`
` , May,200
`Oniine: www. iop. org/joumals/pmb
`
`Published by MPPubnshing on behalf oftheinsiituie
`ofPhyacsandEngMeennghiMedwme
`
`University of Arkansas
`1 Libraries, FayetteviHe
`i PERIODICALS ROOM
`54:10
`QEReceived on: 05—29—09
`3 Physics in medicine &
`
`{bioiogy
`
` £53? Publishing
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`PhysiCs in Medicine 8: Biology
`Volume 54 Number 10 21 Mayzoog
`
`Exact image reconstruction with triple-source saddle-curve cone-beam scanning
`Yang Lu, Jun Zhao and Ge Wang
`Dosimetry for the MRI accelerator: the impact of a magnetic field on the response of a
`Farmer NE2571 ionization chamber
`'
`IMeijsing, B W Raaymakers, A J E Raaijmakers, J G M Kok. L Hogeweg, B Lin and
`J J W Lagendijk
`
`A micro-machined retro-reflector for improving light yield in ultra-high-rcsolution gamma
`cameras
`
`Jan W T Hcemskerk, Marc A N Korevaar, Rob Kreuger, C M Li gtvoet, Paul Schotanus and
`Freek J Beekman
`
`Development of 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-Vim Lee
`Optimizing leaf widths for a multileaf collimator
`Weijie Cui and Jianrong Dai
`Electrical impedance spectroscopy as a potential tool for recovering bone porosity
`C Bonifasi-Lista and E Cherkaev
`
`Evaluation of a compartmental model for estimating tumor hypoxia via FMISO dynamic
`PET imaging
`Wenli Wang, Jens-Christoph Georgi, Sadek A Nchmeh, Manoj Narayanan, Timo Paulus,
`Matthieu Ba], Joseph O’Donoghue, Pat B Zanzonico, C Ross Schmidtlein, Nancy Y Lee and
`John L Humm
`
`Reduction of the number of stacking layers in proton uniform scanning
`Shinichiro Fujitaka, Taisuke Takayanagi, Rintam 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
`trauscranial magnetic stimulation
`R Salvador, P C Miranda, Y Roth and A Zangen
`Audio frequency in viva optical coherence elastography
`Steven G Adie, Brendan F Kennedy, Julian J AImstrong,-Sergey A Alexandrov and
`David D Sampson
`’
`Computed tomography dose assessment for a 160 mm wide, 320 detector row, cone beam
`CT scanner
`J Gcleijns, M Salvadé Artclls, 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 Lautamiiki, Martin A Lodge, Frank M Bengel and
`Benjamin M W Tsui
`,
`The influence of a novel transmission detector on 6 MV x-ray beam characteristics
`Sankar Venkatararnan, Kyle E Malkoske, Maxtin Jensen, Keith D Nakonechny, Ganiyu Asuni
`and Boyd M C McCurdy
`
`297 1
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`2993
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`3003
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`3015
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`3031
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`3051
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`3063
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`3083
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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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`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-energy radiation monitoring based on radio-fluorogenic co-polymerization. 1: small
`volume in situ probe
`J M Warman‘ M P de Haas and L H Luthjens
`An ultrasound cylindrical phased array for deep heating in the breast: theoretical design
`using heterogeneous models
`IF Bakker, M M Paulides, I M Obdeijn, G C 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 Dilmam'an, L Faulconer, T Liu, C Parham, E D Pisano and Z Zhong
`Development and validation of a beam model applicable to small fields
`P Caprile and G H Hartmann
`
`Evaluation of registration strategies for multi-modality images of rat brain slices
`Christoph Palm, Andrea Vieten, Dagmar Salber and Uwe Pietrzyk
`
`NOTES
`
`Development of a remanence measurement-based SQUID system with in-depth resolution
`for nanoparticle imaging
`Song Ge, Xiangyang Shi, James R Baker Jr, Mark M Banaszak H011 and Bradford G Orr
`Modeling time variation of blood temperature in a bioheat equation and its application to
`temperature analysis due to RF exposure
`Akimasa Hirata and Osamu Fujiwara
`The reproducibility of a HeadFix relocatahle fixation system: analysis using the stereotactic
`coordinates of bilateral incus and the top of the crista galli obtained from a serial CT scan
`Etsuo Kunieda, Yohci Oku, Junichi Fukada, Osamu Kawaguchi, Hideyuki Shiba, Atsuya Takeda
`and Atsushi Kubo
`
`HIHHIHEHHHIIHIHIHIliHlHIillHIHIHHHHIHIHIIIHMIIIHHIIIll”)
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`PHYSICS )N MEDICINE AND BIOLOGY
`IOP PUBLISHING
`
`
`Phys. Med. Biol. 54 (2009) 3051—3062
` rm mm may be maimed bvcowrgm law [Yule 17 us (069)
`
`
`
`doi: l 0.] 088/0031-9155/54/10/006
`
`Optimizing leaf widths for a multileaf collimator
`
`Weijie Cui and Jianrong Dai
`
`Department of Radiation Oncology, Cancer Hospital (Institute), Chinese Academy of Medical
`Sciences, Beijing 100021, People‘s Republic of China
`
`E—mail: jianrong_dai@yahoo.com
`
`Received 23 December 2008, in final form 9 March 2009
`Published 27 April 2009
`Online at stacks.inp.org/PMB/54/305l
`
`Abstract
`
`The multilcaf collimator (MLC) is becoming a standard accessory of modern
`linac in shaping radiation fields. However, for a given target (projection),
`the radiation field shaped by an MLC has a stepwise boundary and is not
`identical to the desired field that exactly conforms to the target. That means
`there are always under—blocked and/ or over-blocked areas. The total area of
`discrepancy depends on MLC leaf widths. The purpose of this study is to
`develop an optimization model for determining leaf widths so that the total
`area of discrepancy between MLC—shaped fields and the desired ones can be
`minimized. The optimization model regards leaf widths as variables, the total
`area of discrepancy between MLC-shaped fields and the desired fields as an
`objective function, and the total width of all leaves as a constraint. A problem
`described by the model is solved with the hybrid of a simulated annealing
`technique (ASA, Lester lngber, 1993) and a gradient technique (DONLPZ,
`P Spellucci, 2001). The performance of the optimization model was evaluated
`on 634 target fields continuously selected from the patient database of a
`treatment planning system. The lengths 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 MLC with optimal leaf widths and a conventional MLC
`with the same number of leaf pairs. Optimal leaf widths were obtained for an
`MLC with 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 of MLC—shaped fields to the treatment targets
`when the number of leaf pairs does not change, or reduce the number of leaf
`pairs without sacrifice of field conformity.
`
`(Some figures in this article are in colour only in the electronic version)
`
`00314)]55/09/103051+l2$3000 © 2009 Institute of Physics and Engineering in Medicine Printed in the UK
`
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`1. Introduction
`
`W Cui and I Dai
`
`The multileaf collimator (MLC) is becoming the standard beam-limiting device of a modern
`accelerator, and a number of investigations have been focused on the M LC’s capability to shape
`radiation fields. Due to the physical width of MLC leaves, a field shaped by an MLC has a
`stepwise boundary and cannot exactly match the desired field that has a smooth boundary. The
`conformity between the stepwise boundary and the smooth boundary at least partly dependson
`the width of each leaf. Most earlier investigations found that smaller leaf widths can provrde
`better target conformity and normal tissue sparing (Chem et al 2006, Fiveash et al 2002, Im
`at al 2005). The optimal leaf width proposed by Bortfeld er a] (2000) is ab0ut 1.5;1.8_Inn1
`in the case of a regular single MLC field due to limitations caused by the dose deposmon
`kernel. However, in almost all investigations, an MLC consists of leaves of the same width.
`This seems unreasonable because inner leaves are used more than outer leaves. The leaf width
`arrangement may be optimized to improve MLC’s capability to shape radiation fields.
`In terms of the leaf width arrangement, there are two types of MLCs currently available on
`the market. In one type, each leaf has the same width. Examples include Elekta and Vanans
`40 leaf pair MLC and Siemens 41 leaf pair MLC. (Note that the leaf widths mentioned in thls
`paper are those projected to the isoeenter plane.) In another type, leaves may have different
`widths and the number of leaf widths is 2 or 3. One example of a two—leaf width MLC 15
`Varian’s 60 leaf pair MLC. For this MLC, the leaf width is 0.5 cm for inner 40 leaf pairs and
`1.0 cm for outer 20 leaf pairs. One example of a three-leaf width MLC is Brainlab’s m3 mlnl
`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 seems to be the trend for MLC
`design. Up until now, there have been no explanations for such leaf width arrangements. 0f
`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
`arrangement through minimizing the total area of discrepancy regions between MLC stepwise
`shapes and desired smooth shapes for a group of target fields that serve as a sample of
`desired field population. The minimization problem is solved With a hybrid algorithm of a
`simulated annealing technique (ASA, Lester Ingber, 1993) and a gradient technique (DONLPZ.
`P Spellucei, 2001).
`
`2. Methods
`
`2.1. Optimization model
`
`MLC design involves a large variety of factors (Topolnjak and Heide 2008, 2007) and Vflfies
`with different manufacturers. Our discussion here will focus on the arrangement of leaf
`widths that can be represented by the MLC leaf geometric projections in the isoeenter plane.
`A narrow bar in the isoeenter 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.
`
`To facilitate 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 0f
`conforming the MLC boundary to a desired field shape.
`
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`Optimizing leaf widths for a multileaf collimator
`u:
`
`m
`M
`
`_‘_
`
`3053
`
`e
`3
`‘2
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`>-
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`5
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`31
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`.3
`
`4-5
`
`'15
`-15
`
`.
`
`‘
`-1o
`
`.5
`
`o
`Xaxis (em)
`
`5
`
`1o
`
`_.
`15
`
`.ei. i,” .
`O
`
`1
`
`2
`
`3
`
`5
`4
`Xaxis (cm)
`
`6
`
`7
`
`8
`
`9
`
`Figure 1. The left panel shows the stepwise boundary of the MLC conformed to the smooth
`boundary of a desired field, A part of the left 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.
`
`Due to the MLC being applied to many different target fields, the leaf width arrangement
`should be determined to have the best conformation on the MLC and all 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 sample of all those fields which an MLC will be applied to.
`Accordingly, the optimization objective is to minimize TAD for this group of fields, which
`can be expressed as follows:
`N:
`
`MinZTADi(AW1. AWZ, .
`izl
`
`.
`
`.
`
`, AWN“ 3W1, 3W2. .
`
`.
`
`.
`
`, BWNB, PA; PA'Z, .
`
`. .,
`
`(1)
`I’AZVA,PBi,PB§,...,PB§VB)
`where N, stands for the number of target fields, NA stands for the number of leaves in bank A
`and A W1, A W2, .
`.
`. , AWNA denote widths for NA. Similarly, widths for N3 leaves in bank B
`are B W1, BWz, .
`. , B WNB. Positions for ends of leaves in banks A and B are represented by
`.
`PA"P PAg, .
`.
`PAf'NA and 193;, P35, .
`.
`PBfVH, respectively.
`To make the leaf width arrangement symmetric, the following is assumed: (l) the number
`of leaves in banks A and B are equal: (2) two opposing leaves in 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 number N of leaf pairs is even, then these leaves should have N/2
`different widths. The assumption of symmetry not only simplifies the optimization model
`greatly by reducing the number of variables, but will also make MLC manufacturing easier.
`Now, the optimization objective changes to
`Nr
`
`MinZTAD,(W1, W2, ...,WN/2, PA", PA", .
`i=1
`
`. ., PAi ,PBj, PB§,..., Pij).
`
`(2)
`
`There are three geometric methods to determine leaf positions: the in—field method, the
`out-field method and the cross—field method (Fenwick et al 2004, Frazier er al 1995, Huq
`et al 1995, Mageras 1996, Palta et al 1996, Brahme 1998, Webb 1993, Ma et a] 2000).
`Depending on where the MLC leaf 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
`
`W Cui and I Dai
`
`mean method and the geometric median method. The geometric mean method sets the MLC
`leaf edge to the mean value of the target field boundary within the width of each MLC leaf
`pair. The geometric median method sets the MLC leaf edge to the median value of the target
`field boundary within the width of each MLC leaf pair.
`It can be proven (Yu et al 1995)
`that the geometric mean strategy minimizes the difference between the over-blocked and the
`under-blocked areas while the geometric median strategy minimizes the total areas of the
`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 individually for
`each target field according to the geometric median method, and the final optimization model
`can be expressed as
`
`N;
`
`f = Emma/1, W2, .
`i=1
`
`. ., WW)
`
`N/Z
`
`st. 2 Wj = LW/Z
`['21
`
`Wj>0
`
`Vj:1,2,...,N/2
`
`where LW stands for the width of leaf banks.
`
`2.2. Problem solving
`
`(3)
`
`(4)
`
`(5)
`
`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 determine is the objective value for a set of given variables
`that represent a specific leaf width arrangement To solve this problem, we tried three different
`types of algorithms. The first one is ASA (adaptive simulated annealing algorithm) which uses
`a global optimization mechanism. With this algorithm, we at least have statistical guaranl‘ie
`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
`gradient information of the optimization objective. With this algorithm, we can get an optimal
`point quickly and accurately. However, it may get stuck at a local optimal point. The third
`one is a hybrid algorithm for optimal nesting problems (Li er al 2003) by combining the
`former two. Codes of the former two algorithms are downloaded from internet while the third
`one is written by ourselves by utilizing the former two. We expect that the hybrid algorithm
`can benefit from the advantages of the former two algorithms and avoid their shortcomings-
`Our pretest 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 among all 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 W1, W2, .
`.
`.
`, WW2, which represent N/2 leaf widthS,
`are divided into g groups sequentially (l s g g N /2). The number of variables contained in
`each group is N1, N2, .. .
`, Ng. These parameters could be set to any integer between 1 and
`N/2, but must satisfy the constraint 25:1 NJ» = N/2.
`
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`Optimizing leaf widths for a multileaf collimator
`
`3055
`
`With the symbol SUB TAD” to represent the total area of discrepancy for the ith target
`field and jth group of leaves, the optimization model in equations (3)—(5) is transformed into
`NS
`g
`
`Min
`
`ZSUB 'l‘ADiJ-(ijH, ij+2, .
`i=1 j=1
`
`.
`
`.
`
`, ijmj)
`
`3’
`
`Ni
`
`st. 2 Z W,,J.-,- : LW/Z
`j:| i=1
`
`Wj>0
`
`Vj=1,2....,N/2
`
`(6)
`
`(7)
`
`(8)
`
`where the parameter pj in equations (6) and (7) represents the starting leaf number in the jth
`group of leaves.
`An Optimization problem described by equations (6)48) contains g sub—problems. Each
`of them can be expressed as follows:
`N
`
`Min 2 SUB TADU- (WPJH, WW2, .
`i=1
`
`.
`
`.
`
`, WWNJ.)
`
`N,-
`
`i=1
`
`s.t.
`
`WW = CW]
`
`HIM->0
`
`Vi=1,2,...,N,
`
`(9)
`
`(10)
`
`(11)
`
`where G Wj is the sum of leaf widths in thejth group.
`In the framework of the hybrid algorithm, DONLP2 is used to solve the sub—problems
`described by equations (9)—(ll). Before solving the sub-problems, one must determine
`the parameters GW], G W2, .
`.
`.
`, CW3. 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 shapes of target fields of clinical cases.
`An initial solution of parameters GW] , GWZ, .
`.
`.
`, GWg is then randomly generated using the
`random number generation engine of ASA, and the initial solution is saved as 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 each iteration loop, ASA firstly generates a feasible
`solution for parameters GW1, GWZ, .
`.
`.
`, GWg and then stops to wait for DONLP2 to solve a
`group of g problems in sub—iteration loops. After that, the objective values of sub-problems
`returned from DONLPZ are summed as 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 acceptance criterion (Dai and Que 2004)
`is used to judge whether to accept this solution or not. This iteration process will continue
`until convergence.
`
`2.3. Tm‘getfieldsfor model testing
`
`To test the proposed model, we obtained the optimal leaf width arrangement for a group of
`target fields. This group of target fields was composed of 634 fields that were continuously
`
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`3056 W Cui and .l Dai
`
`
`lnitialile a solution and save
`
`
`this solution as best solution
`
`
`Replace the best solution
`
`with the current one
`
`
`
`
`Read field data
`
`Generate a new solution
`near the current solution
`
`
`
`Calculate objective
`function ofASA
`
`Is current
`solution better
`
`than the best?
`
`Accept current solution
`according to Roltlmann
`acceptance criterion
`
`
`
`Is any convergence
`condition satisfied?
`
`
`
`
`
`
`Figul'c 2. Flowchart for optimizing MLC leaf Width arrangement.
`
`selected from the patient database of a commercial treatment planning system (Pinnacle3,
`ADAC Laboratories, Milpitas, CA, USA). Each field 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
`Pinnacle3 system. The coordinates of all points were exported to 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 movement) of these fields ranged from 20.0 to
`602.7 cmz, 4.0 to 25.9 cm and 3.9 to 38.7 cm, respectively. Their averages were 125.7 :l:
`
`70.0 c1112, 10.7 :l: 3.3 cm and 15.3 :: 6.8 cm, respectively.
`
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`Optimizing leaf widths for a multileaf collimator
`
`3057
`
`3. Results
`
`3.]. Optimal arrangements of leaf widths
`
`Parameters in the optimization model were specified as follows: the leaf bank width LW was
`set to 40 cm as most of the MLCs applied in clinic; the number of leaf pairs N was set to 28, 40
`and 60, respectively, which corresponded to three types of conventional MLCs (Varian MLCs,
`i.e. standard 52 leaf MLC,1 millennium 80 leaf MLC and millennium 120 leaf MLC). When
`N: 28, g was set to 3 and N1, N2, N3 were set to 5, 5, 4, respectively. Similarly, when N:
`40, g was set to 4 and the number of leaves in each group was 5. When N = 60, g was set
`to 5 and the number of leaves in each group was 6. The values for g and N1, N2. N3 were
`determined through the trial -and-e1ror process.
`The three graphs in 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 height of each rectangular bar stands for
`the corresponding leaf’3 width, and its x coordinates stand for the leaf’5 position relative to the
`MLC central axis. From these graphs, we make the following observations:
`
`(1) The most apparent phenomenon is that the width of the outermost leaves is much larger
`than that of the inner leaves. The width of the outermost leaves is 63 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 phenomenon can be explained by the fact that the
`majority of target fields are so small that the outermost leaves are not involved in shaping
`them.
`
`(2) As leaf 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 number of leaf
`pairs is 40, leaf pair nos 20 and 21 (Le. the innermost leaf pairs) have a leaf width of
`0.78 cm whereas leaf pair nos 11 and 29 have a width of 0.46 cm that is the narrowest.
`This phenomenon becomes more apparent as the number of leaf pairs increases. And
`this may be explained by the fact that boundaries of target fields are usually flat near the
`central axis and can be shaped well by wider leaves.
`
`3.2. Analysis of the optimal leafwidth arrangement
`
`The sensitivity of TAD to the small 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 corresponding to the leaf width change was calculated. When
`the selected leafs width was increased by 0.2 mm, the TAD was found to have a maximum
`increase of 0.8% and a minimum increase of 0.08% for the 28-1eaf pair MLC. The maximum
`and minimum increases in TAD for the 40—leaf pair MLC were 0.9% and 0.1%, respectively,
`and those for the 60-leaf pair MLC were 1.4% and 0.1%, respectively. When the selected leaf’s
`width was decreased by 0.2 mm, the TAD was found to have a maximum increase of 0.7%
`and a minimum increase of 0.09% for the 28—1eaf 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 6041eaf pair MLC were 1.5% and 0.08%, respectively.
`It can be concluded that the
`TAD is not very sensitive to small variations of optimization parameters near the optimal leaf
`' For the 52—leaf MLC, we assume that it has a pair of 7 cm leaves on each side of the central 26 pairs of 1 cm leaves
`to make it cover a maximum length of 40 em. Accordingly, the optimized MLC corresponding to this type of MLC
`has 28 pairs of leaves.
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`leafwidth
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`(cm)
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`40'
`6
`
`.15
`
`.10
`
`5
`o
`-5
`off axis chstance (cm)
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`1o
`
`15
`
`20
`
`(b)
`
`—
`-15
`
`~10
`
`5
`0
`5
`off axis distance (cm)
`(0)
`
`10
`
`m
`20
`
`15
`
`5
`
`E 48
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`E 3E“-4(V
`.2 2
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`1 0
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`—20
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`5 ,
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`
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`leafwidth(cm)
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` 0
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`—20
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`-15
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`~10
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`5
`O
`-5
`off axis distance (cm)
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`1D
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`,
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`15
`
`20
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`Figure 3. Optimal arrangements of leaf widths for MLCs: (a) 28 leaf pair MLC, (b) 40 leaf pair
`MLC and (c) 60 leaf pair MLC.
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`Optimizing leaf widths for a multileaf collimator
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`("/0)28leafpairMLC
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`40leafpairMLC
`
`variation
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`relative
`
`TAD
`
`BUleafpairMLC 10
`
`Field length (cm)
`
`20
`Field width (cm)
`
`30
`
`4GB
`200
`Field 5,93 (cmz)
`
`500
`
`Figure 4. Distribution of TAD relative variations between optimized MLCs and conventional
`MLCs for the 634 target fields.
`
`width arrangement. Based on this 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 MLC to a three-level leaf width MLC, one
`solution is to keep the width of the outermost leaf pair unchanged and make the widths of the
`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 cm2 compared with that for the MLC with the optimal leaf width arrangement.
`Comparison of the average total area of discrepancy between MLCs with optimal leaf
`width arrangements and conventional MLCs is listed in table 1. Results show that the
`average area of discrepancy for the optimized MLC with 28 leaf pairs is close to that for
`the conventional 40-1eaf pair MLC (40 cm2 versus 4.2 cm2) while that for the optimized
`MLC with 40 leaf pairs is close to that for the conventional 60—leaf pair MLC (3.0 cm2 versus
`2.8 cmz); 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 radiation fields. In other words, for an MLC, without deteriorating
`the capability of field shaping, the number of leaf pairs can be reduced through the optimal
`arrangement of leaf widths. The values of 11.1%, 28.6% and 25.0% remind us that the
`conventional 28-leaf pair MLC and 60-leaf pair MLC are closer to the optimal design than the
`conventional 40—leaf pair MLC.
`Figure 4 is a scatter diagram showing the distribution of TAD relative variations between
`optimized MLCs and conventional MLCs for the 634 target fields. A negative value for one
`field means that its conformity is improved while a positive Value means the opposite. These
`variations are plotted against three parameters of 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 rows of panels. It implies that the TAD variation of a target
`field has a higher correlation with the field length than the field width or the area. The points
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`Table 1. Comparison of the total area of discrepancy between optimized MLCS and conventional
`MLCs.———_______________
`
`40 leaf pairs
`60 leaf pairs
`28 leaf pairs
`
`Optimized
`Varian
`Optimized
`Varian
`Optimized
`Varian
`MLC type
`standard
`width
`millennium
`width
`millennium
`width
`
`Leaf width (cm)
`1.0
`See
`10
`See
`lnner
`See
`figure 3(a)
`figure 3(b)
`40:05 Outer 20:1.0
`figure 3(a)
`4,2
`4.0
`3.0
`2.3
`2.1
`4.5
`Average TAD (cmz)
`
`
`11.1% 28.6%Reduced by 25.0%M—
`
`
`in the left three panels do not cluster even more tightly to form a curve, mainly because the
`centers of some target 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 below it. This is because we consider that the conventional SZ—leaf MLC has tWenty-six
`1 cm leaf pairs in the center and two extremely wide leaf pairs (7 cm) outside (see footnote 1).
`In all panels, the number of points above the horizontal axis is fewer than those below it. That
`means the minority of target fields are sacrificed to improve the TAD for the majority of target
`fields. The percentages of fields 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
`
`The results 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 represent all fields that are expected to be shaped with an MLC
`with the optimal leaf arrangement. However, no matter how large the sample is, there are
`prediction errors and uncertainties. The leaf width arrangement proposed here is intended for
`a general purpose MLC and we did not distinguish tumor sites when collecting target fields. .
`If the leaf width arrangement is optimized for a specific tumor site, the leaf width design wil l.
`be different and perhaps more suitable for shaping radiation fields. Large centers, where the
`patients are grouped based on the disease to be treated on different machines, may have this
`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 dosimetiic
`effect needs to be evaluated, isodose lines can be solved analytically according to dose models
`and the discrepancy between isodose lines and target fields 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:
`
`DOC: )7, Z) = 1605, y. Z) ® “’0'. y. z)
`
`(12)
`
`where k(x. y, z) is the dose—spread kernel and Ill is the beam fluence distribution at the depth
`2. It is envisioned that the calculated isodose lines will be smoother than the stepwise MLC
`boundary, and the discrepancy between isodose lines and target fields will not be as significant
`as that between MLC and target 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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`However, since the implementation of a dosimetric objective function means evaluating
`the conformity of numerous dose distributions to their corresponding fields iteratively, the
`optimization becomes much more complicated, even impossible.
`The optimal arrangement of leaf widths tells us that each leaf has a different width. One
`concern about this arrangement is whether the wide leaves should exist if the thinnest leaf can
`be manufactured and is rigid enough. An MLC constructed with th