Integrating Delivery Issues in Intensity-Modulated Radiation Therapy Treatment Plan Optimization

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Title:
Integrating Delivery Issues in Intensity-Modulated Radiation Therapy Treatment Plan Optimization
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1 online resource (148 p.)
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english
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Salari,Ehsan
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University of Florida
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Degree:
Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
Romeijn, Hilbrand E
Committee Members:
Smith, Jonathan
Geunes, Joseph P
Palta, Jatinder R

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Subjects / Keywords:
accuracy -- delivery -- efficiency -- intensity -- modulated -- optimization -- planning -- radiation -- therapy -- treatment
Industrial and Systems Engineering -- Dissertations, Academic -- UF
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Industrial and Systems Engineering thesis, Ph.D.
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Abstract:
My Ph.D. dissertation focuses on model and algorithm development to enhance several aspects of an IMRT treatment plan. In particular, it addresses two major clinical issues encountered in treatment delivery, which are efficiency and accuracy. The beam-on-time, as a measure of the delivery efficiency, is incorporated into the IMRT treatment planning problem using a Direct Aperture Optimization approach. This allows for taking the efficiency factor into consideration when designing a treatment plan. Moreover, we have proposed robust and efficient models and solution approaches to account for the dosimetric inaccuracies caused during the treatment delivery. More specifically, we have considered two sources of inaccuracy which may compromise the treatment outcome; (1) the tongue-and-groove effect and (2) the organ motion. To account for the tongue-and-groove effect, we have developed robust models and efficient solution methods to obtain clinically-attractive treatment plans regardless of the exact effect of this source of inaccuracy. Furthermore, to incorporate the uncertainty caused by the organ motion into the IMRT treatment planning problem, we have proposed an entirely new modeling framework and developed solution approaches to obtain high-quality treatment plans while taking the variation of the treatment outcome into account.
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In the series University of Florida Digital Collections.
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Includes vita.
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by Ehsan Salari.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Romeijn, Hilbrand E.
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IwouldliketoexpressmydeepestandeverlastinggratitudetomymentorandacademicfatherEdwinRomeijn.Hiswisdom,dedication,andrigorousthinkinghaveprovedinvaluableoverthecourseofmydoctoralstudies.Iamdeeplyindebtedtohimfortheeffortsheputintodevelopingmyacademicandresearchskills.Hismentorshipextendedfarbeyondthedutiesofanadvisortogenuinesupportinallaspectsofmygraduatecareer.IoweagreatdebtofgratitudetomydoctoralcommitteememberJosephGeunesforhisvaluablesupportandadviceatdifferentstagesofmydoctoralstudies.Hiscalmcriticalthinkinghasneverceasedtoinspireme.IamalsodeeplythankfultohimforhistremendoushelpandsupportduringmygraduatecareerattheUniversityofFlorida.IwouldliketoexpressmydeepestgratitudetoJatinderPaltaforintroducingmetotheclinicalenvironmentanditschallengesattheUniversityofFloridaProtonTherapyInstitute.Iwasveryfortunatetohavehimasmydoctoralcommitteememberandtousehisinsightfulcommentsandsuggestionsinmyresearch.Iamalsoverythankfultohimforhisadviceandinterestinmyacademicandprofessionalsuccess.IamextremelygratefultomydoctoralcommitteememberColeSmith,agreatresearcherandagreatpersonwithasenseofhumorthatwouldalwayscheermeup.Ihavebenetedconsiderablyfromhisconsultationandinsights.Iamalsoverythankfultohimforbeingsuchagreatgraduatecoordinatorandfortheconstanteffortheputsintoit.ThisdissertationhasbeenmadepossiblethroughthehelpandsupportofmyseniorcolleagueChunhuaMen.Iamverygratefultoherfortheprompttechnicalsupportandassistance.SheintroducedmetotheUniversityofFloridaOptimizedRadiationTherapy(UFORT)treatmentplanningsystemandsharedherimplementationexperiencesandcodingskillswithme. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 9 LISTOFFIGURES ..................................... 10 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 12 2ACCOUNTINGFORTHETONGUE-AND-GROOVEEFFECTUSINGAROBUSTDIRECTAPERTUREOPTIMIZATIONAPPROACH ................ 17 2.1IntroductoryRemarks ............................. 17 2.2RobustDAOModel ............................... 20 2.2.1BoundsontheApertureDoseDepositionCoefcients ....... 21 2.2.2TreatmentPlanEvaluationCriteria .................. 22 2.2.3RobustDAOFormulation ........................ 23 2.2.3.1Reformulation1 ....................... 24 2.2.3.2Reformulation2 ....................... 25 2.3SolutionMethod ................................ 27 2.3.1ColumnGenerationAlgorithm ..................... 27 2.3.2PricingProblem ............................. 28 2.3.2.1Formulatingthepricingproblem .............. 28 2.3.2.2Solvingthepricingproblem ................. 29 2.4Results ..................................... 32 2.4.1PatientCases .............................. 32 2.4.2TreatmentPlanEvaluationCriteria .................. 33 2.4.3Tongue-and-GrooveEffectBounds .................. 35 2.4.4ImplementationandResults ...................... 35 2.5ConcludingRemarks .............................. 38 3QUANTIFYINGTHETRADE-OFFBETWEENTREATMENTPLANQUALITYANDDELIVERYEFFICIENCYUSINGDIRECTAPERTUREOPTIMIZATION 46 3.1IntroductoryRemarks ............................. 46 3.2DirectApertureOptimizationProblem .................... 50 3.2.1Introduction ............................... 50 3.2.2TreatmentPlanEvaluationCriteria .................. 50 3.2.3FormulatingtheDAOProblem ..................... 51 3.3SolutionMethod ................................ 53 3.3.1FundamentalResults .......................... 55 3.3.1.1Uniqueness .......................... 55 6

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................... 57 3.3.1.3Maximumbeam-on-timepenalty .............. 58 3.3.1.4Parametricoptimaltreatmentplans ............ 59 3.3.2AlgorithmicFramework ......................... 65 3.3.3ConvexQuadraticEvaluationCriteria ................. 69 3.3.3.1Applicationofthealgorithmicframework .......... 70 3.3.3.2Finiteconvergenceofthealgorithmicframework ..... 74 3.3.4SequentialQuadraticApproximation ................. 76 3.3.5ContinuouslyDifferentiableConvexPenaltyFunctions ....... 78 3.4ComputationalResults ............................. 80 3.4.1ClinicalProblemInstances ....................... 80 3.4.2ImplementationandResults ...................... 82 3.5ConclusionandFutureResearch ....................... 86 4ACCOUNTINGFORTHEINTRAFRACTIONMOTIONUSINGADIRECTAPERTUREOPTIMIZATIONAPPROACH ............................ 95 4.1IntroductoryRemarks ............................. 95 4.2AContinuous-TimeDAOModel ........................ 97 4.3SolutionApproach ............................... 100 4.3.1Caseof2=0 ............................. 101 4.3.1.1Thepricingproblem ..................... 102 4.3.1.2Startingthetreatmentinthesteadystate ......... 104 4.3.1.3StaticDAOmodel ...................... 105 4.3.2Caseof2>0 ............................. 107 4.4ATime-DiscretizedStochasticDAOModel .................. 113 4.5ABranch-and-PriceAlgorithm ......................... 117 4.5.1Bounding ................................ 117 4.5.2BranchingScheme ........................... 119 4.5.3PricingProblem ............................. 119 4.5.4InitialColumns ............................. 120 4.5.5HeuristicandNodeSelectionStrategy ................ 120 4.5.6ColumnManagement ......................... 121 4.5.7EarlyTerminationandIntegralityGap ................. 121 4.6ComputationalResults ............................. 123 4.7FutureResearch ................................ 126 4.7.1Discrete-TimeModel .......................... 126 4.7.2Continuous-TimeModel ........................ 127 4.8ConcludingRemarks .............................. 129 APPENDIX:PRICINGPROBLEMFORTHECONTINUOUS-TIMEDAOMODEL .. 131 A.1Steady-StateCase ............................... 131 A.2Binary-StateCase ............................... 132 A.3Multi-StateCase ................................ 134 7

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....................................... 140 BIOGRAPHICALSKETCH ................................ 148 8

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Table page 2-1Problemdimensions. ................................. 33 2-2LowerandupperboundsfortargetDVHcriteriaobtainedbythenewandtraditionalmodelsfor=0.5mm(in%volume). ....................... 42 2-3LowerandupperboundsfortargetDVHcriteriaobtainedbythenewandtraditionalmodelsfor=5mm(in%volume). ........................ 42 2-4Lowerandupperboundsforcritical-structureDVHcriteriaobtainedbythenewandtraditionalmodelsfor=5mm(in%volume). ............. 42 2-5LowerandupperboundsfortargetDVHcriteriaobtainedbythenewandtraditionalmodelsfor=3mm(in%volume). ........................ 43 2-6Lowerandupperboundsforcritical-structureDVHcriteriaobtainedbythenewandtraditionalmodelsfor=3mm(in%volume). ............. 43 2-7LowerandupperboundsfortargetDVHcriteriaobtainedbythenewandtraditionalmodelsfor=1mm(in%volume). ........................ 44 2-8Lowerandupperboundsforcritical-structureDVHcriteriaobtainedbythenewandtraditionalmodelsfor=1mm(in%volume). ............. 44 2-9LowerandupperboundsfortargetDVHcriteriaobtainedbytheconservativeandexactrobustmodelsfor=3mm(in%volume). .............. 45 2-10Lowerandupperboundsforcritical-structureDVHcriteriaobtainedbytheconservativeandexactrobustmodelsfor=3mm(in%volume). ....... 45 4-1ResultsobtainedfromsolvingthestochasticDAOmodelfor2=0withinatimelimitof3600secusingaCGoptimalitygapof=3%andcomparingwiththestaticDAOmodel. ............................. 130 4-2ResultsobtainedfromsolvingthestochasticDAOmodelfor2>0withinatimelimitof3600secusingaCGoptimalitygapof=5%andcomparingwiththestaticDAOmodel. ............................. 130 4-3ComparingtheresultsobtainedfromthestochasticandstaticDAOmodelswithrespecttobothevaluationcriteriaIandII. .................. 130 9

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Figure page 1-1MLCtongue-and-groovearchitecture ........................ 15 2-1Shadedbeamletsontheboundaryoftheaperturearepartiallyblockedbytheexposedleaftongues. ................................ 21 2-2Thenetworkmodelforthepricingproblemunderrow-convexityconstraints. .. 30 2-3Leafsettingsattwoadjacentrows. ......................... 32 2-4Isodosecurves(dashedlines)for73.8Gy,54Gy,and30GyonatypicalCTslicecorrespondingtooptimaltreatmentplansobtainedforcase1. ....... 40 2-5Lower(dashed)andupper(solid)boundsontheDVHsoftheoptimaltreatmentplanobtainedbythetraditionalandrobustmodelforcase5using=3mm. 41 3-1InnerandouterenvelopesobtainedbytheNISEtechniquetoapproximatetheParetofrontier. .................................. 54 3-2(;)isaconcavepiecewise-linearfunction. .................. 71 3-3yk()characterizedbythegeneralalgorithmicframeworkiscontinuouspiecewiselinear. ......................................... 76 3-4Pareto-efcientfrontierobtainedforclinicalcancercases. ............ 88 3-5DVHcriteriaassociatedwithtargetcoverageandsalivaryglandssparingevaluatedatPareto-efcienttreatmentplansforthehead-and-neckcancercase. ..... 89 3-6DVHcriteriaassociatedwithtargetcoverageandrectumandbladdersparingevaluatedatPareto-efcienttreatmentplansfortheprostatecase. ....... 90 3-7DVHcurvesassociatedwithPareto-efcienttreatmentplansatdifferentlevelsofbeam-on-time(BOT)(inminutes)forthehead-and-neckcase. ........ 91 3-8DVHcurvesassociatedwithPareto-efcienttreatmentplansatdifferentlevelsofbeam-on-time(BOT)(inminutes)fortheprostatecase. ............ 92 3-9Isodosecurves(dashedlines)for60and73.8GyonatypicalCTslicecorrespondingtoPareto-efcienttreatmentplansfortheprostatecase. ............. 93 3-10InnerandouterenvelopesobtainedusingtheNISEtechnique.(a)and(b)head-and-neckcase,(c)and(d)prostatecase. .................. 94 A-1Lowerenvelopeobtainedbysweepingacrossthetimeinterval. ......... 134 A-2Thepropagabilityofbeingateachstatepf(t)(f2F)foracyclicCTMarkovchainstartingfromf=1withidenticaltransitionrates. .............. 139 10

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MyPh.D.dissertationfocusesonmodelandalgorithmdevelopmenttoenhanceseveralaspectsofanIMRTtreatmentplan.Inparticular,itaddressestwomajorclinicalissuesencounteredintreatmentdelivery,whichareefciencyandaccuracy.Thebeam-on-time,asameasureofthedeliveryefciency,isincorporatedintotheIMRTtreatmentplanningproblemusingaDirectApertureOptimizationapproach.Thisallowsfortakingtheefciencyfactorintoconsiderationwhendesigningatreatmentplan.Moreover,wehaveproposedrobustandefcientmodelsandsolutionapproachestoaccountforthedosimetricinaccuraciescausedduringthetreatmentdelivery.Morespecically,wehaveconsideredtwosourcesofinaccuracywhichmaycompromisethetreatmentoutcome;(1)thetongue-and-grooveeffectand(2)theorganmotion.Toaccountforthetongue-and-grooveeffect,wehavedevelopedrobustmodelsandefcientsolutionmethodstoobtainclinically-attractivetreatmentplansregardlessoftheexacteffectofthissourceofinaccuracy.Furthermore,toincorporatetheuncertaintycausedbytheorganmotionintotheIMRTtreatmentplanningproblem,wehaveproposedanentirelynewmodelingframeworkaswellassolutionapproachestoobtainhigh-qualitytreatmentplanswhiletakingthevariationofthetreatmentoutcomeintoaccount. 11

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AccordingtotheAmericanCancerSociety( AmericanCancerSociety 2010 ),eachyeararound1.5millionnewpatientsarediagnosedwithcancerintheUnitedStates.Externalradiationtherapyisoneofthemostcommonlyusedtreatmentmodalitiesforcancerbenetingmorethan25%ofcancerouspatients.InthistreatmentmodalitybeamsofradiationareusedtoeradicatethediseasebydamagingtheDNAinthecellnuclei.Inparticular,regionsthatarediagnosedtocontainthediseaseanditspossiblespread(theso-calledtargets)areirradiated.However,theradiationbeamkillsbothcancerousandnormalcellsalongitspathinthepatient'sbody.Thereby,theradiationtreatmentmustbecarefullyplannedsothataclinically-prescribedradiationdoseisdeliveredtotargetswhilesparingnormalcellsinnearbyorgansandtissues(theso-calledcriticalstructures)tothegreatestextentpossible.Therefore,multiplebeamsfromseveraldirectionsareusedsothattheirintersectionprovidesahighdosetotargets.Onthecontrary,regionscoveredbyasinglebeamoronlyafewbeamsreceivemuchlowerradiationdosesallowingforsparingthefunctionalityofthecriticalstructures. Patientsreceiveradiationtherapyusingaclinicalradiation-deliverydevicecalledlinearacceleratorwhichcanrotatearoundthepatient.TechnologicaladvancementshaveledtotherapiddevelopmentandwidespreadclinicalimplementationofaradiationdeliverytechniqueknownasIntensity-ModulatedRadiationTherapy(IMRT).InIMRT,theheadofthelinearacceleratorisequippedwithaMultileafCollimator(MLC)system,consistingofrowsofleafpairs,whichallowsfordynamicallyshapingtheradiationbeambyblockingpartofit.EachcongurationoftheMLCleavesiscalledanaperture.WiththehelpofanMLC,apertureswithawidevarietyofcomplexshapescanbeformed. InIMRT,eachorientationoftheacceleratorheaddenesarectangularbeam,eachofwhichisconceptuallydiscretizedintoacollectionofbeamletswhichhaveindividuallyadjustableintensities.Inparticular,thecollectionofallbeamletintensitiesformthe 12

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AnIMRTtreatmentplanforanindividualpatientconsistsofacollectionofaperturesalongwiththeirassociatedintensitiestobedeliveredbytheMLC.ThequalityofanIMRTtreatmentplanisevaluatedbyconsideringthecorrespondingdosedistribution.Inparticular,inclinicalpracticeacollectionoftreatmentplanevaluationcriteria,whichareallfunctionsofthedosedistribution,areusedtomeasurethetreatmentquality.ThegoalinIMRTtreatmentplanningisthentondahigh-qualitytreatmentplanwithrespecttothetreatmentplanevaluationcriteria.Forthatpurpose,IMRTtreatmentplanningproblemhasbeentraditionallydecomposedintoseveralsubproblemswhicharesolvedsequentially.Themajorsubproblemsare(1)determiningthenumberandorientationoftheradiationbeams,(2)determiningtheuencemapforeachradiationbeam,and(3)decomposingtheuencemapsintoacollectionofdeliverableapertures.Morespecically,beamorientationoptimization(BOO)istheproblemofselectingthebestsubsetofbeamanglestobeusedinthetreatmentplan,amongthesetofallpossiblebeamangles.Furthermore,giventheoptimalsetofbeamdirections,theuencemapoptimization(FMO)isconcernedwithndingtheoptimaluencemapsforthosebeamdirections.Finally,giventheoptimaluencemapforeachbeamdirection,theleafsequencing(LS)istheproblemofdecomposingeachuencemapintoacollectionofdeliverableapertureswiththeircorrespondingintensities.AmorerecentapproachtoIMRTtreatmentplanningisaperturemodulation,theso-calledDirectApertureOptimization(DAO).Givenasetofbeamdirectionstobeusedinthetreatmentplan,DAOintegratestheFMOandLSproblemsanddirectlysolvesfortheaperturesshapes 13

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LeavesincommercialMLCshaveatongue-and-groovearchitecturetohelpreducetheinterleafradiationleakage(Figure 1-1 ).However,theexposedleaftonguesontheboundaryofthedeliveredaperturesmayundesirablyblockorscatterpartoftheradiationpotentiallycausingunderdosingofthetargets,thiseffectisknownasthetongue-and-grooveeffect.Ithasbeenshownthatthetongue-and-grooveeffectcanbeclinicallysignicantcausingdeviationfromtheplanneddosedistributionaslargeas10%.Ontheotherhand,duetothesmallsizeoftheleaftongues,itisdifcultforbeamlet-baseddosecalculationalgorithmstoaccountforthetongue-and-grooveeffect.InChapter2wedevelopedarobustDAOapproachtoIMRTtreatmentplanningwhichaccountsforthedosimetricinaccuraciescausedbythetongue-and-grooveeffect.Inparticular,weobtainedlowerandupperboundsonthedosedistributiondeliveredtothepatient.WethenformulatedtheIMRTtreatmentplanningproblemasarobustDAOproblemanddevelopedanefcientsolutionapproachusingthecolumngenerationtechnique.OurrobustDAOapproachobtainstreatmentplanswhichareclinicallyattractiveregardlessoftheexacteffectoftheMLCtongue-and-groovearchitecture.ThischapterappearedinMedicalPhysics( Salarietal. 2011 ). Beam-on-timeistheamountoftimethatthelinearacceleratorisdeliveringradiationtothepatient.Itisaveryimportantmeasureofthedeliveryefciencysincetreatmentplanswithverylongbeam-on-timesmaycausebiologicalcomplicationstothepatient.Moreover,sincethepatientisrequiredtobeimmobilizedduringtheradiationtreatment, 14

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MLCtongue-and-groovearchitecture alongbeam-on-timecausesinconveniencetothepatient.Finally,timeefciencyisakeyfactorconsideredbyallradiationtherapyfacilities,impactingthepatientthroughputaswellascosts.Hence,itisdesirabletoavoidtreatmentplanswithlongbeam-on-times.Ontheotherhand,ifthebeam-on-timeistooshortthenthetreatmentoutcomemaybepoorparticularlyfortumorswithcomplexanatomy.Therefore,thereisatrade-offbetweenthebeam-on-timeandthetreatmentplanquality.Beam-on-timecanbeexplicitlyexpressedintermsoftheapertureintensities.Thus,usingaDAOframeworkinChapter3wedevelopandtestanewsolutionapproachtoIMRTtreatmentplanningwhichincorporatesthebeam-on-timeasameasureofthedeliveryefciencyintothetreatment-planoptimizationstage.Morespecically,weformulatedtheDAOproblemasabi-criteriaoptimizationproblemanddevelopedanexactsolutionmethodtocharacterizethecorrespondingPareto-efcientfrontier.Theproposedapproachcanprovideclinicianswiththetrade-offinformationforeachpatientcasesothattheycandesignaclinically-attractiveandatthesametimeefcienttreatmentplan.ThischapterhasbeenacceptedforpublicationinINFORMSJournalonComputing( SalariandRomeijn 2011 ). 15

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Chuietal. 1994 ; Dengetal. 2001 ; Galvinetal. 1993 ; Mohan 1995 ; Luanetal. 2006 ; Siochi 2009 ; SykesandWilliams 1998 ; Wangetal. 1996 ).Ontheotherhand,accuratelyestimatingthedosimetriceffectsofthetongue-and-groovearchitectureisadifculttask,especiallywiththecommonlyusedbeamlet-baseddosemodels.Thegoalofthisstudyistodevelop,implement,andtestarobustmethodthattakesdosimetricinaccuracieswithrespecttotheMLCarchitectureintoaccountexplicitly.Wewillachievethisbyemployingupperandlowerboundsonthedosedistributiondeliveredtothepatient,andtailoringtheoptimizationmodeltondatreatmentplanthatisclinicallyattractivewithrespecttothesebounds(i.e.,onethatisofhigh-qualityregardlessoftheexacteffectoftheMLCtongues).Bytestingourapproachontenclinicalcasesofhead-and-neckcancerweshowthatourapproachissuccessful,inthesensethattightdosedistributionboundscanbeachievedevenundercoarseandeasilyobtainableboundsonthedosimetriceffects.Incontrast,atreatmentplanoptimizationapproachthatdoesnottakethedosimetricinaccuraciesintoaccountyieldsdosedistributionboundsthatareloose,makingithardtoaccuratelyassessthetreatmentplanquality. 17

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Shepardetal. ( 1999 )and RomeijnandDempsey ( 2008 ).Morespecically, Leeetal. ( 2000 2003 )studiedmixedintegerprogrammingapproaches; HamacherandKufer ( 2002 )and Kuferetal. ( 2003 )proposedamulti-criteriaapproachtotheproblem;and Romeijnetal. ( 2003 2006 )developedconvexprogrammingmodels.Formally,theLSproblemistodetermineasetofdeliverableaperturesfordeliveringauencemapthatisoptimalwithrespectto,typically,beam-ontime,numberofaperturesused,ortotaltreatmenttime.Iftheobjectiveistominimizebeam-ontimeandanyrow-convexapertureisdeliverabletheLSproblemisefcientlysolvable( AhujaandHamacher 2005 ; Bortfeldetal. 1994 ; Kamathetal. 2003 ; Siochi 1999 ).Inaddition, Baataretal. ( 2005 ), Bolandetal. ( 2004 ), Kamathetal. ( 2004a ), DaiandHu ( 1999 ),and Siochi ( 1999 )studiedtheproblemunderadditionalMLChardwareconstraints.Incontrast,as Baataretal. ( 2005 )showed,theproblemofdecomposingauencemapintotheminimumnumberofaperturesisNP-hard.Thishasledtothedevelopmentofalargenumberofheuristicsforsolvingthisproblem,suchas Baataretal. ( 2005 ), DaiandZhu ( 2001 ), Siochi ( 1999 ),and XiaandVerhey ( 1998 ).Additionally, Engel ( 2005 ), Kalinowski ( 2005b a ),and LimandChoi ( 2006 )developedheuristicstominimizethenumberofapertureswhileconstrainingthetotalbeam-ontimetobeminimal.Finally, Tasknetal. ( 2010b )proposedanintegerprogrammingapproachtominimizethetotaltreatmenttime.Thetongue-and-grooveeffecthasbeenwidelyaddressedinliterature 18

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VanSantvoortandHeijmen ( 1996 )and Webbetal. ( 1997 )haveproposedLSalgorithmswhichreducethetongue-and-grooveeffectfordynamicdelivery,and Queetal. ( 2004 )and Kamathetal. ( 2004b )haveincorporatedthetongue-and-grooveeffectintheLSproblemforthestaticdelivery.Finally, Kamathetal. ( 2004c )compareddifferentLSalgorithmsforstaticdeliverywithrespecttothetongue-and-grooveeffectaswellasthetotaltreatmenttime. Amajorissuewiththetraditionaltwo-stagemethodisthatthedosedeliveredtoapatientnotonlydependsontheuencemapsbutalsoontheactualshapeoftheaperturesused.Toaddressthisissue,anintegratedapproachtotheFMOandLSproblems,usuallyreferredtoasaperturemodulationorDirectApertureOptimization(DAO)hasbeenproposed.DAOexplicitlysolvesforapertureshapesandintensitiesratherthanbeamletintensities( Shepardetal. 2002 ; Bednarzetal. 2002 ; Preciado-Waltersetal. 2006 ; Romeijnetal. 2005 ; Menetal. 2007 ).Incontrastwiththetraditionalmethod,DAOexplicitlyincorporatestheshapeoftheapertureswhileoptimizingfortheapertureintensities. Ingeneral,treatmentplansobtainedwithDAOusemuchfewerapertures(infact,thisisconsideredtobeoneoftheattractivesideeffectsofDAOsinceitreducestreatmenttimesandbeam-on-times).Therefore,onecouldarguethatthetongue-and-grooveerrorismoresignicantwhenusingDAOsincemuchfeweraperturesareused.However,thesmallernumberofaperturesalsoimpliesthatDAOuencemapstendtobemuchsmootherthanonesobtainedwithtraditionaltwo-stageapproaches,andsmootheruencemapscanbeexpectedtosufferlessfromthemostsevereinaccuraciescausedbythetongue-and-grooveeffect,namelythoseclosetothe 19

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Earletal. ( 2006 )say:Ifthetongue-and-grooveeffectisnotaccountedforintheplanning,anunderestimationoftheabsolutedoseisobservedforDAOIMRTplans.Themagnitudeoftheunderestimationisdependentupontheapertureshapes.Thissuggeststhatthetreatmentplanoptimizationmodelshouldmakethecorrespondingdeterminationandndauencemapandcorrespondingcollectionofaperturesthatyieldahigh-qualitytreatmentplandespitethetongue-and-groovearchitectureoftheMLC.Inthisstudy,weaddressthisbydeveloping,implementing,andtestingarobustDAOmodelthatexplicitlytakesintoaccountthetongue-and-grooveeffectandthecorrespondinginaccuraciesofthedosecalculation.

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Thisexpressiononlyprovidesanapproximationtotheaperturedosedepositioncoefcients( BjarngardandSiddon 1982 ; Clarkson 1941 ; Menetal. 2007 ; Sanz 2000 ).Inthisstudy,wewillfocusonthefactthatEquation( 2 )ignoresthepresenceofthetongue-and-groovearchitectureoftheleaves.Inparticular,beamletsontheboundaryofaperturek2Kthatexposeatonguearepartiallyblocked;Figure 2-1 illustratesthisissue.Thiswillcauseinaccuraciesinevaluatingthedosedistributionzjj2VwhenusingtheaperturedosedepositioncoefcientsinEquation( 2 ).Infact,theright-handsideofEquation( 2 )isanupperboundonDkjsinceiteffectivelyassumesthatnoleaftongueispresent;intheremainder,wewilldenotethisupperboundby 2 ),potentiallycausingunderdosingoftargetvoxels. Figure2-1. Shadedbeamletsontheboundaryoftheaperturearepartiallyblockedbytheexposedleaftongues. 21

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Boyeretal. 2001 ),itisdifcultforcommonlyusedapproximatedosemodelstoaccuratelyaccountforitseffectonthebeamletdosedepositioncoefcients.OurapproachwillthereforebetoobtainalowerboundforthesecoefcientsbyoverestimatingthewidthofthetonguetoallowforanaccurateestimationofDij.Clearly,whenischosentobelargerthanthewidthofthetongue,D 22

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Romeijnetal. 2004 ).Moreover,weassumethatthesetLcanbepartitionedintotwosubsetsL=L 2.2.1 ( Ben-TalandNemirovski 1998 2002 ).Thisisanoptimizationproblemformulatedintermsofalldeliverableaperturesandtheirassociatedintensities:minimizemaxD DX`2L`G`Xk2KDkyk!subjectto(R)yk0k2K 23

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Theobjectivefunctionof(R)hasamathematicallyinconvenientandcumbersomeform.Intheremainderofthissectionwewillthereforeproposetworeformulationsofthismodelastractableconvexoptimizationproblems.Therstisareformulationthatusesthemathematicalpropertiesofthetreatmentplanevaluationcriteriatoderiveaconservativeboundontheobjectivefunction.Thesecondoneisequivalentto(R),butappliesonlyifthetreatmentplanevaluationcriteriaareconvexvoxel-basedpenaltyfunctions.Bothreformulationsrelyonnewdecisionvariablesz Inotherwords,Equations( 2 )and( 2 )providelowerandupperboundsonthedosedistribution.

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ItiseasytoseethattheobjectivefunctionsatisesmaxD DX`2L`G`Xk2KDkyk!X`2L`maxD DG`Xk2KDkyk!. Sinceforeach`2L DG`Xk2KDkyk!=G`Xk2KD DG`Xk2KDkyk!=G`Xk2K 25

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Nowletusevaluate,foreachvoxel,thepenaltyfunctionatboththeupperandlowerboundofthevoxeldose,andassociatethemaximumofthesetwopenaltiestothisvoxel.ThisyieldsthefollowingDAOmodel: minimizeXj2VmaxFjz zj=Xk2K 2 )wecannowexactlyreformulatetheobjectivefunctionasfollows:maxD DXj2VFjXk2KDkjyk!=Xj2VmaxD DjFjXk2KDkjyk! DjFjXk2KDkjyk!=max(FjXk2KD 26

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2.3.1ColumnGenerationAlgorithm InthefollowingwederivethemathematicalformofthepricingsubproblemanddevelopanefcientalgorithmforsolvingthisproblemtooptimalityincasetheonlydeliverabilityconstraintsoftheMLCaretheso-calledrow-convexityconstraints.DependingonthemanufactureroftheMLC,aperturescanbesubjecttootherdeliverabilityconstraintsaswell,andthealgorithmdescribedinAppendixB.2caneasilybeextendedtoaccountfor:(1)interdigitationconstraintsand(2)connectednessconstraints. 27

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2.3.2.1Formulatingthepricingproblem 2.3.1 anddevelopasolutionapproachtosolveit.Let 2 )( 2 ).InordertobeabletoaccommodatebothofthetworeformulationsinSection 2.2.3 wewillactuallystudyaslightlymoregeneralmodelwhich,withaslightabuseofnotation,allowsforthetreatmentplanevaluationcriteriatobeafunctionofbothz Bazaraaetal. 2006 )arenecessaryandsufcientconditionsforoptimalityof(P).Assumingforconveniencethattheobjectivefunctionisdifferentiabletheycanbewrittenasfollows:z zj=Xk2K zjj2Vk=Xj2VD Hiriart-Urruty ( 1978 ).Theanalysisintheremainderofthissectionremainsessentiallyunchanged.)Anysolutionofthesystemabovecanbecharacterizedbyavectorofapertureintensitiesy0;thisvector 28

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Romeijnetal. ( 2005 )showthatthepricingproblemdecomposesbybeamletrow,sothatwecanndtheoptimalsolutiontothepricingproblembyindependentlyndinganoptimalpairofleafsettingsforeachbeamletrow.However,whenthetongue-and-groove 29

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Romeijnetal. ( 2005 )forsolvingthepricingproblemunderinterdigitationconstraints.Inparticular,consideraxedbeamwithabeamletgridofdimensionsmn(i.e.,mrowsandncolumns).Wethencreateanetworkwhereeachnodecorrespondstoapotentialleafsettinginaparticularbeamletrow.Inotherwords,atypicalnodeischaracterizedas(r,c1,c2),whererindicatesthebeamletrowand(c1,c2)representtherightmostbeamletblockedbytheleftleafandtheleftmostbeamletblockedbytherightleafinrowr(r=1,...,m;c1=0,...,n;andc2=1,...,n+1withc1
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2-3 ).Representingthebeamletinrowrandcolumncby(r,c),thecostassociatedwiththearcfromnode(r,c1,c2)tonode(r+1,c01,c02)isequaltoc01Xc=c1+1Xj2VD(r,c)j^ Ahujaetal. 1993 ). Finally,wenotethatunderinterdigitationconstraints(theleftleafofarowcannotoverlapwiththerightleafofanadjacentrow)andconnectednessconstraints(therows 31

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Leafsettingsattwoadjacentrows. inwhichatleastonebeamletisexposedareconsecutive,relevantiftheleftandrightleavescannotentirelyblockabeamletrowandbackupjawsarerequired)thepricingproblemcanbesolvedinaverysimilarwayandwiththesamerunningtimeusingslightmodicationsofthealgorithmsforthesecasespresentedin Romeijnetal. ( 2005 ). 2.4.1PatientCases 2-1 showstheproblemdimensionsforthetencases. EachcasecontainstwoPlanningTargetVolumes,PTV1andPTV2,withprescriptiondosesof73.8Gyand54Gy,respectively.Todeterminetheclinicalqualityofthetreatmentplansobtainedbyourmethodweemploythefollowingclinicaldose-volumehistogram(DVH)criteria(fromthetreatmentplanningprotocolusedintheDepartmentofRadiationOncologyattheUniversityofFlorida): 32

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DVHconstraintsonadditionalcriticalstructures,suchasforexampleopticnervesandchiasm,werealwayseasilysatisedinourexperimentssowehaveomittedthemfromourresultsandanalysis. Table2-1. Problemdimensions. 11481317,10885,0172131,28223,998104,298381,32036,288189,2344111,47138,609195,11351293515,91686,25561369213,78358,6367101,04421,241102,2628101,00518,60984,36991082214,52071,87310121,72140,198148,294

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whereEquation( 2 )quadraticallypenalizesunderdosingbelowtheunderdosingthresholdT 2 )quadraticallypenalizesoverdosingabovetheoverdosingthreshold 2.2.3.2 wehavethefollowingvoxel-basedpenaltyfunctions:Fj(zj)=(s,)F

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2 )and( 2 ).Forthatpurpose,wecalculatedthedosedepositioncoefcientsfortheactualbeamletsaswellasforthereducedbeamletsfromwhichastripofwidthwasremoved.IncommercialMLCsthetongue-and-grooveoffsetoftheleafcanbeassmallas0.5mm( Boyeretal. 2001 ).WethereforegeneratedvaluesofDijfor2f0,0.5,1,3,5g(whereismeasuredinmm)andsolvedtheDAOmodelforeachofthesevalues.Sinceitishardtoaccuratelyquantifytheeffectofaverysmallvalueofwedonotadvocateusing=0.5inaclinicalsetting.However,weusedthisvaluetoassesstheimportanceofevensmalldeviationsfromtheideal(i.e.,withoutleaftongue)dosedepositioncoefcients.Next,wecomparedthequalityofthetreatmentplansforthecoarserboundsobtainedusingthelargervalues2f1,3,5gwiththetreatmentplanobtainedwiththetraditionalmodel(=0). Alemanetal. 2010 ).Wemanuallytunedthemodelparameters(i.e.,theunderdosingandoverdosingthresholdsaswellastheweightsassociatedwiththeevaluationcriteria)basedontwoofthecases.Wethenusedthissetofparameterstosolve(differentvariantsof)theproblemforalltenpatientcases.AlltheexperimentswereperformedinMATLAB2009bona2.33GHzIntelCore2Duoprocessorwith2GBofRAMusingWindowsoperatingsystem. Asmentionedearlierinthissection,weuseourcolumngenerationalgorithmtosolvetheinstancesof(P),andateachiterationofthealgorithm,wesolvethepricingproblemtodetermineifthereexistsanypromisingaperturewhichcanimprovethetreatmentplan.Asthealgorithmprogressesandthenumberofaperturesexplicitly 35

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Table 2-2 comparestheDVHcriteriaassociatedwithtargetcoverageobtainedbytherobustandtraditionalDAOmethodsusing=0.5forthreeclinicalcases.ForallcasesandDVHcriteria,weshowboththelowerandupperboundonthepercentvolumeofastructurethatreceivesatleastthespecieddose.However,duetotherealisticsizeofthetonguewidth,thelowerboundsaremostrepresentativeoftherealizeddelivereddosetothepatient.TheresultsthereforeclearlyshowthattreatmentplansobtainedbythetraditionalDAOmodelexhibitsignicantunderdosingofPTV1andalso,toasomewhatlesserextent,ofPTV2.Inparticular,theactualpercentvolumeofPTV1receivingatleasttheprescribeddose(73.8Gy)inthetraditionalDAOmodel(thatignoresthetongue-and-grooveeffect)variesfrom79%whilethecorrespondingvaluesintherobustDAOmodelare95%.Thishighlightsthepotentialriskofunderdosingthetargetwhenignoringthetongue-and-grooveeffectduringthetreatmentplanningphase.Inparticular,Figure 2-4 illustratestheisodosecurvesonatypicalCTsliceforclinicalcase1.Morespecically,isodoselinescorrespondingtothelowerandupperboundsonthedosedistributionobtainedbythetraditionalandrobustmodelsareseparatelyshowninthegure.Forthetraditionalmodel,thelowerboundonthedosedistributionyieldsseveralcoldspotsinPTV1.However,thelowerboundobtainedbytherobustmodelhassignicantlyfewercoldspotsandyieldsabettertargetcoverage. IntheremainderofthissectionwewillfocusontherobustDAOmodelwithmoreconservativeboundsonthedosedistributionobtainedwithlargervalues. 36

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2-3 2-8 comparetheresultsofourrobustDAOmethodthataccountsforthetongue-and-grooveeffectwithresultsofthetraditionalDAOmodelontenclinicalcasesfor=1,3,and5. Withrespecttotumorcoverage,therobustDAOmodelprovidestightDVHboundscomparedtothetraditionalDAOmodel.IncontrasttothetraditionalDAOmodel,therobustDAOmodeliscapableofmaintainingacceptableDVHlowerboundsontargetcoverageinthemajorityofthepatientcases,evenforlargevaluesof.ItisimportanttonotethatthewidthoftheMLCleaftonguesis(much)smallerthanthevaluesofthatweusedinourexperiments.Thismeansthatthelowerboundscalculatedforbothmodelscanbeexpectedtobequiteloose,especiallyfor=3or5.Thismeansthateveninthefewcaseswherethetargetcoveragelowerboundsfallslightlyshortofwhatisdesired,theactualcoverageshouldbeclinicallyacceptable.ThisargumentofcoursealsoappliestothetraditionalDAOmodel.However,forthismodelthelowerboundsaresofarawayfromwhatisclinicallyacceptablethatwecannotconcludethatthetreatmentplansobtainedwithouttakingtongue-and-grooveeffectsintoaccountareclinicallyacceptable.Inparticular,employingtherobustDAOmodelcansignicantlyimprovetheuncertaintiesinPTV1andPTV2coverageduetothetongue-and-grooveeffect.Forexample,withtherobustDAOmodeland=3,thelowerboundonPTV1targetcoverageattheprescriptiondoseof73.8Gyisatleast95%in8outof10cases,and92%and94%intheothertwocases,respectively.Forthetraditionalmodel,thecorrespondinglowerboundsarebelow10%inallcases. Withrespecttostructuresparing,forlargervaluesofthetraditionalDAOmodelprovidessmallerupperboundsonthecorrespondingDVHcriteriathantherobustDAOmodel.ThiscanbeexplainedbythefactthattoensureadesirabletargetcoverageforallaperturedosedepositioncoefcientsintherangeD D(i.e.,adesirablelowerboundontargetcoverage),therobustDAOmodelisforcedtoselectaperturesthatprovidemoredosetocriticalstructures.Howeverpleasenotethat,inthemajorityofthe 37

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Figure 2-5 illustratestheDVHsoftheoptimaltreatmentplansobtainedbytherobustandtraditionalDAOmodelsforcase5with=3mm.Foragivenstructure,thesolidanddashedlinesrepresenttheupperandlowerboundsonDVHvalues,respectively.Theupperboundscorrespondtoanidealizeddosedistributionthatignoresthepresenceoftheleaftongues,whilethelowerboundsindicatehowmuchtheactualdelivereddosedistributionmaydeviatefromtheoptimizedone. Finally,wealsoevaluatedtheabilityofthemoreconservativemodel(Pc)inprovidinghigh-qualityrobusttreatmentplansthataccountforthetongue-and-grooveeffect.Asnotedbefore,whenusingtreatmentplanevaluationcriteriathatarevoxel-basedpenaltyfunctions(asdescribedinSection 2.4.2 )wedonotneedtousethemoreconservativemodelbutcanderiveanexactrobustmodel.However,itisstillinterestingtocomparethetwoapproachessincethemoreconservativeoneshouldbeusedifothercriteriaaredesired.TheseexperimentsshowedthatthedifferencesbetweentheconservativeandtheexactrobustmodelsintermsofDVHboundsisnegligible.Weillustratetheresultsofthetwomodelsfor=3inTables 2-9 and 2-10 38

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39

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Isodosecurves(dashedlines)for73.8Gy,54Gy,and30GyonatypicalCTslicecorrespondingtooptimaltreatmentplansobtainedforcase1. 40

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BRobustmodel Lower(dashed)andupper(solid)boundsontheDVHsoftheoptimaltreatmentplanobtainedbythetraditionalandrobustmodelforcase5using=3mm. 41

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LowerandupperboundsfortargetDVHcriteriaobtainedbythenewandtraditionalmodelsfor=0.5mm(in%volume). 1(100,100)(95,98)(0,2)(0,0)(99,99)(95,96)(100,100)(79,96)(0,1)(0,0)(98,99)(92,95)5(100,100)(100,100)(0,0)(0,0)(98,99)(96,97)(100,100)(92,100)(0,0)(0,0)(98,99)(94,96)6(100,100)(96,99)(0,0)(0,0)(99,99)(95,97)(100,100)(84,97)(0,0)(0,0)(99,99)(93,96) LowerandupperboundsfortargetDVHcriteriaobtainedbythenewandtraditionalmodelsfor=5mm(in%volume). 1(100,100)(91,98)(0,7)(0,0)(97,100)(92,98)(0,100)(0,96)(0,1)(0,0)(39,99)(22,95)2(100,100)(93,100)(0,11)(0,0)(100,100)(100,100)(1,100)(0,97)(0,3)(0,0)(95,100)(85,100)3(100,100)(96,99)(0,1)(0,0)(99,100)(96,98)(2,100)(0,99)(0,0)(0,0)(50,100)(36,97)4(100,100)(100,100)(0,0)(0,0)(98,99)(95,97)(0,100)(0,100)(0,0)(0,0)(31,99)(15,96)5(100,100)(97,100)(0,0)(0,0)(98,99)(94,97)(0,100)(0,100)(0,0)(0,0)(39,99)(21,96)6(100,100)(93,97)(0,0)(0,0)(99,99)(94,97)(0,100)(0,97)(0,0)(0,0)(41,99)(23,96)7(100,100)(97,99)(0,0)(0,0)(99,99)(96,98)(0,100)(0,99)(0,0)(0,0)(53,99)(32,97)8(100,100)(98,100)(0,0)(0,0)(98,99)(94,98)(0,100)(0,100)(0,0)(0,0)(46,99)(30,96)9(100,100)(94,100)(0,2)(0,0)(98,99)(94,97)(0,100)(0,99)(0,0)(0,0)(48,98)(28,95)10(100,100)(100,100)(0,0)(0,0)(100,100)(100,100)(25,100)(0,100)(0,0)(0,0)(92,100)(30,100) Lowerandupperboundsforcritical-structureDVHcriteriaobtainedbythenewandtraditionalmodelsfor=5mm(in%volume). 1(25,28)(21,21)(48,59)(72,78)(0,0)(0,0)(0,0)(9,21)(10,18)(14,37)(49,66)(0,0)(0,0)(0,0)2(76,77)(94,94)(100,100)(100,100)(0,2)(0,0)(1,1)(67,75)(88,91)(100,100)(100,100)(1,3)(0,1)(0,1)3(21,21)(18,18)n/an/an/a(0,0)(0,0)(17,21)(15,19)n/an/an/a(0,0)(0,0)4(10,10)(2,2)(60,61)(19,22)(0,0)(0,0)(0,0)(5,9)(0,2)(49,57)(9,19)(0,0)(0,0)(0,0)5(39,45)(0,0)(47,56)(48,54)(0,0)(0,0)(0,0)(28,43)(0,0)(26,47)(22,43)(0,0)(0,0)(0,0)6(28,30)(41,43)n/an/a(6,6)(0,0)(0,0)(19,27)(22,37)n/an/a(0,5)(0,0)(0,0)7(0,0)(49,49)(32,38)(100,100)(0,0)(0,0)(0,0)(0,1)(38,48)(4,25)(100,100)(0,0)(0,0)(0,0)8(29,30)(7,8)(100,100)(61,69)(0,0)(0,0)(0,0)(18,32)(0,4)(96,100)(41,59)(0,0)(0,0)(0,0)9(2,2)(45,47)(43,48)(100,100)(0,0)(0,0)(0,0)(0,3)(26,41)(19,41)(83,95)(0,0)(0,0)(0,0)10(7,7)(47,48)(27,32)(100,100)(0,0)(0,1)(0,0)(25,27)(44,50)(27,36)(97,100)(0,0)(0,0)(0,0)

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LowerandupperboundsfortargetDVHcriteriaobtainedbythenewandtraditionalmodelsfor=3mm(in%volume). 1(100,100)(92,98)(0,4)(0,0)(98,100)(93,97)(1,100)(0,96)(0,1)(0,0)(76,99)(57,95)2(100,100)(95,100)(0,8)(0,0)(100,100)(100,100)(27,100)(1,97)(0,3)(0,0)(100,100)(98,100)3(100,100)(97,100)(0,0)(0,0)(100,100)(97,99)(56,100)(1,99)(0,0)(0,0)(84,100)(56,97)4(100,100)(100,100)(0,0)(0,0)(98,99)(95,97)(17,100)(0,100)(0,0)(0,0)(80,99)(41,96)5(100,100)(99,100)(0,0)(0,0)(98,99)(94,97)(24,100)(0,100)(0,0)(0,0)(78,99)(51,96)6(100,100)(94,99)(0,0)(0,0)(99,100)(95,97)(3,100)(0,97)(0,0)(0,0)(79,99)(54,96)7(100,100)(96,100)(0,0)(0,0)(99,99)(96,98)(10,100)(0,99)(0,0)(0,0)(84,99)(64,97)8(100,100)(97,100)(0,0)(0,0)(98,99)(94,98)(13,100)(0,100)(0,0)(0,0)(79,99)(57,96)9(100,100)(95,100)(0,2)(0,0)(98,99)(94,97)(5,100)(0,99)(0,0)(0,0)(79,98)(61,95)10(100,100)(100,100)(0,0)(0,0)(100,100)(100,100)(95,100)(9,100)(0,0)(0,0)(100,100)(100,100) Lowerandupperboundsforcritical-structureDVHcriteriaobtainedbythenewandtraditionalmodelsfor=3mm(in%volume). 1(24,25)(19,20)(40,51)(68,73)(0,0)(0,0)(0,0)(15,21)(14,18)(24,37)(58,66)(0,0)(0,0)(0,0)2(75,75)(92,93)(100,100)(100,100)(1,3)(1,1)(1,1)(70,75)(90,91)(100,100)(100,100)(0,3)(0,1)(0,1)3(21,21)(18,18)n/an/an/a(0,0)(0,0)(19,21)(17,19)n/an/an/a(0,0)(0,0)4(10,11)(1,2)(59,60)(19,20)(0,0)(0,0)(0,0)(8,9)(0,2)(53,57)(11,19)(0,0)(0,0)(0,0)5(42,45)(0,0)(48,56)(44,48)(0,0)(0,0)(0,0)(35,43)(0,0)(34,47)(30,43)(0,0)(0,0)(0,0)6(32,33)(41,44)n/an/a(4,5)(0,0)(0,0)(21,27)(28,37)n/an/a(1,5)(0,0)(0,0)7(0,1)(48,49)(33,38)(100,100)(0,0)(0,0)(0,0)(0,1)(43,48)(11,25)(100,100)(0,0)(0,0)(0,0)8(29,30)(7,8)(100,100)(63,69)(0,0)(0,0)(0,0)(23,32)(2,4)(98,100)(49,59)(0,0)(0,0)(0,0)9(2,2)(42,45)(40,44)(100,100)(0,0)(0,0)(0,0)(0,3)(33,41)(28,41)(90,95)(0,0)(0,0)(0,0)10(9,11)(47,49)(30,33)(99,100)(0,0)(0,0)(0,0)(25,27)(46,50)(30,36)(98,100)(0,0)(0,0)(0,0)

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LowerandupperboundsfortargetDVHcriteriaobtainedbythenewandtraditionalmodelsfor=1mm(in%volume). 1(100,100)(94,98)(0,3)(0,0)(99,99)(94,97)(99,100)(30,96)(0,1)(0,0)(96,99)(88,95)2(100,100)(96,99)(0,6)(0,0)(100,100)(100,100)(100,100)(56,97)(0,3)(0,0)(100,100)(100,100)3(100,100)(97,100)(0,0)(0,0)(100,100)(97,99)(100,100)(67,99)(0,0)(0,0)(99,100)(91,97)4(100,100)(100,100)(0,0)(0,0)(98,98)(95,97)(100,100)(50,100)(0,0)(0,0)(98,99)(90,96)5(100,100)(99,100)(0,0)(0,0)(98,99)(95,97)(100,100)(50,100)(0,0)(0,0)(97,99)(89,96)6(100,100)(95,99)(0,0)(0,0)(99,100)(95,97)(100,100)(35,97)(0,0)(0,0)(98,99)(89,96)7(100,100)(98,100)(0,0)(0,0)(99,99)(96,98)(100,100)(36,99)(0,0)(0,0)(98,99)(91,97)8(100,100)(99,100)(0,0)(0,0)(99,99)(96,97)(100,100)(44,100)(0,0)(0,0)(98,99)(89,96)9(100,100)(98,100)(0,0)(0,0)(98,99)(95,97)(100,100)(37,99)(0,0)(0,0)(96,98)(89,95)10(100,100)(100,100)(0,0)(0,0)(100,100)(100,100)(100,100)(91,100)(0,0)(0,0)(100,100)(100,100) Lowerandupperboundsforcritical-structureDVHcriteriaobtainedbythenewandtraditionalmodelsfor=1mm(in%volume). 1(22,23)(19,20)(38,41)(66,66)(0,0)(0,0)(0,0)(19,21)(16,18)(31,37)(65,66)(0,0)(0,0)(0,0)2(74,74)(91,91)(100,100)(100,100)(2,2)(1,5)(1,1)(74,75)(91,91)(100,100)(100,100)(1,3)(0,1)(0,1)3(21,21)(19,19)n/an/an/a(0,0)(0,0)(20,21)(18,19)n/an/an/a(0,0)(0,0)4(9,10)(1,1)(58,58)(18,18)(0,0)(0,0)(0,0)(9,9)(1,2)(56,57)(17,19)(0,0)(0,0)(0,0)5(42,43)(0,0)(46,49)(39,41)(0,0)(0,0)(0,0)(40,43)(0,0)(41,47)(38,43)(0,0)(0,0)(0,0)6(28,29)(38,41)n/an/a(5,7)(0,0)(0,0)(24,27)(34,37)n/an/a(1,5)(0,0)(0,0)7(0,0)(48,48)(24,26)(100,100)(0,0)(0,0)(0,0)(0,1)(47,48)(21,25)(100,100)(0,0)(0,0)(0,0)8(29,30)(3,4)(100,100)(59,60)(0,0)(0,0)(0,0)(29,32)(3,4)(100,100)(56,59)(0,0)(0,0)(0,0)9(3,5)(41,43)(38,41)(97,97)(0,0)(0,0)(0,0)(2,3)(39,41)(39,41)(95,95)(0,0)(0,0)(0,0)10(13,13)(48,48)(30,30)(97,97)(0,0)(0,0)(0,0)(26,27)(49,50)(35,36)(100,100)(0,0)(0,0)(0,0)

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LowerandupperboundsfortargetDVHcriteriaobtainedbytheconservativeandexactrobustmodelsfor=3mm(in%volume). 1(100,100)(92,98)(0,4)(0,0)(98,99)(93,97)(100,100)(92,98)(0,4)(0,0)(98,100)(93,97)2(100,100)(95,99)(0,7)(0,0)(100,100)(100,100)(100,100)(95,100)(0,8)(0,0)(100,100)(100,100)3(100,100)(97,100)(0,0)(0,0)(100,100)(97,99)(100,100)(97,100)(0,0)(0,0)(100,100)(97,99)4(100,100)(100,100)(0,0)(0,0)(98,99)(95,97)(100,100)(100,100)(0,0)(0,0)(98,99)(95,97)5(100,100)(95,100)(0,0)(0,0)(98,99)(94,97)(100,100)(99,100)(0,0)(0,0)(98,99)(94,97)6(100,100)(95,98)(0,0)(0,0)(99,99)(95,98)(100,100)(94,99)(0,0)(0,0)(99,100)(95,97)7(100,100)(96,100)(0,0)(0,0)(99,99)(96,98)(100,100)(96,100)(0,0)(0,0)(99,99)(96,98)8(100,100)(97,100)(0,0)(0,0)(98,99)(97,98)(100,100)(97,100)(0,0)(0,0)(98,99)(94,98)9(100,100)(95,100)(0,2)(0,0)(98,99)(94,97)(100,100)(95,100)(0,2)(0,0)(98,99)(94,97)10(100,100)(100,100)(0,0)(0,0)(100,100)(100,100)(100,100)(100,100)(0,0)(0,0)(100,100)(100,100) Lowerandupperboundsforcritical-structureDVHcriteriaobtainedbytheconservativeandexactrobustmodelsfor=3mm(in%volume). 1(23,25)(20,21)(40,51)(68,70)(0,0)(0,0)(0,0)(24,25)(19,20)(40,51)(68,73)(0,0)(0,0)(0,0)2(73,74)(92,93)(100,100)(100,100)(2,4)(1,1)(1,1)(75,75)(92,93)(100,100)(100,100)(1,3)(1,1)(1,1)3(21,21)(18,19)n/an/an/a(0,0)(0,0)(21,21)(18,18)n/an/an/a(0,0)(0,0)4(9,10)(2,2)(60,60)(18,18)(0,0)(0,0)(0,0)(10,11)(1,2)(59,60)(19,20)(0,0)(0,0)(0,0)5(43,46)(0,0)(48,53)(46,50)(0,0)(0,0)(0,0)(42,45)(0,0)(48,56)(44,48)(0,0)(0,0)(0,0)6(28,31)(43,45)n/an/a(4,4)(0,0)(0,0)(32,33)(41,44)n/an/a(4,5)(0,0)(0,0)7(0,1)(48,49)(28,35)(100,100)(0,0)(0,0)(0,0)(0,1)(48,49)(33,38)(100,100)(0,0)(0,0)(0,0)8(29,30)(8,9)(100,100)(60,66)(0,0)(0,0)(0,0)(29,30)(7,8)(100,100)(63,69)(0,0)(0,0)(0,0)9(3,5)(43,45)(40,45)(100,100)(0,0)(0,0)(0,0)(2,2)(42,45)(40,44)(100,100)(0,0)(0,0)(0,0)10(10,11)(47,48)(37,37)(98,100)(0,0)(0,0)(0,0)(9,11)(47,49)(30,33)(99,100)(0,0)(0,0)(0,0)

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Webb 2001 ; Bortfeld 2006 ).Inparticular,eachcongurationoftheMLCleavesiscalledanaperture.AnIMRTtreatmentplanconsistsofacollectionofaperturestobeformedbytheMLCalongwiththeirassociatedintensities.Beam-on-timeistheamountoftimethatthelinearacceleratorisdeliveringdosetothepatient,whichcanbeexpressedasthesumofallapertureintensities.Beam-on-timeisanimportantaspectofatreatmentplan.Firstly,longbeam-on-timescancauseinconveniencetothepatientsincetheyarerequiredtobeimmobilizedwhilereceivingradiation.Furthermore,totalbodydose(orintegraldose)anditsbiologicalcomplicationsarehighlydependentonthebeam-on-time( HallandWuu 2003 ; Hall 2006 ).Finally,timeefciencyisakeyfactorconsideredbyallradiationtherapyfacilities,impactingthenumberofpatientsthatcanbetreatedaswellascosts.Ontheotherhand,obtainingaclinicallyacceptabletreatmentplan,especiallyfortumorswithcomplexanatomy,requiresrelativelylongbeam-on-times( Mohanetal. 2000 ).ThegoalofthispaperistodevelopandtestanewsolutionapproachtoIMRTtreatmentplanningwhichincorporatesthebeam-on-timeasameasureofthedeliveryefciencyintothetreatment-planoptimizationstage.Thiswillallowforexplicitlyquantifyingthetrade-offbetweenthebeam-on-timeandthequalityofthetreatmentplan.Theproposed 46

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IMRTtreatmentplanoptimizationistraditionallyperformedintwosequentialstages:(1)uencemapoptimization(FMO)and(2)leafsequencing(LS).Inparticular,theradiationheadateachbeamdirectiondenesarectangularbeam,whichisconceptuallydiscretizedintoasetofsmallbeamlets.TheFMOproblemtheninvolvesdeterminingtheoptimalintensitiesforallbeamlets;formodelingandsolutionapproachestoFMOwerefertothereviewpapersby Shepardetal. ( 1999 )and RomeijnandDempsey ( 2008 ).Giventheoptimaluencemap,theLSproblemdecomposestheuencemapforeachbeamintoasetofdeliverableaperturesthatisoptimalwithrespectto,typically,beam-on-time,numberofaperturesused,ortotaltreatmenttime.Minimizingbeam-on-timeintheLSproblemhasbeenextensivelystudiedintheliterature.Inparticular,ifanyrow-convexapertureisdeliverable,severalpolynomial-timealgorithmshavebeenproposedtosolvetheproblem( AhujaandHamacher 2005 ; Bortfeldetal. 1994 ; Kamathetal. 2003 ; Siochi 1999 ).Inaddition, Baataretal. ( 2005 ), Bolandetal. ( 2004 ), DaiandHu ( 1999 ), Kamathetal. ( 2004a ),and Siochi ( 1999 )studiedtheproblemunderadditionalMLChardwareconstraints.However,postponingthebeam-on-timeminimizationtotheLSstagepreventsusfromstudyingthedesiredtrade-offsincethetreatmentqualityisdeterminedbytheuencemapwhichisxedatthisstage.Hence,somestudieshaveaddressedthebeam-on-timeminimizationintheFMOstage.Inparticular,ithasbeenempiricallyshownthatdecomposingsmootheruencemapsresultsinshorterbeam-on-times( Mohanetal. 2000 ; Webbetal. 1998 ).Basedonthisobservation, Craftetal. ( 2007 )usethemaximumvariationinbeamletintensitiesatMLCrowsasanapproximatemeasureofthebeam-on-time.TheythenformulatetheFMOproblemasamulti-criteriaoptimizationproblemwhichminimizesthisapproximatemeasurealongwithothermeasuresoftreatmentplanquality.Moreover,recently Jinetal. ( 2010 )developedaquadraticprogrammingmodel 47

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AperturemodulationorDirectApertureOptimization(DAO)isarelativelynewapproachtoIMRTtreatmentplanningwhichintegratestheFMOandLSproblems.Moreformally,DAOexplicitlysolvesforapertureshapesandintensitiesratherthanbeamletintensities( Bednarzetal. 2002 ; Preciado-Waltersetal. 2006 ; Romeijnetal. 2005 ). Shepardetal. ( 2002 )and LudlumandXia ( 2008 )compareIMRTtreatmentplansobtainedusingDAOandthetraditionaltwo-stagemethod,andshowthatDAOproducescomparabledoseconformitywhilesignicantlyimprovingthetreatmentefciency.Inparticular, LudlumandXia ( 2008 )and Menetal. ( 2007 )reportthattoachieveasimilartreatmentquality,thebeam-on-timerequiredbythetraditionaltwo-stagemethodismorethantwiceaslongasthebeam-on-timerequiredbyDAO.Additionally,alongtreatmenttimecanundesirablyincreasethesensitivityofthetreatmentoutcometodosimetricinaccuraciescausedduringthedelivery( Shepardetal. 2002 ; Romeijnetal. 2005 ).Furthermore,uencemapdecompositionduringtheLSstageofthetraditionalmethodrequiresdiscretizationwhichmaycompromisethetreatmentqualityaswell.Finally,inordertoadequatelymodelmanyimportantcharacteristicsofanIMRTtreatmentplan,knowledgeoftheshapeandintensityoftheaperturesemployedinthattreatmentplanisrequired.Incontrastwiththetraditionaltwo-stageapproach,DAOallowsforexplicitlyincorporatingtheseaspectsintothetreatment-planoptimizationstage.Forexample, Menetal. ( 2007 )exploitedthistodevelopaDAOapproachthataccountsfortransmissioneffects(i.e.,dosethatistransmittedthroughtheMLCleaves),and Salarietal. ( 2011 )developedarobustDAOapproachtoaccountforthedosimetricinaccuraciescausedbytheMLCarchitectureknownasthetongue-and-grooveeffect. ToaccountforthedeliveryefciencyinIMRTtreatment-planoptimizationstage,weemploythebi-criteriaoptimizationapproachwhichiswidelyusedwhenatrade-off 48

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Ehrgott ( 2005 ).Inthispaper,weproposeabi-criteriaDAOmodeltoexplicitlyincorporatethebeam-on-timeasameasureofthedeliveryefciencyintotheIMRTtreatmentplanoptimization.Takingadvantageofthestructureoftheproblem,wedevelopanexactsolutiontechniquetoobtaintheassociatedsetofPareto-efcientsolutions.However,someclassesoftreatmentplanevaluationcriteriamaynotsatisfythemathematicalassumptionsrequiredbytheexactmethodoritmaybecomputationallyprohibitivetoapplytheexactmethodtothem.Therefore,anapproximatesolutiontechniqueisalsodevelopedwhichisapplicabletomoreclassesofevaluationcriteria.ThismethodsequentiallyemploysourexactalgorithmtocloselyapproximatesegmentsofthePareto-efcientfrontier.Finally,usingthesetofPareto-efcientsolutionsweinvestigatetheeffectofthebeam-on-timeonthetreatmentplanquality. Theoutlineoftheremainderofthepaperisasfollows.InSection2weformulateaconvexbi-criteriamodeltoincorporatethebeam-on-timeintotheDAOproblem.InSection3weusethepropertiesoftheproposedmodeltoobtainsomefundamentalresultswhicharethenemployedtodevelopanalgorithmicframeworktogeneratethePareto-efcientfrontierassociatedwiththebi-criteriamodel.InSection4wepresentthecomputationalresultsobtainedbyapplyingthealgorithmtoclinicalcancercases,andinvestigatethetrade-offbetweenthebeam-on-timeandthetreatmentplanquality.Finally,inSection5weconcludethepaperanddiscussfutureresearchdirections. 49

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3.2.1Introduction 50

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Romeijnetal. 2004 ). Asmentionedearlier,thebeam-on-timecanbeexpressedasthesumofallapertureintensities,whichwerepresentbythefunctionH:RjKj!R:H(y)=Xk2Kyk 51

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3 )evaluatethedosedistributionwhileConstraints( 3 )ensurethattheapertureintensitiesarenonnegative.Itiseasytoseethatanynonnegativevectoryuniquelydeterminesadosedistributionz.Wewillthereforeoftenrefertoyasthetreatmentplan. WecanthenusetheconceptofPareto-efciencytocharacterizethesetoftreatmentplansthatneedtobeconsideredinquantifyingthetrade-offbetweenthetwocriteria(i.e.,GandH).Inparticular,considertreatmentplan^ywhichyieldsdosedistribution^z.^yisaPareto-efcienttreatmentplanifitisnotdominatedbyanyothertreatmentplan.Moreformally,ifatreatmentplan,sayywithdosedistributionz,existssuchthateitherG(z)G(^z)andH(y)
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RuzikaandWiecek 2005 ).Thesetechniquescanbeclassied(basedonthestructureoftheapproximationfunctions)into0thorder(asingleParetopoint)to3rdorder(piecewisecubic)approximations.Amongthese,1storderapproximationsarethemostpopularoneswherethePareto-efcientfrontierisapproximatedbyapiecewiselinearfunction.Thesetechniquesmainlyemployweighted-sumand-constraintmethods( ChankongandHaimes 1983 ; Ehrgott 2005 )inaniterativemannertogeneratePareto-efcientsolutions.ThePareto-efcientfrontieristhenapproximatedusingtheobtainedPareto-efcientsolutions.Inparticular,non-inferiorsetestimation(NISE)techniquewasoriginallyproposedby Cohonetal. ( 1979 )togenerateanapproximaterepresentationofthePareto-efcientfrontierforbi-criteriaconvexproblems( ChankongandHaimes 1983 ).Itisa1st-ordersandwichapproximationwhichusespiecewiselinearcurvestoconstructinnerandouterenvelopesonthePareto-efcientfrontier.Inparticular,aninnerenvelopecomesfromtheconvexhullofanitecollectionofgeneratedPareto-efcientpointswhileanouterenvelopeisprovidedbythelinesegmentsthatsupportthePareto-efcientfrontieratthesepointsasshowninFigure 3-1 TheNISEmethodrstgeneratesthetwoendpointsontheParetofrontierbyminimizingeachobjectivefunctionindividually.Atthisstep,theinitialsandwich 53

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InnerandouterenvelopesobtainedbytheNISEtechniquetoapproximatetheParetofrontier. approximationisatrianglespannedbetweenthetwoendpointsandtheidealpoint(whichisobtainedbyintersectingthetwoperpendicularlinespassingthroughthetwopoints).GiventwoadjacentPareto-efcientpoints,NISEthenemploystheweighted-summethodtogenerateanewPareto-efcientpointusingaweightequaltothenegativeslopeoftheinnerenvelopebetweenthetwoadjacentpoints.Hence,newPareto-efcientpointsaregeneratedandtheenvelopesareupdatedaccordinglyinaniterativemanner.Finally,NISEterminateswhenthemaximumdistancebetweentheinnerandouterenvelopesiswithinauser-speciedthreshold. Therearetwomajorissuesinapplyingtheapproximationtechniquestoourbi-criteriaoptimizationmodel(P).Inparticular,supposethatwewouldusetheweighted-summethodtoiterativelyobtainPareto-efcientsolutions.Theninstancesof(P())for0needtobesolvedindividuallyateachiteration.Regardlessofthemanufacturer-dependentMLCconstraints,thetotalnumberofdeliverableaperturesthatneedtobeconsideredin(P())(i.e,thecardinalityofsetK)isverylarge( Romeijnetal. 2005 ).Therefore,solvingeachinstanceof(P())requiresemployingacolumngenerationapproach;inthisapproach,weimplicitlyconsideralldeliverableaperturesandsequentiallygeneratethemasneeded( Romeijnetal. 2005 ; Menetal. 2007 ).However,employingthisapproachforeachindividualinstancecanbecomputationallyveryexpensive.Weproposeanexactapproachthatavoidsthisissuebysolving 54

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IntheremainderofthispaperwedevelopasolutionmethodwhichobtainsthefamilyofPareto-efcienttreatmentplansasafunctionofthebeam-on-timepenalty.Inparticular,wewillsequentiallyreducethevalueofandcharacterizesegmentsofthePareto-efcientfrontier.Tothisend,werstshowthat,undermildconditions,theoptimaldosedistributionto(P())isuniqueforeach2R+,sayz().Next,wederiveanexplicitexpressionforthe(nite)smallestvalueof,say0,forwhichitisoptimalnottotreat.Inotherwords,forsufcientlylargebeam-on-timepenalty0wehavethatz()=0.Then,startingwith0,weiterativelydetermineintervalsforforwhichz()canbeexplicitlycharacterized. 55

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Proof. Jansenetal. 1997 ).Since,byAssumption 1 ,Gisstrictlyconvex,theobjectivefunctionin(P())isstrictlyconvexaswellandconsequently(P())hasauniqueoptimalsolution. 56

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Inotherwords,Y(z)isthesetofalltreatmentplansthatdeliverzwithminimalbeam-on-time.Thefollowingtheoremsaysthattherealwaysexistsatreatmentplanthatdeliversagivendosedistributionwithminimalbeam-on-timeusingasetofindependentapertures: Proof. 57

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3.2 weconcludethat,forall2R+,theoptimaldosedistributionz()canbedeliveredwithminimalbeam-on-timeusingasetoflinearlyindependentapertures.Notethatthisimmediatelyyieldsanupperboundonthenumberofaperturesrequiredtodeliveranydeliverabledosedistribution: Bazaraaetal. 2006 ).Associatingdualmultipliers=(j;j2V)>withConstraints( 3 )and=(k;k2K)>withConstraints( 3 ),theKKTconditionsfor(P())canbeexpressedasfollows:zj=Xk2KDkjykj2V NotethatEquations( 3 )( 3 )canalternativelybeexpressedinvector-notationasz=D>y,=rG(z),and=D+1where1isacolumnvectorwhoseelementsareallequalto1. SolvingthesystemofnonlinearEquations( 3 )( 3 )yieldsanoptimalsolutionto(P())foragiven2R+.Inparticular,anyvectory()0(where0isavectorwhoseelementsareallequalto0)isacandidatesolutiontothatsystem.Thisvector 58

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Proof. 3 )( 3 ).Clearly,Equation( 3 )impliesthatz()=0.Since,byAssumption 1 ,Gisdifferentiable,accordingtoEquation( 3 )wehavethat()=rG(0),and()isdenedandnite.Moreover,accordingtoEquation( 3 )()=DrG(0)+1.Sincey()=0,thenEquation( 3 )issatised.Nowlet0beasdenedinthetheorem,thenKKTcondition( 3 )and,inparticular,()0issatisedwhenever0.Therefore,forthesevaluesofthesolution(y(),z(),(),())isanoptimalsolutionto(P()). WewillnowturntotheproblemofcharacterizingthefamilyofPareto-efcienttreatmentplansandstudyhowz()changesaschanges. 3.1 thecorrespondingoptimaldosedistributionz()isunique.Moreover,let()bethevectorofdualmultipliersinKKTconditions( 3 )( 3 )for(P()).Notethat()isuniquesincez()isunique.ThefollowingdenitioncharacterizesadesirablepropertyofasubsetK()Kofapertures: 3.2 guaranteestheexistenceofabasicsetofaperturesK()forany2R+.Inparticular,givenanoptimaltreatmentplany2Y(z())forwhichthesetof 59

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whereisthechangingparameter.Fornotationalconvenience,inthecontextoftheparametricproblem(PU(;)),vectorywillrepresenttherestrictedvector(yk:k2K())>. Wenextstudytheoptimalsolutionstotheparametricproblem(PU(;))aswellascorrespondingoptimaldosedistributionsandtreatmentplanstotheoriginalproblem.Werstshowthevalidityofthefollowinglemma: Proof. uwhere uk

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ThefollowingtheoremprovidessufcientconditionsonGunderwhichsolutionstotheparametricunconstrainedproblem(PU(;))canbecharacterizedascontinuouslydifferentiablefunctions: Proof. AnysolutiontothissystemofnonlinearequationswithjK()jequationsandunknownvariablesisastationarypointof(PU(;)).Morespecically,theJacobianmatrixofEquations( 3 )canbewrittenasfollows:J(y)=D0@r2G0@Xk2K()D>kyk1A1AD> 61

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1 ).TheGlobalInverseMappingTheorem( OhtsukiandWatanabe 1969 ; WuandDesoer 1972 )nowsaysthatthereexistsauniquecollectionofcontinuouslydifferentiablefunctionsyk(;):R!Rfork2K()thatsatisesEquations( 3 )astheright-hand-side(i.e.,)changesinR.Hence,(yk(;):k2K())>isthestationarypointof(PU(;)).Finally,sincetheHessianoftheobjectivefunction(i.e.,J(y))ispositivedeniteeverywhere,thestationarypointistheglobalminimumof(PU(;)). Thenexttheoremlinkstheparametricunconstrainedproblem(PU(;))withouroriginalproblemandshowshowtheoptimalsolutionto(PU(;))canbeusedtoobtaintheoptimalsolutionto(P())forsufcientlysmallvaluesof. Proof. 3 )( 3 )for(P()).Inparticular,substitutingyinKKTcondition( 3 )yieldsz=Xk2KD>kyk=Xk2K()D>kyk(;), 3 )yields=rG(z)=rG0@Xk2K()D>kyk(;)1A.

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3 )arek=Dk+=DkrG0@Xk02K()D>k0yk0(;)1A+k2K. 3 )(complementaryslackness)issatised. Finally,weneedtoensurethaty,0.Inparticular,werstconsiderthespecialcase=0,andshowthatat=0,yk(0;)0fork2K(),andk(0;)0fork2K.NotethatK()isabasicsetofapertures.Hence,k()=0fork2K()andasaresultDkrG(z())+=0k2K(). Moreover,z()isdeliverableusingonlyaperturesinK().Hence,forsomey2RjK()j+wehavez()=Xk2K()D>kyk. 3 )yieldsDkrG0@Xk2K()D>kyk1A+=0k2K()

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3 )when=0,thatisDkrG0@Xk2K()D>kyk(0;)1A+=0k2K() 3.4 thesolutiontothissystemisuniqueforagiven,onecanconcludethatyk(0;)0fork2K(),andz()=Pk2K()D>kyk(0;).Furthermore,at=0,foraperturek2Kwehavek(0;)=DkrG0@Xk02K()D>k0yk0(0;)1A+=DkrG(z())+=(), 3 )(nonnegativity)aswell,henceyisoptimalto(P()). Startingfromagivenbeam-on-timepenaltyandforsufcientlysmall,Theorem 3.5 usestheparametricunconstrainedproblem(PU(;))toobtaintheoptimaldosedistributionz()aswellasatreatmentplany2Y(z())thatdeliversz()withminimalbeam-on-time.Therefore,atagivenbeam-on-timepenalty,onecanlocallycharacterizethePareto-efcientfrontierusingTheorem 3.5 .Inthenextsection,weemploythefundamentalresultsfromthissectiontodevelopanalgorithmic 64

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3.3.1 ,whichcharacterizetheoptimalintensityandthedualmultiplierassociatedwithaperturek2Kasthebeam-on-timepenaltychanges,respectively.RecallfromTheorem 3.3 thatz(0)=0,henceanoptimaltreatmentplanandabasicsetofaperturesforbeam-on-timepenalty0canbeobtainedasy(0)=0andK(0)=;,respectively.Therefore,wewillstartwith=0andinitializetheiterationcounterofthealgorithmatm=0.Then,atiterationm,weperformthefollowingsteps: 1. Find1=inf:0andmink2K()yk(;m)<02=inf:0andmink2Kk(;m)<0 2. Setm+1=mmin.Ifmin=1,thensetk=argmink2K(m)yk(1;m)K(m+1)=K(m)nfkg;

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Setz(m+1)=Xk2K(m)D>kyk(min;m)yk(m+1)=yk(min;m)fork2K(m)0fork2KnK(m). Proof.

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67

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Proof. 68

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ThiswillbeusedinSection 3.3.3.2 toshowniteconvergenceofthealgorithmwhenappliedtotheclassofconvexquadraticevaluationcriteria. InStep1ofthealgorithm,wemayhavemin=0.Inthiscase,thealgorithmtakesadegeneratestepandupdatesthebasicsetofapertureswithoutreducingthecurrentbeam-on-timepenalty.Therefore,itispossiblethatthealgorithmgoesthroughasequenceofdegenerateiterationswherethebasicsetofaperturesismodiedduringeachiterationwithouttakingapositivesteplength.However,sincethereareonlyanitenumberofbasicsetsofaperturesforagivenbeam-on-timepenalty,employinganappropriateanti-cyclingrulewillpreventthealgorithmfromcycling. Itisclearthat,ingeneral,Steps1and2ofthealgorithmcanbecomputationallyprohibitive.Moreover,theassumptionoftwicedifferentiabilityofGmaynotbesatisedundersomeclinicallyrelevanttreatmentplanevaluationcriteria.Therefore,intheremainderofthissection,wewillrststudythespecialcaseofconvexquadraticevaluationcriteriaforwhichwecanexplicitlysolvetheparametricunconstrainedoptimizationproblems.Next,wewillproposeamethodthatusesourgeneralalgorithmicframeworktocloselyapproximatethePareto-efcientfrontierusingasequenceofquadraticapproximationstoG.Finally,wewilldiscussacommonly-usedobjectivefunctionthatisbasedonpiecewisequadraticvoxel-basedpenaltyfunctionstoillustratethismethod.

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RomeijnandDempsey 2008 ).Themostelementaryandclinicallyusedpenaltyfunctionsarequadraticones,inwhichcaseGcanbeexpressedasG(z)=1 2z>Qz+q>z+q0 3 )intheproofofTheorem 3.4 reducestothefollowinglinearsystem:Dk0@QXk02K()D>k0yk0+q1A+=k2K(). whereDisajK()jjVjsubmatrixofDconsistingofonlytherowscorrespondingtok2K(),1isthecolumnvectorofonesofsizejK()j,andDQD>1kisthekthrowofDQD>1.Notethatyk(;)fork2K()arelinearfunctionswithrespectto.Hence,thevalueof1inStep1ofthealgorithmicframeworkcanbeobtainedusingaratiotestasfollows:1=mink2K()(DQD>)1k1<0DQD>1kDq+1 DQD>1k1.

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Sinceyk(;)fork2K()arelinearfunctionsasshowninEquation( 3 ),itiseasytoseethatsubstitutingtheminEquation( 3 )willyieldlinearfunctionswithrespecttoasfollows:k(;)=1DkQD>DQD>11()+DkDkQD>DQD>1Dqk2K. Nowtoobtain2,letusdenefunction(;):R!Ras(;)=mink2Kk(;). 3-2 ).Moreover,sincek(;)=0fork2K(),wehavethat(;)0,andsincek(0;)=k()fork2Kandk()=0fork2K(),thenwehavethat(0;)=0(proofofTheorem 3.5 ).Therefore,itiseasytoseethat2correspondstothesmallestbeyondwhich(;)isnegative. Figure3-2. 71

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UsingtheexpressioninEquation( 3 )wecanexpressk(;)fork2KinEquation( 3 )intermsofbeamletdosedepositioncoefcients.Inparticular,letusdenej(;):R!Rforj2Vasj(;)=QjXk2K()D>kyk(;)+qjj2V 3 )canberewrittenasfollows:k(;)=Dk(;)+=Xi2AkXj2VDijj(;)!+k2K.

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InordertodeterminemininStep1ofthealgorithmicframework,weneedtoobtain2onlyif21sinceotherwisemin=1.Thus,werstevaluate(1;)bysolvingthepricingproblemforeachindividualbeamdirectionb2Bat=1.If(1;)=0,thenclearly21andmin=1;otherwise22[0,1].Inthatcase,wecanemployabinarysearchalgorithmtoobtain2.Inparticular,westartwiththeinitialintervalcontaining2(i.e.,I0=[0,1]).Wethenevaluate(;)atthemidpoint=1 DependingontheMLCmanufacturerandtheparticulardeliverytechniqueused,aperturescanbesubjecttovarioushardwareconstraints.Inparticular,therearefourcommonsetsofhardwareconstraints: (C1) (C2) (C3) 73

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Romeijnetal. ( 2005 )providepolynomial-timealgorithmsforsolvingthepricingproblemunder(C1)(C3).Inparticular,supposethateachbeamdirectionisdiscretizedintoabeamletgridofdimensionsmn.Theythenshowthatunder(C1)thepricingproblemforaparticularbeamdirectioncanbesolvedinO(mn)time.For(C2)and(C3)theyformulatethepricingproblemasashortestpathproblemonanappropriatelydenednetworkforwhichthereisaone-to-onecorrespondencebetweenpathsfromthesourcenodetothesinknodeanddeliverableapertures.Theirshortest-pathalgorithmyieldstheoptimalsolutioninOmn4time.Finally, Menetal. ( 2007 )provideasolutionmethodthatsolvesthepricingproblemunder(C4)inOm2ntime. 3.7 ).Moreover,usingabasicsetofaperturesat,thealgorithmemploysyk(;)tolocallycharacterizeyk().Recallthatforthisparticularclassofevaluationcriteria,yk(;)isalinearfunctionwithrespectto.Moreover,foragivenbeam-on-timepenalty,thereareonlyanitenumberofbasicsetsofapertures(i.e.,K()).Inparticular,anupperboundcanbeobtainedbysimplyconsideringalllinearly-independentsubsetsofaperturesKK.Nowsupposeforeverylinearly-independentsubsetKKsuchthatk2K,weobtainyk(;0)asshowninEquation( 3 ),sothatweobtainacollectionoflinescorrespondingtoaperturek.Notethatthiscollectioncontainsnitelymanylines.Atiterationmofthegeneral 74

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3-3 illustratespartofapossiblepath. Duetocontinuityofyk(),thebreakpointscanonlyoccuratintersectionpointsofthelinesinthecollection.Hence,thereisanintersectionpointassociatedwitheachiterationofthealgorithmicframework.Supposethesteplength(i.e.,mininStep1ofthealgorithm)ispositiveateachiteration,sothatthealgorithmtakesonlynondegeneratesteps.Inthatcase,wedonotvisitthesameintersectionpointtwiceduringthecourseofthealgorithm.Asaresult,sincethereareonlyanitenumberofintersectionpoints,thealgorithmtakesnitelymanysteps.Wehavethereforeshownthatthefollowingresultholds: 75

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3.4 guaranteesthatthefamilyofoptimalsolutionstotheparametricunconstrainedproblemcanbecharacterizedwhenGistwicedifferentiableandr2Gisstrictlypositivedeniteeverywhere.However,forsomeclassesofevaluationcriteria,thesystemofnonlinearEquations( 3 )maynotyieldananalyticalsolution,oritmightbecomputationallyprohibitivetoderivethesolution.Inthatcase,functionsyk(;)fork2K()andconsequentlyk(;)fork2Karenotavailable.Hence,inthissection,weproposeanapproximatemethodthatwerefertoasSequentialQuadraticApproximation(SQA).ThismethodisbasedonourgeneralalgorithmicframeworkaswellastheanalyticalresultsforquadraticfunctionsGinSection 3.3.3 .SQAalsorequiresGtobetwicedifferentiablebutr2Gneedsonlytobepositivedeniteeverywhere.Itisbasedonsequentiallyapproximating(P())withaquadraticprogrammingproblem,wherethegeneralalgorithmframeworkisemployedtotheapproximation. Supposethat,foragiven2R+,theoptimaldosedistributionz()(alongwithatreatmentplany2Y(z())forwhichthesetofpositive-intensityaperturesislinearlyindependent)isgiven.SQAthenstartsbyconstructingtherstandsecond-orderTaylorseriesexpansionsofGaroundz=z()whicharedenotedbyGLandGQ, 76

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2(zz())>r2G(z())(zz())+rG(z())>(zz())+G(z()). TheresultsinSection 3.3.3 canthenbeusedtoobtainfunctionsyk(;)fork2K()thatareoptimalto(PQU(;)).Itisimportanttonotethatthesesolutionsarenotnecessarilyoptimalto(PU(;)).However,evaluatingthetrueobjectivefunctionatthesesolutionsdoesofcourseyieldanupperboundontheoptimalsolutionvaluesto(PU(;)).Moreover,duetoconvexityofG,evaluatingtheobjectivefunctionvalueof(PLU(;))atthissolutionsyieldsalowerboundonG.Moreimportantly,evaluatingtheindividualcomponentsoftheobjectivefunctionof(PQU(;))and(PLU(;))yieldslocalupperandlowerboundsontheactualPareto-efcientfrontier,respectively.SQAwillkeeptrackoftheseupperandlowerboundsasasurrogatetoidentifythereductioninbeam-on-timepenalty(i.e.,)forwhichtheapproximation(PQU(;))deviatesfrom(PU(;)).Inparticular,ifthedifferencebetweentheseboundsexceedsauser-speciedthreshold(say)atacertainbeam-on-timepenalty0,SQAoptimizesthecorrespondingproblem(P(0)).Hence,ateachiterationofSQAweobtainanapproximatesegmentofthePareto-efcientfrontieralongwithanexactPareto-efcienttreatmentplanat0. Moreformally,startingwith0anditerationcounterm=0,SQAperformsthefollowingstepsatiterationm: 77

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FindtheapproximatefunctionsGLandGQatz(m). 2. Applythestepsofthegeneralalgorithmicframeworkto(PQ())startingwithm,z(m),y(m),andK(m),untilthelowerandupperboundsontheoptimalvalueof(P())differbymorethan. 3. Letm+1bethenalbeam-on-timepenaltyobtainedinStep2.Solve(P(m+1))startingwiththeinitialsolutionprovidedby(PQ(m+1))toobtainz(m+1),y(m+1),andK(m+1)(forexampleusingthecolumngenerationalgorithmdevelopedby Romeijnetal. ( 2005 )). 3.3.3 :G(z)=Xj2VFj(zj) wherewjandw+jarenonnegativeweightsandTjisanonnegativethresholdvalue(j2V)( RomeijnandDempsey 2008 ).Inthissection,wewillsimplyassumethateachFjiscontinuouslydifferentiableandconvex.Moreover,weassumethatforallj2V,ifwj=0,thenTj=0todisallowatspotsinthecorrespondingpenaltyfunction. Theweightedsumoftheevaluationcriteriawillthereforebeaconvexpiecewisequadraticfunction.Inparticular,functionGispiecewisequadraticifitsdomainisaunionofnitelymanyconvexpolyhedra,oneachofwhichthefunctionisgivenbyaquadraticexpression( RockafellarandRoger 1998 ).Moreformally,letusdenotethedomainoffunctionGbydomG,andthesetofconvexpolyhedraformingdomGbyR.Polyhedronr2RanditsinteriorarethendenotedbyP(r)andintP(r),respectively.Rhasthe 78

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2z>Q(r)z+q(r)>z+q(r)0. ThemaindifcultyinemployingtheSQAmethodtoconvexpiecewisequadraticevaluationcriteriastemsfromthelackoftwicedifferentiabilityofGontheboundarypoints.Inparticular,thesecond-orderTaylorexpansionofGisnotwell-denedeverywhere.SupposeatiterationmofSQA,z(m)2Tr2RP(r)forsomeRR(i.e.,z(m)isaboundarypoint).Thenacollectionofquadraticrepresentationsexistatz=z(m)asfollows:G(z)=1 2z>Q(r)z+q(r)>z+q(r)0r2R. 3 )( 3 )for(P(m)).Moreover,sinceGisdifferentiableatz(m),thenwehaverG(z(m))=Q(r)z(m)+q(r)r2R.

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3 )( 3 )for(PQ(m))andisoptimalto(PQ(m)).Therefore,theSQAmethodcanpickanyofthequadraticrepresentationsinRtoobtainGQ. 3.4.1ClinicalProblemInstances Foxetal. 2006 ). Thehead-and-neckcasecontainstwoPlanningTargetVolumes,PTV1andPTV2,withprescriptiondosesof73.8Gyand54Gy,respectively.TheprostatecasecontainsonlyasinglePlanningTargetVolumePTVwithprescriptiondoseof73.8Gy.Dose-volumehistogram(DVH)isacommonlyusedtoolinclinicalpracticetoevaluatethequalityofatreatmentplan.Foragiventargetorcriticalstructure,thishistogramspeciesthefractionofitsvolumethatreceivesatleastacertainamountofdose.Thus, 80

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Moreover,theDVHcriteriaforprostatecancercasesare: DVHcriteriaforadditionalcriticalstructures,suchasbrainstem,spinalcord,skin,opticnervesandchiasminhead-and-neckcasesandfemoralheadinprostatecases, 81

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Recently,theclinicalfeasibilityofusingconventionaljawsratherthanMLCfordeliveringIMRThasbeeninvestigatedforseveraldiseasesites( Kimetal. 2007 ; Earletal. 2007 ; Menetal. 2007 ; MuandXia 2009 ; Tasknetal. 2010a ).Inparticular,ithasbeenshownthatthisdeliverytechnique,theso-calledjaws-onlyIMRT,iscapableofconstructingcomplexdosedistributionsusingsolelyrectangularapertures.TheseaperturescanbeformedusingconventionaljawswithouttheneedtooperateexpensiveMLC.AlthoughthedevelopmentofDAOhasmadejaws-onlyIMRTmorepractical,theaveragebeam-on-timeforthisdeliverytechniqueisabouttwiceaslongasthebeam-on-timerequiredforMLCdelivery( MuandXia 2009 ).Hence,deliveryefciencyisacriticalfactorinthesuccessfulapplicationofthisIMRTdeliverytechniqueinparticular.Therefore,inourexperimentalresultsweusethisdeliverytechniqueasourproofofconcept. 3.3 ).AllexperimentswereimplementedandperformedinMATLAB7.9.0(2009b)ona2.33GHzIntelCore2Duowith2GBofRAMunderWindowsXPoperatingsystem.Ittakes35and75minutestoapproximatethePareto-efcientfrontierforthehead-and-neckandprostatecase,respectively.TheparameterinStep3oftheSQAmethodissetto0.01.NotethatdetermineshowcloselythePareto-efcientfrontierisestimated,and=0.01wasfoundtoyieldanappropriateestimationforbothcancercases. Figure 3-4 illustratestheapproximatedPareto-efcientfrontierforbothcancercases.ThePareto-efcientfrontierconsistsoftwodistinguishableparts;intherst 82

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Toinvestigatethisfurther,westudytheclinicalqualityofthePareto-efcienttreatmentplansasthebeam-on-timeincreases.Inparticular,weevaluatetheDVHcriteriaforthetargetcoverageaswellasthecritical-structuresparingatdifferentPareto-efcienttreatmentplansobtainedbytheSQAmethod.Figures 3-5 and 3-6 illustratetheDVHcriteriaassociatedwiththehead-and-neckandprostatecancercase,respectively.Initially,thetargetDVHcriteria(i.e.,theDVHcriteriacorrespondingtoPTV1andPTV2inthehead-and-neckcaseandPTVintheprostatecase)substantiallyimproveasthebeam-on-timeincreases,andbeyondacertainbeam-on-timetheyrelativelyconverge.Onthecontrary,thecritical-structureDVHcriteria(i.e.,theDVHcriteriacorrespondingtosalivaryglandsinthehead-and-neckcaseandthebladderandrectumintheprostatecase)initiallydeteriorateasthebeam-on-timeincreases,andafteranearlypeak,theystartimprovinguptoacertainbeam-on-timevaluebeyondwhichtheybecomerelativelyxed.ThecollectionofPareto-efcienttreatmentplanswithbeam-on-timevalueslargerthan4.8and3minutessatisfyalltheDVHcriteriaforthehead-and-neckandprostatecase,respectively. Menetal. ( 2007 )presentaDAOmodelthatdoesnotaccountforthedeliveryefciency.Thetreatmentplansobtainedusingtheirmodelrequireabeam-on-timeofatleast6.1and4.8minutestosatisfytheDVHcriteriaforthehead-and-neckandprostatecase,respectively.However,theSQAmethodcanachievePareto-efcienttreatmentplanswith20%shorterbeam-on-timetosatisfytheDVHcriteria. FromthecollectionofPareto-efcienttreatmentplansthatsatisfytheDVHcriteriaforeachcancercase,Figure 3-8 illustratesandcomparestheDVHcurvesassociated 83

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3-5 and 3-6 .Morespecically,asthebeam-on-timeincreases,thetargetDVHcriteriaaremetpriortothecritical-structureDVHcriteria,andonceasatisfactorytargetcoverageisachieved,furtherincreasesinbeam-on-timewillmostlyimprovethecritical-structuresparing. Figure 3-9 illustratestheisodosecurvescorrespondingtoPareto-efcienttreatmentplansobtainedatdifferentbeam-on-timevaluesonaCTslicefortheprostatecase.APareto-efcienttreatmentplanwithalowbeam-on-timeof0.7minutespendstheentireavailablebeam-on-timeatasinglebeamdirectionwhichhasthelargest 84

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Inadditiontobeam-on-time,thenumberofaperturesemployedinatreatmentplanplaysanimportantroleinthedeliveryefciency.ThenumberofaperturesusedinPareto-efcienttreatmentplansareshowninFigures 3-5D and 3-6D .Inparticular,toobtainPareto-efcienttreatmentplansthatsatisfytheDVHcriteria,90and75aperturesareusedforthehead-and-neckandprostatecancercaseatbeam-on-timeof4.8and3minutes,respectively.Furthermore,apositivecorrelationexistsbetweenthenumberofaperturesusedinthePareto-efcienttreatmentplansandthecorrespondingbeam-on-timevalues. TheNISEtechniquecanbeemployedtoapproximatethePareto-efcientfrontierof(P).However,amajordrawbackofthistechniqueisthelargeamountofcomputationaleffortrequired.Inparticular,NISEtakesaround75minutesforthehead-and-neckcaseandaround5hoursfortheprostatecasetoapproximatethePareto-efcientfrontierusingthesamenumberofpointsgeneratedbytheSQAmethod(i.e.,38forthehead-and-neckcaseand36fortheprostatecase).Morespecically,ateachiterationoftheNISEtechniqueaninstanceof(P())foragiven2R+issolvedusingthecolumngenerationalgorithm.Solving(P())forsmallerbeam-on-timepenaltiesrequiresrelativelymorecomputationaleffort.Therefore,NISEinitiallyspendsalargeamountofcomputationaleffort(16%ofthetotalcomputationtime)tosolve(P()) 85

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Additionally,inapplyingthecolumngenerationtechniquetoinstancesof(P()),duetotheslowconvergenceofthismethod,weemploytheDVH-convergenceruleproposedin Menetal. ( 2007 )asthestoppingcriteria.Inparticular,weterminatethecolumngenerationprocedurewhenthechangesinDVHcriteriawithinthelastveiterationsspanlessthanauser-speciedthreshold,say.Inthehead-and-neckcancercase,=0.025%forthetargetcoverageand=0.25%fortheglandssparing,andintheprostatecase,=0.05%forthetargetcoverageand=0.5%forthebladderandrectumsparing.However,terminatingthecolumngenerationalgorithmusingtheDVH-convergencerulecausesinaccuracyinestimatingtheinnerandouterenvelopesonthePareto-efcientfrontier.Figure 3-10 illustratesdifferentpartsoftheNISEenvelopesforthehead-and-neckandprostatecase.Inparticular,forshorterbeam-on-times,Pareto-efcientpointsobtainedbytheSQAmethodliewithintheenvelopes,whereas,forrelativelylongerbeam-on-times,thesepointsliebelowtheNISEouterenvelope. 86

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Animportantaspectofthedeliveryefciencyisthenumberofaperturesemployedinthetreatmentplan.Futureresearchcanextendthisworkbyquantifyingthetrade-offbetweenthetreatmentplanqualityandthenumberofaperturesrequiredtodeliverthetreatmentplan. 87

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Bprostatecase Pareto-efcientfrontierobtainedforclinicalcancercases. 88

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BPTV2coverage CSalivaryglandssparing DNumberofaperturesemployed DVHcriteriaassociatedwithtargetcoverageandsalivaryglandssparingevaluatedatPareto-efcienttreatmentplansforthehead-and-neckcancercase. 89

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BBladdersparing CRectumsparing DNumberofaperturesemployed DVHcriteriaassociatedwithtargetcoverageandrectumandbladdersparingevaluatedatPareto-efcienttreatmentplansfortheprostatecase. 90

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BDVHcurvesatBOTof4.8(solid)and9.8(dashed) CDVHcurvesatBOTof9.8(solid)and14.4(dashed) DVHcurvesassociatedwithPareto-efcienttreatmentplansatdifferentlevelsofbeam-on-time(BOT)(inminutes)forthehead-and-neckcase. 91

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BDVHcurvesatBOTof3(solid)and5.9(dashed) CDVHcurvesatBOTof5.9(solid)and15(dashed) DVHcurvesassociatedwithPareto-efcienttreatmentplansatdifferentlevelsofbeam-on-time(BOT)(inminutes)fortheprostatecase. 92

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BBeam-on-timeof1.8 CBeam-on-timeof3 DBeam-on-timeof5.9 Isodosecurves(dashedlines)for60and73.8GyonatypicalCTslicecorrespondingtoPareto-efcienttreatmentplansfortheprostatecase. 93

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B C D InnerandouterenvelopesobtainedusingtheNISEtechnique.(a)and(b)head-and-neckcase,(c)and(d)prostatecase. 94

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LangenandJones 2001 ).Inparticular,wedistinguishbetweentwotypesofmotion:(1)interfractionmotionwhichreferstochangesinpatientgeometrythattakesplacebetweentreatmentfractions,causedbythefactthatpatientscannotbeperfectlyrepositionedattheacceleratorforeachfraction,andtheinternalorgansofthepatientscanmoveorchangebetweenfractions;and(2)intrafractionmotionwhichreferstothemotionofthepatient'sbodyduringafraction,forexample,duetobreathing.TheimpactoforganmotiononIMRTdosedeliveryhasbeenclinicallystudiedfordifferenttumorsitesandbothstep-and-shootanddynamicdeliverytechniques( Yuetal. 1998 ; Huangetal. 2002 ; Georgeetal. 2003 ; Bortfeldetal. 2004 ). Incurrentpractice,theseerrorsareusuallyaccountedforbyarticiallyexpandingallrelevantstructuresinthepatientbysomemargin( WambersieandLandberg 1999 ).Inparticular,thevolumeidentiedtocontainthediseaseanditspossiblespreadiscalledclinicaltargetvolume(CTV).Thisvolumeisthenexpandedbyasafetymargintoformtheplanningtargetvolume(PTV).ToensurethedeliveryoftheprescribeddosetoCTV,PTVisthenirradiatedwiththeprescriptiondose.Sinceinthisapproachalargerregionisirradiated,surroundinghealthytissuesareharmed.Alternatively,wecanincorporatetheorganmotionuncertaintyintothetreatmentplanoptimizationstage. 95

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Men 2009 ; UnkelbachandOelfke 2005 ; Baumetal. 2006 )aswellasrobustoptimizationapproaches( Chuetal. 2005 ; OlafssonandWright 2006 )tothisproblem.Inaddition,toaddresstheintrafractionmotionuncertainty, Bortfeldetal. ( 2002 )and Unkelbach ( 2006 )formulatedtheFMOproblemasastochasticoptimizationproblem,and Chanetal. ( 2006 )and Heathetal. ( 2009 )haveproposedrobustFMOapproaches. ThemaindifcultywithmodelingtheintrafractionmotionisthedynamiccharacterofIMRTtreatmentsmakingthecurrentFMOapproachesinadequate.Sinceitispossiblethatthemovementsofthepatientconspirewiththemotionofthemultileafcollimator(MLC),thedelivereddosedistributionmaydiffersignicantlyfromtheplannedone(oftencalledtheinterplayeffect).Asanextremeexample,breathingmotionmaymovethetargetinandoutofthedynamicallychangingeldofradiationinsuchawaythatpartsofthetargetaremissedentirely,orseverelyunderdosedoroverdosed.Severalresearchershaveinvestigatedtheinterplayeffectonthedelivereddosedistribution.Inparticular, Bortfeldetal. ( 2002 )reportthatinhighly-fractionatedIMRTdelivery(30fractions),theinterplayeffectontheexpecteddosedeliveryisnegligible.However, Jiangetal. ( 2003 )and Secoetal. ( 2007 )reportthatignoringtheinterplayeffectcancausenon-negligiblebiologicalimpactwhenahighdoserateorapertureswithlownumberofmonitorunits(oftheorderofthemotionperiod)areemployedinthetreatmentplan.Therefore,toaccountfortheintrafractionmotioninthetreatmentplanoptimizationstage,weneedamodelingframeworkthatincorporatestheassociateduncertaintyinitsentirety. Theonlyapproachthatcanexplicitlyaccountforintrafractionmotionuncertaintiesisonethatdirectlymodelsthesequenceofaperturesthatisdeliveredduringaparticularfraction.Inotherwords,arigorousapproachtothisproblemrequirestheuseofmodels 96

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Inthischapter,weaccountfortheintrafractionmotioninthetreatmentplanoptimizationstageusingaDAOframework.Inparticular,wedescribethestateofthepatientgeometryovertimeusingastochasticprocess.Moreover,tokeeptrackofthemovementoftheMLCleavesateachrow,weassociateacollectionofbinaryvariablestoallpossibleleafcongurationsinthatrowovertime.Usingtheabovemodelingframework,wethenformulatethetreatmentplanoptimizationproblemasastochasticbinaryquadraticprogrammingproblemanddevelopacolumngenerationmethodtosolvetherelaxationofthemodel.Wethenusethecolumngenerationalgorithminabranch-and-boundframeworktosolvethediscreteproblem. (t);t0g,withstatespaceF.Moreover,weletpf(t)=Prf (t)=fgrepresenttheprobabilityofbeinginstatef2Fattimet0.Finally,weassumethatforeachstateofthepatientgeometrythedosedepositionratesareobtainableandxed. WeconsideradynamicIMRTdeliveryinwhichthepatientisirradiatedusingasetofbeamdirections,denotedbyB.Weassumethatthegantryspendsatotaloftbtimeunitsatbeamdirectionb2B.WeletRbdenotethesetofMLCrowswhenitispositionedatbeamdirectionb2BandR=Sb2BRbdenotethesetofallMLCrows.Moreover,weletCdenotethesetofallpossibleleaf-paircongurationsateachMLCrow.Toevaluatethedosedepositedinthepatient'sbody,targetsandotherrelevant 97

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(t);t0gisinitializedagain.Therefore,thetreatmentatbeamdirectionb2Btakesplaceduringatimewindowof[0,tb]. IntheabsenceofMLC-manufacturerconstraints( Romeijnetal. 2005 ),MLCleavesateachrowcanindependentlymove.TokeeptrackofthemovementsoftheMLCleavesatrowr2R,wedenedynamicapertureAr:[0,tb]!f0,1gjCjasfollows:Arc(t)=8><>:1,ifcongurationcisusedatrowrattimet;0,otherwise. Sinceateachpointintimeonlyasingleleafcongurationcanbeused,Pc2CArc(t)=1(t2[0,tb]).A=(Ar:r2R)thenshowsthedynamicapertureusedforeachMLCrow.TheDAOprobleminvolvesidentifyingtheoptimaldeliverabledynamicapertureforeachMLCrowr2R. Dependingontheparticularstateofthepatientgeometryf2F,thedosedepositionratesmaychange.Therefore,weletDfrcjdenotetherateofdosedepositioninvoxelj2VfromMLCrowr2Rwhentheleavesareincongurationc2Candthepatientgeometryisinstatef2F.Hence,thetotaldosereceivedbyvoxelj2VfromdynamicapertureA,canbeexpressedasarandomvariableasfollows:z (t)rcjArc(t)dt. Moreover,weletEz

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(t)rcjArc(t)dtj2V whereVTrepresentsthesetofalltargetvoxels.Fj:RjVj!R(j2V)areconvexquadraticvoxel-basedpenaltiesexpressedasFj(zj)=s 4 )describethedosedistribution,andConstraints( 4 )and( 4 )ensurethatA=(Ar:r2R)aredeliverabledynamicapertures.(~P)isaninniteprogrammingproblemwithaninnitenumberofconstraintsandbinaryvariables. Nowsupposeforthestochasticprocessf (t):t0gunderconsideration,thereareonlynitely-manydeliverabledynamicaperturesforMLCrowr2R,indexedbym2Mr(forinstance,incaseofusingadiscrete-timestochasticprocess).Wecanthenassociateacollectionofbinarydecisionvariablesymr2f0,1g(m2Mrr2R)withdynamicaperturesm2Mr(r2R)andreformulatetheDAOproblemasmin1Xj2VFjEz

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(t)rcjAmrc(t)dt!ymrj2V whereConstraints( 4 )and( 4 )ensurethatexactlyonedynamicapertureischosenforeachMLCrowr2R.Inthenextsection,weshowthatthisformulationleadstoabranch-and-pricealgorithminwhichwesolvetherelaxationof(P)usingacolumngenerationmethod. Forsomeclassesofstochasticprocesses,wehavetoconsideranuncountably-innitenumberofdeliverabledynamicaperturesforeachMLCrow.Inthatcase,itisessentiallyimpossibletoenumerateandindexalldynamicaperturescorrespondingtoeachMLCrow.Hence,forsuchstochasticprocesseswecannotformulatetheDAOproblemas(P).However,usinganynitesubsetofdynamicaperturesyieldsarestrictedversionof(~p)whichcanbeformulatedas(P).Therefore,tosolvetherelaxationof(~p),weusetheresultprovedby Dantzig ( 1960 )sayingthatacolumngenerationprocedurethatemploystherelaxationoftherestrictedversionasthemasterproblem,eitherndstheoptimalsolutiontothefullprobleminnitelymanyiterations,orconvergestotheoptimalsolutioninthelimitprovidedthatthereexistsanondegeneratebasicsolutiontothemasterproblem. 100

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(t)rcjArc(t)dt. (t)rcjAmrc(t)dt!ymrj2V RelaxingConstrains( 4 )tothefollowingnonnegativityconstraintsymr0m2Mr,r2R yieldsaconvexoptimizationproblemwithlinearconstraints,whichwedenoteby Bazaraaetal. 2006 ).Byassociatingdualmultipliersj(j2V),r(r2R),andmr(m2Mr,r2R),withConstraints( 4 ),( 4 ),and( 4 ),respectively,wecan 101

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(t)rcjAmrc(t)dt!ymrj2V (t)rcjAmrc(t)dt!j+rm2Mr,r2R Notethatanysolutiony,,z,,totheKKTconditions( 4 )( 4 )canbecharacterizedbyyandonlysinceydeterminesz,zdetermines,andnallyyanddetermines.Nowsupposey,istheoptimalsolutiontoarestrictedproblemobtainedfrom (t)rcjAmrc(t)dt!j+r

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(t)rcjj. 4 ),wecanformulatethepricingproblemforMLCrowr2RasminZtb0Xc2Crc(t)Arc(t)dt+rsubjectto( 103

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4 )asfollows:rc(t)=Xj2VED (t)rcjj=Xf2FXj2VDfrcjj!pf(t). Sincethestateprobabilitydistributionpf(t)(f2F)dependsonthestartingprobabilitydistribution(i.e.,pf(0)(f2F)),inthefollowingwerstconsiderthespecialcaseinwhichthetreatmentstartswhenthepatientgeometryisinitssteadystate.Wethenestablishaconnectionbetween( 4.2 ,weconsiderdecisionvariablesoftheformArc(tt0)(t2[t0,t0+tb],r2R,c2C)inthepricingproblem( 4 )for 104

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4 )canbewrittenaszj=Xb2BXr2RbXm2MrZt0+tbt0Xc2CED (t)rcjAmrc(tt0)dt!ymr=Xb2BXr2RbXm2MrZtb0Xc2CED (t+t0)rcjAmrc(t)dt!ymr (t+t0)rcj!Amrc(t)dt!ymr=Xb2BXr2RbXm2MrZtb0Xc2Climt0!1ED (t+t0)rcj!Amrc(t)dt!ymr

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(t+t0)rcj=Xf2FDfrcjpf, 106

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wheretheobjectivefunctionevaluatesthequalityofthetreatmentplanbasedontheaveragedosedistribution.Equation( 4 )measurestheexpectedvalueofthetotaldosereceivedbyvoxelj2V,andConstraint( 4 )ensuresthatatotaltreatmenttimeoftbtimeunitsisspentatbeamdirectionb2B. ThetraditionalDAOmodel( Romeijnetal. 2005 ; Menetal. 2007 )isconcernedwithndingtheoptimalcollectionofaperturesalongwiththeirassociatedintensivestodeliverthedesireddosedistributiontothepatient.IntheabsenceofMLC-manufacturerconstraints,theproblemcanbedecomposedoverMLCrowsandformulatedintermsofallleaf-congurationintensities.Giventheoptimalleaf-congurationintensities,wecanthenformtheoptimalcollectionofaperturesbysimplycombiningtheseleafsettings.Therefore,(ST)representsthetraditionalDAOformulationifwelimitthetreatmenttimeatbeamdirectionb2Btotbtimeunits. 4 )in(P)yieldsthefollowingrelaxedproblem:min1Xj2VFjEz

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(t)rcjAmrc(t)dt!ymrj2V Notethatz 4 )and( 4 ),theKKTconditionsfor(PR)canbeexpressedasmr=1Xj2V@FjEz NotethatsimilartotheKKTconditionsderivedfor( 4 )canberewrittenas@FjEz (t)rcjAmrc(t)dt. 108

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4 ),wedeneD (t)rcjAmrc(t)dt. NowsupposejisthejointprobabilitydistributionassociatedwithD 4 ),wecanrewritethesecondtermintheobjectivefunctionof(PR)associatedwithvoxelj2VTasEFjz Flanders 1973 ),torewritethesecondterminEquation 109

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4 )associatedwithvoxelj2VTasfollows:@EFjz @ymrZDjFjD>jyj(Dj)dDj!=ZDj@ @ymrFjD>jyj(Dj)dDj=ZDj@FjD>jy (t)rcjAmrc(t)dt!. Inthefollowing,werstobtaintherstandsecondtermsinEquations( 4 )and( 4 )forconvexquadraticpenalties,andthenformulatethepricingproblem.Inparticular,Equation( 4 )canbeobtainedas@FjEz (t0)r0c0jAm0r0c0(t0)dt0!ym0r0Tj1AZtb0Xc2CED (t)rcjAmrc(t)dt!=2Ztb0Xc2C0@Xb02BXr02RbXm02Mr0Xc02CZtb00ED (t0)r0c0jED (t)rcjAm0r0c0(t0)ym0r0dt01AAmrc(t)dt2TjZtb0Xc2CED (t)rcjAmrc(t)dt. 110

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4 )forthisclassofevaluationcriteriacanbeobtainedas:@EFjz (t)rcjAmrc(t)dt!=2Ztb0Xc2C0@Xb02BXr02RbXm02Mr0Xc02CZtb00ED (t0)r0c0jD (t)rcjAm0r0c0(t0)ym0r0dt01AAmrc(t)dt2TjZtb0Xc2CED (t)rcjAmrc(t)dt. Nowsupposey,istheoptimalsolutiontotherestrictedversionof(PR)whereymr=0(m2Mrn whichshowsifcongurationc2Cisusedatrowr2Rattimet2[0,tb].Notethatsinceycanbeafractionalsolution,thenqrc(t)2[0,1].Usingqrc(r2R,c2C)wecanfurthersimplifyEquations( 4 )and( 4 ).Forthatpurpose,letrcj(t)=2Xb02BXr02Rb0Xc02CZtb00ED (t0)r0c0jED (t)rcjqr0c0(t0)dt02TjED (t)rcj (t0)r0c0jD (t)rcjqr0c0(t0)dt02TjED (t)rcj. Hence,@FjEz

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Wenextreformulate(G)toobtainabeamletrepresentationoftheproblem.Forthatpurpose,letIbbethesetofallbeamletsinbeamdirectionb2B.WethenletDfijbethedosedepositedperunittimeinvoxelj2Vfrombeamleti2Ib(b2B)whenthepatientgeometryisinstatef2F.Moreover,weletIrcdenotethesetofexposedbeamletswhenleavesareincongurationc2Catrowr2R.WecanthenapproximatethedosedepositionratesusingDfrcj=Xi2IrcDfijr2R,c2C,j2V, sothatwecanwriteD (t)rcj=Xi2IrcD (t)ijr2R,c2C,j2V 112

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(t)rcj=EXi2IrcD (t)ij!r2R,c2C,j2V=Xi2IrcED (t)ij. Wenextdenebeamlet-basedij(t)(i2I,j2V)and!ij(t)(i2I,j2V),similartoEquations( 4 )and( 4 ),asfollows:ij(t)=2Xb02BXr02Rb0Xc02CZtb00ED (t0)r0c0jED (t)ijqr0c0(t0)dt02TjED (t)ij (t0)r0c0jD (t)ijqr0c0(t0)dt02TjED (t)ij. SubstitutingEquation( 4 )inEquation( 4 )usingEquation( 4 )(similarly,Equation( 4 )inEquation( 4 )usingEquation( 4 ))weobtainrcj(t)=Xi2Ircij(t)rcj(t)=Xi2Irc!ij(t). Therefore,onecanuseEquations( 4 )and( 4 )todeterminethelowerenvelope(thiswillbeaddressedinSection 4.4 ). (t):t0gisasemi-Markovprocess,whichwouldallowforgeneralprobabilitydistributionsofthetimespentateachstateofthepatientgeometry.However,forreasonsoftractabilityweconsiderMarkovprocesses. 113

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Khameneetal. 2007 ; Muetal. 2008 ; Kaletetal. 2010 ). Inthissectionwestudyadiscrete-timemodelinwhichweassumethatthestateofthepatientgeometrycanonlychangeatequispacedpointsintime.Moreformally,timeisdiscretizedintotimeperiodsoflength,indexedby2Z+whereZ+=f0,1,2,...g.WeassumethatthestateofthepatientgeometryaswellastheMLCleafcongurationsarexedduringeachtimeperiodandcanonlychangeattheendoftheperiod.Wethenconsideradiscrete-timeMarkovchaintorepresentthestochasticprocessf (t):t0gthatdescribesthechangesinthepatientgeometry.Inparticular,weletpf1f2=Prf ((+1))=f2j ()=f1gf1,f22F,pf()=Prf ()=fgf2F NowconsiderMLCrowr2Rb(b2B),sinceleafcongurationatthisrowcanonlychangeattheendofeachtimeperiod,weredenethenotionofdynamicapertureAr:Tb!f0,1gjCjfordisceret-timesettingsasArc()=8><>:1,ifcongurationcisusedatrowrduringtimeperiod;0,otherwise. 114

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4 )describingthetotaldosereceivedbyvoxelj2Vasfollows:z ()rcjArc(). Thediscrete-timeversionof(~P)canberewrittenasmin1Xj2VFjEz ()rcjArc() SincethereexistonlynitelymanydynamicaperturesoftheformAr(r2R),(~PD)isastochasticbinaryquadraticprogrammingprobleminnite-dimensionalspace.Inparticular,ifMr(r2R)containsallsuchdynamicapertures,wecanthenalternativelyuseformulation(P)torepresentthefullproblem(~PD).Hence,wecanusethecolumngenerationalgorithmdevelopedforsolving(P)forthetime-discretizedmodelaswell.Inparticular,westudythepricingproblem(G)forthethismodel.WestartbyconsideringadiscreteversionofEquation( 4 ),thatisqrc()(2Tb,r2Rb,c2C,b2B)whereArc(t)issubstitutedwithArc(t).WethendenethediscreteformofEquations( 4 ) 115

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4 )(i.e.,ij()and!ij())asfollows:ij()=2Xb02BXr02Rb0Xc02CX02Tb0ED (0)r0c0jED ()ijqr0c0(0)2TjED ()ij (0)r0c0jD ()ijqr0c0(0)2TjED ()ij. NotethatED ()ijandED (0)r0c0jareobtainedbyconditioningonthestateoftheMarkovchainattimeand0,respectively,andED (0)r0c0jD ()ijisobtainedbyconditioningonthejointstatesatand0.Bysummingoverthecorrespondingvoxelswethenobtaini()=Xj2Vij() Thus,forMLCrowr2Rb(b2B)andusingEquations( 4 )and( 4 )wecanreformulate(G)forthediscrete-timemodelasfollows:minX2TbXc2CXi2Irc1i()+2i()!Arc()+rsubjectto(GD)Xc2CArc()=12Tb (GD)isabinaryoptimizationprobleminnite-dimensionalspace.Itiseasytoseethat(GD)canbedecomposedovertimeperiods2Tb.Furthermore,foragiventimeperiod2Tb,Equations( 4 )and( 4 )enforcethatexactlyoneleafcongurationischosen.Eachleafcongurationc2Ccanbeuniquelyexpressedintermsofitsexposedbeamlets,i.e,Irc.Inparticular,letIrrepresentthesetofallbeamletsinMLCrowr2R,Irc2Ircanthenbeshownasabinaryvectoru2f0,1gjIrjwhereuiindicates 116

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Bentley 1984 )wherenisthecardinalityofIr,i.e,thenumberofbeamletsinMLCrowr. Fornotationalconvenience,letusdenotetheobjectivefunctionof(PR)byF.SupposeF(y)istheoptimalobjectivevalueofthefullproblem(i.e.,(PR)).Moreover,lety,,bethesolutiontotheKKTconditions( 4 )( 4 )correspondingtotherestrictedmasterproblem.Therefore,F(y)istheoptimalobjectivevalueofthe 117

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4 )wecansubstituterF(y)asfollows:rF(y)>(yy)=Xr2RXm2Mrmrr(ymrymr)=Xr2RXm2MrmrymrXr2RXm2MrmrymrXr2RXm2Mrrymr+Xr2RXm2Mrrymr=Xr2RXm2Mrmrymr 4 )andyandybothsatisfyEquation( 4 ).Finally,theoptimizationproblemminy(F(y)+Xr2RXm2Mrmrymr:Xm2Mrymr=1,r2R).

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Sinceusingourrelaxation-basedheuristicwecaneasilyndintegersolutions(whicharefoundtobeofgoodqualityintheexperiments),wechoseanode-selectionruletomainlyreducetheintegralitygap.Therefore,wedeterminetheorderaccordingwhichthenodesareselectedbyusingthebest-boundrule. 4.5.1 121

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Supposethecolumngenerationisusedtosolvetherelaxedproblemattreenoden2N.Atiterationkofthecolumngenerationalgorithm,lowerandupperboundsontheoptimalobjectivevalueoftherelaxedproblem,denotedbyF(k)LBandF(k)UB,areobtained.Wethendenethecolumngeneration(CG)optimalitygap,denotedby(k),asfollows:(k)=F(k)UBF(k)LB whereisauser-speciedoptimalitygap. Inexploringthebranch-and-boundtree,theminimumoptimalobjectivevalueoftherelaxedproblemamongallleafnodes,theso-calledbestbound,usuallyservesasalowerboundontheobjectivevalueoftheoptimalsolution.However,ifweterminatethecolumngenerationalgorithmpriortooptimality,theexactoptimalobjectivevalueoftherelaxedproblemisnotknownanymore.Hence,inabranch-and-pricesettingwehavetoconsideradifferentlowerbound.Morespecically,letNLdenotethesetofallleafnodesinthebranch-and-boundtree.Furthermore,supposeforleafnoden2NL,FnLBisthenallowerboundobtainedontheoptimalobjectivevalueoftherelaxedproblemusingthecolumngenerationmethod.WethendenethebestboundasFLB=minn2NLFnLB. SinceFLBisalowerboundontheoptimalobjectivevalueoftherelaxedproblematallleafnodes,itisavalidlowerboundontheobjectivevalueoftheoptimalsolutionaswell.Moreover,weletFIbethebestintegersolutionfoundwhileexploringthebranch-and-boundtree.Clearly,FIisanupperboundontheobjectivevalueofthe 122

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Weconsideredatreatmenttimeof1minuteperbeamdirectionwhichwasdiscretizedinto30timeperiods(2secondseach).Moreover,wemadethesimplifyingassumptionthattheMLCleaveshaveaninnitespeed.Inotherwords,regardlessoftheleafcongurationatthecurrenttimeperiod,allleafcongurationsareaccessibleforthenexttimeperiod.Toreplicatetheintrafractionmotion,weconsideredarigidbodymotioninwhichthepatientgeometrytransitionsbetweentwostatesF=f1,2g.Todenethesecondstate,weperturbedthelocationofthepatientby5mmineachcoordinatedirection.WemodeledtheresultingchangesinthepatientgeometryasaMarkovchainwithatransitionprobabilityofp2f0.9,0.93,0.95,0.99g.Forinstance,ifthepatient 123

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Whensolvingtherelaxationof(P)usingthecolumngenerationmethod,arelatively-largeamountofcomputationaleffortisrequiredtoevaluatethesecondtermcorrespondingtotargetvoxelsintheobjectivefunctionof(P)aswellasEquation( 4 )inthepricingproblem.Therefore,weestimatethesetermsbyconsideringonlyafractionoftargetvoxels(5%)andthenscalingtheobtainedresults,accordingly. ToquantifytheadvantagesofusingthestochasticDAOmodel,wecomparetheresultsobtainedfromthismodelwiththeresultsofthestaticDAOmodel.NotethatthestaticDAOmodelassumesthatthetreatmentstartswhenthepatientgeometryisinitssteadystate.However,thestochasticDAOtreatmentplansstartwhenthepatientgeometryisinaparticularstate.Therefore,tocomparethequalityofthetwotreatmentplansweneedtoevaluatethequalityofthestatictreatmentplansusingtheobjectivefunctionof(P).However,thetreatmentplansobtainedfromthestaticDAOmodelarenotintheformofadynamicaperture.Inparticular,therearetwopropertiesthatrequiretobexed.Firstly,leaf-congurationintensitiesinEquation( 4 )obtainedfromthestaticDAOmodel(ST)arefractionalandneedtobediscretizedtoconformtoourtimediscretization.Therefore,giventhestaticDAOtreatmentplan,weroundtheintensitiestothenearestmultipleofthelengthofthetimeperiod.Whilerounding,weensurethatthetreatmenttimespentateachbeamdirectionb2Bremainstbtimeunits.Secondly,thestaticDAOmodelignoresthesequenceofleafcongurationsdeliveredateachMLCrow.Thus,wegenerate100dynamicaperturesfromastaticDAOtreatmentplanbyrandomlyperturbingthesequenceofcongurationsateachMLCrowindividually.Wethenevaluatetheobjectivefunctionof(P)forall100dynamicaperturesandreporttheminimumandmaximumvalues.WhilecomparingthedynamicaperturesobtainedfromthestochasticandstaticDAOmodels,weassumethatbothtreatmentplansstartwhenthepatientgeometryisinthesamestate,andthatthepatientgeometrytransitions 124

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Tables 4-1 4-3 showtheresultsobtainedfromemployingourbranch-and-pricealgorithmtotheprostatecancercaseaswellastheresultsobtainedfromthestaticDAOmodel.Inparticular,Table 4-1 showstheresultsobtainedfromthestochasticDAOmodelforthespecialcaseof2=0andcomparesthemwiththeresultsfromthestaticDAOmodel.Therstcolumn(p)showsthetransitionprobabilityoftheMarkovchainusedtomodelthechangesinthepatientgeometry.Thesecondcolumn(Bestinteger)reportsthesmallestobjectivevalueamongtheintegralsolutionsobtainedfromthetreewithinatimelimitof3600sec.Thethirdcolumn(Bestbound)showsthelargestlowerboundontheoptimalobjectivevalueobtainedwithinatimelimitof3600sec.Thefourthcolumn(Root)reportsthelowerboundobtainedontheoptimalobjectivevalueoftherelaxationproblemattherootnodeaswellasthecomputationaltimerequiredbythecolumngenerationalgorithmatthisnode.Thefthcolumn(Gap)reportstheintegralitygapdenedinEquation( 4 ).Thenextcolumn(objectivevalue)showstheminimumandmaximumobjectivevaluesamong100dynamicaperturesobtainedfromthestaticDAOmodel.Finally,thelastcolumn(Imprv.)reportstherelativeimprovementintheobjectivevalueof(P)obtainedfromthestochasticDAOmodel(Bestinteger)comparedtothestaticDAOmodel(minObjectivevalue).Table 4-2 showsthesameresultsobtainedwhen2>0.Moreover,Table 4-3 comparesthebestintegersolutionobtainedfrom(p)using2>0withthesolutionobtainedfromthestaticDAOmodelthathastheminimumobjectivevalue.Inparticular,thesolutionsarecomparedwithrespecttotherstandsecondtermsoftheobjectivefunctionof(P),whichwecallevaluationcriteriaIandII,respectively. Table 4-1 showsanimprovementof13%intheobjectivevalue(evaluationcriteriaI)obtainedasaresultofemploying(P).Table 4-2 showsanimprovementof3%intheobjectivevalue.Sincecomputingthesecondtermintheobjectivefunction 125

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4-3 showsanimprovementof18%intheevaluationcriteriaII(i.e.expecteddeviationofthedosedistributionfromthedesiredoneintargetvoxels)whenusingthestochasticmodel.However,thismaycomeattheexpenseofdegradingevaluationcriteriaI(i.e.,qualityofthetreatmentmeasuredbasedontheexpecteddosedistribution)forp2f0.9,0.93g.Finally,incaseofusingthestochasticmodel,bothevaluationcriteriaIandIItendtobesmallerforahighertransitionprobability.Thiscanbejustiedsimilartothecaseof2=0, 126

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ThecurrentstochasticDAOmodel,andinparticular,thediscrete-timeversion,isbasedontheassumptionthatMLCleaveshaveaninnitespeed.Hence,leafcongurationsindifferenttimeperiodsareindependent.However,inclinicalsettings,MLCleafmotionsarerestricted.Inparticular,theMLCleafspeedisaround2cm/sec(1.5.5cm/sec( Boyeretal. 2001 )).Therefore,tohaveamorerealisticmodel,thisassumptionneedstoberelaxed.Morespecically,ideallywewouldliketolimitthesetofdeliverabledynamicaperturesforeachMLCrowtotheonesrespectingtheleafmotionrestriction.Notethatinthepresenceofleafmotionrestriction,thepricingproblemforeachMLCrowcannotbedecomposedovertimeperiodsanymoresinceleafcongurationsinconsecutivetimeperiodswillbedependent.Hence,weneedtodevelopadifferentsolutionapproachthattakesthisdependenceintoaccount( Romeijnetal. 2005 ). 127

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Furthermore,inSection 4.3.2 wederivedtheKKTconditionsforthegeneralstochasticDAOmodel(i.e.,when2>0),andshowedhowtondtheoptimalsolutiontothepricingproblembyndingthecorrespondinglowerenvelope(i.e.,minc2C(1rc(t)+2rc(t))(t2[0,tb])).Butthisisonlythecasewhenclosed-formexpressionsforrc(r2R,c2C)andrc(r2R,c2C)arebothgiven.IntheAppendixwederivedtheclosed-formexpressionforrc(r2R,c2C).Therefore,wealsoneedtoobtaintheclosed-formexpressionofrc(t)whichisyettobestudied.Moreover,similartothespecialcaseof2=0,weneedtodevelopasolutionmethodtoobtainthelowerenvelope. InSection 4.4 wediscussedabranch-and-pricealgorithmforthediscrete-timemodelthatemploysthecolumngenerationmethoddevelopedtosolvetherelaxedproblemateachnodeofthebranch-and-boundtree.Forthecontinuous-timeDAOmodel,wewouldliketoconsiderasimilarbranch-and-pricealgorithm.However,sincethemodelisinacontinuous-timesetting,wecannotusethebranchingschemeproposedforthediscrete-timeversion.Morespecically,inthediscrete-timemodel,wechosetobranchonthefractionalvariablesxiindicatingwhetherbeamleti2Ib(b2B)isexposedduringtimeperiod2Tb(b2B)ornot.However,thisbranchingschemecausesanextremelyimbalancedbranch-and-boundtreewhenappliedtoacontinuoussetting.Therefore,deninganappropriatebranchingschemewhichalsopreservesthestructureofthepricingproblemisextremelycriticaltothesuccessofabranch-and-pricealgorithmforthecontinuous-timeproblem. 128

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ResultsobtainedfromsolvingthestochasticDAOmodelfor2=0withinatimelimitof3600secusingaCGoptimalitygapof=3%andcomparingwiththestaticDAOmodel. 0.902196.862101.132079.461644.62527.002611.5913.10.932087.742024.592001.482173.12532.572646.8817.60.952046.211953.011929.001644.82521.112669.3218.80.971943.151863.721839.411684.32535.862670.6323.40.991946.201730.701697.9621412.52565.552760.8224.1 ResultsobtainedfromsolvingthestochasticDAOmodelfor2>0withinatimelimitof3600secusingaCGoptimalitygapof=5%andcomparingwiththestaticDAOmodel. 0.903202.652176.662146.9474047.13318.913715.683.50.933005.962106.462079.7473342.73141.423718.664.30.952683.462045.622019.3570031.23054.203512.3812.10.972514.761947.441917.2859929.12944.763589.6814.60.992691.521807.901766.2795448.92872.103832.546.3 ComparingtheresultsobtainedfromthestochasticandstaticDAOmodelswithrespecttobothevaluationcriteriaIandII. 0.902775.662526.082571.33-9.9426.98792.831144.3646.10.932599.532531.112587.84-2.7406.43610.311130.8333.40.952437.242544.622611.334.2246.23509.58901.0551.70.972307.172545.442654.589.4207.59399.32935.1048.00.992456.282586.192702.145.0235.24285.911130.4117.7

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Inthissectionwedevelopacolumngenerationmethodfor (t);t0gasacycliccontinuous-timeMarkovchainwithstatespaceFforwhichfdenotestherateofleavingstatef2F.Moreover,weletpf1f2(t)=Prf (t+s)=f2j (s)=f1g (t)=fgrepresenttheprobabilityofbeinginstatef2Fattimet0.Dependingonthetreatmentstartingtime,thestateprobabilitiespf(t)(f2F)maychange.Therefore,weconsiderthepricingproblemforthreepossiblecasesinwhichthetreatmentstartswhenthepatientgeometryisin(1)thesteadystate,(2)aparticularstateandtheCTMarkovprocesshasonly2states,and(3)aparticularstateandtheCTMarkovprocesshasmultiplestates.

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Ross 2004 ).Usingthelimitingprobabilitydistribution,wecanobtainEquation( 4 )forthesteadystateasrc=Pj2VPf2F1fDfrcjj 4.3.1.2 isapplicabletothiscase.Wenextstudythecaseinwhichthetreatmentstartswhenthepatientgeometryisinaparticularstate. Ross 2004 ):p1(t)=2 4 )yieldsrc(t)=Xf2FXj2VDfrcjj!pf(t)=Xj2VD1rcjj!2

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A )wouldbesubstitutedwithPj2VD2rcjj. Similartothesteady-statecase,wecansolvethepricingproblemforMLCrowr2Rbydecomposingitoverthetimeinterval[0,tb].Moreformally,weneedtodeterminethelowerenvelope(i.e.,minc2Crc(t))overtheinterval[0,tb].Sincerc(r2R,c2C)areallexponentialfunctionswiththesameexponent,everytwoofthemcanintersectatmostonce.Hence,wecanobtainthelowerenvelopebysimplysweepingacrosstheintervalandidentifyingtheintersectionpoints.Morespecically,werstdeterminec0=argminc2Crc(0).Wethenintersectrc0(t)withrc(t)(c2C,c6=c0),anddeterminetheintersectionpointwiththesmallesttvalue,sayt1,rc1(t1)(ifthereisnointersectionpoint,thenrc0isclearlythelowerenvelopeovertheentiretimeinterval).Nowstartingwitht1,rc1(t1)wecontinuetheaboveprocedureiterativelytosweepacrosstheinterval[0,tb](Figure A-1 ).UsingtheobtainedlowerenvelopewecandeterminetheoptimaldynamicapertureArbyconsideringthecorrespondingleafcongurationsateachpointinthetimeinterval. 133

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Lowerenvelopeobtainedbysweepingacrossthetimeinterval. ThetransitionprobabilitiesforaCTMarkovchaincanbeobtainedusingthemethodofintegralequations( Birolini 2007 )thatformsasystemofequationsbyconditioningonthersttransitionasfollows:pf1f1(t)=ef1t+Xf2Ff6=f1Zt0f1fef1tp1(tx)dx wheref1f2isthetransitionratefromstatef1tof2,andf1istherateofleavingstatef1.NotethatweareconsideringacyclicCTMarkovprocess,andasaresultf1f1+1=f1f12Ff1f2=0f26=f1+1,

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A )( A )canbefurthersimpliedinthecontextofourapplicationasfollows:pf1f1(t)=ef1t+Zt0f1ef1tpf1+1f1(tx)dxpf1f2(t)=Zt0f1ef1tpf1+1f2(tx)dxf16=f2. Toobtainpf1f1(t)andpf1f2(t),wethenusetheLaplaceinversetransformasfollows:pf1f1(t)=L1(~pf1f1(s))f12F Inparticular,letusdenethefollowingpolynomialscorrespondingtothenumeratoranddenominatorof~pf1f1(s),respectively:Rf1f1(s)=Yf2Ff6=f1(s+f)Qf1f1(s)=Yf2F(s+f)Yf2Ff,

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A )wehavepf1f2(t)=1 A-2 illustratespf(t)(f2F)foracyclicCTMarkovchainthatstartsfromstatef=1withjFj=10. Therefore,startingfromstatef=1attimet=0,theprobabilityofbeingateachstatef2Fisasfollows:pf(t)=1 4 )andusingthepropertythateverylinearcombinationofSineorCosinefunctionswiththesamefrequencyisaCosinefunctionwiththesamefrequencybutadifferentphaseshiftandamplitude,wecanobtainthe 137

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Similartothebinary-statecase,thepricingproblemcanbedecomposedoverthetimeintervalt2[0,tb].Therefore,tosolvethepricingproblemforthemulti-statecase,weneedtoobtainthelowerenvelope(minc2Crc(t))overthetimeinterval.However,incontrasttothethebinary-statecase,inthemulti-statecaserc(r2R,c2C)arenotnecessarilymonotoneduetotheexistenceoftheCosineterm(Figure A-2 ),andasaresultanypairofthesefunctionsmayhaveseveralintersectionpoints.Thus,thesolutionmethoddevelopedforthebinary-statecasecannotbedirectlyemployedtothemulti-statecase.Furtherresearchisrequiredtodevelopanefcientsolutionapproachforthepricingproblemofthemulti-statecase. 138

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Thepropagabilityofbeingateachstatepf(t)(f2F)foracyclicCTMarkovchainstartingfromf=1withidenticaltransitionrates. 139

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EhsanSalariwasborninMashhad,Iranin1981.Heholdsabachelor'sdegreeinIndustrialEngineeringfromAmirkabirUniversityofTechnologyandamaster'sdegreeinSystemsEngineeringfromSharifUniversityofTechnology,Tehran,Iran.HereceivedhisPh.D.inIndustrialandSystemsEngineeringfromtheUniversityofFloridaintheSummerof2011.HethenjoinedtheRadiationOncologyDepartmentatMassachusettsGeneralHospitalandHarvardMedicalSchoolasapostdoctoralresearchfellow.Ehsan'sresearchinterestsareinOperationsResearchapplicationstohealth-caresystemsandparticularlycancercare. 148