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61 1.Considerthebasecase, p =0.FromCorollary 5 andthefactthattheplans ( I il 0 ;I ir 0 ) ; 0 i n ,aregeneratedusingAlgorithmDMLCSINGLEPAIR,itfollowsthat S (0)istrue. 2.Assume S ( p )istrue.SupposeAlgorithmDMLCINTERDIGITATIONndsane xt violationandmodiestheschedule N ( p )to N ( p +1).Supposethatthenextviolation occursbetweentherightleafofpair u ,positionedat x v ,andtheleftleafofpair t Wemodifypair t 'splanfor x x v ,toeliminatetheviolation.Allotherplansinthe scheduleremainunaltered.Therefore,toestablish S ( p +1)itsucestoprovethat I 0 tl ( x ) I tl ( p +1) ( x ) ;x x v (3.4) I 0 tr ( x ) I tr ( p +1) ( x ) ;x x v (3.5) Weneedproveonlyoneofthesetworelationshipssince I 0 tl ( x ) I 0 tr ( x )= I tl ( p +1) ( x ) I tr ( p +1) ( x ) ;x 0 x x m .Wenowconsiderpair t 'splanfor x x v .Notethat the( p +1)thviolationmayeitherbeaType1orType2violation.Wes howthat Equation 3.4 istrueinbothcases.This,inturn,impliesthat S ( p +1)istruewhenever S ( p )istrueandhencecompletestheproof.Notethatin( I tl ( p +1) ;I tr ( p +1) ),theleaves moveatmaximumspeedbetweenadjacentsamplepoints.So,it issucienttoshow Equation 3.4 forsamplepoints x v (a)The( p +1)thviolationisaType1violation. From S ( p )itfollowsthat I 0 ur ( x v ) I urp ( x v ).So,therightleafofpair u leaves x v noearlierin I 0 ur thanitdoesin I urp .Fromthisandthefactthat F satisesthe interdigitationconstraint,weconcludethatleafpair t cannotbeginitssweepat therightof x v .Thisobservationtogetherwiththefactthatin( I tl ( p +1) ;I tr ( p +1) ) theleavesmoveatthemaximumvelocityfrom x v to x 0 = xStart ( t;p )implies that ^ I 0 tl ( x 0 ) ^ I tl ( p +1) ( x 0 )and ^ I 0 tr ( x 0 ) ^ I tr ( p +1) ( x 0 ),where ^ I denotesanarrival time.Now,fromLemma 16 ,weget I 0 tr ( x 0 ) ^ I 0 tr ( x 0 ) ^ I tr ( p +1) ( x 0 )= I tr ( p +1) ( x 0 ). So I 0 tl ( x 0 )= I 0 tr ( x 0 )+ I t ( x 0 ) I tr ( p +1) ( x 0 )+ I t ( x 0 )= I tl ( p +1) ( x 0 ).Fromthisand thefactthattheleftleafofpair t movesatthemaximumvelocitybetween
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62 x v and x 0 ,itfollowsthatEquation 3.4 holdsforall x between x v and x 0 .To provethatEquation 3.4 holdsforallsamplepointstotherightof x 0 (andso holdsforall x between x 0 and x m ),considerasamplepoint x w thatistothe rightof x 0 .Let I 0 = I 0 tl ( x 0 ) I tl ( p +1) ( x 0 ) 0andlet I 1 beasinAlgorithm DMLCINTERDIGITATION.Deneanewplan( I 00 tl ;I 00 tr )forleafpair t asbelow I 00 tl ( x )= 8><>: undefinedx
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63 (Lemma 12 byieldsEquation 3.9 onlyfor x x v and x isasamplepoint.From thisandthefactthattheleftleafmovesatmaximumvelocity in I tl between adjacentsamplepoints,wegetEquation 3.9 forall x x x v .) FromEquation 3.9 ,weget I 0 tl ( x ) I tl ( p +1) ( x ) I 0 tl ( x v ) I tl ( x v )+ I tl ( x ) I tl ( p +1) ( x ) ;x x v (3.10) FromthedenitionsofType1andType2modicationsandthew orkingofAlgorithmDMLCINTERDIGITATION,itfollowsthat I tl ( p +1) ( x ) I tl ( x )= I tl ( p +1) ( x v ) I tl ( x v ) ;x x v (3.11) FromEquations 3.10 3.11 and 3.8 ,weget I 0 tl ( x ) I tl ( p +1) ( x ) I 0 tl ( x v ) I tl ( p +1) ( x v ) 0 ;x x v (3.12) Therefore, I 0 tl ( x ) I tl ( p +1) ( x ) ;x x v (3.13) Lemma20 FortheexecutionofAlgorithmDMLCINTERDIGITATION (a) O ( n ) Type1violationscanoccur. (b) O ( n 2 m ) Type2violationscanoccur. (c)Let T max betheoptimaltherapytimefortheinputmatrix.Thetimecom plexityis O ( mn + n min f nm;T max g ) Proof: (a)ItfollowsfromLemma 17 thateachleafpaircanbeinvolvedinatmostone Type1violationaspair t ,i.e,thepairwhoseproleismodied.Hence,thenumberof Type1violationsis n (b)WerstobtainaboundonthenumberofType2violationsat axed x v .Let u t beas inAlgorithmDMLCINTERDIGITATION.Notethat u ischosentobetheleastpossibleindex.Let u i bethevalueof u inthe i thiterationofAlgorithmDMLCINTERDIGITATION at x v t i isdenedsimilarly.Let u maxi = max j i f u j g .If t i = u i 1,itispossiblethat u i +1 = t i = u i 1and t i +1 = u i 2.Notethatinthiscase, t i +1 6 = u i = u i +1 +1.Next,
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64 itispossiblethat u i +2 = u i 2and t i +2 = u i 3 (again t i +2 6 = u i 1= u i +2 +1).In general,onemayverifythat t i = u i +1ispossibleonlyif u maxi = u i .If t i = u i +1,then u i +1 t i = u i +1,sincetheviolationbetween u i and t i hasbeeneliminatedandnoproles withanindexlessthan t i havebeenchangedduringiteration i at x v .Itisalsoeasyto verifythat t i =1 ;u i =2 ) u i +1 u maxi ;u maxi +2 >u maxi .Fromthisand t i 2f u i +1 ;u i 1 g it followsthat u maxi + u maxi >u maxi .Weknowthat u max1 1.Itfollowsthat u max2 2, u max4 3, u max7 4andingeneral, u max( i ( i +1) = 2)+1 i +1.Clearly,forthelastviolation(say j th)at x v u maxj n andforthistobetrue, j = O ( n 2 ).SothenumberofType2violationsat x v is O ( n 2 ).Since x v hastobeasamplepoint,thereare m possiblechoicesforit.Hence,the totalnumberofType2violationsis O ( n 2 m ). (c)Sincetheinputmatrixcontainsonlyintegerintensityv alues,eachviolationmodication raisestheproleforonepairofleavesbyatleastoneunit.H ence,if T max istheoptimal therapytime,noprolecanberaisedmorethan T max times.Therefore,thetotalnumberof violationsthatAlgorithmDMLCINTERDIGITATIONneedstor epairisatmost nT max Combiningthisboundwiththoseof(a)and(b),weget O ( min f n 2 m;nT max g )asabound onthetotalnumberofviolationsrepairedbyAlgorithmDMLC INTERDIGITATION.By properchoiceofdatastructuresandprogrammingmethodsit ispossibletoimplementAlgorithmDMLCINTERDIGITATIONsoastorunin O ( mn + n min f nm;T max g )time. NotethatLemma 20 providestwoupperboundsofonthecomplexityofAlgorithm DMLCINTERDIGITATION: O ( n 2 m )and O ( n max f m;T max g ).Inmostpracticalsituations, T max
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65 (b)Whenthealgorithmterminates,nointerdigitationviol ationsremainandthenal scheduleisfeasible.FromLemma 19 ,itfollowsthatthenalscheduleisoptimalin therapytimeforunidirectionalschedules. 3.2Conclusion Wehavepresentedmathematicalformalismsandrigorouspro ofsofleafsequencing algorithmsfordynamicmultileafcollimationthatmaximiz eMUeciency.Theseleafsequencingalgorithmsexplicitlyaccountforleafinterdigi tationconstraint.Wehaveshown thatouralgorithmsobtainfeasiblesolutionsthatareopti malintreatmentMUs.Furthermore,ouranalysisshowsthatunidirectionalleafmove mentisatleastasecientas bidirectionalmovement.Thusthesealgorithmsarewellsu itedforcommonuseinDMLC beamdelivery.
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CHAPTER4 ELIMINATIONOFTONGUEANDGROOVEUNDERDOSAGE DeliveredofIMRTwithMLCinthestepandshootmodeusesmu ltiplestaticMLC segmentstoachieveintensitymodulation.Thesidesofeach leafofaMLChaveaprotruding tongueorastepononesidethattsintoasimilargrooveofth eadjacentleaf.Thisresults indierentradiologicalpathlengthsacrossdierentpart softheleaves.Galvinetal. (1993a)rstdescribedthatthedierentradiologicalpath lengthsmanifestthemselvesas varyingdosesinaplaneperpendiculartotheleafmotion.Th elowdoseregionbetweentwo adjacentleaveswasclassiedasthetongueandgrooveee ct.InanIMRTtreatmentusing anMLC,thetongueandgrooveeectoccurswhenthetongue, orthegrooveorbothfor themosttimeduringtreatmentdeliverycovertheoverlappi ngregionbetweentwoadjacent pairsofleaves.Aspointedoutbymanyinvestigators,theto ngueandgroovearrangement alwaysresultsinunderdosagesofasmuchas1025%inthetre atmenteldsinbothstatic anddynamicmultileafcollimation(Galvinetal.1993a,Gal vinetal.1993b,Chuietal. 1994,Mohan1995,Wangetal.1996,SykesandWilliams1998). Severalrecentpublications(vanSantvoortandHeijmen199 6,Webbetal.1997,ConveryandWebb1998,Dirkxetal.1998,XiaandVerhey1998)hav eshownthatthetongueandgrooveeectcanbesignicantlyreducedbysynchroniz ationoftheleaves.However, thecostofleafsynchronizationisusuallyanincreaseinth etotalnumberofsubelds andmonitorunits.vanSantvoortandHeijmen(1996)propose analgorithmtoeliminate tongueandgrooveeectsforDMLCtreatmentplans.Althou ghtheynotethattheiralgorithmincreasesthenumberofmonitorunits,theydonotex aminetheoptimalityor suboptimalityoftheplanstheyobtain.Werecentlypublish edapaper(Kamathetal. 2003)thatgavemathematicalformalismsandrigorousproof sofleafsequencingalgorithms forsegmentalmultileafcollimation,whichmaximizeMUec iency.Weprovedthatourleaf sequencingalgorithmsthatexplicitlyaccountforminimum leafseparationobtainfeasible unidirectionalsolutionsthatareoptimal.Wenowextendth atworktodevelopalgorithms 66
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67 thatexplicitlyaccountforleafinterdigitationandtheto ngueandgrooveeectandare optimalinMUeciencyforunidirectionalschedules.Wesho walsothatthealgorithmof vanSantvoortandHeijmen(1996)obtainsoptimaldynamicmu ltileafcollimationtreatment schedules. 4.1AlgorithmwithInterdigitationandTongueandGroove Constraints 4.1.1TongueandGrooveUnderdosageEect Figure 4{1 showsabeamseyeviewoftheregiontobetreatedbytwoadjac entleafpairs, t and t +1.Considertheshadedrectangularareas A t ( x i )and A t +1 ( x i )thatrequireexactly I t ( x i )and I t +1 ( x i )MUstobedelivered,respectively.Thetongueandgroove overlaparea betweenthetwoleafpairsoverthesamplepoint x i A t;t +1 ( x i ),iscoloredblack.Letthe amountofMUsdeliveredin A t;t +1 ( x i )be I t;t +1 ( x i ).Ignoringleaftransmission,thefollowing lemmaisaconsequenceofthefactthat A t;t +1 ( x i )isexposedonlywhenboth A t ( x i )and A t +1 ( x i )areexposed. xx i1ii+1 t+1 I t I t, t+1 AAA tt, t+1t+1 x I Figure4{1:Tongueandgrooveeect Lemma21 I t;t +1 ( x i ) min f I t ( x i ) ;I t +1 ( x i ) g 0 i m 1 t
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68 4.1.2Algorithms Kamathetal.(2003)presentanalgorithmthatgeneratesasc hedulethatsatises interpairminimumseparationconstraint.Thescheduleis optimalintherapytime.However,itdoesnotaccountforthetongueandgrooveeect.I nthissection,wepresenttwo algorithms.AlgorithmTONGUEANDGROOVEgeneratesminimum therapytimeunidirectionalschedulesthatarefreeoftongueandgrooveund erdosageandmaybeusedfor MLCsthatdonothaveainterdigitationconstraint.Algorit hmTONGUEANDGROOVEIDgeneratesminimumtherapytimeunidirectionalschedule sthatarefreeoftongueandgrooveunderdosagewhilesimultaneouslysatisfyingthein terdigitationconstraintandisfor MLCsthathaveaninterdigitationconstraint. Thefollowinglemmaprovidesanecessaryandsucientcondi tionforaunidirectional scheduletobefreeoftongueandgrooveunderdosageeect s. Lemma22 Aunidirectionalscheduleisfreeoftongueandgrooveund erdosageeectsif andonlyif, (a) I t ( x i )=0 or I t +1 ( x i )=0 ,or (b) I tr ( x i ) I ( t +1) r ( x i ) I ( t +1) l ( x i ) I tl ( x i ) ,or (c) I ( t +1) r ( x i ) I tr ( x i ) I tl ( x i ) I ( t +1) l ( x i ) 0 i m 1 t
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69 all.So, I t;t +1 ( x i )=max f 0 ;I ( t;t +1) l ( x i ) I ( t;t +1) r ( x i ) g (4.2) where I ( t;t +1) r ( x i )=max f I tr ( x i ) ;I ( t +1) r ( x i ) g and I ( t;t +1) l ( x i )=min f I tl ( x i ) ;I ( t +1) l ( x i ) g From 4.1 and 4.2 ,itfollowsthat I t;t +1 ( x i )= I ( t;t +1) l ( x i ) I ( t;t +1) r ( x i )(4.3) Considerthecase I t ( x i ) I t +1 ( x i ).Supposethat I tr ( x i ) >I ( t +1) r ( x i ).Itfollowsthat I ( t;t +1) r ( x i )= I tr ( x i )and I ( t;t +1) l ( x i )= I ( t +1) l ( x i ).Nowfrom 4.3 ,weget I t;t +1 ( x i )= I ( t +1) l ( x i ) I tr ( x i )
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70 pair t +1exposesregion A t;t +1 ( x i ),orviceversa,whenever I t ( x i ) 6 =0and I t +1 ( x i ) 6 =0. Notethatifeither I t ( x i )or I t +1 ( x i )iszerothecontainmentisnotnecessary.Wewillrefer tothenecessaryandsucientconditionofLemma 22 asthe tongueandgrooveconstraint condition .Schedulesthatsatisfythisconditionwillbesaidtosatis fythetongueandgrooveconstraint.vanSantvoortandHeijmen(1996)presen tanalgorithmthatgenerates schedulesthatsatisfythetongueandgrooveconstraintf orDMLC. XiaandVerhey(1998)claimthateveryschedulethatviolate stheinterdigitationconstraintalsoviolatesthetongueandgrooveconstraint.W edemonstratewithacounterexamplethatthisisnotnecessarilythecase.Theintensityma trixshowninFigure 4{2 (a)can beexposedinasinglesegmentasshowninFigure 4{2 (b).Thesegmentisfreeoftongueandgrooveconstraintviolations,whileitclearlyviolat estheinterdigitationconstraint. 050000 500050 50 (a)(b) 0 0 00 0 00 0 Figure4{2:Counterexample.Theintensitymatrixshownin( a)canbetreatedusing asinglesegmentwith50MUsasshownin(b).Areasshadeddark arecoveredbyleft leavesandthoseshadedlightarecoveredbyrightleaves.Ar easnotshadedareexposed. Interdigitationconstraintviolationoccursthoughthere isnotongueandgrooveviolation. Eliminationoftongueandgrooveeect. NotethattheschedulegeneratedbyAlgorithmMULTIPAIRmayviolatethetongueandgrooveconst raint.Iftheschedulehas notongueandgrooveconstraintviolations,itisthedesi redoptimalschedule.Ifthereare violationsintheschedule,weeliminateallviolationsoft hetongueandgrooveconstraint startingfromtheleftend,i.e.,from x 0 .Toeliminatetheviolations,wemodifythoseplans oftheschedulethatcausetheviolations.Wescantheschedu lefrom x 0 alongthepositive x directionlookingfortheleast x w atwhichthereexistleafpairs u t t 2f u 1 ;u +1 g thatviolatetheconstraintat x w .Afterrectifyingtheviolationat x w welookforother violations.Sincetheprocessofeliminatingaviolationat x w ,mayattimes,leadtonew
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71 violationsat x w ,weneedtosearchafreshfrom x w everytimeamodicationismadeto theschedule.However,wewillproveaboundof O ( n )onthenumberofviolationsthat canoccurat x w .Aftereliminatingallviolationsataparticularsamplepo int, x w ,wemove tothenextpoint,i.e.,weincrement w andlookforpossibleviolationsatthenewpoint. Wecontinuethescanningandmodicationprocessuntilnoto ngueandgrooveconstraint violationsexist.AlgorithmTONGUEANDGROOVE(Figure 4{3 )outlinestheprocedure. AlgorithmTONGUEANDGROOVE 1. x = x 0 2.While(thereisatongueandgrooveviolation)do3.Findtheleast x w x w x ,suchthatthereexistleafpairs u u +1,thatviolatethe tongueandgrooveconstraintat x w 4.Modifythescheduletoeliminatetheviolationbetweenle afpairs u and u +1. 5. x = x w 6.EndWhile Figure4{3:Obtainingascheduleunderthetongueandgroo veconstraint Let M =(( I 1 l ;I 1 r ) ; ( I 2 l ;I 2 r ) ;:::; ( I nl ;I nr ))betheschedulegeneratedbyAlgorithm MULTIPAIRforthedesiredintensityprole.Let N ( p )=(( I 1 lp ;I 1 rp ) ; ( I 2 lp ;I 2 rp ) ;:::; ( I nlp ;I nrp ))bethescheduleobtainedafterStep 4 of AlgorithmTONGUEANDGROOVEisapplied p timestotheinputschedule M .Notethat M = N (0). Toillustratethemodicationprocessweuseexamples.Toma kethingseasier,weonly showtwoneighboringpairsofleaves.Supposethatthe( p +1)thviolationoccursbetween theleavesofpair u andpair t = u +1at x w .Notethat I tlp ( x w ) 6 = I ulp ( x w ),asotherwise, either(b)or(c)ofLemma 22 istrue.Incase I tlp ( x w ) >I ulp ( x w ),swap u and t .Now, wehave I tlp ( x w )
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72 Thenewplanforpair t ,( I tl ( p +1) ;I tr ( p +1) )isasdenedbelow. If I ulp ( x w ) I tlp ( x w ) I urp ( x w ) I trp ( x w ),then I tl ( p +1) ( x )= 8><>: I tlp ( x ) x 0 x
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73 x w I trp I I tr(p+1) tl(p+1) tlp urp ulp I I I I x Figure4{5:Tongueandgrooveconstraintviolation:case 2(closeparalleldottedandsolid linesegmentsoverlap,theyhavebeendrawnwithasmallsepa rationtoenhancereadability) Since( I tl ( p +1) ;I tr ( p +1) )diersfrom( I tlp ;I trp )for x x w thereisapossibilitythat N ( p +1)isinvolvedintongueandgrooveviolationsfor x x w .Sincenoneoftheother leafprolesarechangedfromthoseof N ( p )notongueandgrooveconstraintviolations arepossiblein N ( p +1)for x
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74 2.Assume S ( p )istrue.SupposeAlgorithmTONGUEANDGROOVEndsanextvio lationandmodiestheschedule N ( p )to N ( p +1).Supposethatthenextviolation occursbetweenleafpairs u and t t 2f u 1 ;u +1 g .Hence, I tlp ( x w ) I urp ( x w ) I trp ( x w ).So I 0 tl ( x w ) I tl ( p +1) ( x w ). Itremainstobeprovedthat I 0 tl ( x i ) I tl ( p +1) ( x i ), ww I 0 tl ( x v )
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75 I 00 tl ( x i )= 8><>: I tl 0 ( x i ) i
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76 Lemma24 Aschedulesatisesthetongueandgrooveidconstrainti itsatisesthetongueandgrooveconstraintandtheinterdigitationconstraint Proof: Itisobviousthatthetongueandgrooveidconstraintsub sumesthetongueandgrooveconstraint.Ifaschedulehasaviolationoftheinter digitationconstraint, 9 i;t I ( t +1) l ( x i )
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77 respectively.AlgorithmTONGUEANDGROOVEIDisshowninFi gure 4{6 .Allnotation usedinthealgorithmandtherelateddiscussionintheremai nderofSection 4.1.2 isalso thesameasthatusedinSection 4.1.2 andcorrespondsdirectlytotheusageinAlgorithm TONGUEANDGROOVE.AlgorithmTONGUEANDGROOVEID 1. x = x 0 2.While(thereisatongueandgrooveidviolation)do3.Findtheleast x w x w x ,suchthatthereexistleafpairs u u +1,thatviolatethe tongueandgrooveidconstraintat x w 4.Modifythescheduletoeliminatetheviolationbetweenle afpairs u and u +1. 5. x = x w 6.EndWhile Figure4{6:Obtainingascheduleunderboththeconstraints Lemma25 Let F =(( I 0 1 l ;I 0 1 r ) ; ( I 0 2 l ;I 0 2 r ) ;:::; ( I 0 nl ;I 0 nr )) beanyunidirectionalschedulefor thedesiredprolethatsatisesthetongueandgrooveid constraint.Let S ( p ) ,bethefollowingassertions. (a) I 0 il ( x ) I ilp ( x ) 0 i n;x 0 x x m (b) I 0 ir ( x ) I irp ( x ) 0 i n;x 0 x x m S ( p ) istruefor p 0 Proof: Theproofisbyinductionon p 1.Considerthebasecase, p =0.FromCorollary 1 andthefactthattheplans ( I il 0 ;I ir 0 ) ; 0 i n ,aregeneratedusingAlgorithmSINGLEPAIR,itfollowsthat S (0)istrue. 2.Assume S ( p )istrue.SupposeAlgorithmTONGUEANDGROOVEIDndsanext violationandmodiestheschedule N ( p )to N ( p +1).Supposethatthenextviolation occursbetweenleafpairs u and t t 2f u 1 ;u +1 g .AsintheproofofLemma 23 weonlyneedproveeitherEquation 4.8 orEquation 4.9 tocompletethisproof.We completetheproofforthefollowingthreecasesthatareexh austive. case1: I t ( x w ) 6 =0and I u ( x w ) 6 =0. TheremainderoftheproofforthiscaseisthesameasthatofL emma 23 case2: I t ( x w )=0. Inthiscase, I tlp ( x w )= I trp ( x w ).Since I ulp ( x w ) I urp ( x w ),wehave I ulp ( x w )
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78 I tlp ( x w ) I urp ( x w ) I trp ( x w ).ThemodicationprescribedbyEquation 4.7 is applicable.Notethatif I urp ( x w ) I trp ( x w )= I ulp ( x w ) I tlp ( x w ),Equation 4.6 isthesameasEquation 4.7 .Inparticular, I tr ( p +1) ( x w )= I trp ( x w )+ I urp ( x w ) I trp ( x w )= I urp ( x w )(4.13) Since I t ( x w )=0, I tr ( p +1) ( x w )= I tl ( p +1) ( x w )(4.14) FromEquations 4.13 and 4.14 I urp ( x w )= I tl ( p +1) ( x w )(4.15) Since F satisestheinterdigitationconstraint,theleftleafofp air t doesnot pass x w beforetherightleafofpair u passes x w .So, I 0 tl ( x w ) I 0 ur ( x w )(4.16) From S ( p )andEquation 4.15 ,weget, I 0 ur ( x w ) I urp ( x w )= I tl ( p +1) ( x w )(4.17) Equations 4.16 and 4.17 yield I 0 tl ( x w ) I tl ( p +1) ( x w ) 0(4.18) Lemma 2 bimplies, I 0 tl ( x ) I tl ( x ) I 0 tl ( x w ) I tl ( x w ) ;x x w (4.19) Subtracting I tl ( p +1) ( x )fromEquation 4.19 ,andrearrangingtermsweget I 0 tl ( x ) I tl ( p +1) ( x ) I 0 tl ( x w ) I tl ( x w )+ I tl ( x ) I tl ( p +1) ( x ) ;x x w (4.20) FromEquations 4.6 and 4.7 andtheworkingofAlgorithmTONGUEANDGROOVEID,itfollowsthat I tl ( p +1) ( x ) I tl ( x )= I tl ( p +1) ( x w ) I tl ( x w ) ;x x w (4.21)
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79 FromEquations 4.20 4.21 and 4.18 ,weget I 0 tl ( x ) I tl ( p +1) ( x ) I 0 tl ( x w ) I tl ( p +1) ( x w ) 0 ;x x w (4.22) Therefore, I 0 tl ( x ) I tl ( p +1) ( x ) ;x x w (4.23) case3: I u ( x w )=0. Theproofissimilartothatofcase2. 4.1.3EcientImplementationoftheAlgorithms Intheremainderofthissectionwewilluse`algorithm'tome anAlgorithmTONGUEANDGROOVEorAlgorithmTONGUEANDGROOVEIDand`violation 'tomeantongueandgrooveconstraintviolationortongueandgrooveid constraintviolation(dependingon whichalgorithmisconsidered)unlessexplicitlymentione d. Theexecutionofthealgorithmstartswithschedule M at x = x 0 andsweepstothe right,eliminatingviolationsfromtheschedulealongthew ay.Themodicationsappliedto eliminateaviolationat x w ,prescribedbyEquations 4.6 and 4.7 ,modifyoneoftheviolating prolesfor x x w .Fromtheunidirectionalnatureofthesweepofthealgorith m,itisclear thatthemodicationoftheprolefor x>x w canhavenoconsequenceonviolationsthat mayoccuratthepoint x w .Thereforeitsucestomodifytheproleonlyat x w atthe timetheviolationat x w isdetected.Themodicationcanbepropagatedtotherighta s thealgorithmsweeps.Thiscanbedonebyusingan( n m )matrix A thatkeepstrackof theamountbywhichtheproleshavebeenraised. A ( j;k )denotesthecumulativeamount bywhichthe j thleafpairproleshavebeenraisedatsamplepoint x k fromtheschedule M generatedusingAlgorithmMULTIPAIR.Whenthealgorithmha seliminatedallviolations ateach x w ,itmovesto x w +1 tolookforpossibleviolations.Itrstsetsthe( w +1)thcolumn ofthemodicationmatrixequaltothe w thcolumntorerectrightwardpropagationofthe modications.Itthenlooksforandeliminatesviolationsa t x w +1 andsoon.
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80 Theprocessofdetectingtheviolationsat x w meritsfurtherinvestigation.Weshow thatifonecarefullyselectstheorderinwhichviolationsa redetectedandeliminated,the numberofviolationsateach x w ,0 w m willbe O ( n ). Lemma26 Thealgorithmcanbeimplementedsuchthat O ( n ) violationsoccurateach x w 0 w m Proof: Theboundisachievedusingatwopassschemeat x w .Inpassonewecheck adjacentleafpairs(1 ; 2) ; (2 ; 3) ;:::; ( n 1 ;n ),inthatorder,forpossibleviolationsat x w .In passtwo,wecheckforviolationsinthereverseorder,i.e., ( n 1 ;n ) ; ( n 2 ;n 1) ;:::; (1 ; 2). Soeachsetofadjacentpairs( i;i +1),1 i
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81 Proof: (a)Lemma 27 providesapolynomialupperbound( O ( n m ))onthecomplexity ofAlgorithmsTONGUEANDGROOVEandTONGUEANDGROOVEID.Th eresult followsfromthis. (b)WhenAlgorithmTONGUEANDGROOVEterminates,notongueandgrooveviolationsremain.FromthisandLemma 23 ,itfollowsthattheschedulegeneratedby AlgorithmTONGUEANDGROOVEisoptimalintherapytimeforun idirectional schedulesfreeoftongueandgrooveviolations. (c)WhenAlgorithmTONGUEANDGROOVEIDterminates,notong ueandgrooveid violationsremainandfromLemma 24 thenalschedulesatisesthetongueandgrooveandinterdigitationconstraints.FromthisandLemm a 25 ,itfollowsthatthe schedulegeneratedbyAlgorithmTONGUEANDGROOVEIDisopt imalintherapy timeforunidirectionalschedulesfreeofbothtypesofviol ations. Theorem21 TheschedulegeneratedbythealgorithmofvanSantvoortand Heijmen(1996) isfreeofinterdigitationandtongueandgrooveconstrai ntviolationsandisoptimalintherapytimeforunidirectionalDMLCscheduleswiththisproper ty. Proof: SimilartothatofTheorem 20 (c). 4.2ExperimentalValidation ThealgorithmswerevalidatedonaVarian2100C/Dwith120l eafMLC(Varian MedicalSystems,PaloAlto,CA).Theintensitymapsofa7e ldheadandneckplan fromacommercialinversetreatmentplanningsystem(CORVU S5.0,NOMOSCorporation, Cranberry,PA)weresequencedusingAlgorithmMULTIPAIR,w hichoptimizestheMU eciency,andAlgorithmTONGUEANDGROOVEID,whichelimin atesthetongueandgrooveeectandinterdigitation.Theintensitymapshavea bixelsizeof1cmx1cm anda20%intensitystep.Figure 4{7 showsthelmmeasurementoftheruencemaps oftheAPeld.Thetongueandgrooveeectisreadilyseeni nFigure 4{7 (a),whileit iscompletelyeliminatedinFigure 4{7 (b)usingAlgorithmTONGUEANDGROOVEID. Table 4{1 comparesthenumberofsegmentsandtheMUecienciesofallt hreealgorithms. TheMUeciencyisdenedastheratioofthemaximumruenceof intensitymodulated eldperMUtotheruenceofanopeneldperMU.Comparedtothe leafsequenceswithno
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82 constraints,theconsiderationoftongueandgroovecorr ectionincreasedboththenumberof segmentsandMUs,withanaverageincreaseof21%and19%,res pectively,forthe7intensity mapsconsideredhere.Withtheadditionaleliminationofin terdigitation,theincreaseswere 25%and24%,respectively.Examinationofallthesubeldso ftheleafsequencesgenerated withAlgorithmTONGUEANDGROOVEIDveriedthatnointerdi gitationconstrainthas beenviolated. Figure4{7:FilmmeasurementoftheAPeld(eldID1inTable 1)ofaseveneldheadand neckplan.Theoptimizedleafsequencesweregeneratedwith out(AlgorithmMULTIPAIR, (a))andwithtongueandgrooveidcorrection(Algorithm TONGUEANDGROOVEID, (b)). 4.3ComparisonwithAlgorithmofQueetal.(2004)) Recentlyanewalgorithmtoeliminatetongueandgroovee ectsinstepandshootdeliveryhasbeenproposed(Queetal.2004).ThealgorithmofQ ueetal.(2004)isdesignedto eliminatetongueandgrooveeect.Althoughthisalgorit hmeliminatestongueandgroove eectonall1000randommatricestriedinQueetal.(2004),n oproofthatthealgorithm eliminatestongueandgrooveeectonallpossiblematric eshasbeenprovided.Further,it isnotknownwhetherornotthealgorithmofQueetal.(2004)m inimizestherapytime. WeanalyzethealgorithmofQueetal.(2004)andshowthatiti salwayssuccessfulin eliminatingtongueandgrooveeect;thegeneratedleafs equenceisalsofreeofinterdigitation.Wealsoperformatheoreticalandexperimentalcomp arisonofthisalgorithmwith ouralgorithms(Kamathetal.(2004)).
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83 Table4{1:ComparisonofthenumberofsegmentsandMUecien cyof thethreeleafsequencingalgorithms(MULTIPAIR,TONGUEAN DGROOVEand TONGUEANDGROOVEID)for7intensitymapsofaheadandneckt reatmentplangeneratedfromacommercialtreatmentplanningsystem.Theperce ntincreasesinthenumberof segmentsandMUsforAlgorithmsTONGUEANDGROOVEandTONGUE ANDGROOVEIDwithrespecttoAlgorithmMULTIPAIRarealsoshown.Theav eragepercentincreases inthenumberofsegmentsare21%and25%,respectively.Thea veragepercentincreases inthenumberofMUsare19%and24%,respectively. FieldID 1 2 3 4 5 6 7 MULTIPAIR #ofSegments 11 8 13 15 14 10 10 MUEciency 0.47 0.63 0.40 0.35 0.37 0.40 0.51 TONGUEANDGROOVE #ofSegments 14 10 15 21 14 13 11 MUEciency 0.37 0.51 0.35 0.25 0.37 0.40 0.40 %Segment#increase 27 25 15 40 0 30 10 %MUincrease 26 24 15 37 0 0 28 TONGUEANDGROOVEID #ofSegments 14 11 16 21 14 14 11 MUEciency 0.37 0.47 0.33 0.25 0.37 0.37 0.37 %Segment#increase 27 38 23 40 0 40 10 %MUincrease 26 36 22 37 0 7 38 4.3.1AnalysisoftheAlgorithmofQueetal.(2004) InthissectionweanalyzethealgorithmofQueetal.(2004). Theyusethe`sliding window'methodproposedbyBortfeldetal.(1994b)togenera teatentativesegment.They thensearchthroughtherightleafpositionstodetermineth eleftmostrightleafposition andpositionallrightleavesatthatposition.Thisdenest herstsegmentoftheleaf sequence.Theresidualintensitymatrixiscalculatedandt heprocessisrepeated.To obtainthe`slidingwindow'leafsequenceforeachleafpair ,horizontallinesaredrawnat unitintensitylevelstointersecttheintensityprolefor thatleafpair.Theleftandrightleaf positionsaredeterminedfromtheseintersectionsandares ortedfromlefttorighttogive thenalunidirectionalleafsequence.Theprocessisrepea tedforallleafpairs.Forthecase wheretheintensitylevelsinthemapgeneratedbytheoptimi zerareintegers,itispossible toshowthatthealgorithmsofMaetal.(1998)andKamathetal .(2003)(Algorithms SINGLEPAIRandMULTIPAIRforoneandmultipleleafpairsres pectively)willyieldthe
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84 sameleafsequenceasthatobtainedusingthe`slidingwindo w'methodofBortfeldetal. (1994b). Notethatthediscreteintensityprolethatneedstobedeli vered, I ,isoutputfrom theoptimizer.Let n bethenumberofleafpairsand m bethenumberofsamplepointsfor eachleafpair(i.e.,foreachrowoftheprole).Wedenoteth erowsof I by I 1 ;I 2 ;:::I n .Let I t ( x i )denotethenumberofMUsthatneedtobedeliveredatsamplep oint i ( i thcolumn) ofleafpair t ( t throw). Lemma28 ThealgorithmofQueetal.(2004)generatesunidirectional schedules. Proof: Duringeachiteration,thenextsegmentgeneratedusingthe `slidingwindow' methodissuchthattheleftleavesarepositionedattheleft mostnonzerosamplepoint (i.e.,theleast i suchthat I t ( x i ) >I t ( x i 1 ),where I t ( x 1 )=0)foreachrow t intheresidual matrix I .Therightleavesarepositionedattherstcolumnsofthere spectiverowswhere thereisadecreaseinintensityprole(i.e.,theleast j forwhich I t ( x j )
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85 I 4 x 5 x x x x x x x x x 1 8 7 6 3 2 0 Figure4{8:Leafpositions:Theleftleafwillbepositioned at x 2 ,i.e.,itwillshield x 0 and x 1 .Therightleafwillbepositionedat x 6 andwillshield x i i 6. Proof: Let I 0 tl ( x i )and I 0 tr ( x i ),respectively,bethenumberofMUsdeliveredwhentheleft andrightleavesofpair t pass x i intheschedulegeneratedbythealgorithmofQueetal. (2004).Intheschedulegenerated,allrightleavespasspoi nt x i ,0 i m (duringtheirleft torightmovement)afterexactlythesamenumberofmonitoru nits(MUs)aredelivered. So I 0 tr ( x i )= I 0 ( t +1) r ( x i ),0 i m ,1 t
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86 time,say 0r ( x i ),whichisequaltothemaximumofthetimesforwhicharightl eafstops at x i intheschedulegeneratedbyAlgorithmMULTIPAIR,i.e., 0r ( x i )=max j f jr ( x i ) g Thetherapytimefortheschedulegeneratedbythealgorithm ofQueetal.(2004)is therefore T 0 tng id = P mi =0 0r ( x i )= P mi =0 max j f jr ( x i ) g .Sinceeach jr ( x i ),0 i m 1 j
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87 4.3.2Results WeimplementedAlgorithmsTONGUEANDGROOVEandTONGUEANDG ROOVEID(Kamathetal.2004)andthealgorithmofQueetal.(2004). Forperformancecomparison,weusedtwoseparatedatasets.Therstsetconsistedo fthreeclinicalIMRTplans with7,5and7beams,respectively.Thersttwoplanshada20 %ruencestepandlastplan hada10%ruencestep.Table 4.3.2 givesthetotalMUsandnumberofsegmentsrequired foreachofthe19beamsinthe3clinicalplans.Onourclinica ldataset,thealgorithmof Queetal.(2004)generatedscheduleswith24timesasmanyM Usandsegmentsasdid thealgorithmsofKamathetal.(2004).Theseconddatasetco nsistedof100,000randomly generated15 15matrices.Theintensityvaluesinthesematriceswereran domintegers from0to10.TheaverageMUsandsegmentsforschedulesgener atedusingthethree algorithmsforthissetandtheirrespectivestandarddevia tionsareshowninTable 4.3.2 Onthisset,thealgorithmofQueetal.(2004)generatedsche duleswithabout2.5times asmanyMUsandsegmentsasdidthealgorithmsofKamathetal. (2004).Notethatin bothcasesthenumberofMUsandsegmentsintheschedulesgen eratedusingAlgorithm TONGUEANDGROOVEID(Kamathetal.2004)areonlyslightlyg reaterthaninthose generatedusingAlgorithmTONGUEANDGROOVE(Kamathetal.2 004). 4.4Conclusion Wehavedescribedmathematicalformalismandrigorousproo fsofleafsequencingalgorithmsforsegmentalmultileafcollimation,whichmaximiz eMUeciencywhilecompletely eliminatingthetongueandgrooveunderdosage.Eventhou ghithasbeenshownthatfor amultipleeldIMRTplan( 5),thetongueandgrooveeectontheIMRTdosedistributionisclinicallyinsignicant(Dengetal.2001)duetot hesmearingeectofindividual elds,yetitstillcanbeproblematicforasmallnumberofe ldsandforthepatientsetup withminimaluncertainty.Comparedtotheunconstrainedle afsequencingalgorithms,the presentedmethodsyieldleafsequences,whichdecreasesth eMUeciencyalittle.Butthey completelyovercometongueandgrooveunderdosages.One ofthemethodsalsoeliminates leafinterdigitation.Mostimportantly,mathematicalpro ofsshowthatthesealgorithms areoptimalinMUeciencyforunidirectionalschedules.We havealsoprovedthatthe algorithmofQueetal.(2004)generatesschedulesthataref reeofthetongueandgroove
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88 Table4{2:NumberofMUsandsegmentsgeneratedfor19clinic alintensitymodulatedbeamsfrom3IMRTplansusingalgorithmsA(Algorithmo fQueetal.2004), B(AlgorithmTONGUEANDGROOVE(Kamathetal.2004))andC(Al gorithm TONGUEANDGROOVEID(Kamathetal.2004)).Beams112havea 20%ruencestep, whilebeams1319havea10%ruencestep. Beamnumber A B C MUsSegments MUsSegments MUsSegments 1 78038 28014 28014 2 52026 20010 22011 3 76033 30015 32016 4 84040 42021 42021 5 74032 28014 28014 6 78034 26013 28014 7 64029 26011 28011 8 150074 38019 42021 9 86043 24012 24012 10 150067 42020 42020 11 166078 42021 44022 12 84039 28014 28014 13 88078 28025 28024 14 1080102 30030 34033 15 107090 31027 32026 16 100090 34031 39036 17 89071 34029 34028 18 99075 31029 31030 19 101084 33030 33030 Table4{3:AveragenumberofMUsandsegmentsgeneratedover asetof100,000 random15 15matricesusingalgorithmsA(AlgorithmofQueetal.2004) B(AlgorithmTONGUEANDGROOVE(Kamathetal.2004))andC(Al gorithm TONGUEANDGROOVEID(Kamathetal.2004)).Therespectives tandarddeviations arealsoshown.Theintensityvaluesinthematriceswereran domlygeneratedintegersfrom 0to10. A B C MUsSegments MUsSegments MUsSegments Average 114.3111.6 47.545.7 48.246.4 StandardDeviation 6.16.0 3.43.0 3.53.0
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89 eectandinterdigitation.Ouranalysisshowsthatthealgo rithmofQueetal.generates schedulesthatmayrequireupto n timesthetherapytimerequiredbythatforanoptimal leafsequencefreeoftongueandgrooveeectandinterdig itation,where n isthenumberof involvedleafpairs.Inexperimentswithclinicalandrando mlygenerateddatasetswend thatthealgorithmofQueetal.(2004)generatesschedulest hatrequire2to4timesthe therapytimerequiredbytheschedulesgeneratedbyouralgo rithms.
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CHAPTER5 ALGORITHMSFORSPLITTINGLARGEFIELDS 5.1Introduction Thedeliveryofabuttingsubeldsthatresultfromthespli tofalargeeldoften resultsinlongerdeliverytimes,poorMUeciency,andeld matchingproblems.Dogan etal.(2003)pointoutthattheuncertaintiesinleafandcar riagepositionscauseerrorsin thedelivereddose(hotorcoldspots)alongthematchlineof theabuttingsubelds.They observeddosedierencesofupto10%alongtheeldsplitlin ewhenthesplitlinecrossed throughthecenterofthetargetforalltheelds.Theproble mofdosimetricperturbation alongtheeldsplitlinehasbeenaddressedinseveralrecen tpublications(Wuetal.2000, Hongetal.2002,Doganetal.2003).Thesolutionsincludeda utomaticfeatheringofspliteldsbymodifyingthesplitlinepositionforeachgantrypo sition(Hongetal.2002,Dogan etal.2003)orbydynamicallychangingradiationintensity intheoverlapregionofthesplit elds.Noneoftheeldsplittingtechniquesreportedinthe literaturehasaddressedthe issueoftreatmentdeliveryandMUeciency.Webelievethat itisequallyimportantto addressthisissue. Ouroptimaleldsplittingalgorithmswithandwithoutfeat heringmaybeintegrated intoourpreviouslydevelopedleafsequencingalgorithmst ooptimallyaccountforinterdigitationandtongueandgrooveeectofsomemultileafcoll imators.Weproviderigorous mathematicalproofsthattheproposedschemesforeldspli ttingareoptimalinMUeciency.Experimentalresultsshowthatouroptimaleldspl ittingalgorithmwithoutfeatheringreducestotalMUsbyupto26%onclinicalcasesandupto 63%onsyntheticcases comparedtoacommercialplanningsystemthatalsosplitse ldswithoutfeathering. 5.2FieldSplittingWithoutFeathering 5.2.1OptimalFieldSplittingforOneLeafPair Inthissectionwedeviateslightlyfromourearliernotatio nandassumethatthesample pointsare x 1 ;x 2 ; ... ;x m ratherthan x 0 ;x 1 ; ... ;x m .Allothernotationremainsunchanged. 90
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91 DeliveringaproleusingOneeld. Let I bethedesiredintensityprole.The problemofdeliveringtheexactprole I usingasingleeldhasbeenextensivelystudied. Maetal.(1998)providean O ( m )algorithmfortheproblemsuchthatthetherapytimeof thesolutionisminimized,where m isthenumberofsamplepoints.Kamathetal.(2003) alsodescribethealgorithm(AlgorithmSINGLEPAIR)andgiv eanalternateproofthatit obtainsaplan( I l ;I r )withoptimaltherapytimefor I ,where I l and I r denotetheleftand rightleafmovementproles,respectively.Theoptimalthe rapytimefor I isgivenbythe followinglemma.Lemma29 Let inc 1 ;inc 2 ;:::;incq betheindicesofthepointsatwhich I ( x i ) increases, i.e., I ( x inci ) >I ( x inci 1 ) .Thetherapytimefortheplan ( I l ;I r ) generatedbyAlgorithm SINGLEPAIRis P qi =1 [ I ( x inci ) I ( x inci 1 )] ,where I ( x inc 1 1 )=0 AlgorithmSINGLEPAIRcanbedirectlyusedtoobtainplanswh en I isdeliverable usingasingleeld.Let l betheleastindexsuchthat I ( x l ) > 0andlet g bethegreatest indexsuchthat I ( x g ) > 0.Wewillassumewithoutlossofgeneralitythat l =1.Sothe widthoftheproleis g samplepoints,where g canvaryfordierentproles.Assuming thatthemaximumallowableeldwidthis w samplepoints, I isdeliverableusingoneeld if g w ; I requiresatleasttwoeldsfor g>w ; I requiresatleastthreeeldsfor g> 2 w Thecasewhere g> 3 w isnotstudiedasitneverarisesinclinicalcases.Theobjec tiveof eldsplittingistosplitaprolesothateachoftheresulti ngprolesisdeliverableusinga singleeld.Further,itisdesirablethatthetotaltherapy timeisminimized,i.e.,thesum ofoptimaltherapytimesoftheresultingprolesisminimiz ed.Wewillcalltheproblemof splittingtheprole I ofasingleleafpairinto2proleseachofwhichisdeliverab leusing oneeldsuchthatthesumoftheiroptimaltherapytimesismi nimizedasthe S 2(single pair2eldsplit)problem.Thesumoftheoptimaltherapytim esofthetworesulting prolesisdenotedby S 2( I ). S 3and S 3( I )aredenedsimilarlyforsplitsinto3proles. Theproblem S 1istrivial,sincetheinputproleneednotbesplitandisto bedelivered usingasingleeld.Notethat S 1( I )istheoptimaltherapytimefordeliveringtheprole I inasingleeld.FromLemma 29 andthefactthattheplangeneratedusingAlgorithm SINGLEPAIRisoptimalintherapytime, S 1( I )= P qi =1 [ I ( x inci ) I ( x inci 1 )].
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92 Splittingaproleintotwo. Supposethataprole I issplitintotwoproles.Let j betheindexatwhichtheproleissplit.Asaresult,wegettw oproles, P j and S j P j ( x i )= I ( x i ),1 i
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93 x 5 x 6 P (x) 3 P (x) 4 S (x) 3 S (x) 4 x 4 31 P (x) 4 2 (a) I x x 1 x 2 x 3 x S (x) xxxx 3456 I(x ) 3 ^ (b) (c) (d) (e) 3 4 x 56 x x 4 I(x ) 4 I(x ) 4 ^ 4 x S (x) I(x ) 31 2 2 x I x 1 x 2 x 3 x I(x ) I(x ) I(x ) I 1 xx 3 I(x ) P (x) 3 I(x ) Figure5{1:Splittingaprole(a)intotwo.(b)and(c)showt heleftandrightproles resultingfromasplitat x 3 ;(d)and(e)showtheleftandrightprolesresultingfroma splitat x 4
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94 theprole I issplitintotwoasdeterminedbyAlgorithm S 2,theleftandrightprolesare deliveredusingseparateelds.Thetotaltherapytimeis S 2( I )= S 1( P j )+ S 1( S j ),where j isthesplitpoint. Splittingaproleintothree. Supposethataprole I issplitintothreeproles.Let j and k j
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95 improvesoptimaltherapytime.Foranupperboundontherati o,notethat S 1( I ) min f I ( x j 1 ) ;I ( x j ) g sinceatleastmin f I ( x j 1 ) ;I ( x j ) g MUsarerequiredtodeliver I So S 2( I ) 2 S 1( I ).TheexampleofFigure 5{2 showsthattheupperboundistight. Theprole I has2 w samplepoints,i.e.,ithasawidth2 w x .Soithastobesplit exactlyat x w +1 .Theresultingleftandrightproleseachhaveanoptimalth erapy timeequaltothatof I x w w I Figure5{2:TightupperboundforLemma32a (b) S 3( I )= S 1( I )+min f I ( x j 1 ) ;I ( x j ) g +min f I ( x k 1 ) ;I ( x k ) g ,where j and k areas inAlgorithm S 3.Clearly, S 3( I ) =S 1( I ) 1.Also, S 1( I ) min f I ( x j 1 ) ;I ( x j ) g and S 1( I ) min f I ( x k 1 ) ;I ( x k ) g .Therefore, S 3( I ) 3 S 1( I ).Onceagaintheupper boundistightasshownintheFigure 5{3 .Theproleshownhaswidth3 w x and needstobesplitat x w +1 andat x 2 w +1 .Eachoftheresultingproleshasoptimal therapytimeequalto S 1( I ). w w w x I Figure5{3:TightupperboundforLemma32b (c)From(a)and(b), S 3( I ) S 1( I )and S 2( I ) 2 S 1( I ).So S 3( I ) =S 2( I ) 0 : 5. S 3( I ) =S 2( I )=0 : 5onlyif S 3( I )= S 1( I )and S 2( I )=2 S 1( I ).Suppose
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96 that S 3( I )= S 1( I ).Thenthereexistindices j;k suchthatmin f I ( x j 1 ) ;I ( x j ) g + min f I ( x k 1 ) ;I ( x k ) g =0,i.e.,min f I ( x j 1 ) ;I ( x j ) g =0andmin f I ( x k 1 ) ;I ( x k ) g =0. Thisandthefactthat I ( x 1 ) 6 =0 ;I ( x g ) 6 =0impliesthattheprolehasatleast twodisjointcomponentsseparatedbyasamplepointatwhich thedesiredintensity iszero.Samplepointsinthetwodisjointcomponentscannot beexposedatthe sametimeandsotheredoesnotexistapoint x i suchthat I ( x i )= S 1( I ).So S 2( I )= S 1( I )+min g w 0 : 5.Figure 5{4 showsanexamplewheretheratiocanbemade arbitrarilycloseto0.5.Inthisexample, S 1( I )= I 2 .Theprolehasawidthof 2 w x andthereforeneedstobesplitat x w +1 .Theresultingproleseachhavean optimaltherapytimeof S 1( I )sothat S 2( I )=2 S 1( I ). S 3( I )= S 1( I )+2 I 1 andso S 3( I ) S 1( I )as I 1 0. I w x 1 2 I I jk x x ww x Figure5{4:TightlowerboundforLemma32c Toobtainanupperboundnotethatthebestsplitpointfor S 2(say x j )isalways apermissiblesplitpointfor S 3.Byselectingthisasoneofthetwosplitpointsfor S 3,wecanconstructasplitintothreeprolessuchthattheto taltherapytimeof prolesresultingfromthissplitis S 2( I )+min f I ( x k 1 ) ;I ( x k ) g ,where k isthesecond splitpointdeningthatsplit.Sincemin f I ( x k 1 ) ;I ( x k ) g S 1( I ) S 2( I ),thetotal therapytimeofthesplit 2 S 2( I ).So S 3( I ) =S 2( I ) 2.Theratiocanbearbitrarily closeto2asdemonstratedinFigure 5{5 .Onecanverifythatfortheprole I inthis example, S 3( I ) =S 2( I ) 2as I 1 0.
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97 I x x I 1 1 Figure5{5:TightupperboundforLemma32c Lemma 32 tellsusthattheoptimaltherapytimescanatmostincreaseb yfactorsof2and 3,respectively,asaresultofasplittingasingleleafpair proleinto2and3.Also,the optimaltherapytimeforasplitinto2canbeatmosttwicetha tforasplitinto3andvice versa.5.2.2OptimalFieldSplittingforMultipleLeafPairs Theinputintensitymatrix(say I )fortheleafsequencingproblemisobtainedusing theinverseplanningtechnique.Thematrix I consistsof n rowsand m columns.Each rowofthematrixspeciesthenumberofmonitorunits(MUs)t hatneedtobedelivered usingoneleafpair.Denotetherowsof I by I 1 ;I 2 ;:::;I n .Forthecasewhere I isdeliverableusingoneeld,theleafsequencingproblemhasbeenw ellstudiedinthepast.The algorithmthatgeneratesoptimaltherapytimeschedulesfo rmultipleleafpairs(Algorithm MULTIPAIR)appliesalgorithmSINGLEPAIRindependentlyto eachrow I i of I .Without lossofgeneralityassumethattheleastcolumnindexcontai ninganonzeroelementin I is 1andthelargestcolumnindexcontaininganonzeroelementi n I is g .If g>w ,theprole willneedtobesplit.Wedeneproblems M 1, M 2and M 3formulipleleafpairsasbeing analogousto S 1, S 2and S 3forsingleleafpair.Theoptimaltherapytimes M 1( I ), M 2( I ) and M 3( I )arealsodenedsimilarly. Splittingaproleintotwo. Supposethataprole I issplitintotwoproles.Let x j bethecolumnatwhichtheproleissplit.Thisisequivalent tosplittingeachrowprole
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98 I i ,1 i n ,at j asdenedforsingleleafpairsplit.Asaresultwegettwopro les, P j (left)and S j (right). P j hasrows P 1 j ;P 2 j ;:::;P n j and S j hasrows S 1 j ;S 2 j ;:::;S n j Lemma33 Suppose I issplitintotwoprolesat x j .Theoptimaltherapytimefordelivering P j and S j usingseparateeldsis max i f S 1( P i j ) g +max i f S 1( S i j ) g Proof: Theoptimaltherapytimeschedulefor P j and S j areobtainedusingAlgorithm MULTIPAIR.Thetherapytimesaremax i f S 1( P i j ) g andmax i f S 1( S i j ) g respectively.Sothe totaltherapytimeismax i f S 1( P i j ) g +max i f S 1( S i j ) g FromLemma 33 itfollowsthatthe M 2problemcanbesolvedbyndingtheindex j 1
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99 Algorithm M 3solvesthe M 3problem. Algorithm M 3 (1)Computemax i f S 1( P i j ) g +max i f S 1( M i ( j;k ) ) g +max i f S 1( S i k ) g for1
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100 splitismax i f S 1( P i j ) g +max i f S 1( M i ( j;k ) ) g +max i f S 1( S i k ) g max i f S 1( P i j ) g +2 max i f S 1( S i j ) g 2 M 2( I ).So M 3( I ) =M 2( I ) 2. Notethattheexamplesusedtoshowtightnessofboundsinthe proofofLemma 32 canalsobeusedtoshowtightnessofboundsinthiscase. Lemma 35 tellsusthattheoptimaltherapytimescanatmostincreaseb yfactorsof2and 3,respectively,asaresultofsplittingaeldinto2and3.A lso,theoptimaltherapytime forasplitinto2canbeatmosttwicethatforasplitinto3and viceversa.Thesebounds giveusthepotentialbenetsofdesigningMLCswithlargerm aximalaperturesothatlarge eldsdonotneedtobesplit.Tongueandgrooveeectandinterdigitation. Algorithms M 2and M 3maybeextendedtogenerateoptimaltherapytimeeldswitheliminat ionoftongueandgrooveunderdosageand(optionally)theinterdigitationconstrain tontheleafsequences.Kamathet al.(2004)presentalgorithmsfordeliveringanintensitym atrix I usingasingleeldwith optimaltherapytime,whileeliminatingthetongueandgr ooveunderdosage(Algorithm TONGUEANDGROOVE)andalsowhilesimultaneouslyeliminati ngthetongueandgroove underdosageandinterdigitationconstraintviolations(A lgorithmTONGUEANDGROOVEID).Denotetheseproblemsby M 1 0 and M 1 00 respectively( M 2 0 M 2 00 M 3 0 and M 3 00 are denedsimilarlyforsplitsintotwoandthreeelds).Let M 1 0 ( I )and M 1 00 ( I ),respectively, denotetheoptimaltherapytimesrequiredtodeliver I usingtheleafsequencesgeneratedby thesealgorithms.Tosolveproblem M 2 0 weneedtodetermine x j where M 1 0 ( P j )+ M 1 0 ( S j ) isminimizedfor g w
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101 etal.2000).Toillustratetheproblemweuseanexample.Con siderthesingleleafpair intensityproleofFigure 5{6 a.Duetowidthlimitations,theproleneedstobesplit. Supposethatitissplitat x j .Furthersupposethatthelefteldisdeliveredaccurately andthattherighteldismisalignedsothatitsleftendispo sitionedat x 0j ratherthan x j Duetoincorrecteldmatchingtheactualproledeliveredm aybe,forexample,eitherof theprolesshowninFigure 5{6 borFigure 5{6 d,dependingonthedirectionoferror.In Figure 5{6 b,theregionbetween x 0j and x j getsoverdosedandisa hotspot .InFigure 5{6 d, theregionbetween x j and x 0j getsunderdosedandisa coldspot Onewaytopartiallyeliminatetheeldmatchingproblemist ousethe`feathering' technique(Wuetal.2000).Inthistechnique,thelargeeld isnotsplitatonesample pointintotwononoverlappingelds.Insteadtheprolest obedeliveredbythetwoelds resultingfromthesplit,overlapoveracentral featheringregion .ThebeamsplittingalgorithmproposedbyWuetal.(2000)splitsalargeeldwithfea thering,suchthatinthe featheringregionthesumofthespliteldsequalsthedesir edintensityprole.Figure 5{7 a showsasplitoftheproleofFigure 5{6 withfeathering.Figures 5{7 cand 5{7 dshowthe eectofeldmatchingproblemonthesplitwithfeathering. Theextentofeldmismatches isthesameasthoseinFigures 5{6 band 5{6 d,respectively.Notethatwhiletheprole deliveredinthecasewithfeatheringisnottheexactprole either,thedeliveredproleis lesssensitive tomismatchcomparedtothecasewhenitissplitwithoutfeat heringasin Figure 5{6 .Inotherwords,thepurposeoffeatheringistolowerthemag nitudeof maximum intensityerror e inthedeliveredprolefromthedesiredproleoverallsamp lepointsin thejunctionregion. Inthissection,weextendoureldsplittingalgorithmstoi ncorporatefeathering.In ordertodoso,wedeneafeatheringschemesimilartothatof Wuetal.(2000).However, therearetwodierencesbetweenthesplittingalgorithmwe proposeandthealgorithmof Wuetal.(2000).First,ourfeatheringschemeisdenedforp rolesdiscretizedinspace andinMUsasistheprolegeneratedbytheoptimizer.Second ,thefeatheringschemewe proposedenestheprolevaluesinthefeatheringregion,w hichiscenteredatsomesample pointcalledthe splitpoint forthatsplit.Thusgivenasplitpoint,ourschemewillspec ify howtosplitthelargeeldwithafeatheringregionthatisce nteredatthatpoint.Thesplit
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102 x xx xx x' xx' e e (a) (b) I x'xx (c) (d) j jj jj jj I II Figure5{6:Fieldmatchingproblem:Theprolein(a)isthed esiredprole.Itissplitinto twoeldsat x j .Duetoincorrecteldmatching,theleftendofrighteldis positionedat point x 0j insteadof x j andtheeldsmayoverlapasin(c)ormaybeseparatedasin(d) In(c),thedottedlineshowstheleftproleandthedashedli neshowstherightprole. (b)showstheseprolesaswellasthedeliveredproleinthi scaseinbold.In(d),the leftandrighteldsareseparatedandtheirtwoprolestoge therconstitutethedelivered prole,whichisshowninbold.Thedeliveredprolesinthes ecases,varysignicantlyfrom thedesiredproleinthejunctionregion. e isthemaximumintensityerrorinthejunction region,i.e.,themaximumdeviationofdeliveredintensity fromthedesiredintensity.
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103 II x xx xx x' x I e e j x' j j j j j (d) (c) (b) xx I (a) Figure5{7:Exampleofeldsplittingwithfeathering:(a)s howsasplitoftheproleof Figure 5{6 withfeathering.Thedottedlineshowstherightpartofthel eftproleand thedashedlineshowstheleftpartoftherightprole.Thele ftandrightprolesare shownseparatelyin(b).(c)and(d)showtheeectofeldmat chingproblemonthesplit withfeathering.Theextentofeldmismatchesin(c)and(d) arethesameasthosein Figure 5{6 bandFigure 5{6 d,respectively,ie.,thedistancesbetween x j and x 0j arethe sameasinFigure 5{6 .Notethatthemaximumintensityerror e reducesinbothcaseswith feathering.
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104 pointtobeusedintheactualsplitwillbedeterminedbyaspl ittingalgorithmthattakes intoaccountthefeatheringscheme.Incontrast,Wuetal.(2 000)alwayschoosethecenter oftheintensityproleasthesplitpoint,astheydonotopti mizethesplitwithrespectto anyobjective. Westudyhowtosplitasingleleafpairproleintotwo(three )eldsusingourfeathering schemesuchthatthesumoftheoptimaltherapytimesofthein dividualeldsisminimized. Wewilldenotethisminimizationproblemby S 2 F ( S 3 F ).Theextensionofthemethods develpedforthemultipleleafpairsproblems( M 2 F and M 3 F )isstraightforwardandis thereforenotdiscussedseparately.5.3.1SplittingaProleintoTwo Let I beasingleleafpairprole.Let x j bethesplitpointandlet P j and S j be theprolesresultingfromthesplit. P j isa leftprole and S j isa rightprole of I .The featheringregionspans x j and d 1samplepointsoneithersideof x j ,i.e.,thefeathering regionstretchesfrom x j d +1 to x j + d 1 P j and S j aredenedasfollows. P j ( x i )= 8>>>><>>>>: I j ( x i )1 i j d d I j ( x i ) ( j + d i ) = 2 d e j d>>><>>>>: 01 i j d I j ( x i ) P j ( x i ) j dg w ) j g w + d .Theserangerestrictionson j leadtoanalgorithmfor the S 2 F problem.Algorithm S 2 F ,whichsolvesproblem S 2 F ,isdescribedbelow.Note
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105 thatthe P i sand S i scanallbecomputedinasinglelefttorightsweepin O ( d )timeateach i .SothetimecomplexityofAlgorithm S 2 F is O ( dg ). Algorithm S 2 F (1)Find P i and S i usingEquations 5.1 and 5.2 ,for g w + d i w d +1. (2)Splittheeldatapoint x j where S 1( P j )+ S 1( S j )isminimizedfor g w + d j w d +1. 5.3.2SplittingaProleintoThree Supposethataprole I issplitintothreeproleswithfeathering.Let j and k j
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106 that g 2 w +3 d 1 j w d +1andthat g w + d k 2 w 3 d +2.Also, k j +1+2( d 1) w ) k j w 2 d +1.Usingtheserangesfor j and k ,wearrive atAlgorithm S 3 F ,whichcanbeimplementedtosolveproblem S 3 F in O ( dg 2 )time. Algorithm S 3 F (1)Find P j M ( j;k ) and S k usingEquations 5.3 5.4 and 5.5 ,for g 2 w +3 d 1 j w d +1, g w + d k 2 w 3 d +2and k j w 2 d +1. (2)Splittheeldattwopoints x j x k ,where S 1( P j )+ S 1( M ( j;k ) )+ S 1( S j )isminimized, subjectto g 2 w +3 d 1 j w d +1, g w + d k 2 w 3 d +2and k j w 2 d +1. 5.3.3TongueandgrooveEectandInterdigitation Thealgorithmsfor M 2 F and M 3 F maybefurtherextendedtogenerateoptimal therapytimeeldswitheliminationoftongueandgrooveu nderdosageand(optionally)the interdigitationconstraintontheleafsequencesasisdone foreldsplitswithoutfeatheringin Section 5.2.2 .Thedenitionsofproblems M 2 F 0 ( M 3 F 0 )and M 2 F 00 ( M 3 F 00 ),respectively, forsplitsintotwo(three)eldsaresimilartothosemadein Section 5.2.2 forsplitswithout feathering. 5.4Results TheperformanceoftheAlgorithms M 2, M 3, M 2 F and M 3 F wastestedusing27 clinicalruencematrices,eachofwhichexceededthemaximu mallowableeldwidth w =14, with d =2forfeathering.Theruencematricesweregeneratedwitha commercialinverse treatmentplanningsystem(CORVUSv5.0,NOMOSCorp.,Sewic kley,PA)forveclinical cases.Algorithm M 2 F wasusedwhentheprolewidthwas 2 w 2 d +1=25and algorithm M 2wasusedwhenevertheprolewidthwas 2 w =28.Algorithms M 3 and M 3 F wereusedinallcases.TheoptimalMUsforthespliteldswer ecalculated assumingthatthespliteldsineachcasearedeliveredbyse quencingleavesusingAlgorithm MULTIPAIR.Table 5{1 displaystheresultingtotalMUsfortheeldsplitsobtaine dusing thefouralgorithms.AlsoshownarethetotalMUsobtainedus ingtheeldsplitlinesas givenbythecommercialtreatmentplanningsystem( C ( I )).TheMUsarenormalizedto giveamaximumpixelvalueof100ofaruencemap.Thepercentd ecreaseinMUsof min f M 2( I ) ;M 3( I ) g asaresultofoptimaleldsplittingover C ( I )isalsoshowninthelast
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107 column.MUreductionsofupto26%areseen.Inabout30%ofthe casesthereduction wasover20%.TheaveragedecreaseinMUsisfoundtobeabout1 1%forthe27ruence matrices.Notethattheapplicationofoptimalsplittingal gorithmswithfeatheringcan reduceMUascomparedtotheoptimalalgorithmswithoutfeat heringasaresultofthe reductioninintensityvaluesinthefeatheringregioninea cheldresultingfromthesplit. Weobservethat M 2 F ( I )
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108 Table5{1:TotalMUsforveclinicalcases Matrix( I ) Width C ( I ) M 2( I ) M 3( I ) M 2 F ( I ) M 3 F ( I ) %MUdecrease 1 15 280 280 330 290 335 0 2 15 310 310 350 318 350 0 3 16 300 260 340 260 310 13.3 4 16 400 300 370 290 353 25 5 16 350 350 380 360 394 0 6 16 340 310 310 325 382 8.8 7 16 390 310 360 310 338 20.5 8 16 350 320 340 320 357 8.6 9 16 400 300 370 310 365 25 10 16 440 350 390 322 398 20.5 11 16 320 320 360 310 362 0 12 16 400 300 340 280 360 25 13 17 380 280 320 285 323 26.3 14 18 280 240 260 240 280 14.3 15 20 320 320 380 320 375 0 16 22 400 360 400 360 400 10 17 22 320 320 360 300 360 0 18 24 540 480 520 480 500 11.1 19 24 540 500 500 490 500 7.4 20 24 460 420 460 420 425 8.7 21 24 520 520 540 525 545 0 22 24 560 520 520 505 505 7.1 23 25 360 360 380 340 380 0 24 26 560 440 460 430 21.4 25 29 520 480 445 7.7 26 29 580 440 440 24.1 27 32 560 480 470 14.3
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109 5.5Conclusion Wehavedevelopedalgorithmstosplitlargeintensitymodu latedeldsintotwoor threesubelds.Suchaworkaroundneedstobeimplemented forMLCsthathavea maximumleafspreadlimitation,whichimposesaeldwidthl imitation.Wehavepresented algorithmsthatsplitlargeeldsintononoverlappingsub eldsalongoneortwocolumns. Alsopresentedarealgorithmsthatspliteldswithfeather ing.Featheringofsplitelds helpsreducetheeectoftheeldmatchingproblemthatoccu rsintheeldjunctionregion duetouncertaintiesinsetupandorganmotion.Wehaveshown thatouralgorithmsresult ineldsplitsforwhichtheMUeciencyisoptimal.Applicat ionofouroptimaleld splittingalgorithmswithoutfeatheringtoclinicaldatar educedtotalMUsbyupto26% andonsyntheticdataupto63%comparedtoacommercialplann ingsystemthatalsosplits eldswithoutfeathering.Wehavealsoshownthatouralgori thmscaneasilybeextended tospliteldsresultinginmaximalMUeciencywhentheMLCm odelissubjecttothe interdigitationconstraintand/orthetongueandgroove eectistobeeliminated.
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CHAPTER6 CONCLUSION Wehavepresentedasystematicstudyofleafsequencingalgo rithmsformultileafcollimation.Algorithmsarepresentedforsequencingleavesw ithoutanyconstraints,with theintrapairmaximumseparationconstraintandwiththei nterpairminimumsepration constraintforSMLCandwithoutanyconstraints,withthein trapairmaximumseparation constraintandwiththeinterdigitationconstraintforDML C.Alsopresentedarealgorithms thateliminatethetongueandgrooveunderdosage(andopt ionallytheinterdigitationconstraint)forSMLCandacomparisonofthesealgorithmswitha recentlypublishedalgorithm thatalsoeliminatesthetongueandgrooveeect.Finally ,algorithmsaredeveloped,that splitalargeintensitymodulatedeldintotwoorthreesub elds.Wehaveshownthat allthesealgorithmsobtainfeasiblesolutionswheneverth eyexist.Further,thesolutions generatedarealwaysoptimalintherapytimeforunidirecti onalschedules.Thealgorithms developedareapplicabletosomeofthepopularcommerciall yavailabledeliverysystems. 110
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BIOGRAPHICALSKETCH SrijitKamathcompletedhishighschooleducationfromKend riyaVidyalaya,IIT Madras,India,in1996.In2000hereceivedaB.Tech.degreei ninformationtechnologyfromtheUniversityofMadras,India.Hethenmovedtothe UniversityofFlorida(UF) forgraduatestudies.SinceAugust2000,hehasbeenaResear chAssistantintheComputerandInformationScienceandEngineeringDepartment. HisworkatUFhasledto vejournalpapersandtwopatentapplications.HeearnedaM .S.in2002andwillreceive aPh.D.in2005,bothfromUF. 115



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AL;GO!. i iT li' FOR UEN'CING MUUT:IILEAF ( OLL;IMATrORS ByS SREIJIT KAMAIITH AE~i 1. il iliON JPF F. UNTED TO THE :RADU!~.i i SCHOOL OF T rii;T UNIVERSITY OF` FLOCRIDA INr PARTI'`AL FU7LFil i OF T iilT Ri' UlIREMENTSI` FOR T ii;T D>EGREEI OF` DO('C)TOR OF PHILOSOPHY i~iii ii OFt i i:.01 i )A C Dedicated to : . parents, who have\ awaiyvs encoura~gedl me to become a scientist. First and .. :, my utmost gratitude goes to my advisor, Sartaj Sahni, for the immeasurably valuable guidance a~nd support, he has given me during the :1 >d of my graduate study. Hle initroduecd mei to this challengingg computationa l i 1 ** in : He ha~s patientlly absorbed all my ideas, proofread every mat(hem2t~ical1 analysis that I have perforrmed, and gently nudged me in the most, i :. ..1 occasion. i i ~judgemnent and experience have proved to be crucial in the success of this work. I could not have asked i : a. better a~dvisor. I feel t~hat he is one of the best computer science educators in Ithe world. It has been a privilege and a~n honor to work wczith him. Iamrr very th~ankfuli to Meec~cra. ::   for the several insights she h~as provided on algorithmic: problems through her teachingf in aind out of class and fo~r her general enthusiasm El. ?! solving t~ha~t ha~s ledi to many absorbing discussions. Serving as t~he coordiinat~or Sthe algorithms and '1: seminar, which was started with her encouragement and initiative, ha~s also been an excellent learning : .. fr e go to Joona~than Li for collaborating on the problems and for providing clinical data. TIli .: 1 go also to Jlatindier P:alta, Anand IRanga: .1 and 'r y I Ranka for taking the time to serve on myS committees. I 1 7::: A?.::::. Jain, Amnita I i Mohit Dhhawtan, Haibin Lu, Jimin Yin aind We~tncheng Lu made I '= : a gfoodi place to work. Intense preparation fo~r tIhe doctoral q : i :: examm wa~s :i i thanks to latle night study sessions with VBl Z Mania~n a~nd Pomnpi Diplan. I would like to thank all my friends ( '..:" too many to namee) their great I :::i : and support during the course of ni graduate study, with? special thanks to fellow students Subi, M~Ianas, .1==:i, Andrew, P:ranav andii I Ti work would not have been possible without, the constant, encouragement. I have received i ::: my parents aind i .. uncle N~a .:. who ha~s also proviided numerous insights on gra~duatei study and wvor~k in the U~nited States. TABLE OF CONTENTS page ACK(NOWLEDGMENTS .......... . .. .. iv LIST OF TABLES ......... . .. .. vii LIST OF FIGURES ......... . .. viii ABSTRACT ............. .......... .. x CHAPTER 1 NTRODUCTION..........1 1.1 Problem Description ......... .. 1 1.2 MLC Models and Constraints . ..... .. 4 1.3 Prior Work ............. ..... .... 5 1.4 Dissertation Outline ......... .. 7 2 SEQUENCING OF SEGMENTED MULTILEAF COLLIMATORS .. .. 8 2.1 Methods ............. ........... 8 2.1.1 Discrete Profile ......... .. 8 2.1.2 Movement of Leaves . ... .. .. 8 2.1.3 Optimal Unidirectional Algorithm for one Pair of Leaves .. .. 11 2.1.4 Bidirectional Movement ... .. .. .. . 16 2.1.5 Algorithm Under Maximum Separation Constraint Condition 20 2.1.6 Algorithm Under InterPair Minimum Separation Constraint .. 23 2.2 Conclusion ......... . .. 34 3 SEQUENCING OF DYNAMIC MULTILEAF COLLIMATORS .. .. .. 36 3.1 Methods ............. ........... 36 3.1.1 Movement of Leaves . ..... .. 36 3.1.2 Maximum Velocity Constraint ... ... .. .. 38 3.1.3 Optimal Unidirectional Algorithm for one Pair of Leaves .. .. 38 3.1.4 Minimum Separation Constraint .. .. 42 3.1.5 Bidirectional Movement ... .. .. .. . 46 3.1.6 Algorithm Under Maximum Separation Constraint Condition 51 3.1.7 Algorithm Under Interdigfitation Constraint .. .. .. .. .. 55 3.2 Conclusion ......... . .. 65 4 ELIMINATION OF TONGUEANDGROOVE UNDERDOSAGE .. .. .. 66 4.1 Algorithm with Interdigfitation and TongueandGroove Constraints .. 67 4.1.1 TongueandGroove Underdosage Effect ... .. .. 67 4.1.2 Algorithms ......... .. .. 68 4.1.3 Efficient Implementation of the Algorithms .. .. .. .. 79 4.2 Experimental Validation . ... ... .. 81 4.3 Comparison with Algorithm of Que et al. (2004)) .. .. .. .. 82 4.3.1 A!! 11, i of the Algorithm of Que et al. (2004) .. .. .. .. .. 83 4.3.2 Results ......... . .. 87 4.4 Conclusion ......... . .. 87 5 ALGORITHMS FOR SPLITTING LARGE FIELDS ... .. .. .. 90 5.1 Introduction ......... . .. .. 90 5.2 Field Splitting Without Featheringf .... .. . 90 5.2.1 Optimal Field Splitting for One Leaf Pair .. .. .. .. 90 5.2.2 Optimal Field Splitting for Multiple Leaf Pairs .. .. .. .. 97 5.3 Field Splitting with Feathering ..... ... .. 100 5.3.1 Splitting a Profile into Two .... ... .. 104 5.3.2 Splitting a Profile into Three .... ... . .. 105 5.3.3 Tongueandgroove Effect and Interdigfitation .. .. .. .. .. 106 5.4 Results. ............ ............ 106 5.5 Conclusion . ...... . .. 109 6 CONCLUSION ............ ............ 110 REFERENCES ............. .............. 111 BIOGRAPHICAL SK(ETCH ......... .. .. 115 LIST OF TABLES Table page 41 Comparison of the number of segments ..... .... . 83 42 Number of MUs and segments ....... ... .. 88 43 Average number of MUs and segments ..... .... . 88 51 Total MUs for five clinical cases . .... .. 108 LIST OF FIGURES Figure page 11 A linear accelerator ......... . 2 12 A multileaf collimator ......... ... 3 13 Interpair minimum separation constraint .... .. 4 14 Cross section of leaves ......... ... 5 21 Geometry and coordinate system . ..... .. 9 22 Discretization of profile ......... ... 9 23 Leaf t! li.* br~!y during <\llC delivery ..... .... . 10 24 Obtaining a unidirectional plan . ..... .. 12 25 A profile and its plan ......... ... .. 13 26 Minimum separation constraint violation .... .. . 15 27 Bidirectional movement ......... .. .. 17 28 Bidirectional movement under minimum separation constraint .. .. .. 19 29 Bidirectional movement under maximum separation constraint .. .. .. 20 210 Obtaining a plan under maximum separation constraint .. .. .. .. .. 21 211 Maximum separation constraint violation .... .. .. 21 212 Obtaining a schedule ......... .. .. 24 213 Obtaining a schedule under the constraint .... .. .. 25 214 Eliminating a violation ......... .. .. 26 215 Eliminating a violation ......... .. .. 30 216 Intensity profiles of adjacent leaf pairs ..... .... . 31 217 Profiles violating interpair constraint ..... .. . 32 31 Leaf t! .1.* b r~!y during DMLC delivery ..... .... . 37 32 Obtaining a unidirectional plan . ..... .. 40 33 Obtaining a unidirectional plan with minimum separation constraint .. 44 34 Minimum separation constraint violation .... .. . 44 35 <\!LlC plan: feasible; DMLC plan: infeasible ... .. .. 47 36 Bidirectional movement ......... .. .. 48 37 Bidirectional movement under minimum separation constraint .. .. .. 50 38 Bidirectional movement under maximum separation constraint .. .. .. 52 39 Obtainingf a plan under maximum separation constraint .. .. .. .. .. 52 310 Maximum separation constraint violation .... .... .. 53 311 Obtainingf a schedule ......... .. .. 56 312 Obtainingf a schedule under the constraint .... .... .. 57 313 Eliminating a Typel violation ....... ... .. 58 314 Eliminating a Type2 violation ....... ... .. 59 41 Tongueandgroove effect ......... .. .. 67 42 Counterexample ......... .. .. 70 43 Obtainingf a schedule under the tongueandgroove constraint .. .. .. 71 44 Tongueandgroove constraint violation: casel ... .. .. .. 72 45 Tongueandgroove constraint violation: case2 ... .. .. .. 73 46 Obtainingf a schedule under both the constraints ... .. .. .. 77 47 Film measurement of the AP field . .... .. 82 48 Leaf positions ......... . .. .. 85 49 Worstcase example ......... .. .. 86 51 Splitting a profile (a) into two ....... ... .. 93 52 Tight upper bound for Lemma 32a . .. .. 95 53 Tight upper bound for Lemma 32b ...... .... . 95 54 Tight lower bound for Lemma 32c ...... .... . 96 55 Tight upper bound for Lemma 32c ...... .... . 97 56 Field matching problem ......... .. .. 102 57 Example of field splitting with featheringf ... . .. 103 58 Comparison of the field split line ...... .... .. 107 Abstract of D~issertation Presented to the Graduate"~ School of the Uj,:1 11 of FI : in P~artial i ::11111::: : of the P. i::' :.. for the Degree of Doctor o P':il AL;GO iiili lilOR i 11 i = :IG i iii 1 '<1'COiii il'`C1 By SaullI KiAMbATH AUrCusTi 2005 S. Sa: i .1 Sahni ' I.'..: Dllepartmencrt: Clomputer anid Ilnformnation Science and Elngineer~ing In. I i .. radiiation t !E. ;v.i fo~r cancer tIreatment., it, is diesirable t~o deliver high doses of raldiat ion to the (tlarget, (umor, while permnitting a lown dosage on the surrl :::. : elh tissues. In recent years, thel .1 ... of intensity mnodulated radiation therapy (lil ) has made this 1ll : be delivered by several t~echniques. 'i : delivery of I' i l. ii with a mulileaf ... i : ( i C) :.. I the delivery of radliation from several beamr orientation. ':  intensity. , ..=1 for each beamn direction is desc~rib~ed as a. ~IvCI sequenc31Ce, wh'ich' is: deve~lo~rped u1sing a leaf sequencingf algorithm. I:::i Itant considerations in developing a leaf sequence fo~r a desired int~ensitly 1includte maximizing t~he monitor unit (MU_) efficiency (equivalentlyl mninimnizing the beamon time) and mninimnizing the total trea~tmecnt time sub' to the leaf mnovemrent constraints of the Ii .C1 model. In this work, we present a systematic study of the optimization of leaf sequencing algorithms and provide rigorous mathematical proofs of optimizedl leaf sequence settings in terms of MUi rr: . under most commron leaf mnovemrent constra~ints that include mninimnum and mnaximnum leaf separationt leaif i. : :~ ~ :: :: lndl rlonguealndg~rr lroove. We also deve(lopr aITlgoithms to sp~lit large int~ensityi modulated fields intlo twio or three ; CHAPTER INTRODUCTION 1.1 Problem Description The objective of radiation therapy for cancer treatment is to deliver high doses of radiation to the target tumor, while permitting a low dosage on the surrounding healthy tissues. For example, for head and neck tumors, it is necessary for radiation to be delivered so that the exposure of the spinal cord, optic nerve, salivary glands or other important structures is niinintized. In recent years, this has been made possible due to the develop nient of confornial radiation therapy. In confornial therapy, treatment is delivered using a set of radiation beams which are positioned such that the shape of the dose distribution I .!!l .! .!." with the shape of the tumor. This is typically achieved by positioning beams of varying shapes front different directions so that each beam approximately irradiates the section of the tunior visible from its direction and avoids the organs at risk in the vicinity of the tumor. Intensity modulated radiation therapy (IMRT) is the stateoftheart in confornial radi ation therapy. IMRT permits the iint, neityi of radiation beant to be varied across a treatment area, thereby improving the dose conformity. Radiation is delivered using a medical linear accelerator (Figure 11). A rotating gantry containing the accelerator structure can rotate around the patient who is positioned on an adjustable treatment couch. Delivery of IMRT is possible by several techniques. In conipensatorbased IMRT, the beant is modulated with a preshaped piece of material called the conipensator (modulator). The degree of modulation of the beam varies depending on the thickness of the material through which the beant is attenuated. The computer determines the shape of each modulator in order to deliver the desired beam. This type of modulation requires the modulator to be fabricated and then manually inserted into the tray mount of a linear accelerator. In toniotherapybased IMRT, the linear accelerator travels in multiple circles all the way around the gantry ring to deliver the radiation treatment. The beant is colliniated to a narrow slit and the iintl, nityi of the beant is modulated during the gantry movement around the patient. Care must be taken to ensure that adjacent circular arcs do not overlap and thereby do not overdose tissues. This type of delivery is referred to as serial toniotherapy. A modification of serial toniotherapy is helical toniotherapy. In helical toniotherapy, the treatment couch moves linearly (continu ously) through the rotating accelerator gantry. So each time the accelerator contes around, it directs the beam on a slightly different plane on the patient. In AILCbased IMRT the accelerator structure is equipped with a computer controlled mechanical device called a niultileaf colliniator (AILC, Figure 12) that shapes the radiation beam, so as to deliver the radiation as prescribed by the treatment plan. The MLC may have up to 120 movable leaves that can move along an axis perpendicular to the beam and can be arranged so as to shield or expose parts of the a~natonly during treatment. The leaves are arranged in pairs so that each leaf pair forms one row of the arrangement. The set of allowable MLC leaf configurations may be restricted by leaf niovenient constraints that are manufacturer and/or model dependent. Figure 11: A linear accelerator (the figure is from http://www.lexnied .cont/ niedicaL~services/IMRT .htni) The first stage in the treatment planning process in IMRT is to obtain accurate three dimensional anatomical information about the tunior and its surroundings. This is achieved Figure 12: A multileaf collimator (the figure is from http://www.lexmed .com/ medicaL~services/IMRT .htm) using computed tomography (CT) and magnetic resonance (MR) imaging. An ideal dose distribution would ensure perfect conformity to the target volume while completely sparing all other tissues. However, such a distribution is impossible to realize in practice. Therefore, minimum dose targets for tumors and tolerable doses for critical structures are prescribed and an objective function that measures the quality of a plan is developed subject to these dose based constraints. Next, a set of beam parameters (beam angles, profiles, weights) that optimize this objective are determined using a computer program. This method is called !!ni. !n planning" since resultant dose distribution is first described and the best beam parameters that deliver the distribution (approximately) are then solved for. It is to be noted that inverse planning is a general concept and its implementation details vary vastly among various systems. Following the inverse planning in MLCbased IMRT, the delivery of radiation intensity profile for each beam direction is described as a MLC leaf sequence, which is developed using a leaf sequencing algorithm. Important considerations in developing a leaf sequence for a desired inltloneityi profile include maximizing the monitor unit (MU) efficiency (equivalently minimizing the beamon time) and minimizing the total treatment time subject to the leaf movement constraints of the MLC model. Finally, when the leaf sequences for all beam directions are determined, the treatment is performed from L2 L3 R2 R3 various beam angles sequuentially using computer control. In this work, we develop optimized leaf sequuencing algorithms for various MLC models. 1.2 MLC Models and Constraints The purpose of the leaf sequuencing algorithm is to generate a sequence of leaf positions and/or movements that faithfully reproduce the desired intensity map once the beam is delivered, taking into consideration any hardware and dosimetric characteristics of the de livry1. I.The two most common methods of IMRT delivery with computercontrolled MLCs are the segmental multileaf collimator (N\llC) and dynamic multileaf collimator (DMLC). In WllC, the beam is switched off while the leaves are in motion. In other words, the delivery is done using multiple static segments or leaf settings. This method is also frequently referred to as the I. pI and I!s .. l or "stop and I!s .. l method. In DMLC the beam is on while the leaves are in motion. The beam is switched on at the start of treatment and is switched off only at the end of treatment. The fundamental difference between the leaf sequences of these two delivery methods is that the leaf sequence defines a finite set of beam shapes for M:\!LC and ft 0 i r. 11. 4 of opposing pairs of leaves for DMLC. In practical situations, there are some constraints on the movement of the leaves. The minimum separation constraint requires that opposing pairs of leaves be separated by at least some distance (Smin) at all times during beam delivery. In MLCs this constraint is applied not only to opposing pairs of leaves, but also to opposing leaves of neighboring pairs. For example, in Figure 13, L1l and R1, L2 and R2, L3 and R3, L1l and R2, L2 and R1, L2 and R3, L3 and R2 are pairwise subject to the constraint. The case with Smin = 0 is called interdigitation constraint and is applicable to some MLC models. Wherever this constraint applies, opposite adjacent leaves are not permitted to overlap. Figure 13: Interpair minimum separation constraint In most commercially available MLCs, there is a tongueandgroove arrangement at the interface between adjacent leaves. A cross section of two adjacent leaves is depicted in Figure 14. The width of the tongueandgroove region is 1. The area under this region gets underdosed due to the mechanical arrangement, as it remains shielded if either the tongue or the groove portion of a leaf shields it. Radiation Leaf movement Figure 14: Cross section of leaves Maximum leaf spread for leaves on the same leaf bank is one more MLC limitation, which necessitates a large field (intensity profile) to be split into two or more adjacent abutting subfields. This is true for the Varian MLC (Varian Medical Systems, Palo Alto, CA), which has a field size limitation of about 15 cm. The abutting subfields are then delivered as separate treatment fields. This often results in longer delivery times, poor MU efficiency, and field matching problems. 1.3 Prior Work Optimization of the leaf sequencing algorithm has been the subject of numerous inves tigations (for example, Convery and Rosenbloom 1992, Bortfeld et al. 1994a, Dirkx et al. 1998, Ma et al. 1998, Xia and Verhey 1998, Siochi 1999, Langer et al. 2001, Luan et al. 2003, Chen et al. 2004). Treatment delivery with IMRT is not very efficient in terms of MU efficiency, which is defined as the ratio of dose delivered at a point in the patient with an IMRT field to the MU delivered for that field. Typical MU efficiencies of IMRT treatment plans are 3 to 10 times lowver than those i : open/wedge fieldbased conventional treatments plans. i i :..e, total body dose due to increased leakage radciation reaching Ithe patient in an iT i i trea~tmernt is a; r concern (ollowill et al. : *., Intensit~y Modulateid F ation ii ; py Collaboratiive Working C~roup : 1). Lowl M/U efficiency of I i i. i' delivery negatively impacts t~he room shielding design because t~he increased workload (Int~ensitly Modulated T. : 1 il: pyll Collaborative Wo~rking Gr~oup : :1, Muti c al.  11). i~ M1U ( :i .. ; (1 on both the degree of i : ;'".i modultion and the algorithm used to convert the: intensity pattern into a. leaf :: for IE 7.T delivery. It is therefore im portant to design a leaf sequenceing algorithmrr thiat is; ini :. 1 for MIU iE^ .:. ..~~ to minimirrize total body dose to the patie~nt. Foir dynamnic beamn delivery where dose rate is i: :! not modulated, an algorithm that optimizes the M4U setting at a given dose rate also optimizes the treatment time. D~ynamnic~ :i i:: : :: algorithmns with the leaves in motion during radiation delivery have been 1 ".. .1 (Convery and Ii..i I .. 1 ':; Spirou and ('.. 1 ),an lte (van Santvoort and Hleit: ; 1 *= .: D Iirkx et ali. 1998) to eimirinate the tongueand groovec underdosag e etffects. Similar leaf sequlenci ng algorithmrrs have also beenr developed for the segrmental mu~lt~ileaf i : ( :: = ) delivery methods (Borlitfeld et alj. 1994a., Bortfeli~d at al. i;~ i ,: M/a t al. i~~ ?( ia and '.' 1: 1. i' 01 I c  , Engel :  alinowsksiii: L~i et al. ::: ). ... of these studies did not consider .. 1. movemecnt constraints. Such? leaf sequencingf algorithms are i 1 : i for certain I of MLC designs. Foir other of MLC designs, notably the .. ..: : (Siemnens !.. =. 1 Systemrs, Inrc., 1lselin, NJ~) MLIIC: design (D~as et al. : ) and 7 1 ( 1 O Cncologi y Systems~ Inc., NUorcross, GA) MPLC design (Jordan aind Wliaiiitms 1' i ), other mechanical constraints need to be taken into consideration when designing the leaf settings .. both dynamic and iC( delivery. ii : minimnum leaf .1 i 0 constraint, for example, was ::li incorporated into the design of: ... (Cc;;  and We~bb 1 ). A general description and echara2cteristic s of somer MLIC7 models can b~e found in Xia. a~nd Ve~r~rhey ( ::). 1.4 D~issertation Outline? In this work, wie a syslemnatic stludyi the optimnizat~ion of sequ~encing algorithms. Til dissertation is organized as follows. In chaptor 2, weC present loaf : ing algorithms the: .' i i 0: beam delivery and provide: rigorous / of optimized leaf sequence settings in terms of MU<~ ? 'i :: under various : i movement constraints. Prac tical leaf mnovement constraints that are considered include thle mninimnu m leaf i..: Jion constraint a~nd minimum int~er ! separation constraint (leaf i: `.~:  constraint.). qi uestion of whether bi*1 i : .i leaf movemnen t : increase the M'U ii: :: i when ed with i.. .' .. : .. leaf movement 4 .1 is aliso addressed. In cha~pteir 3, we de velop : i sequencing algforithmns i : DM~LC beam d i. A~lgorit~hms are presented to sequence: leaves wczith maximum : separation constraint and Ithe leaf interdigitation c~on straint. In chlapte~r 4, we~ sttudy tongiueandgir oov:e :: for SMC7. We provide bounds on th~e maximnum extent to which tongfuealndgroove effetct can be li: .::. .1 and give neces sary; and sufficient? c~onditlions for a un~idirectional leaf sequetnce to attlain ti~he bound, Wei~ then present algorithms that generate i .i1 sequences that climrinate the tongueandgr~oove effect and optionally satisfy the interdigitation constraint. : also .. ... e our a~lgorithmlrs to a. recentliy :i: i 1 leaf sequencing algorithmr th~at also ::::: t~ongueandgroovee ulnderdosage. Thl'ie : 11. of .1 j arge T .!.:. ti r mnodduatted fields into two or three sub~fields is discussed in cha~pte~r 5. All our algorithms generation unidircectionali leaf mnovemennt schedules and are i 1 to bec < ': 1:.1 in M/Us for :: .: l i i: :. .1 schedules. CHAPTER 2 SEQUENCING OF SEGMENTED MULTILEAF COLLIMATORS In this chapter, we present a systematic study of the optimization of leaf sequenc ing algorithms for the M:\! LC beam delivery and provide rigorous proofs of optimized leaf sequence settings in terms of MU efficiency under various leaf movement constraints. Prac tical leaf movement constraints that are considered include the minimum leaf separation constraint and minimum interleaf separation constraint (leaf interdigitation constraint). The question of whether bidirectional leaf movement will increase the MU efficiency when compared with unidirectional leaf movement only is also addressed. We first introduce the notation that will be used in the remainder of this work. 2.1 Methods 2.1.1 Discrete Profile The geometry and coordinate , r. ill used in this study are shown in Figure 21. We consider delivery of profiles that are piecewise continuous. Let I(z) be the desired intensity profile. We first discretize the profile so that we obtain the values at sample points zo,wi,z2, ...,zm. I(z) is assigned the value I(zi) for zi < z < zi 1, for each i. Now, I(zi) is our desired inltlonityi profile. Figure 22 shows a piecewise continuous function and the corresponding discretized profile. The discretized profile can be efficiently delivered with the M:\!LC method. However, a WllC sequence can be transformed to a dynamic leaf sequence by allowing both leaves to start at the same point and close together at the same point, so that they sweep across the same spatial interval. We develop our theory for the W11LlC delivery. 2.1.2 Movement of Leaves In our I!! 11, R~ we assume that the leaves are initially at the left most position to and that the leaves move unidirectionally from left to right. Figure 23 illustrates the leaf ft i l, b r~!y during <:\!LC delivery. Let Il(zi) and Ir(zi) respectively denote the amount of monitor units (MUs) delivered when the left and right leaves leave position zi. Consider I . Radiation Beams Radiation Source Right Leaf Left Leaf 'ff x, Figure 21: Geometry and coordinate 1 r nl xo x; x, xo x, Figure 22: Discretization of profile the motion of the left leaf. The left leaf begins at to and remains here until Iz(zo) MUs have been delivered. At this time the left leaf is moved to zl, where it remains until I;(zl) MUs have been delivered. The left leaf then moves tO 23 where it remains until Ig(23) MUS have been delivered. At this time, the left leaf is moved tO 26, where it remains until Ig(26) MUs have been delivered. The final movement of the left leaf is to my, where it remains until Iz(27) = Imaz MUs have been delivered. At this time the machine is turned off. The total therapy time, TT(It, Ir), is the time needed to deliver Imax MUs. The right leaf starts at z2; mOVeS tO 24 when Ir (z2) MUs have been delivered; moves to as when Ir (24) MUS have been delivered and so on. Note that the machine is off when a leaf is in motion. We make the following observations: Ir3 II I~~ I l X I I ~ I ll$X X X Figure 23: Leaf trajectory during 9llC delivery 1. All MUs that are delivered along a radiation beam along me before the left leaf passes zi fall on it. The greater the a value, the later the leaf passes that position. Therefore Iz(zi) is a nondecreasing function. 2. All MUs that are delivered along a radiation beam along me before the right leaf passes zi are blocked by the leaf. The greater the a value, the later the leaf passes that position. Therefore Ir (zi) is also a nondecreasing function. From these observations we notice that the net amount of MUs delivered at a point is given by I;(zi) I,(zi), which must be the same as the desired profile I(zi). 2.1.3 Optimal Unidirectional Algorithm for one Pair of Leaves Unidirectional movement. When the movement of leaves is restricted to only one di rection, both the left and right leaves move along positive a direction, from left to right (Figure 21). Once the desired intensity profile, I(zi) is known, our problem becomes that of determining the individual "iil. 10; pr ol. / ~~I to be delivered by the left and right leaves, I and I, such that I(i) = II(i) I,(zi), O < i < m (2.1) We refer to (I, 4) as the treatment plan (or simply plan) for I. Once we obtain the plan, we will be able to determine the movement of both left and right leaves during the therapy. For each i, the left leaf can be allowed to pass zi when the source has delivered Ih(as) MUs. Also, we can allow the right leaf to pass zi when the source has delivered I,(as) MUs. In this manner we obtain unidirectional leaf movement /pr oll.~ for a plan. Algorithm. From Equation 2.1, we see that one way to determine It and I, from the given target profile I is to begin with Ih(zo) = I(zo) and I,(zo) = 0; examine the remaining zis from left to right; increase It whenever I increases; and increase I, whenever I decreases. Once It and I, are determined the leaf movement profiles are obtained as explained in the previous section. The resulting algorithm is shown in Figure 24. Figure 25 shows a profile and the corresponding plan obtained using the algorithm. Ma et al. (1998) shows that Algorithm SINGLEPAIR obtains plans that are optimal in therapy time. Their proof relies on the results of Spirou and Chui (1994), Stein et al. (1994) and Boyer and Strait (1997). We provide a much simpler proof below. Theorem 1 Algorithm SINGLEPAIR obtains plans that are optimal in therapy time. Proof: Let I(zi) be the desired profile. Let incl, inc2, ..., inck be the indices of the points at which I(as) increases. So minel, zinc2, *inck are the points at which I(z) increases (i.e., I(zinci) > I(zinci1)). Let ai = I(2inci) I(2inci1) Suppose that (IL, IR) is a plan for I(zi) (not necessarily that generated by Algorithm Algorithm SINGLEPAIR Iz(zo) = I(zo) I,(zo) = 0 For j = 1 to m do If (Ij)> I r(j_1) (j (j Else Iz(j) = Iz(j_1) End for Figure 24: Obtaining a unidirectional plan SINGLEPAIR). From the unidirectional constraint, it follows that IL(2i) and IR(2 ) are nondecreasing functions of z. Since I(as) = IL(as) IR(as) for all i we get ai = (IL(2inci) In(2inci)) (IL(2incil) In(2incil)) = (IL(2inci) IL(2incil)) (In(2inci) In(2incil)) I L inci) L 2inci1)* Summing up Ai, we get Li [I(2inci) I(2inci)] < C i[IL(2inci) IL(2inci)] = TT(IL, In) Since the therapy time for the plan (It, I,) generated by Algorithm SINGLEPAIR is Cl, [I(zinci) I(zinci1)], it follows that TT(I I, ) is minimum. Corollary 1 Let I(zi), O < i < m be a desired prll.~II Let Iz(zi), and I,(zi), O < i < m be the left and right leaf tll.~II generated by Algorithm SINGLEPAIR. Iz(zi) and I,(as), O < i < m define optimal therapy time unidirectional left and right leaf let..I;I. for I(zi), S< i Proof: Follows from Theorem 1 m In the remainder of this paper, (Ig, I,) is the optimal treatment plan for the desired profile I. Properties of the optimal treatment plan. The following observations are made about the optimal treatment plan (It, I,) generated using Algorithm SINGLEPAIR. Lemma 1 At each zi at most one of the /; r oll. It and I, changes (increases). I1 rl x, x Figure 25: A profile and its plan Lemma 2 Let (IL, IR) be any treatment plan for I. (a) a(zi) = IL(2i) 1h(2i) = IR(2i) Ir(2i) > 0,0 < i < m. (b) a(as) is a nondecreasing function. Proof: (a) Since I(as) = IL(as) IR(a) = I (2i) I(zi),IL(2i) I (2) = IR(as) r(as). Further, from Corollary 1, it follows that IL(2i) > 4(zi),0 O i < m. Therefore, A(zi) > 0, O (b) We prove this by contradiction. Suppose that A(z,) > A(z, 1) for some n, O I n < m. Consider the following three all encompassing cases. Case 1: II(z,) = II(z,+l) Now, IL(2n) = 1h(22) + A(an2) > 1h(z, 1) + A(z, +1) = IL(z, +1). This is not possible because IL is a nondecreasing function. Case 2: I,(z,) = I,(z,+l) Now, IR(2n) = Ir(2n) + a(zn) > I,(z, 1) + a(2, 1) = IR(z, 1). This contradicts the fact that IR is a nondecreasing function. Case 3: II(z,) II(z,+l) and I,(z,) f &(zl) From Lemma 1 it follows that this case cannot arise. Therefore, A(zi) is a nondecreasing function. m Theorem 2 If the optimal plan (It, ) violates the minimum separation constraint, then there is no plan for I that does not violate the minimum separation constraint. Proof: Suppose that (I, I,) violates the minimum separation constraint. Assume that the first violation occurs when II MUs have been delivered. From the unidirectional movement constraint, it follows that the left leaf has just been positioned at my (for some j, 0 < j < m) at this time and that the right leaf is at zk, such that zk, zj is less than the permissible minimum separation. Figure 26 illustrates the situation. We prove the theorem by contradiction. Let (IL, IR) be a plan that does not violate the minimum separation constraint. When j = 0, (II, ) has a violation at the initial positioning to of the left leaf. Since the leaves move in only one direction, the violation is when II = 0. When II = 0, the left leaf in (IL, IR) is also at to (because the left leaf must begin at to in all plans; otherwise I(zo) = 0). For (IL, IR) not to have a violation at II = 0, the right leaf must begin to the right of zk, I, at some point p > zk, (nOte Separation between the jawsI xo xj xk Figure 26: Minimum separation constraint violation that p may not be one of the zis). The MUs delivered at zk, by the plan (IL, IR) are IL(2k) IR ~) = L k~) 1 k2) (Corollaryl). Also, I(z() k ~z) r kz) k ~z) (Ir (zk) > 0). So (IL, IR) delivers more than I(zk) MUS at zk, and so is not a plan for I. This contradicts the assumption on (IL, IR). Hence, j 0 . Suppose that j > 0. Now, Iz(zj) > Iz(zj_l). Also, IL(zj) = Iz(zj) + A(zj) and IL(2j_1) = Iz(2j_1) + A(2j_l). Since A(zj) > A(zj_1) (Lemma 2(b)), IL(zj) > IL(zj_i). Therefore, the left leaf is positioned at my at some time during the on cycle of the plan (IL, IR). Let the amount of MUs delivered when the left leaf arrives at zj in IL be I2. Let the right leaf be at z = p at this time. Note that p may not be one of the zis. If p > zk, then In(2k) 2 I. Also, from Lemma 2 we have IL(2k) = 1 k2) k (2) 1~~ k a(j1) I(zk) 2I 1I 1 k~z) 2I I, k) k ~z) 2I. Therefore, IL(2k) IR k) k (2). This contradicts IL(2k) IR() k k() (since (IL, IR) is a plan for I). Therefore, j cannot be > 0 either. So, there is no plan (IL, IR) that does not violate the minimum separation constraint. The separation between the leaves is determined by the difference in a values of the leaves when the source has delivered a certain amount of MUs. The minimum separation of the leaves is the minimum separation between the two profiles. We call this minimum separation Sudmin. When the optimal plan obtained using Algorithm SINGLEPAIR is delivered, the minimum separation is Sudmin(opt)* Corollary 2 Let Sudmin(opt) be the minimum leaf separation in the plan (II, Ir). Let Sudmin be the minimum leaf separation in any (not ii... ~,tale optimal) given unidi rectional plan. Sudmin < Sudmin(opt) * 2.1.4 Bidirectional Movement In this section we study beam delivery when bidirectional movement of leaves is per mitted. We explore whether relaxing the unidirectional movement constraint helps improve the efficiency of treatment plan. Properties of bidirectional movement. For a given leaf (left or right) movement profile we classify any 2coordinate as follows. Draw a vertical line at z. If the line cuts the leaf profile exactly once we will call z a unidirectional point of that leaf profile. If the line cuts the profile more than once, z is a bidirectional point of that profile. A leaf movement profile that has at least one bidirectional point is a bidirectional /pr oll.I All profiles that are not bidirectional are unidirectional prll.~II Any profile can be partitioned into segments such that each segment is a unidirectional profile. When the number of such partitions is minimal, each partition is called a stage of the original profile. Thus unidirectional profiles consist of exactly one stage while bidirectional profiles 1.h I, have more than one stage. In Figure 27, the leaf movement profile, BI, shows the position of the left leaf as a function of the amount of MUs delivered by the source. The leaf starts from the left edge and moves in both directions during the therapy. C'I, ..ly, BI is bidirectional. The movement profile of this leaf consists of stages S1, S2 and S3. In stages S1 and S3 the leaf moves from left to right while in stage S2 the leaf moves from right to left. zj is a bidirectional point of BI. The amount of MUs delivered at my is Ll+L2. In stage S1, when Li amount of MUs have been delivered, the leaf passes my. Now, no MU is delivered at my till the leaf passes over my in S2. Further, L2~ MUs are delivered to my in stages S2 and S3 Thus we have I(zjm) = L1 + L2a. Here, Lil = II, L2~ = 13 1 2* k is a unidirectional point of BI. The MUs delivered at zk, are L3 = 14. Note that the inltloneityi profile It is different from the leaf movement profile BI, unlike in the unidirectional case. I Be UL S L, S3 x, xk x Figure 27: Bidirectional movement Lemma 3 Let (II, Ir) be a plan delivered by the bidirectional leaf movement l; r oll. pair (BI, Br) (i.e., BI and Br are, ,. Ifr..1.: I:I. the left and right leaf movement /roll ~II ) (a) I is nondecreasing. (b) Ir is nondecreasing. Proof: (a)Whenever a point zi, O < i < m, is blocked by the the left leaf, the points zo,zl,...,as_l are also blocked. It follows that Ih(zi) > Ih(zj), O < j < i < m. (b)The proof is similar to (a) From Lemma 3 we note that a bidirectional leaf movement profile B delivers a non decreasing intensity profile. This nondecreasing intensity profile can also be delivered using a unidirectional leaf movement profile (Section 2.1.3). We will call this profile the unidirectional leaf movement fol/.II~ that corresponds to the bidirectional l; roll. B and we will denote it by U to emphasize that it is unidirectional. Thus every bidirectional leaf movement profile has a corresponding unidirectional leaf profile that delivers the same amount of MUs at each zi as does the bidirectional profile. Theorem 3 The unidirectional treatment plan constructed by Algorithm SINGLEPAIR is optimal in 'il.. ,~,Ite time even when bidirectional leaf movement is permitted. Proof: Let BL and BR be bidirectional leaf movement profiles that deliver a desired intensity profile I. Let IL and IR, respectively, be the intensity profiles for BL and BR. From Lemma 3, we know that IL and In are nondecreasing. Also, IL(as) IR(as) = I(zi), 1 < i < m. From the proof of Theorem 1, it follows that the therapy time for the unidirectional plan (Ig, Ir) generated by Algorithm SINGLEPAIR is no more than that of (IL, IR). m Incorporating minimum separation constraint. Let Uz and Ur be unidirectional leaf movement profiles that deliver the desired profile I(zi). Let BI and Br be a set of bidirectional left and right leaf profiles such that Uz and Ur correspond to BI and Br re spectively, i.e., (BI, Br) delivers the same plan as (Ug, Ur). We call the minimum separation of leaves in this bidirectional plan (BI, Br) Sbdmin* Theorem 4 Sbdmin < Sudmin for a bidirectional leaf movement l; roll. pair and its corresponding unidirectional l;*<.Gl. Proof: Suppose that the minimum separation Sudmin occurs when Ims MUs are deliv ered. At this time, the left leaf arrives at my andl the~ bright leafI is poslitione atl Zk. ULet Bj and Ul' respectively, be the profiles obtained when BI and Uz are shifted right by Sudmin. Since Ul' is Uz shifted right by Sudmin and since the distance between Uz and Ur is Sudmin when Ims MUs have been delivered, Ul' and Ur touch when Ims units have been delivered. Therefore, the total MVVDUs delivered (UY' \Ur) at zk is zero. So the total MUs delivered by (BJ, Br) at zk, is also zero (recall that Ul' and Ur, respectively, are corresponding profiles for Bb and Br). This isn't possible if Br is 11.  ; t the ih o ,(oreapli he situa tionof igur 2, te M~ deliver b (B,~ B) atI Z alre (LI + L2)r (mL', + n L'2 + L3) 0) Therefore Bf and Br must touch (or cross) at least once. So Sbdmin < Sudmin Theorem 5 If the optimal unidirectional plan (II, Ir) violates the minimum separation con straint, then there is no bidirectional movement plan that does not violate the minimum separation constraint. B, U, B UI I , 1 %L~,L L, Figure 28: Bidirectional movement under minimum separation constraint Proof: Let BI and Br be bidirectional leaf movements that deliver the required profile. Let the minimum separation between the leaves be Sbdmin. Let the corresponding unidi rectional leaf movements be UI and Ur and let Sudmin be the minimum separation between Uz and Ur. Also, let Smin be the minimum allowable separation between the leaves. From Corollary 2 and Theorem 4, we get Sbdmin < Sudmin < Sudmin(opt) < Smin Incorporating maximum separation constraint. Let UI and Ur be unidirectional leaf movement profiles that deliver the desired profile I. Let Sudmat be the maximum leaf separation using the profiles Ug and Ur and let Sudmaz(opt) be the maximum leaf separation for the plan (It,Ir). Let BI and Br be a set of bidirectional left and right leaf profiles such that UI and Ur correspond to BI and Br, respectively. Let Sbdmaz be the maximum separation between the leaves when these bidirectional movement profiles are used. Theorem 6 Sbdmaz > Sudmat for every bidirectional leaf movement I;I rol. and its cor responding unidirectional movement l; r oll. Proof: Suppose that the maximum separation Sudmat occurs when Ims MUs are deliv ered. At this time, the left leaf is positioned at my and the right leaf arrives at zk. Let BI' and Ul' respectively, be the profiles obtained when BI and Uz are shifted right by Sudmax. Since Uf is UI shifted right by Sudmat and since the distance between UI and Ur is Sudmat when Ims MUs have been delivered, Uf and Ur touch when Ims units have been delivered. Therefore, the total MUs delivered by (Ur, U') at Zk is zero. So the total M/lTs delivered~ by (Br, Bf) at zk, is also zero (recall that Uf and Ur, respectively, are corresponding profiles for Bf and Br). This isn't possible if Br is 11.h I, a to the left of Bf (for example, in the situation of Figure 29, the MUs delivered by (Br, Bf) at sk, are (L': + L'2 + L'3 \ (L L2a) > 0). Therefore Bf and Br must touch (or cross) at least once. So Sbdmaz > Sudmaz U, B, SB, UI U, B, LL, Figure 29: Bidirectional movement under maximum separation constraint 2.1.5 Algorithm Under Maximum Separation Constraint Condition In this section we present an algorithm that generates an optimal treatment plan under the maximum separation constraint. Recall that Algorithm SINGLEPAIR generates the optimal plan without considering this constraint. We modify Algorithm SINGLEPAIR so that all instances of violation of maximum separation (that may possibly exist) are eliminated. We know that bidirectional leaf profiles do not help eliminate the constraint. So we consider only unidirectional profiles. Algorithm. The algorithm is described in Figure 210. Theorem 7 Algorithm MiAXSEPARATION obtains plans that are optimal in therapy time, under the maximum separation constraint. Algorithm MAXSEPARATION 1. Apply Algorithm SINGLEPAIR to obtain the optimal plan (It, I). 2. Find the least value of inltlencity,) I,) such that the leaf separation in (Ig, I,) when II MUs are delivered is > Smax, where Smax is the maximum allowed separation between the leaves. If there is no such li, (II, 1,) is the optimal plan; end. 3. Let my and Zk, respectively, be the position of the left and right leaves at this time (see Figure 211). Relocate the right leaf at a~ such that za zj Smax, when II MUs are delivered. Let AI Il(zj) II I2 1I. Move the profile of I,, which follows z(, up by AI along I direction. To maintain I(z) I(zs) I,(z) for every z, move the profile of It, which follows z(, up by AI along I direction. Goto Step 2. Figure 210: Obtaining a plan under maximum separation constraint A after __ __ __ __ I Sm modification~L I; Before Figure 211: Maximum separation constraint violation Proof: We use induction to prove the t~heoremn. i ~statemnent wez prove, S(ub), is the ~ After Step 3 of the algor~ithmr is i i n times~, thelr resulting _! lin,, I,,): (a) It has no maximum separation violation when I < IgL(ul) M~s are delivered, where I2(r?) is tIhe value of I2 during t~he ut~h iteration of Algorithm 11 : = 'P: : = N`. (b) For plans that satisfy (a), (/7,, In) is optimal in therapy time. 1. Consider ther base ca~se, a2 = 1. Let (II, I) be the 1 :: generated by Algorithmn i GLEPAIR. i.i : 1 >,3 is appied onice, the resulting i" (11:, l) meets the : :1: ...=.1 thiat there is no rnaximnurn separation violation wheicn I < I2(1) MIl~s are delivered by th~e radiation sources. : therapy timet increases by AI, i.e., TT(IIn, Ir) = TT(In: Ir) i aI. Assume,, that~l threl:li is? anotherlll p~lan, (rl:y 17), which satisfies condition (a) of S(1) and TT(II, 1:) < TT7'jli, Ir). We show this assumption leads to a. c< ::i: 1: ndo there is no such plan (It, 1~). L~et my, and be a~s in Algorithmn I M/ A~RATIOCN. We'c consider th~reec cases for thie: 7 .r'.: 1.' >"c bewe r"y)adIiz) (a) If (xy) I~ln~x) 7(1) Since there is no mnaximnumn :i :tion violation when I < [,(1) Mljs are deliv eired, if,( ') > Itljxyg) 111zj) 1,('.Sne (') '(x ( )  rIii(') ri, ( '), we have It (x ) > Ill( '}. Wej now constructl a i : (Ii: n." i) as follows: I"(z (x) =r '1 > (x) AI x: < 2 ' Ify~ ) Al2 Il~x l It(x ) > It( .1 > a a_1) .I .i noocesg So (.s /')' isC) a~ pla for e~li~i IIix g). This contradicts our knowledge that (II, Ir) is the optimal unconstrained plan. This leads to a contradiction as in the previous case. In this case, I',(zj) < Ill(zj) = Iz(zj). This violates Corollary 1. So this case cannot arise. Therefore S(1) is true. 2. Induction step Assume S(u) is true. If there are no more maximum separation violations in the resulting plan, (It,, Im,), then it is the optimal plan. If there are more violations, we find the next violation. Apply Step 3 of the algorithm to get a new plan. Assume that there is another plan, which costs less time than the plan generated by Algo rithm MAXSEPARATION. We consider three cases as in the base case and show by contradiction that there is no such plan. Therefore S(n + 1) is true whenever S(u) is true. Since the number of iterations of Steps 2 and 3 of the algorithm is finite (at most one iteration can occur when the left leaf is at me, O < i < m), all maximum separation violations will eventually be eliminated. Note that the minimum leaf separation of the plan constructed by Algorithm MAXSEP ARATION is min{Sudmin(opt), Sm,, }. From Theorem 7, it follows that Algorithm MAXSEP ARATION constructs an optimal plan that satisfies both the minimum and maximum sepa ration constraints provided that Sudmin(opt) > Smin. Note that when Sudmin(opt) < Smin, there is no plan that satisfies the minimum separation constraint. 2.1.6 Algorithm Under InterPair Minimum Separation Constraint Introduction. We use a single pair of leaves to deliver inltlensityi profiles defined along the axis of the pair of leaves. However, in a real application, we need to deliver intensity profiles defined over a 2D region. Each pair of leaves is controlled independently. If there are no constraints on the leaf movements, we divide the desired profile into a set of parallel profiles defined along the axes of the leaf pairs. Each leaf pair i then delivers the plan for the corresponding inltloneityi profile lilt). The set of plans of all leaf pairs forms the solution set. We refer to this set as the treatment schedule (or simply schedule). In this section, we present leaf sequuencing algorithms for <\llC with and without constraints. The constraints considered are (i) minimum separation constraint and (ii) tongueandgroove constraint and (optionally) interdigitation constraint. We use the term intrapair minimum separation constraint to refer to the constraint imposed on an opposing pair of leaves and interpair minimum separation constraint to refer to the constraint imposed on opposing leaves of neighboring pairs. Recall that, in Section 2.1.3, we proved that for a single pair of leaves, if the optimal plan does not satisfy the minimum separation constraint, then no plan satisfies the constraint. In this section we present an algorithm to generate the optimal schedule for the desired profile defined over a 2D region. We then modify the algorithm to generate schedules that satisfy the interpair minimum separation constraint. Optimal schedule without the minimum separation constraint. Assume we have n pairs of leaves. For each pair, we have m sample points. The input is represented as a matrix with n rows and m columns, where the ith row represents the desired intensity profile to be delivered by the ith pair of leaves. We apply Algorithm SINGLEPAIR to determine the optimal plan for each of the n leaf pairs. This method of generating schedules is described in Algorithm MULTIPAIR (Figure 212). Algorithm MULTIPAIR For(i = 1;i n; i ++) Apply Algorithm SINGLEPAIR to the ith pair of leaves to obtain plan (lil, Iir) that delivers the inltloneityi profile lilt). End For Figure 212: Obtaining a schedule Lemma 4 Algorithm M~ULTIPAIR generates schedules that are optimal in therapy time. Proof: Treatment is completed when all leaf pairs (which are independent) deliver their respective plans. The therapy time of the schedule generated by Algorithm MULTIPAIR is maz {TT(Iu1, IIr), TT(I2, 2r,), TT ulI, Inr)}. From Theorem 1, it follows that this therapy time is optimal. Optimal algorithm with interpair minimum separation constraint. The schedule generated by Algorithm MULTIPAIR may violate both the intra and interpair minimum separation constraints. If the schedule has no violations of these constraints, it is the desired optimal schedule. If there is a violation of the intrapair constraint, then it follows from Theorem 2 that there is no schedule that is free of constraint violation. So, assume that only the interpair constraint is violated. We eliminate all violations of the interpair constraint starting from the left end, i.e., from to. To eliminate the violations, we modify those plans of the schedule that cause the violations. We scan the schedule from to along the positive a direction looking for the least z, at which is positioned a right leaf (say Ru) that violates the interpair separation constraint. After rectifying the violation at z, with respect to Ru we look for other violations. Since the process of eliminating a violation at me, may at times, lead to new violations at my, my < z,, we need to retract a certain distance (we will show that this distance is Smin) to the left, every time a modification is made to the schedule. We now restart the scanning and modification process from the new position. The process continues until no interpair violations exist. Algorithm MINSEPARATION (Figure 213) outlines the procedure. Algorithm MINSEPARATION //assume no intrapair violations exist 1. z = to 2. While (there is an interpair violation) do 3. Find the least me, 2, > 2, such that a right leaf is positioned at z, and this right leaf has an interpair separation violation with one or both of its neighboring left leaves. Let a be the least integer such that the right leaf Ru is positioned at z, and Ru has an interpair separation violation. Let LL denote the left leaf (or one of the left leaves) with which Ru has an interpair violation. Note that L E {u 1, a + 1}. 4. Modify the schedule to eliminate the violation between Ru and Lt. 5. If there is now an intrapair separation violation between Rt and LL no feasible schedule exists, terminate. 6. z = 2v Smin 7. End While Figure 213: Obtaining a schedule under the constraint Let M~ = ((III, 11r), (121 2r,), ul ar,)) be the schedule generated by Algorithm MULTIPAIR for the desired intensity profile. Let N(p) = ((III,, 11rp), (121p 2rp,), ulp arp)) be the schedule obtained after Step 4 of Algforithm MINSEPARATION is applied p times to the input schedule M~. Note that M~ = N(0). Figure 214: Eliminatingf a violation To illustrate the modification process we use an example (see Figure 214). To make things easier, we only show two neighboring pairs of leaves. Suppose that the (p + 1)th violation occurs when the right leaf of pair n is positioned at z, and the left leaf of pair t, L E {u 1, a + 1}, arrives at me, 2, 2, < Smin. Let at = 2, Smin. To remove this interpair separation violation, we modify (Itlp, Itr). The other profiles of N(p) are not modified. The new Illp *O*>. tl(p+1)) is aS defined below. 0o I a n~ m 2 where AI = lurp v,) ftl( n~ 2 1 Itr(p+1)2 .= ,,! IL)(2 t(2), where It(z) is the target profile to be delivered by the leaf pair t. Since Itr(p 1) differs from Itrp for z > z', = 2, Smin there is a p. 11 .11 117r, that N(p+1) has interpair separation violations for right leaf positions z > z', = 2,Smin. Since none of the other right leaf profiles are changed from those of N(p) and since the change in Itl only delays the rightward movement of the left leaf of pair t, no interpair violations are possible in N(p + 1) for z < z', = z, Smin. Itlp 2 I~;!I )()= ma{Itlp(> fl(2 + r One may also verify that since Islo and Isro are nondecreasing functions of z, so also are Itlp and Itrp, P > 0. Lemma 5 Let F = ((I' I' fi), (120 r' ',.,11 nD )) be any feasible schedule for the de sired in.~ 4/.I i. e., a schedule that does not violate the intra or interpair minimum separation constraints. Let S(p), be the following assertions. (a) rlz (s > lilp (2), O < i < n, so < z < sm (b) IL (z) > lirp (2), O < i < n, so < z < sm S (p) is true for p > 0. Proof: The proof is by induction on p. 1. Consider the base case, p = 0. From Corollary 1 and the fact that the plans (lilo, firo), O < i < n, are generated using Algorithm SINGLEPAIR, it follows that S(0) is true. 2. Assume S(p) is true. Suppose Algorithm MINSEPARATION finds a next violation and modifies the schedule N(p) to N(p + 1). Suppose that the next violation occurs when the right leaf of pair n is positioned at z, and the left leaf of pair t arrives at me, 2, 2, < Smin (see Figure 214). Let z', = z, Smin. We modify pair t's plan for z', < z < m,, to eliminate the violation. All other plans in the schedule remain unaltered. Therefore, to establish S(p + 1) it suffices to prove that I'l~) > tl~p1) ) n< 2 Im(2.2) 7 r (s > Itzp1 ( n< 2 < (2.3) Weneed,, prove only one of hesetw re,,,lationships,, sinc I',() I (s) = I,~,! I )(2)  1tr(p+1) 02) in turn, implies that S(p + 1) is true whenever S(p) is true and hence completes the proof. (a) No modification (relative to M~ = N(0)) has been made to pair t's plan for z > z' prior to this. In this case, Itl,() = Itlo() = Itl(),a > '. The situation is illustrated in Figure 214. Since there is no niiinimun separation violation in F, the left leaf of pair t passes 2'( only after the right leaf of pair a passes ',., i.e., lil24) ir2,.)(2.4) Since S(p) is true, fir 2' ) > Zor; ( ) =L (a(,)(2.5) Front Equations 2.4 and 2.5, Itl\ ) I ;>j+1) (a (2.6) tt u' =l \44; 2 l1\(a ), O <2 12' z(27 Sintila~rly, In,,. I L) (2) = L ,, I L)\" (2;)+la )(p2) Itl;>+l) (a'(,, O o 2 since Ist;,>(2 = stl(>) 2 > , f1(p+1) 2') = It1(2) a, 2' IC 2' 2,n (2.9) Front Equations 2.8 and 2.9, we get I, I1) 7) L~ /,, I )() + (Ist (2) + ar) (It(2'() + aI), 2'( < 2' < ak = Icor I1) (2';) + rtl(2') rt,(2 ;), 2' I 2' I 2' (2.10) Subtracting Equation 2.10 front Equation 2.7, ,ila') ltit;>+1) (' 77( ,It(2)4) + 1 fr (2)) (11(24;) lIt/@{;), 2 ;< 2 2'm (.1 From Equations 2.6 and 2.11, liz( (s  Ill~u1, (p+1 1 1 From Lemma 2b, II' (s) Itz (2) > I1(m' )  It (al),2 I' < I < m 2.3 From Equations 2.12 and 2.13, we get Itl(s) > L;.,, I, g(), z', < m (2.14) (b) Some prior modification has been made to pair t's plan for z > z's. There exists a modification at z, such that Itlp2 fl wt(2 < ar <, I m, and there is no a < 2, that satisfies this condition. Note that Itlp ua) < amount of MUs delivered when profile Itlp Z) arriVeS at Zu (SinCe Itlp Z) is a DOndecreasing func tion of 2) < lurp v,) (since there is a minimum separation violation when profile Isrp(2) is at 2,). Therefore, Itlp(S a ftl n urp+ v,? 2)I1() = fl n2 ~ * So, 2( > z' . In this case (see Figure 215), I.,,,g1()= + ar )+AI z' < 2~, Note that, in the example of Figure 215, a prior modification was made to pair t's plan for z > sq. However, Itlp fl qt(2 w*r ,I , We get I,'1(2) > Itl(p+1)() 2 I zj < 2, fOr r88Sons similar to those in the previous case. Also, lizs) > I. (s) = tlp() w1( < 2 < 2m, since S(p) is (c) Some prior modification has been made to pair t's plan for z > z's. However, Itlp fl nt(2 m*r ~I , In this case, Itl(p+1)(2 flt2 n tr p+1)    hhc ~I~ zx 4 x,~ x , Fiur 215: Elmntn ilto (a)~~l I'z (s) Itlp 0 if Ilp nd Irp ave miimfum separ lmiation f violationtenoramntpn(IIs)ht LExaml 1 Weilsraea ntac hr an interpair minimum separation violation dtce Sp INEP obAtiONe using lgithmr sn MUTIAsIR. Tch.dl fo are dshonine F~~igue7 ac f h plans: ((Itz os),Isr the)) a d (Ist+1) ss)Iet+1) F ))) s feasible, ie., there isn intrapair mn miimum separation (Sin =io 7). Howee,p whe OrMINSEPARAIO is applied (fr imliit consderlef paWeirls adte +1 inisolatin) it etect an interpair minimum separation vito oaionm beprtwen I(t+1)z a Is owvr, when I~t1z rives at A = 6Nd is aposiioed ato si = 11. To eliminate this violation, I~t+1)I is positioned at z = 4 (since 11 4 = 7 = Smin) and its Ipo llI. is raised from z = 4. C .;,, <;1:, ,1:1t~+1)r is also raised from z = 4 resulting in the plan (I~t+1)11(2) ),I~t+1)rl(2)). This In..J.I.:; ,tl... t; results in an intrapair violation for pair t +1i, when I~t+1)I1 is at z = 1 and I~t+1)rl is at z = 4. From Lemma 6, there is no feasible schedule. La~ (x) 1 4 6 17 21 29 35 49 X Figure 216: Inltlensityi profiles of adjacent leaf pairs For N(p), p > 0 and every leaf pair j, 1 < j < n, define Ijlp 1) = Ijrp( l 1 Notice that App (i) giVeS the time (in monitor units) for which the left leaf of pair j stops at position zi. Let Ajl,(zi) and Ajrp i) be zero for all me when j = 0 as well as when j = + 1. Lemma 7 For every j, 1 < j < n and every i, 1 < i < m, nApp i> I mtz ajl0 i) a(j1)rp zi + Smin), a(j+1)rp zi + Smin)} (2.15) ItlJ I       1..   I__ (t+1)rl 0 1 4 6 9 11 13 17 21 29 35 47 49 x Figure 217: Profiles violating interpair constraint Proof: The proof is by induction on p. For the induction base, p = 0. Putting p = 0 into the right side of Equation 2.15, we get maz {Ajo (zi), A(jl),o(2i + Smin), a(j+l)ro(2i + Smin)} > aj1o(2i) (2.16) For the induction hypothesis, let q > 0 be any integer and assume that Equation 2.15 holds when p = q. In the induction step, we prove that the equation holds when p = q + 1. Let t, u, and 2, be as in iteration p 1 of the while loop of algorithm MINSEPARATION. Following this iteration, only atlp and atrp are different from Ac,;, _1) and atrp1), respec tively. Furthermore, only Atlp(2w) and Atrp w,), where 2, = 2, Smin may be larger than the corresponding values following iteration p 1. At all but at most one other a value (where a may have decreased), Atlp and atrp are the same as the corresponding values following iteration p 1. Since 2, is the right leaf position for the leftmost violation, the left leaf of pair t arrives at 2, = 2, Smin after the right leaf of pair n arrives at z, = z, + Smin. Following the modification made to L ;, 1), the left leaf of pair t leaves z, at the same time as the right leaf of pair a leaves 2, + Smin. Therefore, Atlp 2w) Iur(p1) w + Smin) = aurp 2w + Smin) The induction step now follows from the induction hypothesis and the observation that n E {t 1, t + 1}. m Lemma 8 For every j, 1 < j < n and every i, 1 < i < m, Ajrpi jp i /i) / 1)(2.17) where ly (a ) = 0. Proof: We examine N(p). The monitor units delivered by leaf pair j at zi are Ijl,(zi)  Ijrp zi) and the units delivered at as_l are Ijlp i1) Ijrp i1). Therefore, ly (i) Ijl i) jr i)(2.18) ly~a _1) Ijl i1 jrp 1)(2.19) Subtracting Equation 2.19 from Equation 2.18, we get Isms lyze1) = (jp i l ) r ) p i1 =ajlp i> ajrp 2i) (2.20) The lemma follows from this equality. Notice that once a right leaf a moves past m,, no separation violation with respect to this leaf is possible. Therefore, z, (see algorithm MINSEPARATION) < zm. Hence, njlp ~i) Ijl0 i), and Ajrp i) Ijr0 zi), sm Smin < zi < 2m, 1 < j < n. Starting with these upper bounds, which are independent of p, on Ajrp i), 2mSmin < 2i < 2m and using Equations 2.15 and 2.17, we can compute an upper bound on the remaining Ajl,(zi)s and Ajrp zi)S (frOm right to left). The remaining upper bounds are also independent of p. Let the computed upper bound on App (i) be Ujl(as). It follows that the therapy time for (rj,,, Ijrp) is at most Tmaz(j) = Co at most Tmax = meazlign{Tmaz(j)} Theorem 8 The following are true of Algorithm M~INSEPARATION: (a) The algorithm terminates. (b) When the algorithm terminates in Step 5, there is no feasible schedule. (c) Otherwise, the schedule generated is feasible and is optimal in 'I: r,ten~ time for uni directional schedules. Proof: (a) As noted above, Lemmas 7 and 8 provide an upper bound, Tmax on the therapy time of any schedule produced by algorithm MINSEPARATION. It is easy to verify that I.,, I L (2) > Iilp(2),0 O i < n, so < z < sm lir(p+1)() >irp(2),0 O i < n, so < z < s and that I~., 1 1, (m',) > Idp u)6 Itr(p+1) (2',) > Itrp u~) Notice that even though a a value (proof of Lemma 7) may decrease at an me, the lilp and lirp ValueS never decrease at any zi as we go from one iteration of the while loop of MINSEPARATION to the next. Since la increases by atleast one unit at atleast one zi on each iteration, it follows that the while loop can be iterated at most mnTmax times. (b) Follows from Lemma 6. (c) If termination does not occur in Step 5, then no minimum separation violations remain and the final schedule is feasible. From Lemma 5, it follows that the final schedule is optimal in therapy time for unidirectional schedules. Corollary 3 When Smin = 0, Algorithm M~inseparation always generates an optimal fea sible schedule. Proof: When Smin = 0, Algorithm Minseparation cannot terminate in Step 5 because the Step 4 modification never causes the left leaf of a leaf pair to cross the right leaf of that pair. The Corollary follows now from Theorem 8. 2.2 Conclusion In conclusion, we presented mathematical formalisms and rigorous proofs of leaf se quencing algorithms for segmental multileaf collimation which maximize MU efficiency. These leaf sequencing algorithms explicitly account for minimum leaf separation constraint and leaf interdigfitation constraint. We have shown that our algorithms obtain all feasible solutions that, are optimnal in t~rea~tment MT~s. Furthermore, our :: 1 shows that1 uni directional leaf move~ment is at least as : as bidirectiona~l movement. ii '..hese: a~lgorithmirs are suited for commron use in ::.C1 barn delive~ry. CHAPTER 3 SEQUENCING OF DYNAMIC MULTILEAF COLLIMATORS Delivery using DMLC is different from that using <\llC. The leaf positions change with respect to time. In terms of the MLC controller it is the change in position with respect to monitor units delivered that is important. The inputs required are the leaf positions at various control points, the fractional number of monitor units to be delivered at each control point, and the total number of monitor units to be delivered for that beam. In this chapter, we present a systematic study of the optimization of leaf sequencing algorithms for the dynamic beam delivery and provide rigorous proofs of optimized leaf sequence settings in terms of MU efficiency under various leaf movement constraints. Practical leaf movement constraints that are considered include the leaf interdigitation constraint. The question of whether bidirectional leaf movement will increase the MU efficiency when compared with unidirectional leaf movement only is also addressed. 3.1 Methods 3.1.1 Movement of Leaves In our I!! 11, i we will assume that I(zo) > 0 and I(zm) > 0 and that when the beam delivery begins the leaves can be positioned ir, .tu rie. We also assume that the leaves can move with any velocity v, vmax I v I vmax, where vmax is the maximum allowable velocity of the leaves and that the leaf acceleration can be infinite. The consequences of assuming infinite leaf acceleration are negligible. Figure 31 illustrates the leaf t l r .1. b!ry during DMLC delivery. In the example, the leaves move from left to right. Let Il(zi) and Ir (as), respectively, denote the amount of Monitor Units (MUs) delivered when the left and right leaves leave position zi. Consider the motion of the left leaf. The left leaf begins at to and remains here until Iz(zo) MUs have been delivered. At this time the left leaf leaves to and is moved to 21, where it remains until I;(zl) MUs have been delivered. The left leaf then moves tO 23 where it remains until Ig(23) MUs have been delivered. At this time, the left leaf is moved to us, where it remains until I,(2s) MUs have been delivered. Then it moves to where it remains until 41( . ) MITs have been delivered. 'i : 1: : ; movement of thle left leaf is to anlo. ... left leaf arrives at1 a to when Imax M~Ts have been delivered. At this tirne the mnac~hine is turned off. Thel~ total i! timeo, TT ~ig, Ir), is theic tirne needed to deliver Imax, M4~s. 'i .. rights leaf starts at. xo aind begins to moove :1 I .i .rtl t ece 22; leaVeS 22 when Ir (n2) M4Ts have been I i i leaveS 24 when Ir (n4) M4~s have been delivered, and so on. Note that the machine is on throughout the treatment. All MIrs that~ are delivered along a radiation bea~m along me, before Ithe left leaf n:: on it. :::i all M~ITs that are delivered alongf a radiation beam along x4 before the right, leaf passes zi, ar~e blockedi by thie i So the net amount of MIUs dieliverecd at a 1= .1 is given '.Il(x4)  Irjmxi), whlich rust be th~e samne as the desired i T (z ). I~x I( r _ Ids Ix) I~xI Xo X, X2 X3 X4 X5 X6 X, X, X9 X ,o X Figure 3 1: ILleaf t 1. i :y diurinig DMLCI delivery Th~eorem 9 The ." r;; are true: ,' pair %r that delivers a discreite: (a) :' left~ f' must reach z~o at some time. (b) The leaf must reach at some? time. (c) lef IJ ! rmust reach at some time. (d) :1 .' i f mu~st reach .. at somecl timecl. Proof: (a) Suppose: that, the: i leaf aliway2s stays to the right of mo: then xo does not, receive any M/I~s: contradicting ou~r !: tha21t I(..) > 0). (b) Similar to that of (a). (c) If the left leaf doesn't reach 2m (i.e., it doesn't go to the right of sm1), from (b), it follows that the region between zm1 and am receives a nonuniform distribution of MUs. Hence the discrete profile can't be accurately delivered. (d) Similar to that of (c). m 3.1.2 Maximum Velocity Constraint As noted earlier, the velocity of leaves cannot exceed some maximum limit (say vme,) in practice. This implies that the leaf profile cannot be horizontal at any point. From Figure 23, observe that the time needed for a leaf to move from zi to ziay is > (asyl zi)/vmax. If # is the flux density of MUs from the source, the number of MUs delivered in this time along a beam is > ##(zi41zi)/vmax. So, 4(as 1)4(as) > Os(as~ 1zi)/vmax # Az/vmax. The same is true for the right leaf profile I,. 3.1.3 Optimal Unidirectional Algorithm for one Pair of Leaves Unidirectional movement. When the movement of leaves is restricted to only one di rection, both the left and right leaves move along the positive a direction, from left to right (Figure 21). Once the desired intensity profile, I(zi) is known, our problem becomes that of determining the individual "iil. 10; pr ol. / ~~I to be delivered by the left and right leaves, It and I, such that I(i) = r (i) I (zi), O < i < m (3.1) Of course, It and I, are subject to the maximum velocity constraint. We refer to (I I, ) as the treatment plan (or simply plan) for I. Once we obtain the plan, we will be able to determine the movement of both left and right leaves during the therapy. For each i, the left leaf can be allowed to pass zi when the source has delivered Iz(zi) MUs. Also, we can allow the right leaf to pass zi when the source has delivered I,(as) MUs. In this manner we obtain unidirectional leaf movement /pr oll.~ for a plan. Some simple observations about the leaf profiles are made below. Theorem 10 In every unidirectional plan the leaves begin at to and end at zm. Proof: Follows from Theorem 9 and the unidirectional constraint. m Lemma 9 In the region between each pair of successive sample points, say zi and me 1, both leaf lpr./.I maintain the same shape, i.e., one is I... ,. 10I a vertical translation of the other. Proof: As explained previously, the input profile is discretized to a square wave I. Since the profile of I is horizontal between successive sample points and since it is equal to It Ir, It and Ir must have the same shape. For example, if the left leaf moves at a constant velocity v between points zi and me 1, so should the right leaf. m Lemma 10 In an optimal plan, both leaves must move at vmax between every successive pair of sample points they move across. Proof: Suppose that in an optimal solution the leaves move between points zi and meil at a possibly varying velocity v(z) < vmax. From Lemma 9, we know that both leaf profiles are the same between zi and me 1. Setting v(z) = vmax results in new leaf profiles whose difference remains the same as before (which is the desired profile I) and total therapy time is lowered. This leads to a contradiction. m Corollary 4 In an optimal plan, no leaf stops at an a that is not one of the zis. Algorithm. From Equation 3.1, we see that one way to determine It and Ir from the given target profile l is to begin from to; set Iz(zo) = I(zo) and Ir(zo) = 0; examine the remaining zis to the right; increase It at zi whenever I increases and by the same amount (in addition to the minimum increase imposed by the maximum velocity constraint); and similarly increase Ir whenever I decreases. This can be done till we reach zm. So the treatment begins with the leaves positioned at the leftmost sample point and ends with the leaves positioned at the rightmost sample point. Once It and Ir are determined the leaf movement profiles are obtained as explained earlier. Note that we move the leaves at the maximum velocity vmax whenever they are to be moved. The resulting algorithm is shown in Figure 32. Figure 23 shows a profile I and the corresponding plan (It,Ir) obtained using Algorithm DMLCSINGLEPAIR. Ma et al. (1998) shows that Algorithm DMLCSINGLEPAIR obtains plans that are optimal in therapy time. Their proof relies on the results of Spirou and Chui (1994), Stein et al. (1994) and Boyer and Strait (1997). We provide a simpler and direct proof below. Algorithm DM4LC .i :i .ilZ IR, Foir j = 1 to m do ii (zj) It (xyl ) 1(xy)  I jzj) + 4,* Ax:/vmax, I,(my) I,(y_1 *am/ Else I,(my)  I,(my_1) I$ y1 , I*y + A/,,lt E5nd for Figure 32: ObtainingS a unidirectional plan Theorem 11 .i t .: DMnLC'SOVG'LEPAIR obtains planrs that arle optimail in :! time. Proof: Let. I(m,) be Ithe desired profile. Let 0 = .: < inrcl < ... < inack be: Ith indices of the points a~t which 1(x:,) increases. So Xminco: wjinLCi:,.,Zick are thei ******i at which I(x) increases (i.e., I(zi,,ce) > I(zi,,ci1), assume that Ijx_1 = 0)). L~et Ai I( ) ( )? i > 0. PEi:4 th~at (lL, Ty) is a _1 for I(xri) (not thelr plan gernerated by Algorithm DML '. \lLEPIR) '.=.I(xy) = II( )  Injxi) for all i we get = (17 (: 1I sni1)(nxni 1)) 1 (ici I(1) ~ n: + Ax/vaZ)  (In(xLinci)  In(xilncii)  '" z/~nar Note th~atf~r~om th~e maximum velocit~y constraint, I"Zll(:rii) IR(:rinc:i1) > Ax/vmass, i > 1. So ln(xinci) In(xinci1) : Ax/vmax o i > 1, andi Ai I II (xines)  fr(xinci1)  Ax/vmazl!,,. Also, Ao = T(zo)  T(2_1) = Tjzo) < II (Zo)  IL(x_1), w~her3 1(x1) = 0. Summingf up A~i, we: get ~ .(ljinci) 1(xLilLCi1j)] I Ei[lL(2inc)  1L(xinci1i)  k.* i Az/vmax,~. Lect S1 o~i [1; (X~inc~i) 1;L sinLCi 1)]. ii : ::, Si > E 0 [Ij~xines) I(wiLCjj)] +k X;: (I Ax C/vmax.r Let Sg' = E[I%(xy) II,(1:j_1)], where the summation is carriedl out over indices j (0 < j < m) suc~h that I(xy~) < I(xy~_l). Thel~re are a. total of m + 1 indiices of whlichl k + 1 do not satisfyi thiis condiition. So there ar~e m1  k: indicees j at which I(zj) < I(xy1). At Ceah of thelCse j, IL(2j) > IL(2j_l) + # A2//vmax. Hence, S2 > (m k) # Az/vmax. Now, we get Sl + S2 0~ [L zi> L 2i1~ =0 C [(inci) I(inci1)] + me A2/vmaz Finally, TT(IL, IR) = I (2m) = IL(2m) IL(2_1) C= Eo IL(as) IL(2i_1)] > E =o[I(2inci)  I(zinci1)] + m* Az/vma = TT(II, Ir). Hence, the treatment plan (Ig, Ir) generated by DMLCSINGLEPAIR is optimal in therapy time. Corollary 5 Let I(zi), O < i < m be a desired prll.~II Let Il(zi) and Ir(zi), O < i < m be the left and right leaf tll.~II generated by Algorithm DM~LCSINGLEPAIR. Iz(zi) and Ir(zi), O < i < m define optimal 'it.. ,~,,ti time unidirectional left and right leaf let..I;I. for Proof: Follows from Theorem 11 m In the remainder of Section 3.1, (II, Ir) is the optimal treatment plan generated by Algorithm DMLCSINGLEPAIR for the desired profile I. Properties of the optimal treatment plan. The following observations are made about the optimal treatment plan (It, Ir) generated using Algorithm DMLCSINGLEPAIR. Lemma 11 At most one of the leaves stops at each me. Lemma 12 Let (IL, IR) be any treatment plan for I. (a) a(zi) = IL(2i) 1h(2i) = IR(2i) Ir(2i) > 0,0 < i < m. (b) a(as) is a nondecreasing function. Proof: (a) SincelI(as) = IL(as) IR(a) = I (as) Ir (z), IL (as) I () = IR(as) Ir (zi). Further, from Corollary 5, it follows that IL(2i) > 4(zi),0 O i < m. Therefore, A(zi) > 0, O (b) We prove this by contradiction. Suppose that A(z,) > A(z, 1) for some n, O I n < m. Consider the following three all encompassing cases. Case 1: Iz(z, 1) = I(za) + # Az/vmax (left leaf does not stop at an 1) Now, IL(2n) = 1h(22) + A(an1) > 1h(z, 21) #. A2/vmaz + A(2n 1) = IL(2n 1) #. * This is not possible because IL(2n 1) > IL(2n) + # s A2/vmaz from the maximum velocity constraint. Case 2: Ir(as 1) = Ir(z,) + # Az/vmax (right leaf does not stop at an 1) Now, IR(2n) = Ir (2n) + A(212) > Ir (2n1) '" Az/vmax + A(z, 1) = IR(z, 1) # s This is not possible because IR(z, 1) > IR(2n) + # A2/vmaz from the maximum velocity constraint. Case 3: Il(z,+ 1) I(z,) + # Az/vmax and Ir(z,+ 1) Ir(z,) + # Az/vmax (both leaves stop at an 1) From Lemma 11 it follows that this case cannot arise. Therefore, A(zi) is a nondecreasing function. m Corollary 6 Let Alzia) (Ar(zi)) and AL(as) (An(as)), ,. ,7. .: Ho lo. denote the amount of time for which the left (right) leaf stops at zi in plans (I I,r) and (IL, IR). Then (a) AL(as) > Az(ws). (b) An(as) > Ar (as). Proof: (a) Suppose that AL(as) < Az(zi). We have the following two cases: Case 1: Both leaves move at the maximum velocity between as_l and me in (IL, IR). We get a(z,) < a(z,_l) contradicting Lemma 12(b). Case 2: In (IL, IR), the leaves do not travel uniformly at the maximum velocity between as_l and me. In this case, transform plan (IL, IR) to a plan (I I'9) as follows. Between as_l and me move the leaves at the maximum velocity. The leaves now arrive at my earlier than they did in (IL, IR) by an amount 6i. Propagate this difference to the right from zi so that I (2j) = IL(2j)6i and I'9(2j) = IR(2j)bi, j > i. Note that this transformation preserves the As, i.e., A',(2j) = AL(2j). Also, the resulting leafprofile, rI and I',silfomapn for I. Let a'(z,) = I (zj) Il(zj) = I' (zj) Ir(zj). Since A',(as) = AL(as) < Aza) and since the leaves move at maximum velocity from zi_l to me in (It, Ir) and (If, I'9), we have a'(z,) < a'(z,_l) contradicting Lemma 12(b). (b) Similar to proof of (a). 3.1.4 Minimum Separation Constraint Some MLCs have a minimum separation constraint that requires the left and right leaves to maintain a minimum separation Smin at all times during the treatment. Notice thlat in1 the planl generated by AIU~lgrithm DMLCU~SING~LEPAIR, the two leaves start and end at the same point. So they are in contact at to and zm. When smin > 0, the minimum separation constraint is violated at to and zm. In order to overcome this problem we modify Algorithm DMLCSINGLEPAIR to guarantee minimum separation between the leaves in the vicinity of the end points (zo and 2m). In particular, we allow the left leaf to be initially positioned at a point zo/ = to smin and the right leaf to be finally positioned at am/ = 2m + Smin. The only changes made relative to Algorithm DMLCSINGLEPAIR are for the movement of the left leaf from zo/ to to and for the right leaf from zm to sm* We define the movement of the left leaf from zo/ to to (and as ; en?~rowil: definition for the right leaf from zm to m,/) to be such that it maintains a distance of exactly Smin from the right leaf at all times. Once the left leaf reaches to it follows the trajectory as before. While this modification results in the exact profile being delivered between to and am it also results in some undesirable exposure to the intervals (zo/,, o) and (zm, s,/). In the remainder of this section we will consider an exposure of this kind permissible, provided the exact profile is delivered between to and zm. Note that for most accelerators (Varian, Elekta) undesirable exposure to the intervals (zo/,, o) and (mm, sm,) can be avoided by positioning the backup jaws at to and am respectively. However, a difficulty arises when the number of monitor units delivered at the time the left leaf reaches to in this new plan (call it (If, II\ )) i greter than 7./o). This would prevent us, fromusing theold plan from to to m,, since the leaf cannot pass to before it arrives there. Observe however, that if the left leaf were to arrive at to any earlier, it would be too close to the right leaf. In the discussion that follows we show that in this and other cases where the original plan violates the constraint, there are no feasible solutions that deliver exactly the desired profile between to and m,, while permitting exposure outside this region. The modified algorithm, which we~ calll DMVLC~MVINS I~V,,1 ~n nNGanLEPAIR, is described in Figure 33. Note that the therapy time of the plan produced by DMLCMINSINGLEPAIR is the same as that of the plan produced by DMLCSINGLEPAIR. Therefore, the modified plan is optimal in therapy time. Theorem 12 (a) Smin O/Umax` > h\o) or' (b I the plan (If ,' If ) enerted by DMLC M~INSINGLEPAIR violates the minimum separation constraint, then there is no plan for I that does not violate the minimum separation constraint. Proof: Suppose~ that (I ,/ I ) violates the minimum separation constraint. Assume that the first violation occurs when II MUs have been delivered. Since there is no violation when less than II MUs are delivered and since the leaves are either stationary or move at the maximum 44 Algorithm DMLCMINSINGLEPAIR 1. Apply Algorithm DMLCSINGLEPAIR 2. Modify the profile of the left leaf from zo/ to to and the right leaf from zm to m,/ to maintain a minimum interleaf distance of Smin. Call this profile (I,', I'). 3. If the number of MUs delivered when the left leaf arrives at to is greater than Iz(zo) there is no feasible solution. End. 4. Else output (I,', I'). There is a feasible solution only if (I', I') is feasible. Figure 33: Obtaining a unidirectional plan with minimum separation constraint velocity, at the time of the violation, it must be the case that the right leaf is stationary at a sample point (say zk) and the left leaf is moving. The violation occurs when the left leaf passes 2' = zk, Smin. Since the left leaf is moving, II = I,'(z') < I'(zk). Figure 34 illustrates the situation. Spoetathrisnterln(I!" htde o ilt h minimum separation constraint. Let If"(m') = It'(z')+A(z') and let I"(zk) k k)*~+~z) From Lemma 12(a), A(z'), A(zk) > 0 and from Lemma 12(b), A(z') < A(zk). Here, we have made use of the fact that in the statement of Lemma 12, we can replace the plan (II, 1r) with the plan (If, I'). Now, I"(zk) kf(' I(~ kz) 1I( az) (I'(zk) 1 ~z)) k)a~, a(z')). Since I'(zk) > I'(z') and A(zk) > a(z'), We get I"(Zk) l "(2'). Therefore there is a minimum separation violation in (I"', I") when the the left leaf passes 2'. Separation between the leaves I xo xo x' xk Xm Xm,, X Figure 34: Minimum separation constraint violation The separation between the leaves is determined by the difference in a values of the leaves when the source has delivered a certain amount of MUs. The minimum separation of the leaves is the minimum separation between the two profiles. We call this minimum sep aration Sudmin. When the optimal plan obtained using Algorithm DMLCSINGLEPAIR is delivered, the minimum separation is Sudmin(opt) * Corollary 7 Let Sudmin(opt) be the minimum leaf separation in the plan (I', I'). Let Sudmin be the minimum leaf separation in any (not ii... whil~ optimal) given unidirec tional plan. Sudmin < Sudmin(opt) * Theorem 13 If Algorithm DM~LCM~INSINGLEPAIR terminates in Step 3, then there is no plan for I that does not violate the minimum separation constraint. Pro:Let (I"', I" ) be a feasible plan (i.e., a plan that delivers I and satisfies the minimum separationl cvlonstaint). ULet (II I'.) be. Ilthe pllanI of Srtep 2i of DMVLC~MVINS ING~LEPA I. FromII Corollary 5, it follows that I"(zo +Smin) > I:((o +Smin)+I"(zo), where Ir(z) is the number of MUs delivered when the right leaf reaches n (note that Ir (z) is the number of MUs If"(zo) > If'(zo) > I('(o + Smin). Also, because DMLCMINSINGLEPAIR terminates in So, (If", I") does not deliver the proper dose at to and so cannot be feasible. m Comparison with SMLC. K~amath et al. (2003) discuss the optimal therapy time algorithm for M \! LC. Their algorithm generates an optimal plan that satisfies the minimum separation constraint, whenever one exists. We prove the existence of profiles for which there are feasible plans using 9llC, but no feasible plans using DMLC. Lemma 13 Let the minimum separation between the leaves in the optimal SM~LC plan be Ssmin. Let the minimum leaf separation in the plan generated by algorithm DM~LC M~INSINGLEPAIR be Sdmin. Then Sdmin < Ssmin Proof: Consider the delivery of a profile I by the optimal <\llC plan of K~amath et al. (2003). Call this plan (If, If) Let IIl (s b the number of M"Ts de~livered~ when the left leaf arrives at a using the plan IfS. I S(z), I,'(z), and I'(z) are defined similarly. Let fl(2k) = 1~s k) ls k~) = 1 kz) 1 kz). r k2) is defined similarly. Note that fl(zk) > 0 iff the left leaf stops at zk, and Fl(2k) gives the amount of MUs delivered while the left leaf is stopped at zk. Let me and my, j > i, be such that Ssmin = zj zi and If (zi) < I, (zj). Such an zi and my must exist by definition of Ssmin. It is easy to see that I,"(zj) I, (zi) ,= E Fr(2k). From this and lf (as) < I, (zj), we get Er (k I i) i)(3.2) k=i+1 Since I(as) = If (as) I, (as) = If (as) + El(z ) I, (as) = It'(zi) + El(zi) I'(as), I'r\ I ( (Ia I'(a )) (3.3) Also, we see that I' (x ) = I' (ws\I'" ~ ,z) + (j F (Zk azlUmaz Izi) 1Is zi)  I'(mi))+] Fr(23) + (j i) # A2/vmaz (from (3.3)) > II (mi) + (J i) Y # us Azmaz (from (3.2)). So, Sdmin < my 2i = Ssmin The following result immediately follows and can be easily verified. Corollary 8 All l; r oll. that have feasible plans using DM~LC have feasible plans using SM~LC. There exist foll.~II for which there are feasible plans using SM~LC, but no feasible plans using DM~LC. Figure 35 shows two plans for an intensityi profile. The TU L IC plan for the profile is fea sible. The corresponding DMLC plan obtained using Algorithm DMLCMINSINGLEPAIR is infeasible. 3.1.5 Bidirectional Movement In this section we study beam delivery when bidirectional movement of leaves is per mitted. We explore whether relaxing the unidirectional movement constraint helps improve the efficiency of treatment plan. Properties of bidirectional movement. For a given leaf (left or right) movement profile we classify any 2coordinate as follows. Draw a vertical line at z. If the line cuts the leaf profile exactly once we will call z a unidirectional point of that leaf profile. If the line cuts the profile more than once, z is a bidirectional point of that profile. A leaf movement profile that has at least one bidirectional point is a bidirectional /pr oll.I All profiles that are Constraint violation XO XO Figure 35: C ( plan: feasible; DMLC: i i. not bidirectional are unidirectional prll.~II Any profile can be partitioned into segments such that each segment is a unidirectional profile. When the number of such partitions is minimal, each partition is called a stage of the original profile. Thus unidirectional profiles consist of exactly one stage while bidirectional profiles 11.h I, have more than one stage. In Figure 36, the bidirectional leaf movement profile, BL, shows the position of the left leaf as a function of the amount of MUs delivered by the source. The movement profile of this leaf consists of stages S1, S2 and S3. In stages S1 and S3 the leaf moves from left to right while in stage S2 the leaf moves from right to left. zj is a bidirectional point of BL. Let IL be the inltlencityi profile corresponding to the leaf movement profile BL. IL(2) gives the number of MUs delivered at a using movement profile BL. The amount of MUs delivered at my is Li + L2~. In stage S1, when II amount of MUs have been delivered, the leaf passes my. Now, no MU is delivered at my till the leaf passes over my in S2. Further, I3 2~ MUs are delivered to my in stages S2 and S3. Thus we have Iz(zj) = Li + L2, where Li = II and L2 = 13 1 2 k is a unidirectional point of BL. The MUs delivered at zk, are L3S = 14. Note that the intensity profile IL is different from the leaf movement profile BL, unlike in the unidirectional case. I, I4 S3 SL, I,~  Figure 36: Bidirectional movement Lemma 14 Let IL and In be the 1. /10 pr l~l. l ~II corresponding to the bidirectional leaf movement l],.61. pair (BL, BR) (i~e., BL and BR are, e.I .1/.:: I:. the left and right leaf movement /pr oll.~ ). Let I(zi) = IL(2i) IR(2i), O < i < m, be the il. i i~ /10~ pr;I l delivered by (BL, BR). Then (a) IL(ziy1) > IL(2i) + # A2//vmax. (b) IR(as 1) > IR(as) + # Az//vmax. Proof: (a) Between the time, 1, the left leaf moves rightward from zi for the last time (since the left leaf ends at m,, such a last right move must occur) and the least time t2, t2 1 t, that the left leaf reaches zi 1 (again, since the left leaf ends at m,, such a 2 exists), sail receives at least # Az/vmax MUs that are not delivered to me. At all other times during the therapy, whenever the left leaf doesn't cover me, it also doesn't cover me 1. Hence, outside the time interval [tl, t2], the number of MUs delivered to meil is atleast as many as delivered to me. Therefore, for the entire therapy, IL(2i+1) > IL(2i)+ *Az/vmax. (b) The proof is similar to that of (a). m From Lemma 14 we note that every bidirectional leaf movement profile (BL, BR) delivers an intensity profile (IL, IR) that satisfies the maximum velocity constraint. Hence, (IL, IR) is deliverable using a unidirectional leaf movement profile (Section 3.1.3). We will call this profile the unidirectional leaf movement /pr oll. that corresponds to the bidirectional pil.Thus every bidirectional leaf movement profile has a corresponding unidirectional leaf profile that delivers the same amount of MUs at each zi as does the bidirectional profile. Theorem 14 The unidirectional treatment plan constructed by Algorithm DM~LC SIN GLEPAIR is optimal in 'it.. ,~,I;*0 time even when bidirectional leaf movement is permitted. Proof: Let BL and BR be bidirectional leaf movement profiles that deliver a desired intensity profile I. Let IL and IR, respectively, be the corresponding inltlensityi profiles for BL and BR. From Lemma 14, we know that IL and In are deliverable by unidirectional leaf movement profiles. Also, IL(as) IR(as) = I(as), 1 < i < m. From the proof of Theorem 11, it follows that the therapy time for the unidirectional plan (Ig, Ir) generated by Algorithm DMLCSINGLEPAIR is no more than that of (IL, IR). m Incorporating minimum separation constraint. Let Uz and Ur be unidirectional leaf movement profiles that deliver the desired profile I(zi). Let BI and Br be a set of bidirectional left and right leaf profiles such that Uz and Ur correspond to BI and Br re spectively, i.e., (BI, Br) delivers the same plan as (Ug, Ur). We call the minimum separation of leaves in this bidirectional plan (BI, Br) Sbdmin. Sudmin is the minimum separation of leaves in (UI, Ur). Theorem 15 Sbdmin < Sudmin for every bidirectional leaf movement I;I rol. pair (BI, Br) and its corresponding unidirectional l *<. Gl. (UI Ur)  Proof: Suppose that the minimum separation Sudmin occurs when Ims MUs are deliv ered. At this time, the left leaf arrives at my andl thel bright IClea is poslitione atl Zk. ULet BJ and Uf respectively, be the profiles obtained when BI and Uz are shifted right by Sudmin. Since Uf is Uz shifted right by Sudmin and since the distance between Uz and Ur is Sudmin when Ims MUs have been delivered, Uf and Ur touch when Ims units have been delivered. Therefre, th totalrC YV M~ILUsdlvrdb (Uf, \Ur) at zk is zero. So the total MUs delivered by (Bj, Br) at zk, is also zero (recall that UI and Ur, respectively, are corresponding profiles for Bb and Br). This isn't possible if Br is 11. ; c t the right of BJ(o eapeih situation of Figure 37, the MUs delivered by (Bf, Br) at zk, are (L1 + L2a) (L', + L2) > 0). Therefore Bf and Br must touch (or cross) at least once. So Sbdmin < Sudmin I, B n ; u X1 Xk X Figure 37: Bidirectional movement under minimum separation constraint Theorem 16 If the optimal unidirectional plan (If I') violates the minimum separation constraint, then there is no bidirectional movement plan that does not violate the minimum separation constraint. Proof: Let BI and Br be bidirectional leaf movements that deliver the required profile. Let the minimum separation between the leaves be Sbdmin. Let the corresponding unidi rectional leaf movements be Uz and Ur and let Sudmin be the minimum separation between Uz and Ur. Also, let Smin be the minimum allowable separation between the leaves. From Corollary 7 and Theorem 15, we get Sbdmin < Sudmin < Sudmin(opt) < Smin Incorporating maximum separation constraint. Let UI and Ur be unidirectional leaf movement profiles that deliver the desired profile I. Let Sudmat be the maximum leaf separation using the profiles Uz and Ur and let Sudmaz(opt) be the maximum leaf separation for the plan (Ig, I) generated by Algorithm DMLCSINGLEPAIR. Let BI and Br be a set of bidirectional left and right leaf profiles such that UI and Ur correspond to BI and Br, respectively. Let Sbdmaz be the maximum separation between the leaves when these bidirectional movement profiles are used. Theorem 17 Sbdmaz > Sudmat for every bidirectional leaf movement l; roll. and its corresponding unidirectional movement p ll.~II Proof: Suppose that the maximum separation Sudmat occurs when Ims MUs are deliv ered. At this time, the left leaf is positioned at my and the right leaf arrives at zk. Let BI' and Uf respectively, be the profiles obtained when BI and Uz are shifted right by Sudmax. Since Uf is Uz shifted right by Sudmat and since the distance between Uz and Ur is Sudmat when Ims MUs have been delivered, Uf and Ur touch when Ims units have been delivered. Therefore, the total MUs delivered by (Ur, Uf) at zk, is zero. So the total MUs delivered by (Br, BJ) at Z is also zero (recall that Uf and Ur, respectively, are corresponding profiles for Bf and Br). This isn't possible if Br is 1.h ., to the left of Bf (for example, in the situation of Figure 38, the M"Ts delivered~ by, (Br Bj ts ae( L'2 L L2a) > 0). Therefore Bf and Br must touch (or cross) at least once. So Sbdmaz > Sudmaz 3.1.6 Algorithm Under Maximum Separation Constraint Condition In this section we present an algorithm that generates an optimal treatment plan under the maximum separation constraint. Recall that Algorithm DMLCSINGLEPAIR U, B, B B; Ur U; X1 4 x Figure 38: Bidirectional movement under maximum separation constraint generates the optimal plan without considering this constraint. We modify Algorithm DMLCSINGLEPAIR so that all instances of violation of maximum separation (that may possibly exist) are eliminated. We know (Theorem 17) that bidirectional leaf profiles do not help eliminate the constraint. So we consider only unidirectional profiles. Algorithm. The algorithm is described in Figure 39. Algorithm DMLCMAXSEPARATION 1. Apply Algorithm DMLCSINGLEPAIR to obtain the optimal plan (It,Ir). 2. Find the least value of inltlencity,) I,) such that the leaf separation in (Ig,Ir) when II MUs are delivered is Smax, where Smax is the maximum allowed separation between the leaves and the leaf separation when II + t MUs are delivered is > Smax, for some positive constant t. If there is no such li, (II, 1r) is the optimal plan; end. 3. From Lemma 10 it follows that when II MUs are delivered, the left leaf is stopped at some my. Let 2' be the position of the right leaf at this time (see Figure 310). Note that 2' may not be one of the sample points zi, j < i < m. Let AI Il(zj) II I2 1I. Move the profile of Ir, which follows z', up by AI along I direction. To maintain I(z) = I;(z) Ir(z) for every 2, move the profile of It, which follows z', up by aI along I direction. Goto Step 2. Figure 39: Obtaining a plan under maximum separation constraint Theorem 18 Algorithm DM~LCMiAXSEPARA TION obtains plans that are optimal in ther apy time, under the maximum separation constraint. 53 Ife oifcto After modificatio x1 xX F o:: 3 10: Maximnum separation constraint violation Proof: We~ usse induction to i :.. the theoremn. 'i statements wre prove, S(nz), is the I .i:19i (iii) of the algorithm is :_ i iT n times, the resulting plan, (la,, In): ii (a.) It has no maximum separation violation when I < Ig(ub) Mijs are delivered, where [2(8) is the V8,1UO Of I2 during theic nith iteration of Algorithmr DMLIIC N= . =A RATO111 >Nr. (b) F~or plans tha~t sa~tisfyi (a2), (Ita, Im.7) is o~trimai'l in '" i time. 1. C~onsider the base case, a = 1. L~et (11, .: ) be the plan generated 1 Algorithm DI) C i::i ..' l i vi After Step (iii) is ii=' once, the resulting i : (Il, 1ry) meets the requirement that there is no ma~ximumn separation violation when I < I,(1) MITs are delivered by the radiation source. T'~ therapy time increases '.AieTTTi r)=Ti l r l A~ssumeo that there is another plan, (I,', 11): which saitlisfies c :.': :(a) of S(1) aind TT(i(, I1) < TT(In, 1,r3). We't showi this assumption leads to a at : `; : =  and so there is no such plan ((Ill I1). L~et xyg and z' be a~s in Algorithm Dn 1) CI ': i i'li iON). ':'. consider three cases for the relationshhip between Ill(xy) a~nd sI;(xy)>. Since there is no maximum separation violation when I < I2(1) MUs are de h1(2') 1,1(2'), we have If1(2') > 1hi(2'). We now construct a plan (I'(, I",) as follows : If,(2) aI z 2 z I",(s)= &(2) 0 I z < 2' CI.~ ~ 7/ / yI(s "i\s = s,\ 0 < x < 2m. Also, If( is nondecreasing and satisfies the maxi;mum ve~loncity cnstraint~ (I/ m)= f m) I>h/m)A 1h(2') > 1h(2' Az) + # A2/vmaz = If((m' Az) + `f sP Az/ vmaz). Smlrly is nondecreasing and satisfies the maximum velocity constraint. So (If:, I",) is a plan for I(zi). Alo TT'( (', = T( 7 I This contradicts our knowledge that (I, I,) is the optimal unconstrained plan. This leads to a contradiction as in the previous case. In this case, If,(zj) < hIl(zj) = Ih(zj). This violates Corollary 5. So this case cannot arise. Therefore S(1) is true. 2. Induction step Assume S(u) is true. If there are no more maximum separation violations in the resulting plan, (4,, Im,), then it is the optimal plan. If there are more violations, we find the next violation. Apply Step (iii) of the algorithm to get a new plan. Assume that there is another plan, which costs less time than the plan generated by Algorithm DMLCMAXSEPARATION. We consider three cases as in the base case and show by contradiction that there is no such plan. Therefore S(n + 1) is true whenever S(u) is true. Since the number of iterations of Steps (ii) and (iii) of the algorithm is finite (for each iteration, the left leaf must be stationary at my and there can be at most one iteration for each such my), all maximum separation violations will eventually be eliminated. When the minimum separation constraint is also applicable, we can use Algorithm DMLCMINSINGLEPAIR in place of Algorithm DMLCSINGLEPAIR in Step (i) of Algo rithm DMLCMAXSEPARATION. Note that in this case the minimum leaf separation of the plan constructed by Algorithm DMLCMAXSEPARATION is min{Sudmin(opt), Smax }. From Theorem 18, it follows that Algorithm DMLCMAXSEPARATION constructs an op timal plan that satisfies both the minimum and maximum separation constraints provided that Sudmin(opt) > Smin. Note that when Sudmin(opt) < Smin, there is no plan that satisfies the minimum separation constraint. 3.1.7 Algorithm Under Interdigitation Constraint Introduction. The interpair minimum separation constraint with Smin = 0 is of special interest and is referred to as the interdigitation constraint. Recall that, in Section 3.1.3, we proved that for a single pair of leaves, if the optimal plan does not satisfy the minimum separation constraint, then no plan satisfies the constraint. In this section we present an algorithm to generate the optimal schedule for the desired profile defined over a 2D region. We then modify the algorithm to generate schedules that satisfy the interdigitation constraint. Note that in our discussion on single pair of leaves (Section 3.1.1), we assumed that I(zo) > 0 and that I(zm) > 0. However, with multiple leaf pairs, the first and last sample points with nonzero intensity levels could be different for different pairs. Hence we will no longer make this assumption. Optimal schedule without the interdigitation constraint. For sequencing of mul tiple leaf pairs, we apply Algorithm DMLCSINGLEPAIR to determine the optimal plan for each of the n leaf pairs. This method of generating schedules is described in Algorithm DMLCMULTIPAIR (Figure 311). Note that since to, am are not necessarily nonzero for any row, we replace to by zl and am by z, in Algorithm DMLCSINGLEPAIR for each row, where 2z and my, respectively, denote the first and last nonzero sample points of that row. Also, for rows that contain only zeroes, the plan simply places the corresponding leaves at the rightmost point in the field (call it zm+l). Algorithm DMLCMULTIPAIR For(i = 1;i n; i ++) Apply Algorithm DMLCSINGLEPAIR to the ith pair of leaves to obtain plan (lil, lir) that delivers the intensity profile lilt). End For Figure 311: Obtaining a schedule Lemma 15 Algorithm DM~LCMULTIPAIR generates schedules that are optimal in 'I: ,**?/* time. Proof: Treatment is completed when all leaf pairs (which are independent) deliver their respective plans. The therapy time of the schedule generated by Algorithm DMLC MUL TIPAIR is maz {TT(I11, IIr), TT(I2, 2r,), TT ulI, Inr)}. From Theorem 11, it follows that this therapy time is optimal. Optimal algorithm with interdigitation constraint. The schedule generated by Al gorithm DMLCMULTIPAIR may violate the interdigitation constraint. Note that no intra pair constraint violations can occur for Smin = 0. So the interdigitation constraint is essen tially an interpair constraint. If the schedule has no interdigitation constraint violations, it is the desired optimal schedule. If there are violations in the schedule, we eliminate all violations of the interdigitation constraint starting from the left end, i.e., from to. To eliminate the violations, we modify those plans of the schedule that cause the violations. We scan the schedule from to along the positive a direction looking for the least z, at which is positioned a right leaf (say R,) that violates the interpair separation constraint. After rectifying the violation at z, with respect to R, we look for other violations. Since the process of eliminating a violation at me, may at times, lead to new violations involving right leaves positioned at me, we need to search afresh from z, every time a modification is made to the schedule. We now continue the scanning and modification process until no interdigitation violations exist. Algorithm DMLCINTERDIGITATION (Figure 312) outlines the procedure. Algorithm DMLCINTERDIGITATION 1. z = zo 2. While (there is an interdigitation violation) do 3. Find the least me, 2, > 2, such that a right leaf is positioned at z, and this right leaf has an interdigfitation violation with one or both of its neighboring left leaves. Let a be the least integer such that the right leaf R, is positioned at z, and R, has an interdigfitation violation. Let Le denote the left leaf with which R, has an interdigitation violation. Note that L E {u 1, a + 1}. In case R, has violations with two adjacent left leaves, we let t = 1. 4. Modify the schedule to eliminate the violation between R, and Lt. 5. z = 2v 6. End While Figure 312: Obtaining a schedule under the constraint Let M~ = ((III, 11r), (121 2r,), ul ar,)) be the schedule generated by Algorithm DMLCMULTIPAIR for the desired intensity profile. Let N(p) = ((Illp IIrp), 21p 2rp),.( I,, up r)) be the schedule obtained after Step 4 of A~lgorithm~nr DMCNTR~Dr/ IGITA`\TIO is applied p times to the input schedule M~. Note that M~ = N(0). To illustrate the modification process we use examples. There are two types of viola tions that may occur. Call them Typel and Type2 violations and call the corresponding modifications Typel and Type2 modifications. To make things easier, we only show two neighboring pairs of leaves. Suppose that the (p + 1)th violation occurs between the right leaf of pair u, which is positioned at me, and the left leaf of pair t, L E {u 1, a + 1}. In a Typel violation, the left leaf of pair t starts its sweep at a point zStart(t, p) > z, (see Figure 313). To remove this interdigitation violation, modify (Itlp, trp) tO (tl(p+1), 1tr(p+1)) aS follows. We let the leaves of pair t start at z, and move them at the maximum ve locity vmax towards the right, till they reach zStart(t, p). Let the number of MUs delivered when they reach zStart(t, p) be II. Raise the profiles Itlp() and Itrp(z), a > zStart(t, p), by an amount II = s (zStart(t, p) z,)/vmax. We get, Itlp~z 1 xI '> zStart(t, p) 1tr(p+1)(2 )2 I t(2), where It(z) is the target profile to be delivered by the leaf pair t. Iul x, xStart(t,p) x Figure 313: Eliminating a Typel violation A Type2 violation occurs when the left leaf of pair t, which starts its sweep from z < ze, passes 2, before the right leaf of pair a passes z, (Figure 314). In this case, I ;,!, I) is as defined below. where AI = lurp v,) lp v(2) = 13 12. Once again, Itr(p+1)(> 1tcl(p+1) Itz where It(z) is the target profile to be delivered by the leaf pair t. In both Typel and Type2 modifications, the other profiles of N(p) are not modified. Since Itr(p+1) differs from Itrp for z > 2, there is a possibility that N(p + 1) has inter pair separation violations for right leaf positions z > z,. Since none of the other right leaf profiles are changed from those of N(p) and since the change in Itl only delays the rightward movement of the left leaf of pair t, no interdigitation violations are possible in N(p + 1) for z < 2,. One may also verify that since Itlo and br~o are feasible plans that satisfy the maximum velocity constrains, so also are y,~ and Itrp, P > 0. Lemma 16 Ijrp(zStart(j, p)) = 0, 1 < j < m, p > 0. Proof: The proof is by induction on p. Let T(p) be the following statement: Ijrp (zStart(j, p)) = 0. Itl x, x Figure 3 14l: li:: ..: oaT'l` i.. violation (close parallel dottedi and solid linre segmeonts c they have been drawn writh a small separation to enhance readabiiliy) *t For the base case, p = 0. (lIyol, lyr) is th~e 1 :: generatled by A~lgorit~hm DMILC SlnG jINGLEPAR and it satisfies thle statedi proper Assume that T(.. .) is true. Fo~r theic (. +1)th violation, we h~ave th~e followingi two cases:  The ( i 1)t~h violation is a Ti 1 violation. A TL. mnodification is ~;;'ii .1 Such a ...... : .= .. results in chang ing thel start position of thel leaves of pair t (as defined in Algorithm DMLZC: II i 1i)IC' i i. ; i ON)) to x:,: whichh becomes xStart(t, p) + 1)) and Isr (+1)(x,) = 0. F~or j 7' to Ir(p+)(x:Start(j ,p + 1)) = lyr,,(xStart(j? p)) = 0 by induction  The: (p 1 ) th violation is a T : violation. AZT i modification is is i Let I be: as in Alg~orithm DM4LC INTER DI>( ~i'I i iSN. Suppose that zSta~tjt, p) < Sincei a T:`. mnodification does not alter the plan for x < x.,,, Itr.jp+,(x:Start(t, p +1)) = Itrp(x:Start(t, pu)) = 0). If zStart(t, p) = n:,:, it must be Ithe case: that xStart(u, p) = xStart(t, p) (as otherwi~W se there is aT'l i violation at zStarstju,p) < x,i). So th~e right 1 I of pair n is not ri.. at Hience, there is no interdiigitation violations at z,. So, the case zStart(t, p) = 2, cannot arise. For j t, the plan is unchanged. So, Ijr(p+1)(zStart(j, p+1)) = Ifrp(zStart(j, p)) = 0 by induction hypothesis. Corollary 9 A Lt;*~ 1' violation in which Itlp v,) = 0 cannot occur. Proof: From the proof of Lemma 16, it follows that whenever there is a Type2 violation, zStart(t, p) < 2,. Hence, Itlp v,) > 0. m Lemma 17 In case of a Typel violation, (Itl, Itrp) iS the Same aS (Itl, Itr0) Proof: Let p be such that there is a Typel violation. Let t, u and v be as in Algo rithm DMLCINTERDIGITATION. If (Itlp, Itr) is different from (Itlo, Itro), leaf pair t was modified in an earlier iteration (say iteration q < p) of the while loop of Algorithm DMLC INTERDIGITATION. Let v(q) be the v value in iteration q. If iteration q was a Typel vio lation, then zStart(t, p) < zStart(t, q + 1) = 2,(,) < 2,. So, iteration p cannot be a Typel violation. If iteration q was a Type2 violation, zStart(t, p) < zStart(t, q) < z,(,) < z,. Again, iteration p cannot be a Typel violation. Hence, there is no prior iteration q, q < p, when the profiles (la, Itr) were modified. m Lemma 18 (a) A Typel n,~~.:.lI;. al... r; eliminates a Typel violation. (b) A 07.1~ 1' n,~~.:.I; .al...r; eliminates a 07.1~ 1' violation. Proof: (a) From Lemma 16, lurp v,) = 0. By changing the start position of leaf pair t to me, we eliminate this violation. (b) Follows from the construction of (I,~, 1, Itr(p+1))* Note that Itlp(2) and Itrp(z) are defined only for z '> zStart(t,p). In the sequel, we adopt the convention that z > Iip(2 (X> rp(2)) is true whenever Alpl( rp ~z>> is undefined, irrespective of whether z is defined or not. Lemma 19 Let F = ((I' I r), 11 I' r),. 0 )) be any feasible schedule for the de sired l; roll.~. i.e., a schedule that does not violate the interdigitation constraint. Let S(p), be the following assertions. (a) IL1(2) > Iilp(2), 0 < i < n, to < z < am S(p) is true for p > 0. Proof: The proof is by induction on p. 1. Consider the base case, p = 0. From Corollary 5 and the fact that the plans (lio, firo), O < i < n, are generated using Algorithm DMLCSINGLEPAIR, it fol lows that S(0) is true. 2. Assume S(p) is true. Suppose Algorithm DMLCINTERDIGITATION finds a next violation and modifies the schedule N(p) to N(p +1). Suppose that the next violation occurs between the right leaf of pair u, positioned at me, and the left leaf of pair t. We modify pair t's plan for z > z,, to eliminate the violation. All other plans in the schedule remain unaltered. Therefore, to establish S(p + 1) it suffices to prove that II'ls >~M Itp1 T M v (3.4) 1sr(2) > Itr(p+1)() v 2 (3.5) We need prove only one of these two relationships since I',(z) Ir(z) = I,~, l)(z)  Itr(p+1)(),z 0 < 2 the (p + 1)th violation may either be a Typel or Type2 violation. We show that Equation 3.4 is true in both cases. This, in turn, implies that S(p+1) is true whenever S(p) is true and hence completes the proof. Note that in (Itl(p+1), >tr(p+1)), the leaves move at maximum speed between adjacent sample points. So, it is sufficient to show Equation 3.4 for sample points > z,. (a) The (p + 1)th violation is a Typel violation. From S(p) it follows that lI,(z,) > lurp v,). So, the right leaf of pair a leaves z, no earlier in I'r than it does in lurp. From this and the fact that F satisfies the interdigitation constraint, we conclude that leaf pair t cannot begin its sweep at the right of 2,. This observation together with the fact that in (I,,. 1), Itr(p+1)) the leaves move at the maximum velocity from z, to z' = zStart(t, p) implies that I' ~') > Itl(p+1)(2') and I' (5') > Itr(pi+1)(Z'), where I denotes an arrival time. Now, from Lemma 16, we get IAr(z') > I$r(z') > Itr(p+1)z) trI,(p+1) (z)* So I'z (2') = Ir (z') + It(2') > Itr(p+1)2) + It2/ = .! I L) (2'). From this and the fact that the left leaf of pair t moves at the maximum velocity between 2, and 2', it follows that Equation 3.4 holds for all a between z, and z'. To prove that Equation 3.4 holds for all sample points to the right of z' (and so holds for all a between to and zm), consider a sample point me, that is to the right of 2'. Let aI' = I'l(z') I,~,. I)(z') > 0 and let II be as in Algorithm DMLCINTERDIGITATION. Define a new plan (I"~, I"~) for leaf pair t as below I"(s = unde fined a < 2' I'z< (s I =x>z I"(s = unde fined a < 2' Note that I'~((') = I'l(2') AI' II = I,,! I )(2') II = Itlp(2') > 0. Sim ilarly, I" (z') > 0. Hence (I'(, I") is a plan for It. Also, Iff(as) = liz(2,) Itlp w,) = 102w,) (Lemma 17). This contradicts Corollary 5. Hence, Ifl(as,) > (b) The (p + 1)th violation is a Type2 violation. The situation is illustrated in Figure 314. Since F satisfies the interdigfitation constraint, the left leaf of pair t does not pass 2, before the right leaf of pair a passes 2,. So, I'z (me) > lu (2v) (3.6) From S(p) and the definition of a Type2 modification, we get, ~r (2v) > lurp zv) =tl(p+1) 2v) (3.7) Equations 3.6 and 3.7 yield I'z me) It~p+) v)> 0(3.8) Lemma 12b implies, I1'l() Irtl() > Itll(2v) Itl (2v),a > z, (3.9) (Lemma 12b yields Equation 3.9 only for z > z, and z is a sample point. From this and the fact that the left leaf moves at maximum velocity in Itl between adjacent sample points, we get Equation 3.9 for all z, a > z,.) From Equation 3.9, we get I1() I ., 1)(s > 1(m ) Itz(m ) +Itz(s) I,, I1) s),a > z, (3.10) From the definitions of Typel and Type2 modifications and the working of Al gorithm DMLCINTERDIGITATION, it follows that I ,., I I) () Itl () = Itl(p+1) 2v) ftl (v), v 1 (3.11) From Equations 3.10, 3.11 and 3.8, we get I' z(2) I ,., I L) (2) > I1~(2,) I ,.! I L) (2,) > 0, a > 2, (3.12) Therefore, IL () >I .,I t)(s) m z,(3.13) Lemma 20 For the execution of Algorithm DM~LCINTERDIGITA TION (a) O(u) Typel violations can occur. (b) O(n2m) 71/** .' viOatliOnS CGH OCCUT. (c) Let Tmax be the optimal 'I: t ten time for the input matrix. The time complexity is O(mn + u s min~nm, Tmax})). Proof: (a) It follows from Lemma 17 that each leaf pair can be involved in at most one Typel violation as pair t, i.e, the pair whose profile is modified. Hence, the number of Typel violations is I n. (b) We first obtain a bound on the number of Type2 violations at a fixed z,. Let n, t be as in AIU~lgrithml DMVLCINTERDIGUI[TATION. Note that n is chosen to be the least possible in dex. Let ui be the value of a in the ith iteration of Algorithm DMLCINTERDIGITATION at 2,. 14 is defined similarly. Let Umax = mazjgi~uj}. If ti = ui 1, it is possible that ni+1 = ti = ui 1 and tis = ui 2. Note that in this case, til f ui = uity + 1. Next, it is possible that ui+2 = ui 2 and ti+2 = ui3 (again ti+2 f i 1 = ui+2 1 ). In general, one may verify that ti = ui + 1 is possible only if ofax = Ui. If ti = Ui + 1, then ui 1 > ti = ui + 1, since the violation between ui and ti has been eliminated and no profiles with an index less than ti have been changed during iteration i at z,. It is also easy to verify that ti = 1, as = 2 > ui > u$"m, age > u$"m. From this and tis E {ui+1, as 1} it follows that urnex > upm. W knowi,,, that afax > 1. It follows that sy"" > 2, afax" > 3, lly""i '> 4 and in general, a x1l)/2)+1 > i + 1. Clearly, for the last violation (say jth) at me, Urnam < n and for this to be true, j = O(n2). So the number of Type2 violations at z, is O(n2). Since 2, has to be a sample point, there are m possible choices for it. Hence, the total number of Type2 violations is O(n2m). (c) Since the input matrix contains only integer intensity values, each violation modification raises the profile for one pair of leaves by at least one unit. Hence, if Taw is the optimal therapy time, no profile can be raised more than T,,, times. Therefore, the total number of violations that Algorithm DMLCINTERDIGITATION needs to repair is at most sham,. Combining this bound with those of (a) and (b), we get O(min~n2m, nTrum}) as a bound onI thel totarl nIumber~ of violllations repaired by A1~~lgrithml DMLCINTERDIGUIUTATION. By proper choice of data structures and programming methods it is possible to implement Al gorithm~nr DMLCINTERDr/ IGITA`\TIO so, as to ru in O(mn + u s min {nm, T,, }) time. Note that Lemma 20 provides two upper bounds of on the complexity of Algorithm DMLCN TE~mrR DIG~rr I\TARTION Or2m) and O(n maz {m, T,,, }). In most practical situa tions, T,,, < nm and so O(n maz {m, T,,, }) can be considered a tighter bound. Theorem 19 The following are true of Algorithm DM~LCINTERDIGITA TION: (a) The algorithm terminates. (b) The schedule generated is feasible and is optimal in therapy time for unidirectional schedules. Proof: (a) Lemma 20 provides a polynomial upper bound (O(n2m)) on the complexity of A~,,:lgorithm r DMCITERT ~~Dr~IGI~TATION The result follows from this. (b) When the algorithm terminates, no interdigitation violations remain and the final schedule is feasible. From Lenina 19, it follows that the final schedule is optimal in therapy time for unidirectional schedules. 3.2 Conclusion We have presented niathentatical fornialisnis and rigorous proofs of leaf sequencing algorithms for dynamic niultileaf colliniation that niaxintize MU efficiency. These leaf se quencing algorithms explicitly account for leaf interdigitation constraint. We have shown that our algorithms obtain feasible solutions that are optimal in treatment MUs. Fur therniore, our .!!l .h~ shows that unidirectional leaf niovenient is at least as efficient as bidirectional movement. Thus these algorithms are well suited for coninon use in DAILC beant delivery. C il T"ER 4 E M IN .iN O nr F TO~NGUE ,AND,GROV UNDERDOSAGE,:;,,,,,,,: Deiiveired of li i' with i i0C in the i. .. shloot mode uses mnultiplei static i i0C segments to achieve intensity modulation. i i sides of each leaf of a. MLC' have a pr, : tonigue or a i i on one side thiat fits into a similar groove of thie ..11 : i leaf. 'i1. results in :: = radiologiical path lenith~s across different parts of the leaves. G~alvin e~t al. (1~~I :` i'. :4 that the diffetrent radiologicall path lengths mnanifest, themselves a~s v doses in a, plane perpendicular to the leaf motion. 'i i low dose region between twvo ad1 : leaves was I. :i: as the tongueandgr~ ooveo ii : In an IMI1'1 treatment using an MILC? the tonguea~ndgroove effetct occurs wrhen the tongue, or Ithe groove or both for the rnost timer during treatments deivery cove~r th~e ove~rla~pping region be~twee~c n twLo :P r e~nt Sof leaves. As iointed out investigators, the tonguecandigr oovec arrangement always resullt~s in underdosagfes of as mulch as 10 inl the treatmnentl : in both static and dynamnic multileaf 1' ::: .: (Galvin: at al. i~ ' " Gav at al ,(1::o l ::' M ohan = : '\: ; .., t al. :: Sykes and '': i I : ). Severali recent publications (van Sanovroort and HT 1::: : 1= :' We~ bb et al. 1: ., Con very a~nd WelTbb 1 : D~irkxi et al1. 1 . ; Xia? and Verhey 1 = ;) have: shown that the tongue andgroovei :: can be significantly reduced by synchr~onizatio n of th~e leaves. However, thle cost of leaf ~synchir'onizationi is :: :: 11 an icrierase in thle total number of sub i and monitor units. van Santvoort a~nd i i : (: ::) ] a.an algorithm to elimrinate tongueandgroove effects for D>MLC treatment Although 11 note that their al gorithm increases Ithe number of monitor units, they do not examine the optimality or subopt~imality of the plans bi obta~in. We recently published a. paper (K~armath et al1. .:: ) that gave rnathernatical .. .1 : : and rigorous proofs of leaf sequencing algorithmns for se~grne~ntal rnuitileaf collimration, which rnaxirnize MUj ~ : ~ ~ We' 1 that our leaf sequencing algorithms thatl explicitly account for minimum leaf separation obtain feasible unidirectional solutions that are optimal. ':' now extend that wTLork to develop a~lgorithmlrs that explicitly account for leaf interdigitation and the tongueandgroove effect and are optimal in MU efficiency for unidirectional schedules. We show also that the algorithm of van Santvoort and Heijmen (1996) obtains optimal dynamic multileaf collimation treatment schedules. 4.1 Algorithm with Interdigitation and TongueandGroove Constraints 4.1.1 TongueandGroove Underdosage Effect Figure 41 shows a '.. I!n , 1 view of the region to be treated by two adjacent leaf pairs, t and t+1. Consider the shaded rectangular areas At(zi) and At+l(zi) that require exactly It(zi) and It+l(zi) MUs to be delivered, respectively. The tongueandgroove overlap area between the two leaf pairs over the sample point zi, At~t+l(zi), is colored black. Let the amount of MUs delivered in At,t+l(zi) be It,t+1(as). Ignoring leaf transmission, the following lemma is a consequence of the fact that At~t+l(z ) is exposed only when both At(zi) and At+l(zi) are exposed. Xi1 Xi Xi+1 It ~ At It,t+1 :::: ::::m : 11: ::::II At, t+1 It+1i i At+1 Figure 41: Tongueandgroove effect Lemma 21 It,t+1(i) < min{It(zi), It+1(zi)}, O < i < m, 1 5 t < n. Schedules in which It,t+1(as) = min{It(zi), It+1(as)} are said to be free of tongueand groove underdosage effects. Unless treatment schedules are carefully designed, it is possible that It,t+1(zi) << min{It(zi), It+1(zi)} for some i and t. For example, in a schedule in which Isr(zi) = 30, Itz (zi) = 50, I~t+1)r (zi) = 50 and Irt+1)z (as) = 60, we have It,t+1(2i) = Itz (2i) I~t+1)r (2i) = 50 50 = 0. Note that in this case, min{It(zi), It+1(zi)} = I(t+1)l(zi) Itl (zi) = 60 50 = 10. It is clear from this example that It~t+1(zi) could be 0 even when min{It(zi), It+1(2i)} is arbitrarily large. 4.1.21 Algoritihms K~ama~th etl al. ( ) an algorithm that greneratles a schedlelt t~hat satisfies interpair mninimnum separation constraint. T1: schedule is optimal in therapy time. Hlow ever, it does not account for the longuea~ndgroove~ : In this section, wre present two algorithms. Algorithm TON\( : i1 ANDC i :OOVE generates minimum therapy time : rectional schiedules that are free of tongueandgroovev underdiosage and : .. 'e used for MiL(' s that do not have a inllterdigitat2(ion cionstlraint,. Algorithm T'ON(': : : \;Dc : =)OVE ID generates minimum i"..: .1 y time ::. `:: : :: 1 schedules that are: free of (onguealnd groovie underdosa~ge 1.. simultaneously sa~tisfying thle interdigitation constraint and is for MLc th~at have an interdigitatlion constraint,. following lemma provides a necessary and suficiient condition for a sechedule to be free of tongiueandgir oov:e underdosa~ge ef~ects. Lemmua 22 A1 ubid'ilr~etionbal scheduled is f~ree~ of '. . ; .7 ..: :! (a) I?(xi) = 0 or I?+l(xyi) = 0, or (b) < I(t+1: ) (+4(x)<. or 0 Proof: It is easy to see that slchedule that satisfie~s theic above conditions is freec of (ongfuealndgroove underdosage: effetcts. So w~hat remains is fo~r us to show that ( sched ule that is free of tongueandgroove underdosage : thle above conditions. Consider :i such schedule. If condition (a) is :i at : i and t, the proof is So assume i and t such that Ii(x4) 0 ( and ~fIt(x4) 0 exist. We need to showr that eitheicr (b) or (c) is true for this valued of i and t.  the schedule is free of tonigueandgroove effects, It,>1(:i) mi {I(x4) It1(x) }> 0(4.1) the unidirectional constraint, it t i that A?;t+i(x4)i : gets exposed when both right leaves pass rs, and it remains exposed till the first of th~e : leaves 2,. Further, if a left leaf i ; a neighiborinig right leaf xi ,.+ 1( ) is niot exposed at all. So, I,,t+1(ws) = max {, 0, I~tt+1)(2i) I~t,t+1)r (a) } (4.2) where I~t,t+1)r (zi ) = max {Itr(i), I~t+1)r (zi)) and I~,t,t+l)lz () = min {It,(i), (,+l), (zi) }. From 4.1 and 4.2, it follows that It~t1(w) = ~t~+1)z(ws It~t1)r as)(4.3) Consider the case It(zi) > It+1(zi). Suppose that Isr(zi) > I~t+1)r(as). It follows that I~t,t+1)r (2i) = Isr (2i) and I~,t,t+l)lz () = I~t+l)lz (a). Now from 4.3, we get It,t+1(2i) = I~t+1)z (2i) Isr (2i) < Irt+1)z (ws) I~t+1)r (2i) = It+1(2i) < It(i) So It~,t+(zi) < min{It(zi), It+1(a))}, which contradicts 4.1. So Isr (2i) < I~t+1)r (as) (4.4) Now, suppose that Itl(i) < I~t+1)l(as). From It(zi) > It+1(zi), it follows that I~t,t+1)l(zi) = rtz(,) and I~t,t+l)r(zi) = I~t+l)r(zi). Hence, from 4.3, we get It,t+1(2i) = Itz (2i) I~t+1)r (2i) < Irt+1)z (ws) I~t+1)r (2i) = It+1(2i) So It~,t+(zi) < min{It(zi), It+1(a))}, which contradicts 4.1. So Itz (i) > I~t+1)z (as) (4.5) From 4.4 and 4.5, we can conclude that when It(zi) > It+1(zi), (b) is true. Similarly one can show that when It+1(zi) > It(zi), (c) is true. m Lemma 22 is equivalent to saying that the time period for which a pair of leaves (say pair t) exposes the region At~t+l(zi) is completely contained by the time period for which pair t + 1 exposes region At~t+l(zi), or vice versa, whenever It(zi) 0 and It+l(zi) 0 . Note that if either It(zi) or It+1(zi) is zero the containment is not necessary. We will refer to the necessary and sufficient condition of Lemma 22 as the tongueandgroove constraint condition. Schedules that satisfy this condition will be said to satisfy the tongueand groove constraint. van Santvoort and Heijmen (1996) present an algorithm that generates schedules that satisfy the tongueandgroove constraint for DMLC. Xia and Verhey (1998) claim that every schedule that violates the interdigitation con straint also violates the tongueandgroove constraint. We demonstrate with a counterex ample that this is not necessarily the case. The intensity matrix shown in Figure 42(a) can be exposed in a single segment as shown in Figure 42(b). The segment is free of tongue andgroove constraint violations, while it clearly violates the interdigitation constraint. 0 0 50 50 50 0 0 50 0 0 (a) (b) Figure 42: Counterexample. The inltlencityi matrix shown in (a) can be treated using a single segment with 50 MUs as shown in (b). Areas shaded dark are covered by left leaves and those shaded light are covered by right leaves. Areas not shaded are exposed. Interdigfitation constraint violation occurs though there is no tongueandgroove violation. Elimination of tongueandgroove effect. Note that the schedule generated by Al gorithm MULTIPAIR may violate the tongueandgroove constraint. If the schedule has no tongueandgroove constraint violations, it is the desired optimal schedule. If there are violations in the schedule, we eliminate all violations of the tongueandgroove constraint starting from the left end, i.e., from to. To eliminate the violations, we modify those plans of the schedule that cause the violations. We scan the schedule from to along the positive a direction looking for the least z, at which there exist leaf pairs n, t, L E {u 1, a + 1}, that violate the constraint at me. After rectifying the violation at z, we look for other violations. Since the process of eliminating a violation at e,, may at times, lead to new violations at e,, we need to search afresh from z, every time a modification is made to the schedule. However, we will prove a bound of O(u) on the number of violations that can occur at me. After eliminating all violations at a particular sample point, e,, we move to the next point, i.e., we increment w and look for possible violations at the new point. We continue the scanning and modification process until no tongueandgroove constraint vioaton e ist A,,:lgorithml~ TONGUEA DGOOVE (Figure 43) outlines the procedure. Algorithm TONGUEANDGROOVE 1. z = zo 2. While (there is a tongueandgroove violation) do 3. Find the least me, 2, > 2, such that there exist leaf pairs n, a + 1, that violate the tongueandgroove constraint at me. 4. Modify the schedule to eliminate the violation between leaf pairs n and n + 1. 5. z = 2w 6. End While Figure 43: Obtaining a schedule under the tongueandgroove constraint Let M~ = ((III, 11r), (121 2r,), ul ar,)) be the schedule generated by Algorithm MULTIPAIR for the desired intensity profile. Let N(p) = ((Illp, IIrp), 21p 2rp),.( I,, ** lpar)) be the schedule obtained after Step 4 of Algorithm TONGUEANDGROOVE is applied p times to the input schedule M~. Note that M = N(0). To illustrate the modification process we use examples. To make things easier, we only show two neighboring pairs of leaves. Suppose that the (p + 1)th violation occurs between the leaves of pair n and pair t = u + 1 at me. Note that Itlp(, w ulp 2w), aS Otherwise, either (b) or (c) of Lemma 22 is true. In case Itlp w,) >ulp(2w,), swap n and t. Now, we have Itlp zw) < ulp w;). 11 the Sequel, we refer to these n and t values as the n and I of Algorithm TONGUEANDGROOVE. From Lemma 22 and the fact that a violation has occurred, it follows that Itrp w) violation, we modify (Itlp, Itr). The other profiles of N(p) are not modified. The new plan for pair t, (I,~, Itr1,(p+1)) is aS defined below. If lulp 2w) ftlp 2w) Iurp 2w) trp 2w), then Itl~p1) fl 0 w(4.6) where AI = lulp2, w ftlp(> Itr(p+1)() I L( t( ), where It(z) is the target profile to be delivered by the leaf pair t. Otherwise, Itr~+1)trp(2 0 w (4.7), where AI' = lurp w, Itrp 2w)* Itl(p+1)(2 =tr(p 1)() + t(2), where It(z) is the target profile to be delivered by the leaf pair t. The former case is illustrated in Figure 44 and the latter is illustrated in Figure 45. Note that our strategy for plan modification is similar to that used by van Santvoort and Heijmen (1996) to eliminate a tongueandgroove violation for dynamic multileaf collimator plans. 1~(p+1) Ir(p+1) Itrp Xw Figure 44: Tongfueandgroove constraint violation: casel Ilp Tr(p+1) t~rp Xw X Figure 45: Tongueandgroove constraint violation: case2 (close parallel dotted and solid line segments overlap, they have been drawn with a small separation to enhance readability) Since (Itl(p+1), I r(p+1)) differs from (Ilip, Itrp) for z > 2, there is a possibility that N(p + 1) is involved in tongueandgroove violations for z > z,. Since none of the other leaf profiles are changed from those of N(p) no tongueandgroove constraint violations are possible in N(p + 1) for z < z,. One may also verify that since Itlo and btro are nondecreasing functions of z, so also are Atlp and Itrp, P > 0. ULemmla 23) LetI F = (()I~ofr)) (20I Ibr).)(ll "' 0r)) be any unidirectional schedule for the desired /poll.~ that il. 6. ~ the tongueandgroove constraint. Let S(p), be the following assertions. (a) IL1(2) > Iilp(2), O < i < n, to < z < am (b) lI,(z) > lirp(2), O < i < n, so < z < sm S(p) is true for p > 0. Proof: The proof is by induction on p. 1. Consider the base case, p = 0. From Corollary 1 and the fact that the plans (lio, liro), O < i < n, are generated using Algorithm SINGLEPAIR, it follows that S(0) is true. 2. Assume S(p) is true. Suppose Algorithm TONGUEANDGROOVE finds a next vio lation and modifies the schedule N(p) to N(p + 1). Suppose that the next violation occurs between leaf pairs n and t, te { u 1, a + 1}. Hence, Itlp it>) < ulp it,). We modify pair t's plan for z > at,, to eliminate the violation. All other plans in the schedule remain unaltered. Therefore, to establish S(p + 1) it suffices to prove that II I~s) I Tl IMp+1) t (4.8) 1~,(2) > Itr(p+1)() it, (4.9) Weneed, prove only one of these two relationships;,, sinc I'(s) r(2) = I,~, 1)(2)  Itr~~~p+1)~I 0 m *O )tr(p+1) (z)). We now consider pair t's plan for z > z, and show that Equation 4.8 is l.h ., true. This, in turn, implies that S(p + 1) is true whenever S(p) is true and hence completes the proof. Suppose that lulp(, It lp(w It tirp it) Itrp it,). Then, lulp it) tIrpz to Itlp it>) Itrp to *O ise. It>z) I Itz,). Clearly, in a schedule F, which is free of tongueandgroove violation between pairs n and I at at,, only the ordering li(' , ler(zwi) < I'sz(2,) < I'l(2z,) is possible (refer Lemma 22) in this scenario (the ex ception being when Iz,(zz,) = It(z,,), in which case all the quantities in the ordering are equal). From this ordering, I'l(zz,) > I'st(zz,). From the induction hypothesis, I'l(2z,) > lulp2, it isl(p+1) i,). From Equation 4.6, Itl(p+1)z) it> telp(, i litl(p+1) it,). Hence, I'l(2z,) > Itl(p+1) it,) when litlp it) ftlp it>) tI rp(w t Itrp i,). A ;, my,?! ril: argument can be presented to show that I'l~z, > Itp1 i when litlp it) ftlp it)>) tirp it) Itrp it,). So I'l(2,,) > I~.,, I )(z,). It remains to be proved that I'l(as) > I,, I t)(zi), w < i I m. Suppose for a contractionon that 3,y > w., I'lz,) < I.,;, I L)(2v). Let aI"l = I'l(2z,) Itlo(zz,). Note that I'z(4,,) > I,,! ,)(4,,) and so AI" > 1t(p+1)2) It>) 10 f>) = ..,! I ,) v,) 10 v)> (from the working of Algorithm TONGUEANDGRO OVE). DefineI aC new pla (ICCI ft I"~) as follows: I'iws = Itlo(zi) i< w I'l(as) aI"l w < i < m I" ws = Isro(i) i< w Note that I'~((a) = I'l(2,) AI" = Itlo(z,) > Itlo(z,_l) = If((z,_l). Similarly, I"(a ) > I"(as_ ) So (I'(,If) is a plan for t. Also, I'~(z,) = I' /,) AI" I liz(2v) I,~,, 9(2,) +Itlo(z,) (since AI" > Itl(p+1)2) v 10 vz) as explained above). From this and our assumption that liz(2,) < Itl(p+1) v,), it follows that I'((z,) < Ito(z,). Since plan (Itlo, Isro) was generated using Algorithm SINGLEPAIR, I'((z,) < Itlo(z,) violates Corollary 1. So our assumption was wrong and hence Equation 4.8 is 1.h ,. true. Elimination of tongueandgroove effect and interdigitation. As we have pointed out, the elimination of tongueandgroove constraint violations does not guarantee elimina tion of interdigitation constraint violations. Therefore the schedule generated by Algorithm TONGUEANDGROOVE may not be free of interdigitation violations. The algorithm we propose for obtaining schedules that simultaneously satisfy both constraints, Algorithm TO>NGUIEANDG ROOVEID,: is '' similar toAloitmTONGUEAND GROOVE. The only difference between the two algorithms lies in the definition of the constraint condition. To be precise we make the following definition. Definition 1 A unidirectional schedule is said to satisfy the tongueandgrooveid con straint if for 0 < i < m, 1 5 t < n. The only difference between this constraint and the tongueandgroove constraint is that this constraint enforces condition (a) or (b) above to be true at all sample points zi including those at which It(zi) = 0 and/or It+l(zi) = 0. Lemma 24 A schedule ,t. G. the tongueandgrooveid constraint iffit ,t. G. the tongue andgroove constraint and the interdigitation constraint. Proof: It is obvious that the tongueandgrooveid constraint subsumes the tongfueand groove constraint. If a schedule has a violation of the interdigfitation constraint, 3i, t, I(t+1)1(zi) < Isr(zi) or Itlziw) < I~t+1)r(as). From Definition 1, it follows that schedules that satisfy the tongueandgrooveid constraint do not violate the interdigitation constraint. Therefore a schedule that satisfies the tongueandgrooveid constraint satisfies the tongfue andgroove constraint and the interdigfitation constraint. For the other direction of the proof, consider a schedule O that satisfies the tongueand groove constraint and the interdigfitation constraint. From the fact that O satisfies the tongueandgroove constraint and from Lemma 22 and Definition 1, it only remains to be proved that for schedule O, (a) Isr (w) < I~t+1)r (2) < I~t+1) (2i) < Itz (2i), or whenever It(zi) = 0 or It+1(zi) = 0, O < i < m, 1 < t < n. When It(zi) = 0, Itz (i) = Isr (a) (4.10) Since O satisfies the interdigfitation constraint, Isr (2i) < I~t+1)1(as) (4.11) and I~t+)r (i) From Equations 4.10, 4.11 and 4.12, we get I~t+1)r(zi) I Isr(zi) = rtl(zi) I I(t+1)l(zi). So (b) is true whenever It(zi) = 0. Similarly, (a) is true whenever It+1(zi) = 0. Therefore, O satisfies the tongueandgrooveid constraint. m Algorithm TONGUEANDGROOVEID finds violations of the tong ueandgrooveid constraint from left to right in exnactly ~lthe samelt mannerllt~ in whIich AI~lgrithm TONGUE ANDGROOVE detects tongueandgroove violations. Also, the violations are eliminated as before, i.e., as prescribed by Equations 4.6 and 4.7 and illustrated in Figures 44 and 45, respectively. Algorithm TONGUEANDGROOVEID is shown in Figure 46. All notation used in the algorithm and the related discussion in the remainder of Section 4.1.2 is also the same as that used in Section 4.1.2 and corresponds directly to the usage in Algorithm TONGUEANDGROOVE. Algorithm TONGUEANDGROOVEID 1. z = zo 2. While (there is a tongueandgrooveid violation) do 3. Find the least me, 2, > 2, such that there exist leaf pairs n, a + 1, that violate the tongueandgrooveid constraint at me. 4. Modify the schedule to eliminate the violation between leaf pairs n and n + 1. 5. z = 2w 6. End While Figure 46: Obtaining a schedule under both the constraints Lemma 25 Let F = ((I(1 r'), (1fe I' ), >, (Inc )) be any unidirectional schedule for the desired prll.II~ that ,t. 6. ~ the tongueandgrooveid constraint. Let S(p), be the fol lowing assertions. (a) IL1(2) > Iilp(2), 0 < i < n, to < z < am (b) lI,(z) > lirp(2), O < i < n, so < z < sm S(p) is true for p > 0. Proof: The proof is by induction on p. 1. Consider the base case, p = 0. From Corollary 1 and the fact that the plans (lio, br~o), O < i < n, are generated using Algorithm SINGLEPAIR, it follows that S(0) is true. 2. Assume S(p) is true. Suppose Algorithm TONGUEANDGROOVEID finds a next violation and modifies the schedule N(p) to N(p +1). Suppose that the next violation occurs between leaf pairs n and t, te { u 1, a + 1}. As in the proof of Lemma 23, we only need prove either Equation 4.8 or Equation 4.9 to complete this proof. We complete the proof for the following three cases that are exhaustive. case 1: It(as) 0 and I,(as) 0 . The remainder of the proof for this case is the same as that of Lemma 23. case 2: It(as) = 0. In this case, Itlp 2w) = Itrp w). Since lulp2, w Irp w), we have lulp ,) Itlp w,) > urp 2w) Itrp2w,). The modification prescribed by Equation 4.7 is applicable. Note that if lurp w,) Itrp zw) = Iulp w,) ftlp2w,), Equation 4.6 is the same as Equation 4.7. In particular, Itr(p+1) 2w) = Itrp 2w> +urp 2w) Itrp 2w) = urp2w,) (4.13) Since It(m,,) = 0, Itr~+1) IL) (s,)(4.14) From Equations 4.13 and 4.14, Isrp(w f Il(p+1) w,) (4.15) Since F satisfies the interdigitation constraint, the left leaf of pair t does not pass me, before the right leaf of pair a passes me,. So, I1(a,) >lur as,)(4.16) From S(p) and Equation 4.15, we get, for as, > lrp ) ; L)w,)(4.17) Equations 4.16 and 4.17 yield I1(as,) Itl(p+1)2w,) > 0 (4.18) Lemma 2b implies, I1l() Itl() > I~1(as,) It1(as,), a > 2, (4.19) Subtracting L a, l) (z) from Equation 4.19, and rearranging terms we get I1(s) Itl(p+1)(2 I \w> 1 (w) f Il f2 Il(p+1) n, (4.20) From Equations 4.6 and 4.7 and the working of Algorithm TONGUEAND GROOVE ID, it follows that rtlcp+1) fl t12 ;.! I L)2w> 1 I (w), > ,( (4.21) From Equations 4.20, 4.21 and 4.18, we get Izl() Itlg+)(2) '> Itllas) I. I ;,!,)(as) > 0, a > 2, (4.22) Therefore, IlI(2) > I ,.!, 1 )(2), 2 > 2, (4.23) case 3: I,(z,) = 0. The proof is similar to that of case 2. 4.1.3 Efficient Implementation of the Algorithms In the remainder of this section we will use 'algorithm' to mean Algorithm TONGUE ANDGROOVE or Algorithm TONGUEANDGROOVEID and 'violation' to mean tongue andgroove constraint violation or tongueandgrooveid constraint violation (depending on which algorithm is considered) unless explicitly mentioned. The execution of the algorithm starts with schedule M~ at z = zo and sweeps to the right, eliminating violations from the schedule along the way. The modifications applied to eliminate a violation at e,, prescribed by Equations 4.6 and 4.7, modify one of the violating profiles for z > z,. From the unidirectional nature of the sweep of the algorithm, it is clear that the modification of the profile for z > z, can have no consequence on violations that may occur at the point me. Therefore it suffices to modify the profile only at z, at the time the violation at 2, is detected. The modification can be propagated to the right as the algorithm sweeps. This can be done by using an (n x m) matrix A that keeps track of the amount by which the profiles have been raised. A(j, k) denotes the cumulative amount by which the jth leaf pair profiles have been raised at sample point zk, from the schedule M generated using Algorithm MULTIPAIR. When the algorithm has eliminated all violations at each e,, it moves to 2,+ to look for possible violations. It first sets the (w+1)th column of the modification matrix equal to the wth column to reflect rightward propagation of the modifications. It then looks for and eliminates violations at z,+l and so on. The process of detecting the violations at z, merits further investigation. We show that if one carefully selects the order in which violations are detected and eliminated, the number of violations at each e,, O < w < m will be O(u). Lemma 26 The algorithm can be implemented such that O(u) violations occur at each e,, O < w Proof: The bound is achieved using a two pass scheme at me. In pass one we check adjacent leaf pairs (1, 2), (2, 3), .. ., (n 1, n), in that order, for possible violations at me. In pass two, we check for violations in the reverse order, i.e., (n 1, n), (n 2, a 1), .. ., (1, 2). So each set of adjacent pairs (i, i + 1), 1 < i < n is checked exactly twice for possible violations. It is easy to see that if a violation is detected in pass one, either the profile of leaf pair i or that of leaf pair i + 1 may be modified (raised) to eliminate the violation. However, in pass two only the profile of pair i may be modified. This is because the profile of pair i is not modified between the two times it is checked for violations with pair i + 1. The profile of pair i + 1, on the other hand, could have been modified between these times as a result of violations with pair i + 2. Therefore in pass two, only i can be a candidate for t (where t is as explained in the algorithm) when pairs (i, i + 1) are examined. From this it also follows that when pairs (i 1, i) are subsequently examined in pass two, the profile of pair i will not be modified. Since there is no violation between adjacent pairs (1,2), (2, 3),...,(i, i +1) at that time and none of these pairs is ever examined again, it follows that at the end of pass two there can be no violations between pairs (i, i + 1), 1< i Lemma 27 For the execution of the algorithm, the time complexity is O(nm). Proof: Follows from Lemma 26 and the fact that there are m sample points. Theorem 20 (a) Algorithms TONGUEANDGROOOVE and TONGUEANDGROOVE ID terminate. (b) The schedule generated by Algorithm TONGUEANDGROOVE is free of tongueand groove constraint violations and is optimal in fl... tuptI~ time for unidirectional schedules. (c) The schedule generated by Algorithm TONGUEANDGROOVEID is free of interdig itation and tongueandgroove constraint violations and is optimal in fl... tuptI~ time for unidirectional schedules. Proof: (a) Lemma 27 provides a polynomial upper bound (O(n am)) on the complexity of Algorithms TONGUEANDGROOVE and TONGUEANDGROOVEID. The result follows from this. (b) When Algorithm TONGUEANDGROOVE terminates, no tongueandgroove viola tions remain. From this and Lemma 23, it follows that the schedule generated by A~,,c~ ~Tlgorth TONGUEADGOOVE is optimal in therapy time for unidirectional schedules free of tongueandgroove violations. (c) When Algorithm TONGUEANDGROOVEID terminates, no tongueandgrooveid violations remain and from Lemma 24 the final schedule satisfies the tongfueand groove and interdigitation constraints. From this and Lemma 25, it follows that the schedule generated by Algorithm TONGUEANDGROOVEID is optimal in therapy time for unidirectional schedules free of both types of violations. Theorem 21 The schedule generated by the algorithm of van Santvoort and Heij~men (1996) is free of interdigitation and tongueandgroove constraint violations and is optimal in ther apy time for unidirectional DM~LC schedules with this /****/** '/// Proof: Similar to that of Theorem 20(c). m 4.2 Experimental Validation The algorithms were validated on a Varian 2100 C/D with 120leaf MLC (Varian Medical Systems, Palo Alto, CA). The inltloneityi maps of a 7field head and neck plan from a commercial inverse treatment planning , r. inl (CORVUS 5.0, NOMOS Corporation, Cranberry, PA) were sequenced using Algorithm MULTIPAIR, which optimizes the MU efficiency, and Algorithm TONGUEANDGROOVEID, which eliminates the tongueand groove effect and interdigitation. The intensity maps have a bixel size of 1 cm x 1 cm and a 'II' inltloneity step. Figure 47 shows the film measurement of the fluence maps of the AP field. The tongueandgroove effect is readily seen in Figure 47(a), while it is completely eliminated in Figure 4 /7(b): using ' Algorithm TONGUEADGOOVEID. Table 41 compares the number of segments and the MU efficiencies of all three algorithms. The MU efficiency is defined as the ratio of the maximum fluence of intensity modulated field per MU to the fluence of an open field per MU. Compared to the leaf sequences with no constraints, the consideration of tongueandgroove correction increased both the number of segments and MUs, with an average increase of 21% and 19l' i respectively, for the 7 inltloneityi maps considered here. With the additional elimination of interdigitation, the increases were ''.' and 2!1' .,i respectively. Examination of all the sub fields of the leaf sequences generated with Algorithm TONGUEANDGROOVEID verified that no interdigfitation constraint has been violated. (a) (b~) Figure 47: Film measurement of the AP field (field ID 1 in Table 1) of a sevenfield head and neck plan. The optimized leaf sequences were generated without (Algorithm MULTIPAIR, (b)). 4.3 Comparison with Algorithm of Que et al. (2004)) Recently a new algorithm to eliminate tongueandgroove effects in step and shoot de livery has been proposed (Que et al. 2004). The algorithm of Que et al. (2004) is designed to eliminate tongueandgroove effect. Although this algorithm eliminates tongueandgroove effect on all 1000 random matrices tried in Que et al. (2004), no proof that the algorithm eliminates tongueandgroove effect on all possible matrices has been provided. Further, it is not known whether or not the algorithm of Que et al. (2004) minimizes therapy time. We .., 1,... the algorithm of Que et al. (2004) and show that it is 1.h .,. successful in eliminating tongueandgroove effect; the generated leaf sequence is also free of interdigfi tation. We also perform a theoretical and experimental comparison of this algorithm with our algorithms (K~amath et al. (2004)). same leaf sequence as that obtained using the 'slidingf window' method of Bortfeld et al. (1994b3). Note that the discrete intensity profile that needs to be delivered, I, is output from the optimizer. Let n be the number of leaf pairs and m be the number of sample points for each leaf pair (i.e., for each row of the profile). We denote the rows of I by Il, I2, In. Let It(zi) denote the number of MUs that need to be delivered at sample point i (ith column) of leaf pair t (tth row). Lemma 28 The algorithm of Que et al. (I',ir),) generates unidirectional schedules. Proof: During each iteration, the next segment generated using the 'sliding window' method is such that the left leaves are positioned at the leftmost nonzero sample point (i.e., the least i such that It(zi) > It(zi_l), where It(z_l) = 0) for each row t in the residual matrix I. The right leaves are positioned at the first columns of the respective rows where there is a decrease in inltlensityi profile (i.e., the least j for which It(zj) < It(zj_1)). For example, for the single row intensity profile of Figure 48, the left leaf will be positioned at z2 and the right leaf will be positioned at 26. The algorithm of Que et al. (2004) repositions all right leaves to the position of the leftmost right leaf thus obtained. During the delivery of this segment, the inltloneityi values in the matrix in the exposed areas decreases, while the other values remain unaltered. Therefore in the new residual matrix, the leftmost nonzero points of rows either remain at the same positions as in the residual matrix of the previous iteration (or the original intensity matrix for the first iteration) or they move to the right. So the left leaves cannot move to the left between successive segments. It is also easy to verify that there can be no index k such that in the updated residual inltloneityi matrix zrk is to the left of the column of right leaf positions in the segment and It(zk) < It k1> for some row. It follows that the right leaves cannot move to the left either. So the leaf movements are unidirectional and from left to right. m Definition 1 and Lemma 24 are from K~amath et al. (2004) and are used in the proof of Theorem 22. Theorem 22 The algorithm of Que et al. (I',ir),) generates schedules free of tongueand groove effect and interdigitation. XO X1 X2 X3 X4 X5 X6 X7 X8 X Figure 48: Leaf positions: The left leaf will be positioned at z2, i.e., it will shield to and zl. The right leaf will be positioned at 26 and will shield me, i > 6. Proof: Let I l(zi) and Ir (zi), respectively, be the number of MUs delivered when the left and right leaves of pair t pass zi in the schedule generated by the algorithm of Que et al. (2004). In the schedule generated, all right leaves pass point zi, O < i < m (during their left to right movement) after exactly the same number of monitor units (MUs) are delivered. So Ir (zi) = I t+,)r (zi), O < i < m, 1 < t < n. From this equality, Lemma 28, Definition 1, and Lemma 24, it follows that the schedule generated by the algorithm of Que et al. (2004) is free of tongueandgroove effect and interdigfitation. m Theorem 23 Let Tengid be the optimal 'il.. ,~,,;*0 time unidirectional leaf sequence that deliv ers an "iil. i10 il ol l.~II I, while eliminating the tongueandgroove effect and interdigitation. The 'it.. r,tya~ time for the schedule generated by the algorithm of Que et al. (.',ie),) is at most usTengid, where n is the number of involved leaf pairs. Further, usTengid is a tight bound, i.e., there exist prll.~II I for which the schedule generated by the algorithm of Que et al. (.',ir),) requires a therapy time of n*Tengid Proof: Let ar (zi) denote the amount of therapy time for which the right leaf of leaf pair j stops at zi in the schedule obtained for I using Algorithm MULTIPAIR (K~a math et al. 2003). The therapy time for the plan of leaf pair j is the sum of times for which its right leaf stops at all sample points, which is p m ~ s. h hrp time of the entire schedule, T, is the maximum of the therapy times of all leaf pairs, i.e., T = maxj( {E o Ajr(zi)}. Clearly, Ttng_44 > T. In the schedule generated by the algo rithm of Que et al. (2004), all the right leaves stop at each zi for the same amount of time, say A (zi), which is equal to the maximum of the times for which a right leaf stops at zi in the schedule generated by Algorithm MULTIPAIR, i.e., A (as) = maxy {Ajr(zi)}. The therapy time for the schedule generated by the algorithm of Que et al. (2004) is therefore T/_4= A(a)= o maxy {Ajr(z))}. Since each Ajr(zi), O < i < m, 1 < j < n can contribute a term to this expression for T/ng at most once, Tingid L=1 io nir(i) < u s maxjy { o nir(zi)} = n *T I n Ttngid. Note that the al gorithm of K~amath et al. (2004) generates schedules that are optimal in therapy time for unidirectional schedules. Hence the algorithm of Que et al. (2004) may generate sched ules requiring up to n times the therapy time required by the schedules generated by the algorithm of K~amath et al. (2004). The above .!! .11 i assumes that leaf pairs are allowed to close within the field as defined by the collimator jaws. This is true for certain designs of MLCs. For MLCs with rounded leafend design, significant radiation transmission through the closed leaf pairs requires them to be moved under the collimator jaws. In this case, both the algorithms of K~amath et al. (2004) and Que et al. (2004) violate the interdigitation constraint, and only tongue and groove effect is eliminated. Figure 49 shows an intensity map with 4 rows for which the algorithm of Que et al. (2004) requires 4 20 = 80 MUs. The map can be delivered using 20 MUs without violating the tongueandgroove constraint and interdigfitation constraint using Algorithm TONGUEANDGROOVEID (K~amath et al. 2004). The example can be generalized for n rows. m 20 0 0 0 0 20 0 0 0 0 20 0 0 0 0 20 Figure 49: Worstcase example. This intensity map can be delivered using 20 MUs using Algorithm TONGUEANDGROOVEID (K~amath et al. 2004). The algorithm of Que et al. (2004) delivers this map using 80 MUs. 4.3.2 Results We implemented Algorithms TONGUEANDGROOVE and TONGUEANDGROOVE ID (K~amath et al. 2004) and the algorithm of Que et al. (2004). For performance compar ison, we used two separate data sets. The first set consisted of three clinical IMRT plans with 7, 5 and 7 beams, respectively. The first two plans had a 'II' fluence step and last plan had a 10% fluence step. Table 4.3.2 gives the total MUs and number of segments required for each of the 19 beams in the 3 clinical plans. On our clinical data set, the algorithm of Que et al. (2004) generated schedules with 24 times as many MUs and segments as did the algorithms of K~amath et al. (2004). The second data set consisted of 100,000 randomly generated 15 x 15 matrices. The intensity values in these matrices were random integers from 0 to 10. The average MUs and segments for schedules generated using the three algorithms for this set and their respective standard deviations are shown in Table 4.3.2. On this set, the algorithm of Que et al. (2004) generated schedules with about 2.5 times as many MUs and segments as did the algorithms of K~amath et al. (2004). Note that in both cases the number of MUs and segments in the schedules generated using Algorithm ~~T l~ TONGEA DGROOVEID (K~amath et al. 2004) are only slightly greater than in those generated using Algorithm TONGUEANDGROOVE (K~amath et al. 2004). 4.4 Conclusion We have described mathematical formalism and rigorous proofs of leaf sequencing algo rithms for segmental multileaf collimation, which maximize MU efficiency while completely eliminating the tongueandgroove underdosagfe. Even though it has been shown that for a multiple field IMRT plan (> 5), the tongueandgroove effect on the IMRT dose distri bution is clinically insignificant (Deng et al. 2001) due to the smearing effect of individual fields, yet it still can be problematic for a small number of fields and for the patient setup with minimal uncer' lnir ,. Compared to the unconstrained leaf sequencing algorithms, the presented methods yield leaf sequences, which decreases the MU efficiency a little. But they completely overcome tongueandgroove underdosages. One of the methods also eliminates leaf interdigitation. Most importantly, mathematical proofs show that these algorithms are optimal in MU efficiency for unidirectional schedules. We have also proved that the algorithm of Que et al. (2004) generates schedules that are free of the tongueandgroove Table 42: Number of MUs and segments generated for 19 clinical intensity modu lated beams from 3 IMRT plans using algorithms A (Algorithm of Que et al. 2004), B (Algorithm TONGUEANDGROOVE (K~amath et al. 2004)) and C (Algorithm TONGUEANDGROOVEID (K~amath et al. 2004)). Beams 112 have a 'II' fluence step, while beams 1319 have a 10% fluence step. Beam number A BC MUs Segments MUs Segments MUs Segments 1 780 38 280 14 280 14 2 520 26 200 10 220 11 3 760 33 300 15 320 16 4 840 40 420 21 420 21 5 740 32 280 14 280 14 6 780 34 260 13 280 14 7 640 29 260 11 280 11 8 1500 74 380 19 420 21 9 860 43 240 12 240 12 10 1 500 67 420 20 420 20 11 1660 78 420 21 440 22 12 840 39 280 14 280 14 13 880 78 280 25 280 24 14 1 080 102 300 30 340 33 15 1070 90 310 27 320 26 16 1000 90 340 31 390 36 17 890 71 340 29 340 28 18 990 75 310 29 310 30 19 1 010 84 330 30 330 30 Table 43: Average number of MUs and segments generated over a set of 100,000 random 15 x 15 matrices using algorithms A (Algorithm of Que et al. 2004), B(Algorithm ~~~T'';"TONGUEAD GROOVEi (K~amath et al. 2004)) and C (Algorithm TONGUEANDGROOVEID (K~amath et al. 2004)). The respective standard deviations are also shown. The intensity values in the matrices were randomly generated integers from Sto 10. A B C: MUs Segments MUs Segments MUs Segments Average 114.3 111.6 47.5 45.7 48.2 46.4 Standard Deviation 6i.1 6.0 3.4 3.0 3.5 3.0 effct, and i:: : l :: Our analysis shows that th~e algorithmn of i`' et, al. generates schedules that miyv require: up to a1 times the therapy time required by tha~t fo~r an optimal I seqcuenc~e free of tongueandgr oovec :: : and inter~digita~tion, where n is the number of involved leaf palirs. In experiments witlh < .' and ralndomnly generatled data sets we~t find that, the alg~rorit~hmn <. (`'. et alj. ( ::: = ) gen31erartet s schedules tlhat require 2 t~o 4 times tihe therapy time required the schedules generated our algorithms. C il T"ER 5 ALGR YI /0u i FOR SPLITTING LARGE FIELDS'""'T" 5.1 Introduction 1 of abutting subfeieds thatl result fromn the ; 1: of a large ( results in longer delivery times, poor ii !I ..., and field matching problems. D~ogan at al. ( : ) point out that the uncertainties in loaf and carriage positions cause errors in thle delivered dlose (hot or cold spots) along Ithe match line of the a~butting i observed dose~l 7 of up to 11= aiongi th~e field split "v whn theo split line crossed through the center of thle target for all thie fields. 'ii.. problems of doshrnetric perturbationi along t~he fieldi split line has been addressed in several recent "!: i: 1 . (W,~u et al. ::: Hiong at al. :  Dogan at al. .T solutions included automatic feathering of i ir' fields 1 mnodif  thei split line position foreach ...i position (11 i ;, et al. :: D ogan et al. :  ) or by d~ : : : ,~ changing radiation intensity in the overlap region of th~e split fields. NJone of thle field splitting techiniques~ reported in thie literature has addressed the issue of trecatmnent delivery a~nd MU; :I7 . We believe th~at it is important to address this issue. O~ur optimnal i splitting algorithms with and without :1 .., 1 be integrated into our previously developed i .1~ sequencing algorithmlrs to < :: i account for interdig itation a~nd tongueiandgr oovec :i: of some multilea~f ii We provide rigorous ma2them atlical proofs t~hat t~he i : schemes for 1 1 !; : are optimal in MUT clency. 7; i..:T: .:.i 1 results showr thiat our (. .1: .' field i. I:: algor~ithmr withiout feathi ering re~duices total M~is '` up to  on i 1 cases a~nd up to on synthetic cases compared to ~ a commercia planning i. .. that, also splits fields without featheringS. 5.2 F~ield Splitting W\~ithout Ferathering 5.2.1 Optirnal Field Splitting for One Leaf Pair In this section we deviate slightly from our : : : notation and assume that1 the: sample points ar~e 2i, 2, .., 4, rather than : 2i, .., 42. All other notation remains unc~hanged.. 