapproximately3mm(El-Bassiounietal.[ 22 ],Brittonetal.[ 12 ],Price[ 53 ]andWangetal.[ 69 ])andwethereforegenerated200scenariosbyperturbingthelocationofthepatientaccordingtoanormaldistributionwithameanof0andastandarddeviationof3mmineachcoordinatedirectionforeachcase.Fortarget,wetestthedosecoverageforindividualscenarios;forcriticalstructures,wechecktheaverageddosedistributionoverallscenarios.Dose-volumecriteriaandconstraintshavebeenestablishedtokeeptoxicityatacceptablelevels.However,PTVistheCTVwithinterfractionandintrafractionmotionmarginsandthetargetinourmodelisCTVwithintrafractionmotionmarginsothetraditional 51

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57 ])(inTable 3-2 )whicharecommonlyusedbyphysiciansandresearchers(Moiseenkoetal.[ 48 ],Rodriguesetal.[ 59 ]andChenetal.[ 16 ]).Finally,wecompareoursolutionswiththeonesobtainedfromthetraditionalmethod.Thewidthoftheset-uperrormarginissettobeabout1.65timesthestandarddeviation(seeAntolakandRosen[ 3 ]).Hence,5mmofmarginineachdirectionhasbeenaddedtothetargetforthetraditionalmethod. Table3-2. RadiationTherapyOncologyGroup(RTOGP0126)criteriaforprostatecancer. RectumBladder OurFMOmodelsemploytreatmentplanevaluationcriteriathatarequadraticone-sidedvoxel-basedpenalties.Wetunedtheproblemparameters(underdoseandoverdosethresholdsandcriteriaweights)bymanualadjustment.Ingeneral,comparedtothetraditionalmodel,weincreasethepenaltyweightstothetargetdosedistributionswhiledecreasethepenaltyweightstothecriticalstructuresdosedistributions.Themodelsweresolvedbyourin-houseprimal-dualinteriorpointalgorithm.AllexperimentswereperformedonaPCwith2.66GHzIntelQuadCPUand4GBofRAM,runningunderWindowsVista.OuralgorithmswereimplementedinMatlab7. 52

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3-1 and 3-2 estimatetheprobabilitythataspeciedpercentageofthetargetreceivedtheprescriptiondosefortwomodels,respectively.Weobservethatthetreatmentqualitiesfromthescenariosoutoftheoptimizationmodelsarenotasgoodasonesincludedintheoptimizationmodelsifweonlyinclude25,50or75scenariosinthemodels.Forinstance,ifweinclude25scenariosintherstoptimizationmodel,forallthese25scenarios,atleast95%oftargetvolumecouldreceivetheprescriptiondose,however,thereareonly21outof25scenarios(i.e.,84%)outoftheoptimizationmodelinwhichatleast95%oftargetvolumecouldreceivetheprescriptiondose.Ifweinclude100scenariosinthemodels,thetreatmentqualitiesobtainedfromthescenariosoutoftheoptimizationmodelsareverysimilarwiththeonesobtainedfromthescenariosintheoptimizationmodels(seeFigure 3-1 and 3-2 ).Bytestingallvecases,weconcludethat100isthe(estimated)minimalnumberofscenarioswhichshouldincludeinbothoptimizationmodelsinordertoobtainthestabletreatmentqualityforallavailablescenarios. Figures 3-3 to 3-7 estimatetheprobabilitythataspeciedpercentageofthetargetreceivedtheprescriptiondoseforScenarios1-100and101-200,respectively,forthevecases.Noremarkabledierencesareobservedbetweenthescenariosinandoutoftheoptimizationmodels.Weobservethat,ingeneral,ourstochasticmodelsobtainbettertargetdosecoveragethanthetraditionalmodeldoes.TakingCase1asanexample,theprobabilityof95%targetcoverageis98%fortherststochasticmodel;theprobability 53

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(b) (c) (d) Figure3-1. TheprobabilitythataspeciedpercentageofthetargetreceivedtheprescriptiondoseforCase1(rststochasticmodel). is99%forthesecondstochasticmodel;whilebysolvingthetraditionalmodel,thatprobabilityisonly91%.Inparticular,thesecondstochasticmodelobtainslightlybettertargetdosecoveragethantherstone.Figure 3-8 and 3-9 showtheDVHsforthecriticalstructuresforthetwostochasticmodelscomparedtothetraditionalmethod,respectively.Table 3-4 liststhecorrespondingDVHvaluesbasedonthecriteria.WeconcludefromFigure 3-8 3-9 andTable 3-4 thatourmethodscouldobtainbettertargetdosecoveragewith(almost)noadditionaldosetothecriticalstructures. Besidesestimatingtheprobabilitythataspeciedpercentageofthetargetreceivetheprescriptiondoseforthetwomodels,wewouldliketocheckanothertwoimportant 54

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(b) (c) (d) Figure3-2. TheprobabilitythataspeciedpercentageofthetargetreceivedtheprescriptiondoseforCase1(secondstochasticmodel). criteria:targetcoldspots(underdosing)andhotspots(overdosing)whichrefertothevolumeoftargetwhichreceiveslessthan93%andmorethan110%ofprescriptiondose,respectively.Table 3-3 showstheaveragedhotspotsforallvecases.Weobservethateventhoughthesecondstochasticmodelobtainbettertargetdosecoveragethanthetraditionalandtherststochasticmodel,ingeneral,itdoesnothavehigherhotspots.Inmostofscenarios,100%oftargetreceiveatleast93%ofprescriptiondoseforallvecasessotherearealmostnocoldspotsforallmodels. Ourstochasticoptimizationmodelstakeindividualdosedistributionsfortargetintoaccountandrobusttreatmentplanscanbeobtainedwhilethetraditionalmethoddoesnot 55

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3-10 and 3-11 showtheDVHsforrstandsecondstochasticmodelscomparedtothetraditionalmethod,respectively.Wecanseethatabout96%and98%oftargetreceivetheprescriptiondose(73.8Gy)bysolvingtherstandsecondstochasticmodels,respectively,whileforthetraditionalmethod,thatvolumeisonly92%. Theamountoftimerequired,onaverage,byouroptimizationalgorithmstondthesolutionsrangesfromabout3{7minutesofCPUtimefortwomodelswhileittakeslessthan2minutestosolvethetraditionalmodel.Moreover,thetimerequiredbythesecondstochasticmodelislessthantherstmodelduetothesmallersizeofthemodel.Table 3-5 recordstheCPUtimeforthesethreemethods. Finally,recallthatthetraditionalmethoddoesnotaccountforinterfractionmotionineithertheoptimizationmodeloritsreportingofthetreatmentplanquality.Hence,theperceivedtreatmentplanqualitymaydierfromtheactualone.Toassessthiseect,wecomparedtheperceivedDVHsreportedfromthetraditionalmodelwiththeactualDVHsthattakeintoaccounttheuncertaintyusingthe200scenariosfromtheinterfractionmotionmodel.TheresultsindicatethattheperceivedDVHsforcriticalstructuresareveryclosetotheactualones(seeexamplesinFigure 3-12 ).However,theperceivedDVHsoverestimatethedosedistributiontothetargets(seeresultsinFigure 3-13 ).Thisfurtherunderscorestheimportanceoftakinginterfractionmotionuncertaintyintoaccountexplicitly. 56

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(b) Figure3-3. TheprobabilitythataspeciedpercentageofthetargetreceivedtheprescriptiondoseforCase1. (a) (b) Figure3-4. TheprobabilitythataspeciedpercentageofthetargetreceivedtheprescriptiondoseforCase2. (a) (b) Figure3-5. TheprobabilitythataspeciedpercentageofthetargetreceivedtheprescriptiondoseforCase3. 57

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(b) Figure3-6. TheprobabilitythataspeciedpercentageofthetargetreceivedtheprescriptiondoseforCase4. (a) (b) Figure3-7. TheprobabilitythataspeciedpercentageofthetargetreceivedtheprescriptiondoseforCase5. Table3-3. Hotspots(%)forthreemodels. Casetraditionalmodelstochasticmodel1stochasticmodel2 11.652.592.07 23.634.554.75 36.414.364.70 41.974.611.46 53.004.012.17 58

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(b) (c) (d) (e) Figure3-8. DVHsforcriticalstructures(rststochasticmodelvs.traditionalmodel). 59

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(b) (c) (d) (e) Figure3-9. DVHsforcriticalstructures(secondstochasticmodelvs.traditionalmodel). 60

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Figure3-10. DVHsforCase1inScenario135(rstmodelvs.traditionalmodel). (a) Figure3-11. DVHsforCase1inScenario135(secondmodelvs.traditionalmodel). 61

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DVHcriteriaforcriticalstructures(%). Case@75Gy@70Gy@65Gy@60Gy@80Gy@75Gy@70Gy@65Gy(15%)(25%)(35%)(50%)(15%)(25%)(35%)(50%) 1Trad.8.3711.9114.7817.400.576.099.5112.30Stoch.17.0510.2513.2515.320.745.889.9713.32Stoch.29.2912.3515.1616.920.6810.4715.0617.97 2Trad.3.996.228.4511.090.826.888.4510.09Stoch.15.538.0911.1114.414.118.7910.6313.63Stoch.210.6513.4316.6420.087.7013.1216.5720.16 3Trad.2.7014.1317.1722.850.7911.3616.8224.95Stoch.10.0013.5718.8325.760.003.7713.2917.06Stoch.22.779.6916.0620.490.002.589.5214.48 4Trad.13.7920.1525.8230.860.424.656.878.71Stoch.113.6420.5025.3330.020.954.406.838.65Stoch.210.5617.9822.9527.400.543.215.607.13 5Trad.11.2019.5225.4129.570.294.456.938.99Stoch.111.2018.0724.6428.880.083.826.398.38Stoch.28.2115.4620.2925.600.142.634.906.96 CPUrunningtime(seconds)forthreemodels(%). Casetraditionalmodelstochasticmodel1stochasticmodel2 145312271 242201149 371467375 467308246 589301251 (a) (b) Figure3-12. PerceivedandactualDVHsforcriticalstructuresforCase1andCase2 62

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(b) (c) (d) (e) Figure3-13. PerceivedandactualDVHsfortarget. 63

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3.3.1Aperture-basedStochasticOptimizationModels 2 ,weusedadirectapertureoptimizationapproachtodesignradiationtherapytreatmentplansforindividualpatientswhichintegratesthebeamlet-basedFMOandleaf-sequencingproblemsintoasingleoptimizationmodel.Wecanapplythisapproachtoourbeamlet-basedstochasticoptimizationmodels.TherelatednotationcanbefoundinChapter 2 .Therstaperture-basedstochasticFMOmodelcanbeformulatedasfollows:minimize1 subjectto whereDskj=Xi2AkDsijforj2VT;s=1;:::;SDkj=Xi2AkDijforj2VC:

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2 ,wederivedthepricingproblemfortheaperture-basedstaticFMOmodel.Similarly,wecanderivethepricingproblemforthesetwoaperture-basedstochasticmodels.NotethattheKarush-Kuhn-Tucker(KKT)optimalityconditionsarestillnecessaryandsucientforoptimalitybecauseoftheconvexityoftheobjectivefunctionandthelinearityoftheconstraintsforthetwomodels.Inourtwostochasticmodels,wecheckthetargetdoseforeachscenariowithadditionalconstraints( 3{1 )and( 3{4 ),respectively,thereforeweneedtodenotenewdualmultipliersassociatedwiththemtoderivetheKKTconditions.Werstderivethepricingproblemfortherstmodel:letusdenotethedualmultipliersassociatedwithconstraints( 3{1 ),( 3{2 )and( 3{3 )bysj(j2VT;s=1;:::;S),!j(j2VC)andk(k2K),thentheKKTconditionscanbewrittenasfollows:sj=1

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2 ,weconcludethatthecurrentsolutionisoptimalifandonlyiftheoptimalsolutiontothefollowingoptimizationproblemminimizek2KbXi2AkSXs=1Xj2VTDsij^sj+Xj2VCDij^!j! 3{4 ),( 3{5 )and( 3{6 )bysj(j2VT;s=1;:::;S),!j(j2V)andk(k2K),thentheKKTconditionscanbewrittenasfollows:sj=1

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2 :PSs=1Pj2VTDsij^sj+Pj2VCDij^!jandPSs=1Pj2VTDsij^sj+Pj2VDij^!jmeasuretheper-unitchangeinobjectivefunctionvalueiftheintensityofbeamletiisincreasedforthetwomodels,respectively. 2 ,bytestingonclinicalhead-and-neckcancercases,weshowedtheecacyofthisapproachcomparedtothetraditionalmethod.TheresultsindicatethatdeliveryeciencyisveryinsensitivetotheadditionoftraditionalMLCconstraints;however,jaws-onlytreatmentrequiresaboutadoublinginbeam-ontimeandnumberofaperturesused.Wealsoshowedtheimportanceofaccountingfortransmissioneects.Inthischapter,wefocusontheeectsofthesetwostochasticmodelswherethetransmissioneectsaretakenintoaccount. 67

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2 ,wedevelopedconvergenceandclinicalstoppingrulesbasedonthetreatmentquality(inparticular,wecheckedtheDVHsineachiteration).NotethetraditionalDVHcriteriaforPTVarenotsuitableforthetargetinourstochasticmodels,wethereforecheckthetargetdosecoverage(byestimatingtheprobabilitythatafractionofthetargetreceivessomecertaindose,fordierentvaluesof)ineachiterationintheconvergencestoppingrule.Ournewtwostoppingrulesare: Tables 3-6 to 3-9 showthenumberofaperturesandbeam-ontimeforthetwostochasticmodelsaccordingtofourdierentMLCconstraints,respectively.Nobigdierencesareobservedwiththesetwostochasticmodels.TakingCase1asanexample,Figure 3-14 showsthetargetdosecoveragefortwostochasticmodelswithtwostoppingrules.Byobservingallvecases,weconcludethatingeneral,thesecondstochasticmodelyieldsslightlybettertargetdosecoverage. 68

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69

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C1:Numberofaperturesandbeam-ontime. Stochasticmodel1Stochasticmodel2Stochasticmodel1Stochasticmodel2 caseClinicalConvergenceClinicalConvergenceClinicalConvergenceClinicalConvergence 1175120422.473.222.543.15 2195940682.373.493.935.15 3374650502.953.143.223.22 4164721392.122.962.252.78 5175022412.333.182.502.91 Average21.250.630.648.02.453.202.893.44 C2:Numberofaperturesandbeam-ontime. Stochasticmodel1Stochasticmodel2Stochasticmodel1Stochasticmodel2 caseClinicalConvergenceClinicalConvergenceClinicalConvergenceClinicalConvergence 1174519342.393.052.432.75 2164522372.293.092.372.77 3203420312.632.852.522.98 4194320362.362.772.262.64 5213724452.102.742.542.84 Average18.640.821.036.42.352.902.422.90 C3:Numberofaperturesandbeam-ontime. Stochasticmodel1Stochasticmodel2Stochasticmodel1Stochasticmodel2 caseClinicalConvergenceClinicalConvergenceClinicalConvergenceClinicalConvergence 1172917332.492.762.412.97 2174627502.222.982.633.06 3233722362.522.802.783.42 4285232592.392.892.242.93 5294433402.562.732.592.90 Average22.841.626.245.62.432.832.533.05 C4:Numberofaperturesandbeam-ontime. Stochasticmodel1Stochasticmodel2Stochasticmodel1Stochasticmodel2 caseClinicalConvergenceClinicalConvergenceClinicalConvergenceClinicalConvergence 1215637752.775.373.916.09 2183725412.404.683.084.84 3275927582.335.873.105.12 4224224572.703.922.905.14 5256333632.835.373.485.29 Average22.651.429.258.82.605.043.295.30

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Targetdosecoveragefortwomodelswithtwostoppingrules. 71

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13 ]andCayirlietal.[ 14 ].However,thereareonlyafewstudies(e.g.,Confortietal.[ 17 ])concerningpatientschedulingintheareaofradiationtherapytreatmentpatientschedulingwhichhasthefollowingspecialconcerns(butnotlimitedto): 72

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Inthischapter,werstdevelopandimplementstrategicoptimizationmodelsandtestthemusingdatafromtheUniversityofFloridaProtonTherapyInstitute(UFPTI)andthemaingoalsareto: Moreover,wehavedenedandmodeledtheon-linepatientschedulingproblemforprotontherapytreatmentsattheUFPTI,andproposedalocalsearchheuristicforsolvingthisproblem.Themaincontributionofthisstudyliesinitspreciseanduniquemodelingapproachwhichhascapturedalloperationaldetailsoftheproblem.Furthermore,onthesolutionside,thelocalsearchheuristicdevelopedallowsforanecient,real-timesolutionoftheproblem. TheUFPTIisthenewestprotonfacilityintheUS,performingitsrstpatienttreatmentinAugust2006.Itisoneofonlyseveralprotontherapyfacilitiesoperatinginthenation,andtheonlyprotontherapyfacilityintheSoutheast.Duetothefactthatthetechnologyisnovelandscarce,ecientpatientschedulingisessential.Todate,theUFPTIhasdeliveredmorethan21,000protontherapytreatments,topatientswithhead-and-neck,skull-base,sarcoma,prostate,pancreasandpituitarycancers.Itisundergreatpressuretoincreasethecapacityduetothelargenumberofpatientsrequiringordesiringtreatment.Currently,theUFPTIhasthreeprotontreatmentrooms,eachequippedwithonegantry,andtreatsthepatientsfrom6:00amto9:00pmforeachbusinessday.Sincemostofthesepatients(about80%)suerfromprostatecancer,theGantry3iscommissionedforthesepatientsonly.Duetoclinicalreasonsandlimitations 73

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4.2.1TheObjectiveFunction 74

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4.2.2.1Thecapacity Ingeneral,thesnouthasbeensetupeverymorningbeforethetreatmentstarts,sothesnoutchangingtimeisnotcountedfortherstpatient.IfwedenotetheavailabletimeperdayforgantrygbyCg,thenthegantrycapacityconstraintsandtheadditionalconstraintsrequiredare:KXk=1ckxtkg+KXk=1ckfkytkgCg(SXs=1Itsg1)t=1;:::;T;g=1;:::;GXk2KsytkgMItsgt=1;:::;T;s=1;:::;S;g=1;:::;GItsg2f0;1gt=1;:::;T;s=1;:::;S;g=1;:::;G: 75

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2TXt=1GXg=1xtk0gdk0TXt=1KXk=1GXg=1xtkg+1 2k0=1;:::;K: 76

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ThenthecapacityconstraintsforanesthesiapatientsareXk2Ka(ckxtkg+ckfkytkg)C0gSXs=1I0tsgt=1;:::;T;g=1;:::;GXk2KsSKaytkgMI0tsgt=1;:::;T;s=1;:::;S;g=1;:::;GI0tsg2f0;1gt=1;:::;T;s=1;:::;S;g=1;:::;G:

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78

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2TXt=1GXg=1xtk0gdk0TXt=1KXk=1GXg=1xtkg+1 2k0=1;:::;KXk2Ka(ckxtkg+ckfkytkg)C0gSXs=1I0tsgt=1;:::;T;g=1;:::;GXk2K2(ckxtkg+ckytkg)Cgt=1;:::;T;g=1;:::;Gxtkg=0t2TNA;k=1;:::;K;g=1;:::;GXk2KsytkgMItsgt=1;:::;T;s=1;:::;S;g=1;:::;GItsg2f0;1gt=1;:::;T;s=1;:::;S;g=1;:::;GXk2KsSKaytkgMI0tsgt=1;:::;T;s=1;:::;S;g=1;:::;GI0tsg2f0;1gt=1;:::;T;s=1;:::;S;g=1;:::;Gxtkg;ytkg0t=1;:::;T;k=1;:::;K;g=1;:::;Gxtkg;ytkg2Nt=1;:::;T;k=1;:::;K;g=1;:::;G

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4.2.3 requiresalotofcomputationaleorttosolve.Wenoticethatthepatientsmixconstraintsareveryhardtosatisfybecausethefeasibleregiondenedbytheseconstraintsisverylimitedandthereforendingafeasiblesolutionisveryexpensive.However,inpractise,theactualpatientmixisallowedtohaveasmalldeviationwiththeplannedone.Wethereforesplitconstraintsinto\hard"and\easy":eliminate\hard"constraints,butpenalizetheviolationinthecostfunction.Thentheobjectivefunctionandrelatedconstraintscanbeformulatedasfollows:maximizeTXt=1KXk=1GXg=1fkytkgKXk=1wkUkwithadditionalconstraints(A)Uk0TXt=1GXg=1xtkgdk0TXt=1KXk=1GXg=1xtkgUk0k0=1;:::;KUk0k=1;:::;K: 80

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1. Setwk=0fork2K. 2. SolvetheLPrelaxationofmodel(A). 3. Ifmink2KjUkj>0,gotoStep4otherwisegotoStep5. 4. 5. SolveMIPmodel(A). 4.2.5.1Inputdata 4-1 showsthepatientclassicationandpatientmix.Inaddition,thefollowingconcernsorconstraintsareappliedtothemodel: 81

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4-2 showstheresultsofthemodel:theaveragenumberoffractionsperdayis100.34.WealsosolvedLPrelaxationmodelbysettingxtkg'sandytkg'scontinuouswhichcouldprovideanupperboundoftheobjectivevalueandtheresultsareshowedinTable 4-2 too:theaveragenumberoffractionsperdayis103.28.Table 4-3 showsthegantryutilizations. 4-4 andtheresultsforScenarios1{20arelistedinTable 4-5 4-5 ). 82

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4-6 basedonthepercentagesofpatientsinCategores1and2(C1andC2),andateachscenariothepatients'percentageinC1+C2areforming100%,90%,70%and60%ofthepatients'population,andtherestisproportionallyassignedtotheothercategoriesbasedonthebasicscenario.ComparedtotheScenario14,wedecreasedthepercentagesofanesthesiapatientsinScenario15becausethereareonlyfourhoursinonegantryavailableforanesthesiapatientseachday.WeobservedthattheaveragefractionsperdaydecreasedwiththedecreasingofpercentagesofCategory1and2becausethetreatmenttimeperfractionistheshortestforthepatientsinthesetwocategories.Notethatthedeviationofpatientmixisveryhigh(7%)duetothetimelimitationofanesthesiapatientsforScenario14.Moreover,thegantryutilizationingantry3forCategory5isverylow(72.9%)becausethegantry3isonlyforthepatientsinCategory1buttherearenosucientsuchpatients.Therefore,wetestedthemodel 83

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4-5 :theGantry3utilizationhasbeenincreasedto95%. 4-3 ,onecannoticethattheaveragegantryutilizationsforThursdayandFridayarelowercomparedtotherestoftheweek.Itcanbejustiedasfollows:thereexistsanadditionaltreatmenttimefortherstfractionandthersttreatmentcanonlybedeliveredbetweenMondayandWednesday.Consequently,thereislessloadongantriesonThursdaysandFridays.Inordertogetmorestabledailyutilizations,wecanletthemodeldeterminethedailyavailabletimeforeachgantry.Inscenario17,wereducethegantryavailabilityonThursdayandFridaybytimeunitsandextendtheavailabilityonMondaytoFridaybyunitsinreturn,andletthemodelndtheoptimal.Notethatinthisscenario,thetotalweeklygantryavailabletimehasbeenincreasedbytimeunits.Inscenario18,wedenethegantryavailabletimeonMondaytoWednesdayby1andonThusdayandFridayby2whilekeepingthetotalgantryavailabletimeunchangedcomparedtothebasicscenario.Inscenario19,wedenotethegantryavailabletimefromMontoFriby1;:::;and5,respectivelywhilekeepingthetotalweeklygantryavailabletimeunchanged.TheresultsareshowninTable 4-7 and 4-8 4-9 showstherevenueperfractiondeliveredtoeachcategory.Itturnedoutthattheoptimalaveragerevenueperdayis$397,106.22withthecorrespondingaveragedailyfractionsof100.15. 84

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InScenario21thenumberofpediatricspatientsismaximizedandthefollowingoperationalimprovementssuggestedbytheUFPTIareincluded: { Category1:3minutes,Category2:2minutes,Category6:15minutes { Category7:15minutes,Category8:30minutes TheresultsarereportedinTable 4-10 Sincepediatricspatientsrequireanesthesiafortreatment,thereisahighcorrelationbetweenanesthesiateamavailabilityandthenumberofpediatricspatientstreated.InScenario22,wevarytheavailabilityofanesthesiateamondierentgantries:4hoursononegantry(14);5hoursontwogantries(25),etc.Moreovertheinputdataisthesameasscenario21withnopatientmixconsiderationandthefollowingconstrainthasbeenaddedtothemodel: 85

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4-11 and 4-12 SinceScenario22onlymaximizesthenumberofpediatricspatients,thereareopportunitiesforimprovingtheaveragedailyfractionsdelivered.InScenario23,wemaximizethetotalnumberoffractionsdeliveredwhileretainingtheoptimalnumberofpediatricspatients.tables 4-13 and 4-14 showthecorrespondingresults. Category8hasalmostnopatientsinScenarios21{23duetothefactthatCategory8hasalongertreatmenttimecomparedtoCategory6,soallpediatricspatientsareselectedfromCategory6bythemodel.WethereforetestthesescenarioswithanadditionalconstraintthatthenumberofpatientsinCategory8isatleastaslargeasthenumberofpatientsinCategory6,i.e.:TXt=1GXg=1xt(k=8)gTXt=1GXg=1xt(k=6)g 4-15 to 4-18 InScenarios23{25wehavenoticedthatexceptforcategories1,6,and8,othercategorieshavenooronlyafewpatients.SoinScenario26,weaddpenaltyifthenumberofpatientsineachcategoryisdierentwiththedesiredpatientmix.Thentheobjectivefunctionandrelatedconstraintscanbeformulatedasfollows:maximizeTXt=1Xk2KaGXg=1xtkgKXk0=1wk0Uk0withthefollowingadditionalconstraints: 86

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4-19 and 4-20 87

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PatientsClassication. 1n184011518651-FieldProstate 2n304011518152-FieldProstate 3n3562220187H&N/BOS 4n4562225253Thorax/Abdomenchordomas 5n3530120103SimpleBrain 6y5530120182PedsBrainwithAnesthesia 7n6030145251CSInoAnesthesia 8y9030145251CSIwithAnesthesia 9n5042130182Lung/AbdomenwithABC/BodyFlX 10n3512120181ConcomitantBoostPatients Theresultsforthebasicscenariomodel. ScenarioAveragefractionPatientmixdeviationCPUtime/day(%)(seconds)(%) BasicIP100.343%6819 BasicLP103.280673 Table4-3. GantryutilizationperdayusingIPmodel(%). GantryMon.Tue.Wed.Thu.Fri.Average 196.796.797.093.993.595.6 293.695.294.990.789.592.8 398.098.098.294.894.896.7 88

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Thedenitionsofscenarios. 1Allowinggantryswitching 2Nogantryspecication 3Changinganesthesiateamavailabilityto22 45-minsnoutchangingtime 57.5-minsnoutchangingtime 610-minsnoutchangingtime 72-minreductionintreatmenttimeperfraction 83-minreductionintreatmenttimeperfraction 95-minreductionintreatmenttimeperfraction 10Scenario9withoutgantryspecialization 11-15Dierentpatientmixes 16Scenario15withoutgantryspecialization 17-19Gantryavailabletime 20Revenuemaximization 21-26Pediatricsmaximization 89

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Sensitivityanalysis. ScenarioAveragefractionPatientmixdeviationGantry1Gantry2Gantry3/day(%)utilizationutilizationutilization Basic100.343969397 1100.693959399 299.112949593 3100.953949398 4100.463919598 599.572939398 6100.182929598 7106.062959391 8109.142959385 9114.282959574 10124.031949496 11126.773959698 12108.493939594 1390.403929494 1482.037928784 1577.483909473 1682.923929295 17100.953959396 18102.314959498 19102.804959598 20101.153939495 90

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Patientmixscenarios(%). ScenarioC1C2C3C4C5C6C7C8C9C10 12731731111111 135713115531131 144911146642242 154911147722152 164911147722152 Table4-7. Scenarios17{19:Gantryavailabletime(minutes). ScenarioMon.Tue.Wed.Thu.Fri. Basic900900900900900 17916916916884884 18912912912882882 19982886894872866 Table4-8. Scenario17{19:Gantryutilization(%). ScenarioMon.Tue.Wed.Thu.Fri.Average Basic96.196.696.793.492.695.1 1795.495.494.294.794.494.8 1894.696.197.494.894.395.4 1993.396.496.797.397.396.2 91

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Scenario20:Revenueperfraction. Table4-10. ResultsforScenario21. Scenario#fx/day#pedGantry1Gantry2Gantry3utilizationutilizationutilization Basic100.34(3,1)969397 2092.80(6,3)969694 Table4-11. Scenario22:Patientmixfordierentanesthesiaavailabilities. Anesth.12345678910 1454.523.71.00.51.54.50.00.51.012.6 2554.425.90.00.00.014.00.00.50.05.2 2856.80.00.00.00.020.30.40.00.022.5 3561.78.32.60.00.023.30.00.00.04.1 Table4-12. Scenario22:Otherstatisticsfordierentanesthesiaavailabilities. Anesth.G1G2G3#ped 1488.191.191.110 2590.891.089.128 2889.891.489.446 3593.588.092.745 Table4-13. Scenario23:Patientmixfordierentanesthesiaavailabilities. Anesth.12345678910 1496.10.00.00.00.03.90.00.00.00.0 2582.00.80.80.00.411.20.00.00.04.8 2880.30.00.00.00.019.70.00.00.00.0 3576.60.00.90.00.019.10.00.00.03.4 Table4-14. Scenario23:Otherstatisticsfordierentanesthesiaavailabilities. Anesth.G1G2G3#ped#fx/day 1487.593.393.310156.6 2596.496.799.528144.9 2896.097.094.246136.3 3597.396.797.145134.9 92

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Scenario24:Patientmixfordierentanesthesiaavailabilities. Anesth.12345678910 1462.48.17.60.00.02.00.02.00.017.9 2568.12.91.51.00.05.40.05.40.015.7 2875.90.00.00.00.09.20.09.20.05.2 3576.60.00.00.00.08.10.08.10.07.2 Table4-16. Scenario24:Otherstatisticsfordierentanesthesiaavailabilities. Anesth.G1G2G3#ped 1484.889.488.68 2591.187.283.622 2890.985.883.636 3588.179.486.132 Table4-17. Scenario25:Patientmixfordierentanesthesiaavailabilities. Anesth.12345678910 1497.00.00.00.00.01.50.01.50.00.0 2591.00.00.00.00.04.50.04.50.00.0 2879.50.00.00.03.78.40.08.40.00.0 3585.30.00.00.00.07.30.07.30.00.0 Table4-18. Scenario25:Otherstatisticsfordierentanesthesiaavailabilities. Anesth.G1G2G3#ped#fx/day 1496.493.899.58167.4 2595.696.199.522148.6 2893.395.298.536124.9 3593.595.497.832128.3 Table4-19. Scenario26:Patientmixfordierentanesthesiaavailabilities. 12345678910 Desiredpat.mixn/a15733n/a1n/a21 1466.516.45.71.71.72.30.62.31.71.1 1565.315.95.12.81.22.81.22.81.71.2 2865.415.00.50.02.16.90.09.60.00.6 3566.815.20.50.02.17.40.07.40.00.5 93

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Scenario26:Otherstatisticsfordierentanesthesiaavailabilities. Anesth.G1G2G3#ped#fx/day 1490.989.592.9(4,4)110.1 1595.592.193.0(5,5)110.1 2895.691.393.2(13,18)110.2 3595.492.192.4(14,14)110.1 4.3.1ProblemDenition 94

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(i) (ii) Inthenextsection,therstoptimizationproblemhasbeenmodeledusingadecompositionapproach.Inmodelingthesecondproblem,wehaveusedthesameapproachwithsomeslightmodications. 4.3.2.1Sequenceandtimingdecomposition 95

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Inaddition,inordertodeterminetherelevantpatientcopiesforacertainday,abinaryindicatorarrayoflengthTisusedtospecifyonwhichdaysduringtheplanninghorizon 96

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Whilecomputingtheweightedaverages,theweightsarechosensothatearlierdaysoftheplanninghorizonaremoreimportantthanlaterdayssincetheformeraremorelikelytobeclosetonal.Inthefollowingwerstformulatethedailytimingoptimizationproblemforagivenunionsequenceandproposeanexactalgorithmtosolveit.Next,wequantifyour 97

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4.3.3.1Timingoptimizerforasingleday

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Furthermore,letdecisionvariablesndenotethestartingtime,andvnandundenotetheviolationofthedesiredtimewindowforpatientcopynondayt.TheLPformulationforthetimingoptimizationproblemfordaytisasfollows:minimizePDI(t)=Xn2N((t;n))(un+vn)subjectto(M1)LRnsnURnn2NtsnLDnunn2NtsnUDn+vnn2Ntsnsnsn+n2Stsnsn1+ckn1+1fn16=ngn2Ntsn;un;vn0n2Nt 99

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Setsn=0;n=1;sn1+ckn1+1fn16=ng;otherwise. 2. Modifysnasfollows: 3. Calculateinfeasibilityindexasthesummationofallrestrictedtimewindowsandrestrictedstartingtimesviolations. 4. IfinfeasibilityindexispositivegotoStep9otherwisegotoStep5. 5. LetQ=NtSt. 6. LetnbetherstelementinQ. 7. Shiftpatientcopyn'sstartingtimeforwardasmuchasittouchesminfLDn;URnckngandprovidedthat: (i) theobjectivefunctionvaluedecreases. (ii) noneofsubsequentpatientcopiesviolatetheirrestrictedtimewindowsorrestrictedstartingtimes. 8. Ifwestopdueto(ii)withn0asthecriticalpatientcopy,setQ=Q\fn0+1;:::;jNtjgandreturntoStep6. 9. Returntheinfeasibilityindex. IfanoptimalsolutiontoM1exists,thenthetimingoptimizerprovidestheoptimaltimingforthecorrespondingday,otherwiseitreturnsthetotalinfeasibilityassociatedwith 100

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Inselectingpatientcopiesforshiftingorswapping,thesequenceoptimizerisbiasedtowardscloserpatientcopiesintheunionsequenceratherthanmoredistantones,sothatperturbationsarewithinasmallerrangeandconsequentlythenewscheduleobtainedhaslocalchangescomparedtothepreviousschedule.Insearchingforagoodunionsequence,sequenceoptimizerhierarchicallyrestoresfeasibility,minimizessnoutchangeindex,andnallyminimizespatientdissatisfactionindex.Givenaninitialsequenceofcopies,sequenceoptimizertriestoobtainfeasibilityrstandimprovetheevaluationcriterianextinfourphasesasfollows: 102

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1. Generatethreerandomnumbersforthestartingandendingcopiesinthesequenceandthelengthoftheshift. 2. Performthetemporarysubsetshift. 3. Runthetimingoptimizerfort=1;:::;Tandndthetotalinfeasibilityovertheplanninghorizon. 4. Calculatethetotalnumberofsnoutchangesrequired. 5. Iftheneighborsequencehaslessinfeasibilitymaketheshiftpermanentandgotostep1. 6. Iftheneighborsequencehasthesameinfeasibilitybutlesssnoutchangesmaketheshiftpermanentandgotostep1. 7. Updatethetrialcounter. 8. Ifterminationconditionissatisedstop. 1. Findanunvisitedrstfractioncopyotherwisestop. 2. Runallpossibleshiftsandchoosetheshiftwiththeleastinfeasibilityindexforonlydayt. 103

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Iftheneighborsequencehaslessinfeasibilityindexondaytmaketheshiftpermanentandgotostep1. 1. Generatethreerandomnumbersforthestartingandendingcopiesinthesequenceandthelengthoftheshift. 2. Performthetemporarysubsetshift. 3. Runthetimingoptimizerfort=1;:::;Tandndthetotalinfeasibilityovertheplanninghorizon. 4. CalculateSCIandPSI. 5. Iftheneighborsequencehaslesstotalinfeasibilitymaketheshiftpermanentandgotostep1. 6. IftheneighborsequencehasthesameinfeasibilitybutlessSCImaketheshiftpermanentandgotostep1. 7. IftheneighborsequencehasthesameinfeasibilityandSCIbutbiggerPSImaketheshiftpermanentandgotostep1. 8. Updatethetrialcounter. 9. Ifterminationconditionissatisedstop. 104

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1. Generatetworandomnumbersfortwoswappingcopiesinthesequence. 2. Performthetemporaryswapping. 3. Runthetimingoptimizerfort=1;:::;Tandndthetotalinfeasibilityovertheplanninghorizon. 4. CalculateSCIandPSI. 5. Iftheneighborsequencehaslesstotalinfeasibilitymaketheshiftpermanentandgotostep1. 6. IftheneighborsequencehasthesameinfeasibilitybutlessSCImaketheshiftpermanentandgotostep1. 7. IftheneighborsequencehasthesameinfeasibilityandSCIbutbiggerPSImaketheshiftpermanentandgotostep1. 8. Updatethetrialcounter. 9. Ifterminationconditionissatisedstop. 1. Add2copies(or4copiesincaseofaBIDpatient)totheunionsequence. 2. Generate2f1;0;1g. 105

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Updatepresencearraysforthenewlyaddedcopiesaccordingto. 4. Generatethreerandomnumbersforthestartingandendingcopiesinthesequenceandthelengthoftheshift. 5. Performthetemporarysubsetshift. 6. Runthetimingoptimizerfort=1;:::;Tandndthetotalinfeasibilityovertheplanninghorizon. 7. Calculatethetotalnumberofsnoutchangesrequired. 8. Iftheneighborsequencehaslessinfeasibilitymaketheshiftpermanentandgotostep2. 9. Iftheneighborsequencehasthesameinfeasibilitybutlesssnoutchangesmaketheshiftpermanentandgotostep2. 10. Undotheupdatemadetopresencearrays. 11. Updatethetrialcounter. 12. Ifterminationconditionissatisedstop. Table4-21. Resultsforschedulinganewpatientondierentgantries. Gantry#ofpatientsSCIPDIRunningtime(seconds) G1702.935%42 G2631.957%53 G3970.06%55 Inallofthethreecasesabove,ourlocalsearchwasabletondhighqualityschedulesincludingthenewpatientinlessthanaminute,forinstance,incaseofschedulingthe 106

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ChunhuaMenwasborninXinjiang,China.Shecompletedherbachelor'sandmaster'sdegreeinpowerengineeringatHohaiUniversityin1997and2000respectively.ThensheworkedatNanjingAutomationResearchInstituteandShanghaiSunrise-powerAutomationCo.,tilltheendof2003thenshemovedtoUSAwithherhusband.In2005,ChunhuastartedherdoctoralstudyattheUniversityofFloridamajoringinindustrialandsystemsengineeringandreceivedherPh.D.in2009.ThenshejoinedtheDepartmentofRadiationOncologyatUniversityofCalifornia,SanDiegoMooresCancerCenterasapostdoctoralfellow. 114