ISSN 0031-9155
`
`
`Online: www.iop.org/jaummalslomb
`
`PublishedbyIOPPublishing on behalfoftheInstitute
`of Physics and Engineering in Medicine
`
`@University of Arkansas
`‘ Libraries, Fayetteville
`4 PERIODICALS ROOM
`954:10
`aReceived on: 05-29-09
`; Physics in medicine &
`
`jy biology
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` f©Publishing
`
`nen 2EELETEENAETTTINETELRDTE
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`Physics in Medicine & Biology
`Volume 54 Number 10 21 May 2009
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`2971
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`2993
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`3003
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`3015
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`3101
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`3113
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`3129
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`3141
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`3161
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`3173
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`Exact imagereconstruction with triple-source saddle-curve cone-beam scanning
`Yang Lu, Jun Zhao and Ge Wang
`Dosimetry for the MRIaccelerator: the impact ofa magnetic field on the response ofa
`Farmer NE2571 ionization chamber
`I Meijsing, B W Raaymakers, A J E Raaijmakers, J G M Kok, L Hogeweg, B Liu and
`JJ W Lagendijk
`
`A micro-machinedretro-reflector for improvinglight yield in ultra-high-resolution gamma
`cameras
`
`Jan W T Heemskerk, Marc A N Korevaar, Rob Kreuger, C M Ligtvoet, Paul Schotanus and
`Freek J Beekman
`
`Developmentof an imaging modality utilizing 2D optical signals during an EPI-fluorescent
`optical mapping experiment
`Phillip Prior and Bradley J Roth
`Beam hardening correction in CT myocardial perfusion measurement
`Aaron So, Jiang Hsieh, Jian-Ying Li and Ting-Yim Lee
`Optimizing leaf widths for a multileaf collimator
`Weijie Cui and Jianrong Dai
`Electrical impedance spectroscopyas a potential tool for recovering bone porosity
`C Bonifasi-Lista and E Cherkaev
`
`Evaluation of a compartmental modelfor estimating tumor hypoxia via FMISO dynamic
`PETimaging
`Wenli Wang, Jens-Christoph Georgi, Sadek A Nehmeh, Manoj Narayanan, Timo Paulus,
`Matthieu Bal, Joseph O’ Donoghue, Pat B Zanzonico, C Ross Schmidtlein, Nancy Y Lee and
`John L Humm
`
`Reduction of the numberof stacking layers in proton uniform scanning
`Shinichiro Fujitaka, Taisuke Takayanagi, Rintaro Fujimoto, Yusuke Fujii, Hideaki Nishiuchi,
`Futaro Ebina, Takashi Okazaki, Kazuo Hiramoto, Takeji Sakae and Toshiyuki Terunuma
`High permeability cores to optimize the stimulation of deeply located brain regions using
`transcranial magnetic stimulation
`R Salvador, P C Miranda, Y Roth and A Zangen
`Audio frequencyin vivo optical coherence elastography
`Steven G Adie, Brendan F Kennedy,Julian J| Armstrong,Sergey A Alexandrov and
`David D Sampson
`Computed tomography dose assessmentfor a 160 mm wide, 320 detector row, cone beam
`CT scanner
`J Geleijns, M Salvadé Artells, P W de Bruin, R Matter, Y Muramatsu and M F McNitt-Gray
`Optimization of Rb-82 PET acquisition and reconstruction protocols for myocardial
`perfusion defect detection
`Jing Tang, Arman Rahmim, Riikka Lautamaki, Martin A Lodge, Frank M Bengel and
`Benjamin M W Tsui
`Theinfluence of a novel transmission detector on 6 MV x-ray beam characteristics
`Sankar Venkataraman, Kyle E Malkoske, Martin Jensen, Keith D Nakonechny, Ganiyu Asuni
`and Boyd M C McCurdy
`
`Bibliographic codes
`CODEN: PHMBA7 54 (10) 2971-3290, N177-204 (2009)
`
`ISSN: 0031-9155
`
`(Continued on inside back cover)
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`N177
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`N189
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`N197
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`(Continuedfrom outside back cover)
`High-energyradiation monitoring based on radio-fluorogenic co-polymerization.I: small
`volumein situ probe
`J M Warman, M P de Haasand L H Luthjens
`An ultrasound cylindrical phased array for deep heating in the breast: theoretical design
`using heterogeneous models
`J F Bakker, M M Paulides, I M Obdeijn, GC van Rhoon and K W A van Dongen
`Experimental validation of a Monte Carlo proton therapy nozzle model incorporating
`magnetically steered protons
`S W Peterson, J Polf, M Bues, G Ciangaru, L Archambault, S Beddar and A Smith
`Dosimetric variation due to CT inter-slice spacing in four-dimensional carbon beam lung
`therapy
`Motoki Kumagai, Shinichiro Mori, Gregory C Sharp, Hiroshi Asakura, Susumu Kandatsu,
`Masahiro Endo and Masayuki Baba
`Characterization of diffraction-enhanced imaging contrast in breast cancer
`T Kao, D Connor, F A Dilmanian, L Faulconer, T Liu, C Parham, E D Pisano and Z Zhong
`Development and validation of a beam model applicable to smallfields
`P Caprile and G H Hartmann
`
`Evaluationof registration strategies for multi-modality images of rat brain slices
`Christoph Palm, Andrea Vieten, Dagmar Salber and UwePietrzyk
`
`NOTES
`
`Developmentof a remanence measurement-based SQUID system with in-depth resolution
`for nanoparticle imaging
`Song Ge, Xiangyang Shi, James R Baker Jr, Mark M Banaszak Holl and Bradford G Orr
`Modelingtime variation of blood temperature in a bioheat equation andits application to
`temperature analysis due to RF exposure
`Akimasa Hirata and Osamu Fujiwara
`The reproducibility of a HeadFix relocatable fixation system: analysis using the stereotactic
`coordinates of bilateral incus andthetop ofthe crista galli obtained from a serial CT scan
`Etsuo Kunieda, Yohei Oku, Junichi Fukada, Osamu Kawaguchi, Hideyuki Shiba, Atsuya Takeda
`and Atsushi Kubo
`
`TAAA
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`minimized. The optimization modelregards leaf widthsas variables, the total
`area of discrepancy between MLC-shapedfields and the desired fields as an
`objective function, and the total width ofall leaves as a constraint. A problem
`described by the model is solved with the hybrid of a simulated annealing
`technique (ASA, Lester Ingber, 1993) and a gradient technique (DONLP2,
`P Spellucci, 2001). The performanceofthe optimization model was evaluated
`on 634 target fields continuously selected from the patient database of a
`treatment planning system.
`Thelengths of these fields ranged from 3.9 to
`38.7 cm and had an average of 15.3 cm. The total area of discrepancy was
`compared between an MLCwith optimal leaf widths and a conventional MLC
`with the same numberofleaf pairs. Optimal leaf widths were obtained for an
`MLCwith total leaf pairs of 28, 40 and 60, respectively, which corresponded
`to three types of conventional MLCs. The optimal leaf width first decreases
`slightly and then nonlinearly increases with the distance away from the central
`line. Compared with the MLC with conventional leaf width arrangement,
`the MLCs with optimal leaf width arrangement reduced the total area of
`discrepancy by 11.1%, 28.6% and 25.0%, respectively. Optimizing leaf widths
`can either improve the conformity ofMLC-shapedfieldsto the treatment targets
`whenthe numberofleaf pairs does not change, or reduce the numberof leaf
`pairs withoutsacrifice offield conformity.
`
`(Somefiguresin this article are in colouronly in the electronic version)
`
`003 1-9155/09/103051+12$30.00 © 2009 Institute of Physics and Engineering in Medicine Printed in the UK
`
`3051
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`3052
`
`1. Introduction
`
`W Cui and J Dai
`
`The multileaf collimator (MLC)is becoming the standard beam-limiting device of a modem
`accelerator, and a numberofinvestigations have been focused on the MLC’s capability to shape
`radiation fields. Due to the physical width of MLCleaves, a field shaped by an MLC has a
`stepwise boundary and cannot exactly match the desiredfield that has a smooth boundary. The
`conformity betweenthe stepwise boundary and the smooth boundaryatleastpartly dependson
`the width of each leaf. Mostearlier investigations found that smaller leaf widths can provide
`better target conformity and normaltissue sparing (Chern et al 2006, Fiveash et al 2002, Jin
`et al 2005). The optimal leaf width proposed by Bortfeld et al (2000) is about 1.5-1.8 mm
`in the case of a regular single MLCfield dueto limitations caused by the dose deposition
`kernel. However, in almostall investigations, an MLC consists of leaves of the same width.
`This seems unreasonable becauseinnerleaves are used more than outer leaves. The leaf width
`arrangement may be optimized to improve MLC’s capability to shape radiation fields.
`In termsofthe leaf width arrangement, there are two types of MLCscurrently available an
`the market. In one type, each leaf has the same width. Examples include Elekta and Varian’s
`40 leaf pair MLC and Siemens 41 leaf pair MLC. (Note that the leaf widths mentionedin this
`paper are those projected to the isocenter plane.) In another type, leaves may have different
`widths and the numberof leaf widths is 2 or 3. One example of a two-leaf width MLCis
`Varian’s 60 leaf pair MLC. For this MLC,the leaf width is 0.5 cm for inner 40leafpairs and
`1.0 cm for outer 20 leaf pairs. One example of a three-leaf width MLC is Brainlab’s m3 min!
`MLC (Topolnjak and Heide 2008). For this MLC, 26 leaf pairs have leaf widths of 3 mm,
`4.5 mm or 5.5 mm, respectively. The second type of MLC seemsto be the trend for MLC
`design. Up until now, there have been no explanations for such leaf width arrangements,OF
`investigations to find the optimal one.
`Here we introduce an optimization model to address the above issue. The model
`assumes that each leaf may have a different width and determines the optimal leaf width
`arrangementthrough minimizingthetotal area of discrepancyregions between MLC stepwise
`shapes and desired smooth shapes for a group oftarget fields that serve as a sampleof
`desired field population. The minimization problem is solved with a hybrid algorithmof a
`simulated annealing technique (ASA, Lester Ingber, 1993) and a gradient technique (DONLP2,
`P Speilucci, 2001).
`
`2. Methods
`
`2.1. Optimization model
`
`MLCdesign involves a large variety of factors (Topolnjak and Heide 2008, 2007) andvaries
`with different manufacturers. Our discussion here will focus on the arrangement of leaf
`widthsthat can be represented by the MLCleaf geometric projections in the isocenter plane.
`A narrowbarin the isocenter plane represents an MLC leaf, and two banks of closely abutting
`bars constitute an idealized MLC. These two banks are arranged face to face and indicated as
`banks A and B.
`Tofacilitate the optimization of leaf width arrangements, one must define an objective
`function to score different leaf width arrangements. Here, we use a geometric objective
`function that is the total area of discrepancy regions (TAD) between MLC stepwise field
`shapes and desired smooth field shapes.
`In figure 1, TAD is illustrated by an example of
`conforming the MLC boundaryto a desired field shape.
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`Optimizing leaf widths for a multileaf collimator
`
`3053
`
`a o
`
`S.
`
`
`
`Yaxis(cm) ooo
`
`an oS
`
`enlarge
`
`Yaxis]
`
`(cm)owkh&w®I
`
`op=oO.
`
`|
`
`nN
`
`o
`
`a
`b
`Xaxis (cm)
`
`o
`
`~
`
`©
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`©
`
`aaa
`
`3
`
`a
`
`oO
`Xaxis (cm)
`
`a
`
`3
`
`a
`
`Figure 1. The left panel shows the stepwise boundary of the MLC conformedto the smooth
`boundary of a desired field. A part of the Jeft panel (enclosed by a rectangle) is enlarged as shown
`in the right panel, and the discrepancy regions between two boundaries are represented by the
`shaded region.
`
`Dueto the MLC beingapplied to many different targetfields, the leaf width arrangement
`should be determined to have the best conformation on the MLCandall those target fields.
`However, the shapes of those fields are impossible to predict, and an optimization process
`would be unfeasible if dealing with too many fields. Therefore, a group of target fields
`are introduced to serve as a sampleofall those fields which an MLC will be applied to.
`Accordingly, the optimization objective is to minimize TAD for this groupoffields, which
`can be expressed as follows:
`Ns
`Min ) TAD; (AW), AW2,..., AWy,, BWi, BWo,..., BWwg, PA}, PAd, >
`i=l
`
`a)
`PA, PB), PBS,..., PB)
`where N,standsfor the numberoftargetfields, N.4 stands for the numberof leaves in bank A
`and AW;, AW2,..., AWy, denote widths for V4. Similarly, widths for Ny leaves in bank B
`are BW, BW2,..., BWwy,. Positions for ends of leaves in banks A and B are represented by
`PA}, PA},..., PA, and PBi, PBS,..., PBy,, respectively.
`To makethe leaf width arrangement symmetric,the following is assumed: (1) the number
`of leaves in banks A and B are equal; (2) two opposingleavesin one pair have the same width,
`(3) leaf pairs with same distance from the central axis have the same leaf width. According
`to these assumptions,if the numberNofleaf pairs is even, then these leaves should have NV./2
`different widths. The assumption of symmetry not only simplifies the optimization model
`greatly by reducing the numberof variables, but will also make MLC manufacturing easier.
`Now,the optimization objective changes to
`Ns
`Min ) TAD; (Wi, Wo, .... Ww2, PA}, PA}, -.., PAly, PBi, PBS, ..., PBy):
`i=1
`
`(2)
`
`There are three geometric methods to determineleaf positions: the in-field method, the
`out-field method and the cross-field method (Fenwick et al 2004, Frazier et al 1995, Huq
`et al 1995, Mageras 1996, Palta et al 1996, Brahme 1998, Webb 1993, Maet al 2000).
`Depending on where the MLCleaf edge is placed to intersect the prescribed field boundary,
`the cross-field method can be divided into two more sophisticated methods:
`the geometric
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`3054
`
`WCui and J Dai
`
`mean methodand the geometric median method. The geometric mean method sets the MLC
`leaf edge to the mean valueofthe target field boundary within the width of each MLC leaf
`pair. The geometric median method sets the MLC leaf edge to the median value ofthe target
`field boundary within the width of each MLCleafpair.
`It can be proven (Yu ef al 1995)
`that the geometric mean strategy minimizes the difference betweenthe over-blocked andthe
`under-blocked areas while the geometric median strategy minimizes the total areas ofthe
`over-blocked and under-blocked regions. All these geometric methods can be used in our
`model. However, since the geometric median method has the minimum TAD with the same
`leaf width arrangement, which is consistent with the criterion to score different leaf width
`arrangements, we use it here. Now each leaf’s position can be determined individuallyfor
`eachtargetfield according to the geometric median method,andthe final optimization model
`can be expressed as
`
`N.
`f =O TAD(M, We, ..., Ww)
`i=l
`
`N/2
`st. DW) =LWw/2
`j=!
`
`Ww; >0
`
`Vj =1,2,...,.N/2
`
`where LWstandsfor the width ofleaf banks.
`
`2.2. Problem solving
`
`(3)
`
`(4)
`
`()
`
`Because there is no analytical formula for the optimization objective represented by
`equation (3), it is difficult to determine the properties of the optimization model represented
`by equations (3)-(5). What we can determineis the objective value for a set of given variables
`that representa specific leaf width arrangement. To solvethis problem,wetried three different
`types of algorithms. Thefirst one is ASA (adaptive simulated annealing algorithm) which uses
`a global optimization mechanism. With this algorithm, weat least have statistical guarantee
`that the optimization process will not get stuck at a local optimal point (Dai and Que 2004).
`The second one is DONLP2 (a sequential quadratic programming algorithm) which uses
`gradientinformation ofthe optimization objective. With this algorithm, we can get an optimal
`point quickly and accurately. However, it may getstuck at a local optimal point. The third
`one is a hybrid algorithm for optimal nesting problems (Li et al 2003) by combining the
`former two. Codesofthe former two algorithms are downloaded from internet while the third
`one is written by ourselves by utilizing the former two. We expectthat the hybrid algorithm
`can benefit from the advantages of the former two algorithms and avoid their shortcomings.
`Ourpre-test validated our expectation. Although the computation time taken by the hybrid
`algorithm was always more than ASA and much more than DONLP2,it always resulted in the
`smallest objective value amongall three algorithms. Therefore, we chose the hybrid algorithm
`for this study.
`To apply this hybrid algorithm, some modifications must be done to the model in
`equations (3)-(5). The N/2 variables W,, Wo,..., Ww/2, Which represent N/2 leaf widths,
`are divided into g groups sequentially (1 << g¢<WN/2). The numberof variables contained in
`each group is Ny, N2,...,.N,. These parameters could beset to any integer between 1 and
`N/2, but mustsatisfy the constraint Dini Nj; = N/2.
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`Optimizing leaf widths for a multileaf collimator
`
`3055
`
`With the symbol SUB TAD;; to representthe total area of discrepancy for theith target
`field and jth group ofleaves, the optimization model in equations (3)-(5) is transformedinto
`Ns
`8
`
`i=] j=l
`
`Min)> )> SUB TAD: (Wp,s1, Woy425 ++ Woy) (6)
`
`g
`Nj
`st. > Yo Wj = LW/2
`j=l i=l
`
`W; >0
`
`Vj =1,2,...,N/2
`
`(7)
`
`(8)
`
`where the parameter p; in equations (6) and (7) represents thestarting leaf numberin the jth
`group ofleaves.
`An optimization problem described by equations (6)-(8) contains g sub-problems. Each
`of them can be expressedasfollows:
`N,
`
`i=l
`
`Min )>SUB TAD;;(Wp,11, Wp,+2s +--+ Wot) (9)
`
`Nj
`S.t. Wi = GW;
`i=l
`
`Wp+i > 0
`
`Vi = 1,2,...,.Nj
`
`(10)
`
`(1)
`
`where GW;is the sum ofleaf widthsin the jth group.
`In the framework of the hybrid algorithm, DONLP2is usedto solve the sub-problems
`described by equations (9)-(11). Before solving the sub-problems, one must determine
`the parameters GW), GW2,...,GW,. These parameters are handled by ASA. The sum
`of minimum objective values of these sub-problems is the objective value of the original
`optimization problem and what the hybrid algorithm needs to minimize.
`Figure 2 is the flowchart of the full optimization process of the hybrid algorithm. The
`starting point of the optimization process is reading shapesoftargetfields of clinical cases.
`Aninitial solution of parameters GW), GW2, ..., GW, is then randomly generated using the
`random numbergeneration engine of ASA, andtheinitial solutionis savedas the best solution.
`Next, the program enters the most time-consuming part, that is an iteration process aiming
`to find the global optimum solution. For eachiteration loop, ASAfirstly generates a feasible
`solution for parameters GW,, GW, ..., GW, and then stops to wait for DONLP?2 to solve a
`group of g problemsin sub-iteration loops. After that, the objective values of sub-problems
`returned from DONLP2are summedas the objective value of the ASA current solution. If the
`current objective value is smaller than that of the best solution, then replace the best solution
`with the current solution. Otherwise, the Boltzmann acceptancecriterion (Dai and Que 2004)
`is used to judge whether to accept this solution or not. This iteration process will continue
`until convergence,
`
`2.3. Targetfields for model testing
`
`To test the proposed model, we obtained the optimal leaf width arrangement for a group of
`target fields. This group oftarget fields was composed of 634 fields that were continuously
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`
`
`3056 W Cui and J Dai
`
`Read field data
`
`Initialize a solution and save
`this solution as best solution
`
`
`
`Generate a newsolution
`near the current solution
`
`
`
`Accept currentsolution
`according to Boltzmann
`acceptancecriterion
`
`
`
`
`
`
`
`
`
`
`
`Calculate objective
`
`function of ASA
`
`
`
`Is current
`
`solution better
`than the best?
`
`Is any convergence
`
`condition satisfied?
`
`
`
`
`
`Figure 2. Flowchart for optimizing MLC leaf width arrangement.
`
`selected from the patient database of a commercial treatment planning system (Pinnacle’,
`ADACLaboratories, Milpitas, CA, USA). Eachfield had a boundary conformal to a treatment
`target with a margin of 0.5 cm. The boundary was defined with about 100 points in the
`Pinnacle* system. The coordinatesofall points were exportedto a text file, and then read into
`the in-house developed optimization program.
`The total 634 target fields came from 19 head-and-neck cases, 68 thorax cases and 17
`abdomen cases. The area, width (field size in the direction of leaf movement) and length
`(field size in the direction perpendicular to leaf movemcnt)ofthesefields ranged from 20.0 to
`602.7 cm?, 4.0 to 25.9 cm and 3.9 to 38.7 cm, respectively. Their averages were 125.7 +
`
`70.0 cm?, 10.7 + 3.3 cmand 15.3 £68 cm,respectively.
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`Optimizing leaf widths for a multileaf collimator
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`3057
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`3. Results
`
`3.1. Optimal arrangements of leaf widths
`
`Parameters in the optimization model werespecified as follows: the leaf bank width LW was
`set to 40 cm as most of the MLCsappliedin clinic; the numberofleaf pairs N wassetto 28, 40
`and 60, respectively, which corresponded to three types of conventional MLCs (Varian MLCs,
`i.e. standard 52 leaf MLC,! millennium 80 leaf MLC and millennium 120 leaf MLC). When
`N= 28, g was set to 3 and N,, No, N3 were set to 5, 5, 4, respectively. Similarly, when N =
`40, g was set to 4 and the numberof leaves in each group was 5. When N = 60, g was set
`to 5 and the numberofJeaves in each group was 6. The values for g and N,, N2, N3 were
`determined throughthetrial-and-error process.
`Thethree graphsin figure 3 show the optimal arrangements of leaf widths obtained from
`computation results for an MLC with 28, 40 and 60 leaf pairs, respectively.
`In each graph,
`rectangle bars represent MLC leaves. The width and heightof each rectangularbarstandsfor
`the corresponding leaf’s width, and its x coordinates standfor the leaf’s position relative to the
`MLCcentral axis. From these graphs, we make the following observations:
`(1) The most apparent phenomenonis that the width of the outermost leaves is much larger
`than thatof the inner leaves. The width of the outermostleaves is 6.3 cm, 5.8 cm and
`4.5 cm for three MLCs, respectively, whereas that of the innermost leaves is 0.96 cm,
`0.78 cm and 0.50 cm, respectively. This phenomenoncan be explainedbythe factthat the
`majority of target fields are so small that the outermost leaves are not involved in shaping
`them.
`(2) Asleaf positions change from the outermost to the innermost, the leaf width decreases
`rapidly. However,
`leaves near the central axis are not the narrowest. Their widths
`are somehow larger than those leaves outside. For example, when the numberof leaf
`pairs is 40, leaf pair nos 20 and 21 (ie. the innermostlcaf pairs) have a leaf width of
`0.78 cm whereas leaf pair nos 11 and 29 have a width of 0.46 cm thatis the narrowest.
`This phenomenon becomes more apparent as the numberofleaf pairs increases. And
`this may be explained by the fact that boundariesoftarget fields are usually flat near the
`central axis and can be shaped well by widerleaves.
`
`3.2. Analysis of the optimal leaf width arrangement
`
`Thesensitivity of TAD to the smail variations of leaf widths is analyzed as follows: first, each
`leaf was selected separately to increase (or decrease) its width by a small value and widths
`of its one or two neighboring leaves were decreased (or increased) by the same amount in
`total; then the variation of TAD correspondingto the leaf width change wascalculated. When
`the selected leal’s width was increased by 0.2 mm, the TAD was found to have a maximum
`increaseof 0.8% and a minimumincrease of 0.08% for the 28-leaf pair MLC. The maximum
`and minimum increases in TAD for the 40-leaf pair MLC were 0.9% and 0.1%,respectively,
`and thosefor the 60-leaf pair MLC were 1.4% and 0.1%, respectively. Whenthe selectedleaf’s
`width was decreased by 0.2 mm, the TAD was found to have a maximumincrease of 0.7%
`and a minimum increase of0.09% for the 28-leaf pair MLC. The maximum and minimum
`increases in TAD for the 40-leaf pair MLC were 1.0% and 0.1%, respectively, and those
`for the 60-leaf pair MLC were 1.5% and 0.08%, respectively.
`It can be concluded that the
`TADis notvery sensitive to small variations of optimization parameters near the optimal leat
`' For the 52-leaf MLC, we assumethatit has a pair of 7 cmleaveson eachside of the central 26 pairs of 1 cm leaves
`to make it cover a maximum length of 40 cm. Accordingly, the optimized MLC correspondingto this type of MLC
`has 28 pairs of leaves.
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`leafwidth(cm)
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`W Cui and J Dai
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`off axis distance (cm)
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`10
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`off axis distance (cm)
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`5-
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`off axis distance (cm)
`
`10
`
`15
`
`20
`
`Figure 3. Optimal arrangements ofleaf widths for MLCs:(a) 28 leaf pair MLC,(b) 40 leafpair
`MLCand(c) 60 leaf pair MLC.
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`Optimizing leaf widths for a multileaf collimator
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`3059
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`(%)28leafpairMLC
`40leafpairMLC
`
`variation
`
`relative
`
`TAD60leafpairMLC a
`
`: 0
`
`20
`Field length (cm)
`
`40
`
`I
`
`; 0
`
`20
`10
`Field width (cm)
`
`|
`
`0
`
`30
`
`«49400
`200
`Field area (cm?)
`
`600
`
`Figure 4. Distribution of TADrelative variations between optimized MLCsand conventional
`MLCsfor the 634targetfields.
`
`width arrangement. Basedonthis finding, modifications can be performed to the optimization
`results so that the curves shown in figure 3 can be simplified. Taking the 40-leaf pair MLC as
`an example, if we need to simplify the optimized MLCto a three-level leaf width MLC, one
`solutionis to keep the width of the outermost leaf pair unchanged and make the widths ofthe
`next four pairs to be their mean value,i.e. 1.35 cm, and the widths of the other leaf pairs are
`also set to their mean value,i.e. 0.59 cm. Consequently, the TAD will be increased by 3.3%
`to 3.1 cm? compared with that for the MLC with the optimal leaf width arrangement.
`Comparison of the average total area of discrepancy between MLCs with optimalleaf
`width arrangements and conventional MLCsis listed in table 1. Results show that the
`average area of discrepancy for the optimized MLC with 28leafpairs is close to that for
`the conventional 40-leaf pair MLC (4.0 cm? versus 4.2 cm’) while that for the optimized
`MLCwith 40leafpairs is close to that for the conventional 60-leaf pair MLC (3.0 cm?versus
`2.8 cm); optimization reduces the total area of discrepancy by 11.1%, 28.6% and 25.0%
`for three types of MLCs, respectively. Therefore, optimizing the leaf width can improve
`MLC’s capability to shape radiationfields. In other words, for an MLC, without deteriorating
`the capability of field shaping, the numberofleaf pairs can be reduced through the optimal
`arrangement of leaf widths. The values of 11.1%, 28.6% and 25.0% remind usthat the
`conventional 28-leaf pair MLCand 60-leaf pair MLC are closer to the optimal design than the
`conventional 40-leaf pair MLC.
`Figure 4 is a scatter diagram showingthe distribution of TADrelative variations between
`optimized MLCsand conventional MLCsfor the 634target fields. A negative value for one
`field meansthatits conformity is improved while a positive value meansthe opposite. These
`variationsare plotted against three parametersof target fields: field length, field width and
`field area. From figure 4, we can see that points in the left three panels cluster together more
`tightly than those in the other two rowsof panels. It implies that the TAD variation ofa target
`field has a higher correlation with the field length than the field width or the area. The points
`
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`3060
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`W Cui and J Dai
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`Table 1. Comparison ofthe total area of discrepancy between optimized MLCsand conventional
`MLCs.oe
`
`40 leaf pairs
`60 leaf pairs
`28leaf pairs
`
`Varian Optimized—Varian Optimized—Varian Optimized
`
`
`
`
`
`
`
`MLCtype millennium—widthstandard width millennium width
`
`
`Leaf width (cm)
`1.0
`See
`1.0
`See
`Inner
`See
`figure 3(a) —40:0.5 Outer 20:1.0_figure 3(c)figure 3(b)
`
`
`4.0
`3.0
`Average TAD (cm?)
`4.5
`4.2
`2.8
`21
`25.0%
`Reduced by
`11.1%
`28.6%
`-_—_-_eseceeee
`
`in the left three panels do not cluster even moretightly to form a curve, mainly because the
`centers of sometarget fields are away from the midline of the MLC. Fields with the same
`boundaries but different positions relative to the midline of the MLC will have different TAD
`variations. In the left three panels, the figure shown in the upper panel is different from the
`two belowit. This is because we considerthat the conventional 52-leaf MLC has twenty-six
`1 cm leafpairs in the center and two extremely wide leaf pairs (7 cm)outside (see footnote1).
`In all panels, the numberofpoints abovethe horizontal axis is fewer than those below it. That
`meansthe minority of targetfields are sacrificed to improve the TADfor the majority oftarget
`fields. The percentagesoffields with improved conformity are 81.7%, 93.0% and 93.5% for
`the 28, 40 and 60 leaf pair MLCs, respectively. Therefore, the optimization improves the
`conformity in a total view.
`
`4. Discussion
`
`Theresults show that optimal leaf arrangements outperform conventional leaf arrangements.
`But the exact performance difference is affected by the sample of target fields. The sample
`should be large enough and can representall fields that are expected to be shaped with an MLC
`with the optimal leaf arrangement. However, no matter how large the sampleis, there are
`prediction errors and uncertainties. The leaf width arrangement proposed hereis intended for
`a general purpose MLC and wedid notdistinguish tumorsites when collecting targetfields. 3
`If the leaf width arrangementis optimized for a specific tumorsite, the leaf width design will.
`be different and perhaps more suitable for shaping radiation fields. Large centers, where the
`patients are grouped based onthe diseaseto be treated on different machines, may havethis
`need,
`Different definitions of objective functions are supported by the proposed optimization
`model. We used a geometric definition that is the total area of discrepancy. If the dosimetric
`effect needs to be evaluated, isodoselines can be solved analytically according to dose models
`and the discrepancy between isodoselines andtargetfields can be analyzed. In the dose model
`for solving the isodose lines using the convolution and sector summation method (Ma et al
`2000), the dose D(x,y, z) at the position (x, y) and the depth z is calculated with the following
`equation:
`
`D(x, yz) = K(x, y,z) @ W(x, y, Z)
`(12)
`where k(x, y, z) is the dose-spread kernel and W is the beam fluencedistribution atthe depth
`z. It is envisioned that the calculated isodoselines will be smootherthan the stepwise MLC
`boundary, and the discrepancy between isodoselines andtargetfields will not be as significant
`as that between MLC andtarget fields. This dosimetric effect can also be included in the
`proposed model as the objective function to make the model closer to clinical interests.
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`Optimizingleaf widths for a multileaf collimator
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`However, since the implementation of a dosimetric objective function means evaluating
`the conformity of numerous dose distributions to their correspondingfieldsiteratively, the
`optimization becomes much more complicated, even impossible.
`The optimal arrangementofleaf widthstells us that each leaf has a different width. One
`concern aboutthis arrangement is whether the wide leaves should existif the thinnest leaf can
`be manufactured andis rigid enough. An MLCconstructed with the thinnest leaves of course
`showsbetter performancein shaping radiationfields, but also has a higher manufacturing cost
`and may increase MLC downtime. Considering all these issues, combining both wide and
`thin leaves in an MCLis a better choice. Another concern is that manufacturing an MLC
`composedofleaves with different widths will not be easy. An MLC composed ofleaves with
`fewerdifferent widthsis preferred. This problem can be dealt with by two measures. Thefirst
`measureis to add constraints in the optimization modelto express the preference. The second
`measure is to assemble neighboring leaves into groups after the optimalleaf arrangementis
`obtained. The leaves in the same groupare set to have the average oftheir origmal widths.
`Since most inner. leaves have similar widths, these leaves can be divided into two to four
`groups. However, groups of leaf widths will discount the improvement of optimized MLCs.
`A trade-off has to be made between reducing the numberof different widths and improving
`the MLC’s capability.
`The current form of the optimization model is only applicable to one kind of MLCthat
`contains two opposing leaf banks, each bank havingtens ofleaves driven by stepper motors.
`However, the model can be adapted to suit other kinds of MLCs,including binary MLCs
`used in the Peacock system (NOMOSInc.) and Hi-Art Tomotherapy machines (Tomotherapy
`Inc.), and MLCs composed offour leaf banks (Acculeaf of Direx Inc.) or even six leaf banks
`(Topolnjak et al 2004).
`
`§. Conclusion
`
`A flexible optimization modelis proposed to determinethe optimal arrangementof leaf widths
`for an MLC. A hybrid algorithm combined by ASA and DONLP2 is developed to solve the
`corresponding optimization problems. Testresul