Multiple local minima in IMRT optimization based on dose–volume criteria

Multiple local minima in IMRT optimization based on dose–volume criteriaQiuwen Wu and Radhe Mohan

Citation: Medical Physics 29, 1514 (2002); doi: 10.1118/1.1485059 View online: http://dx.doi.org/10.1118/1.1485059 View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/29/7?ver=pdfcov Published by the American Association of Physicists in Medicine Articles you may be interested in A novel dose–volume metric for optimizing therapeutic ratio through fractionation: Retrospective analysis of lungcancer treatments Med. Phys. 40, 084101 (2013); 10.1118/1.4812884 Reduced order constrained optimization (ROCO): Clinical application to lung IMRT Med. Phys. 38, 2731 (2011); 10.1118/1.3575416 Speed and convergence properties of gradient algorithms for optimization of IMRT Med. Phys. 31, 1141 (2004); 10.1118/1.1688214 A comparison of physically and radiobiologically based optimization for IMRT Med. Phys. 29, 1447 (2002); 10.1118/1.1487420 Multiple local minima in radiotherapy optimization problems with dose–volume constraints Med. Phys. 24, 1157 (1997); 10.1118/1.598017

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Multiple local minima in IMRT optimization based on dose–volume criteriaQiuwen Wua) and Radhe MohanDepartment of Radiation Oncology, Medical College of Virginia Campus, Virginia CommonwealthUniversity and McGuire VA Hospital, Richmond, Virginia 23298

~Received 25 January 2002; accepted for publication 8 April 2002; published 21 June 2002!

Multiple local minima traps are known to exist in dose–volume and dose-response objective func-tions. Nevertheless, their presence and consequences are not considered impediments in findingsatisfactory solutions in routine optimization of IMRT plans using gradient methods. However,there is often a concern that a significantly superior solution may exist unbeknownst to the plannerand that the optimization process may not be able to reach it. We have investigated the soundnessof the assumption that the presence of multiple minima traps can be ignored. To find local minima,we start the optimization process a large number of times with random initial intensities. Weinvestigated whether the occurrence of local minima depends upon the choice of the objectivefunction parameters and the number of variables and whether their existence is an impediment infinding a satisfactory solution. To learn about the behavior of multiple minima, we first used asymmetric cubic phantom containing a cubic target and an organ-at-risk surrounding it to optimizethe beam weights of two pairs of parallel-opposed beams using a gradient technique. The phantomstudies also served to test our software. Objective function parameters were chosen to ensure thatmultiple minima would exist. Data for 500 plans, optimized with random initial beam weights, wereanalyzed. The search process did succeed in finding the local minima and showed that the numberof minima depends on the parameters of the objective functions. It was also found that the conse-quences of local minima depended on the number of beams. We further searched for the multipleminima in intensity-modulated treatment plans for a head-and-neck case and a lung case. In addi-tion to the treatment plan scores and the dose–volume histograms, we examined the dose distribu-tions and intensity patterns. We did not find any evidence that multiple local minima affect theoutcome of optimization using gradient techniques in any clinically significant way. Our studysupports the notion that multiple minima should not be an impediment to finding a good solutionwhen gradient-based optimization techniques are employed. Changing the parameters for the ob-jective function had no observable effect on our findings. ©2002 American Association of Physi-cists in Medicine. @DOI: 10.1118/1.1485059#

Key words: intensity-modulated radiotherapy, optimization, local minima, dose–volume criteria

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I. INTRODUCTION

In intensity-modulated radiotherapy~IMRT!, the goal of theoptimization system is to find a treatment plan that corsponds to the extremum~minimum or maximum! value ofthe objective function. The value of the objective functionalso termed the ‘‘score’’ of the treatment plan. Realistic ojective functions, e.g., dose–volume-based, that are curreemployed in IMRT optimization, have been shown to contmultiple local extrema.1,2 There is often a concern that thsearch process may get trapped in a local extremum fafrom the best solution possible~the global extremum!, andthat the solution found might not be satisfactory at all.~Con-sidering that objective functions are imperfect, even if tglobal extremum is reached, the plan may not be satisfaconly that it is optimum according to the criteria specified!Thus, it is important to investigate whether and under wcircumstances the existence of such local extremum tmay be an obstacle. In this paper, we confine our investtions to dose–volume-based objective functions, whichthe most commonly used objective functions in curre

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IMRT systems. However, it is likely that similar consideations and conclusions may apply to dose-response-bobjective functions. For dose–volume-based objective futions, the aim is to minimize the treatment plan score, whtypically consists of the weighted sum of variances of dodistributions in each of a number of anatomic structurTherefore, from this point on in this paper, we will use tterm ‘‘minimum’’ instead of ‘‘extremum.’’

Figure 1~a! is a schematic plot of the treatment plan scoas a function of an optimization variable~e.g., beam weight,ray intensity, etc.! illustrating multiple local minima. In thissimple example, two local minimaB andC are equally deep,and either of them would be equally acceptable accordingthe numerical score. In general, however, local minima arediffering depths. When a gradient optimization algorithsearches for a solution, it changes the optimization variabin small steps at a time and examines the score. The sand the magnitudes of the changes depend upon the detive of the objective function with respect to the variableThis process is repeated until the change in score is smthan some specified convergence criteria or the score be

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1515 Q. Wu and R. Mohan: Multiple local minima 1515

to increase. At the point of search termination, the solutfound may correspond to the global minimum or may habeen trapped in another minimum with a significantly pooscore.

It has been recommended that stochastic optimizamethods, such as simulated annealing or its various hspeed variants, be used to overcome the obstacles creatmultiple local minima traps. The simulated annealing aproach, for instance, allows the search process to occasally ‘‘climb up hill’’ or ‘‘tunnel through’’ the peaks intodeeper valleys to enable the search process to escapelocal traps. Stochastic search processes are consideslower than gradient approaches. Furthermore, especialIMRT problems, where the number of variables can be in

FIG. 1. ~a! An objective function illustrating multiple local minima. Therare four valleys in the objective function. A and D are the local minima, aB and C are the global minima. A and D and B and C have the same sand are degenerate.~b! The dose–volume objective functions. Only thpoints with a dose between D1 and D2 are considered in the score evaluatioand points with a dose above D2 are ignored.

Medical Physics, Vol. 29, No. 7, July 2002

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tens of thousands, there is no assurance that a global mmum has been reached, only that the best among the conations examined has been selected. Stochastic technhave been applied for optimizing three-dimensional confmal treatment plans since the early 1990s.3–7 Some investi-gators and at least one commercial IMRT system~NOMOS,Sewickley, PA! have employed stochastic methods to opmize intensity distributions, whereas most other investigatand commercial systems have used gradient techniques.question often asked is whether the latter are missing besolutions or whether the former are unnecessarily applyinmuch more computationally intensive technique.

Whether or not multiple local minima are present awhether they are an impediment to finding suitable solutiodepend on the nature of the objective function and its pareters, constraints, beam configurations, and the anatomyexample, it can be shown that there are no multiple lominima in the objective functions defined purely in termsdose variance. While existence of multiple minima is indpendent of the optimization algorithm, the choice of algrithm has an impact on whether the search process is abavoid the traps. As stated previously, in principle, stochamethods possess a greater potential in this regard. Moand Mageras8,9 explored the use of the fast simulated anneing method and an objective function defined in termsbiological indices to optimize beam weights. They demostrated that multiple minima do exist and that it is necessto use stochastic methods for such situations~beam weightoptimization and biology-based objectives!. Deasy1 investi-gated the multiple minima problems when dose–voluconstraints were added to different objective functions.pointed out that multiple minima may exist and asserted t‘‘possible local minima come not from the nonlinear natuof the objective functions considered, but from the ‘eiththis volume or that volume but not both’ nature of the voume effect.’’ Chui and Spirou2 also showed an example omultiple minima for the beam weight optimization on a twdimensional phantom with two beams. Multiple locminima have also been found to occur in beam anoptimizations.10–12

The optimization process may converge to solutions tare different in one of many different ways. Therefore, itimportant to distinguish among various types of locminima.

Type 1:In this type, the optimization process convergestreatment planscores that are significantly different fromeach other. This type is perhaps the most important sinceplan scores indicate that the solutions found are clearlyferent.

Type 2:In this type, optimization converges to plans thhave similar scores~insignificantly different!, but thedose–volume histograms~DVHs! of anatomic structures are significantly different. This may happen, for example, if thDVHs of two optimized plans for an organ-at-risk~OAR!cross at the point where the dose–volume constraint isfined as yielding the same contribution to the score. BDVHs may be equally acceptable. If they are not, it indicainadequately defined dose–volume criteria. Similarly, if t

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two DVHs are nested and they both satisfy the constraboth of them contribute zero to the score. However, thetimization system is unable to distinguish that the lower lone actually causes less damage.

Type 3:In this type, the scores as well as DVHs are simlar, but thedose distributionsin one or more anatomic structures are significantly different. This happens when eqvolumes in different parts of an anatomic structure recethe same dose. This is the type pointed out by Deasymentioned previously, and is due to the degeneracyDVHs. If DVHs are used to evaluate plans, both solutiowill be equally acceptable. However, if dose distributionsused for plan evaluation, one~or more! of the solutions maybe considered superior compared with others due to thecation of a specific dose level, information that is hiddenDVHs.

Type 4:In this type, scores, DVHs, and dose distributioare similar, butintensity distributionsare different. This canoccur for different reasons. Different beam weights or intesity distributions in different beams could, in principle, combine to produce dose distributions that are very nearlysame. Furthermore, the IMRT optimization process hasherent high frequency noise. Intensity distributions thatbroadly similar but different in detail often produce vesimilar dose distributions. This type of minima is not impotant from the viewpoint of patient dose distributions but mhave an impact on optimum leaf trajectories of multileaf climators to deliver dose distributions.

We should underscore the use of the word ‘‘similar’’ istead of ‘‘same.’’ Considering the fact that convergenceteria have a nonzero value, and that in computer optimizaand dose calculations, all quantities and processes arecrete, it is extremely unlikely that starting with different intial conditions will lead to exactly the same results. The te‘‘similar’’ is used to imply plans that are clinically indistinguishable.

II. METHODS AND MATERIALS

We assume that, if multiple local minima exist, they cbe located by starting from a large number of randomlylected initial positions in the solution space~i.e., beamweights or ray intensities!. For the cases presented in thpaper, except one, we chose to run the optimization pro500 times.~For one case, we ran the optimization syste2000 times.! These numbers were somewhat arbitraryconsidered sufficiently large, yet at the same time smenough to allow completion of this project within a reasoable period of time. Each optimization run was performfor the exact same phantom or patient geometry, beamrangement, and optimization criteria.

An IMRT system, developed in-house and describelsewhere,13 was used in this study. The optimization algrithm is of a gradient type. Normally, the optimization stafrom a set of uniform intensities, where if the ray~beamlet!passes through the target, the relative initial intensity is se1, otherwise to 0. The absolute initial intensity distributionnormalized so that the mean target dose matches the


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After the specified number of optimization runs for eacase studied, we analyzed the results to determine if lominima existed. We examined the final scores first and texamined DVHs, dose distributions, and intensity pattefor a selected subset of plans to determine the type of lominima and to learn more about their impact.

A. Objective function

The details of the dose–volume-based objective functwe used are described in Ref. 13. However, we havecluded the following brief descriptions for completenessdiscussion. We used the following quadratic form for tobjective function:

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for constraint on the OAR.f is the objective function,N isthe number of voxels in the structure, andp is the penalty/weight factor.Di is the computed dose in thei th voxel and isa linear function of the ray weights, which are the variablethe optimization. The criterion is specified as volume receing doseD1 to be less thanV1 , i.e., V (.D1),V1 , it isillustrated in Fig. 1~b!. The step functionH is defined asfollows:

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To implement such a constraint into the objective functioonly the points in the rangeD1 to D2 are considered andcontribute to the score of expression~1!. It is assumed thatpoints with doses greater thanD2 can be ignored. Thismethod is similar to the one suggested by Bortfeld14 andconceptually duplicates the methods implemented in mcommercial and research IMRT systems.

B. Phantom studies

To study the nature and properties of multiple minima,used a simple case of two orthogonal pairs of paralopposed uniform photon beams of 18 MV incident on a symetric phantom of dimension 30330340 cm3 as shown inFig. 2~a!. The phantom studies also served as a test ofsoftware system. Only beam weight optimization was pformed, i.e., there was only one ray per beam. The phanhad a central cubic target volume and an OAR defined bcubic shell surrounding the target. Symmetry is inherendegenerate and the objective function, under certain circstances, can have local minima. That is, for a given solutanother one exists that can be obtained by applying symtry transformation~rotation or mirroring!. By carefully defin-ing constraints, we can ensure that the objective functdoes have multiple minima. To ascertain how to define san objective function, we started with two manually designplans whose DVHs are shown in Fig. 2~b!. The prescription

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required that both Plans 1 and 2 deliver 60 Gy to 95% oftarget volume in 2 Gy per fraction for 30 fractions. In Planall four beams were equally weighted and each beam deered 65 monitor unit~MU! per fraction. In Plan 2, only oneset of parallel-opposed beams~beams 1 and 3, or beamsand 4! were turned on. Both were equally weighted to deliv130 MU each. DVHs for the OAR crossed as shownFig. 2~b!.

Examining the DVHs, it would seem that, limiting voume for high dose to below certain values would leadpreference for plans in which either one parallel-oppopair ~POP! or the other dominates. In practical situations, twould correspond to OARs with large volume effect, fexample, lung, where limiting the volume receiving hig

FIG. 2. ~a! Geometry of the phantom. The phantom is 30330340 cm3 indimension, with the target at the center and a size of 63636 cm3. Theorgan at risk~OAR! is a shell surrounding the target, with the inner aouter side dimensions of 9 and 12 cm, respectively. There are four 18photon beams incidental on the phantom, arranged as shown, paropposed pairs~POP! 1 and 3 and 2 and 4. Each beam has a collimasetting of 8 cm3 8 cm. ~b! The DVHs for Plans 1 and 2. The desiredose–volume constraints are values at the center of the two triangles.


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doses must be restricted to small values, and irradiationtarget with beams incident from the smallest number ofrections is preferred.

For the following discussion, it is important to note thabecause of symmetry and the high energy of the beamscontribution of any one beam to a point in the target orOAR is nearly the same as the contribution of its parallopposed partner. Similarly, so far as the target is concerneither one POP or the other, or any combination of them, wproduce satisfactory dose distribution.

In the example shown in Fig. 2, the POP~either 1 and 3 or2 and 4! in Plan 2 delivers a high dose to about 30%–40%the OAR and a very low dose~,10 Gy! to the remainder ofthe volume. However, in Plan 1, each pair delivers the sa30–40 Gy to most of the OAR volume. Thus, if we werespecify that we desire the volume receiving greater thanGy, for example, to be less than 30%, then in the optimition solution found, either POP 1 and 3 or POP 2 andwould be preferable and would contribute most of the tardose. However, the weaker POP will also contribute a dto the OAR as long as its dose contribution to the OAvolume that it is traversing is below the specified dose cstraint of 20 Gy. This constraint will favor Plan 2 over Pla1, which would lead to the existence of at least twminima—one corresponding to the stronger contributfrom POP 1 and 3 and the other corresponding to the slarly strong contribution from POP 2 and 4. As we shall sthere is a range of values of dose–volume constraintswhich at least two minima exist. There are also rangesdesired dose–volume limits for which only one minimuexists.

Based on these considerations, we selected the followdose–volume criteria for optimization for the schemaphantom: target: volume (60 Gy).95%, penalty520; vol-ume (90 Gy),5%, penalty51; OAR, volume (20 Gy),30%, penalty53. Obviously, both plans satisfy the constraints on the target. However, based on the desired limPlan 2 is preferable to Plan 1 for the OAR, since 40% ofvolume exceeds 20 Gy for the former as compared with 7of the volume that exceeds 20 Gy. Based on the mannewhich objective function is implemented for Plan 1~as dis-cussed previously!, points in a large volume between 30%and 74% and dose in the range of 20–34 Gy are penalyielding a score of 1572, whereas for Plan 2, points betw30% and 40% volume and in the range of 20–60 Gy havscore of 1540. Note that a high penalty for violating ttarget coverage constraint means that optimized plans wdiffer mainly in their OAR dose distributions.

C. Clinical cases

We investigated the existence and impact of multipminima for a head-and-neck case and a lung case. Both cinvolved several co-planar beams, all of which weintensity-modulated. The parameters for dose–volume-baobjective function were similar to the ones used in the clin

The head-and-neck case was chosen because of the hcomplex nature of the anatomy as well as the desired patof dose distribution. We arbitrarily chose the case of a 6

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TABLE I. Dose–volume objective function parameters for the head-and-neck case. PTV5planning target vol-ume, PRV5planning organ at risk volume,p is the penalty/weighting factor.

Structure Normal criteria Strong criteria

PTV1 ~PTV2 and allelectively treated nodes!

Vol(45 Gy).90%, p52Vol(58 Gy),20%, p51

Same as normal criteria

PTV2 ~PTV3 with about 1cm margin!

Vol(59 Gy).92%, p53Vol(64 Gy),10%, p52


PTV3 Vol(69 Gy).95%, p55Vol(76 Gy),10%, p51


PRV–Brainstem Vol(45 Gy),2%, p510Vol(40 Gy),10%, p510

Vol(5 Gy),2%, p510Vol(2 Gy),10%, p510

PRV–Cord Vol(30 Gy),2%, p510Vol(25 Gy),25%, p510

Vol(5 Gy),2%, p510Vol(3 Gy),25%, p510

PRV–LeftParotid Vol(30 Gy),3%, p55Vol(20 Gy),30%, p55

Vol(5 Gy),3%, p55Vol(4 Gy),30%, p55

PRV–Larynx Vol(45 Gy),5%, p55 Vol(5 Gy),5%, p55

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year-old man with stage T2N2bM0 squamous cell carcinoof the right soft palate. He had a mass involving the right spalate and extending inferiorly to the tonsillar fossa asuperior-medially along the soft palate to the uvula, withoinvolvement of the uvula. His right neck contained a 5 c35 cm fixed mass. In the IMRT plan used to treat the ptient, nine 6 MV intensity-modulated photon beams placedequiangular steps were employed. The dose–volume obtive function parameters used are shown in columns 1 anof Table I. The planning target volume~PTV3! is the grossdisease1 margins for positioning uncertainty, PTV2 is thgross tumor volume (GTV)11 cm margins for microscopicextensions1 margin for positioning uncertainty, and PTVincludes PTV21all electively treated nodes. Planning orgat risk volumes~PRV! for cord and brainstem include a 0.cm safety margin, no safety margins were added for leftrotid and larynx. The resolution of intensity matrices w0.5 cm30.5 cm and the total number of beamlets~variables!in all beams was 5716.

The lung case was chosen because of the large voleffect of lung tissue. As mentioned previously, local minimare likely to occur for radiotherapy problems with large voume effect. The patient is a 69-year-old female with lograde leiomyosarcoma of the retroperitoneal metastasithe lung. The patient was treated with 3D-conformal radtion; the IMRT plan was generated later to match the trement prescription. The structures relevant to IMRT optimiztion were the PTV~with a volume of 93 cc!, the right and leftlungs, and the spinal cord~assumed to be a serial structure!.The PTV is located in the right lung. Therefore, the objectfunction parameters for the right lung are less demandthan those for the left lung. Seven coplanar 6 MV photbeams were employed, with gantry angles at 10°, 35°, 1180°, 210°, 240°, and 340°. Beam directions were chosemaximally avoid the left lung and the spinal cord. The ojective function parameters used are listed in Table II.

III. RESULTS AND DISCUSSION

A. Phantom studies

Figure 3~a! shows the frequency distribution histogramfinal score of the 500 optimized plans.~The range of score

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values is very large. Therefore, to make graphical data cparison more transparent, the numerical values of score hafter actually refer to the square root of the objective funtion as described in Wu and Mohan.13! A final score is thevalue of the objective function at the point where the opmization process converges to a minimum. There arepeaks, indicating that at least two local minima exist. Tnarrow high peak is centered on a score of 950, whileupper broad and low peaks are centered around 1300,cating that a Type 1 local minimum has been found. Tscores at convergence are small compared with the scorethe initial randomly assigned beam weights, which were dtributed broadly in the range of 940–19 000. Note that itpossible for the initial score of one run to be lower than tfinal score of another for such a problem with only a fevariables. However, within a given run, the final scorealways better/lower than the initial score.

To examine the solutions in greater detail, we createscatter plot of final scores versus beam 1 weights~in terms ofMU!. Figure 3~b! shows two clusters of scores. The solutioare distributed continuously over the range of beamweights from 0 to approximately 150 indicating that, whithe optimization process converges to two distinct groupssolutions, neither group is correlated to a unique groupbeam 1 weights. Because of symmetry, scatter plots of obeams are similar but are not shown.

As noted previously, in this symmetrical case and for hienergy beams, the optimization system would not be abldistinguish between one or the other member of a paraopposed pair in terms of their dose contribution to the tarand the OAR. Therefore, we made the scatter plot of fi

TABLE II. Dose–volume objective function parameters for the lung caPRV5planning organ at risk volume; PTV5planning target volume.

Structures Normal criteria

PTV Vol(66 Gy).95%, p55Vol(75 Gy),3%, p53

PRV–Cord Vol(42 Gy),2%, p55PRV–RightLung Vol(20 Gy),20%, p53PRV–LeftLung Vol(20 Gy),1%, p53

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scores versus the sum of weights of POP 1 and 3 in Fig. 3~c!.The scatter plot of the POP 2 and 4 sum of weights is simbut is not shown. Now we see three distinct clusters. TPOP 1 and 3 sum of weights for the lower left cluster

FIG. 3. ~a! Final score frequency distribution for the phantom study.~b!Scatter plot of final score vs beam 1 MU.~c! Scatter plot of final score vsPOP 1 and 3 (beam11beam3) MU.


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solutions is approximately 75. Since the total weight offour beams must be approximately 265, the sum of weigfor POP 2 and 4 is 190. Similarly, by symmetry, for the lowright cluster, the sum of weights for POP 1 and 3 is 190 afor POP 2 and 4, it is 75. Thus, there are two minima wthe same score. As discussed previously, these two minwhich were expected because of the dose–volume cstraints imposed, correspond to either one or the other Pdelivering a higher dose to the OAR. Being lowest in scothey correspond to the global minima. A lower score fthese solutions means that, for the specified dose–volconstraints, to reduce the volume exposed to high doses,figurations in which one POP has a significantly lowweight than the other is preferred to the configurationswhich both POPs have nearly the same weights. The narhigh peak of Fig. 3~a! and the lower dense cluster of Fig3~b! contain plan scores for both of these minima.

The existence of the third local minimum at a higher scowith the weights of both POPs in the neighborhood of 1was unexpected. This minimum is a real local trap and cresponds to worse solutions. The broad low peak of Fig. 3~a!and the upper sparse cluster of plans in Fig. 3~b! correspondto this minimum. The reason for the existence of this mimum, which is not evidenta priori, is expounded later in thissection.

Figure 4 shows beam weight scatter plots for POP 2 anMU vs POP 1 and 3 MU. Figure 4~a! depicts initial randomlyassigned weights and Fig. 4~b! is final optimized beamweights. It is interesting to note in Fig. 4~b! that almost allsolutions fall into three clusters on a nearly straight line tcorresponds to the total MU of all beams equal to 265.explained previously, a combined 265 MU from all beamare necessary to fulfill the requirement of 200 cGy for ttarget. The cluster at the position where the POP 1 anweight is 75 MU and the POP 2 and 4 weight is 190 Mcorresponds to the lower left cluster in Fig. 3~c!, and soforth.

To help understand why there are three distinctive minirather than 2, and why the solutions are not continuous althe 265 MU line, we plotted score as a function of beaweights for some special combinations of beam weights~Fig.5! adding up to a total of 265 MU. Note that the treatmeplan scores shown in Fig. 5 are not for optimized plansfor plans designed with a specified set of beam weights.horizontal axis is the sum of POP 1 and 3 weights. FixiPOP 1 and 3 weights fixes POP 2 and 4 weights to 265 mthe POP 1 and 3 weight. There are infinite numbers of wof allocating beam weights between beams 1 and 3~orbeams 2 and 4!. We plotted only three representative curveThe black curve in Fig. 5, which was obtained by fixinbeam weight 1 to be equal that of beam weight 3 and beweight 2 to be equal that of beam weight 4, has only tminima with a peak at the center where the weightsequally divided between the two POPs.

Next, we set beam weight 1 equal to 1/3 of beam wei3 and beam weight 2 equal to 1/3 of beam weight 4,shown in Fig. 5~gray curve!. Notice that a flat region existsaround the point where the POP 1 and 3 weight approxim

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the POP 2 and 4 weight. Remember that the goal ofoptimization is to search for the minimum score. If the opmization process reaches a flat region of this type, it wassume incorrectly that it has reached a minimum sincescore does not change when the beam weights are chan

Continuing further, we set beam weight 3 and beweight 4 equal to 0~dark gray curve in Fig. 5!. In otherwords, the weights are distributed entirely between oneof orthogonal beams. Now, the flat region turns into a lominimum. Examining this curve closely, we find that if thinitial randomly selected beam weights were such thatorthogonal pair of beams had zero or nearly zero weightsthe other orthogonal pair had weights in the approximrange of 120–150 for one of the beams, the optimizatprocess would likely get trapped in this third local minimum

Thus, there are three minima: two of them are degenedue to symmetry with a score in the neighborhood of 9

FIG. 4. Scatter plot of POP 2 and 4 MU vs POP 1 and 3 MUs:~a! initialweights,~b! final weights.


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and the third is a shallow one with a score of around 13Parenthetically, we should note that, although the phanand beams were designed to be symmetrical, the plots areperfectly symmetrical. The slight asymmetry—which is dto discreteness, the finite size of the dose grid, and the rouoff of MU inherent to computer applications—does not afect our conclusions.

From the results described so far, we conclude that lominima exist in beam weight optimization when constrainare chosen appropriately. This exercise also served to pthat our optimization system and the experiment designesearch for local minima are able to accomplish their goal.further understand the characteristics of local minima,also performed the following additional studies.

Figure 6 shows the OAR DVHs for one randomly slected optimized plan from the two degenerate clusters wscores around 950~Plan 3! and one plan from clusters withscores around 1300~Plan 4!. Plan 3 has the POP 1 and 3 anPOP 2 and 4 weights of 181 and 84, respectively, and a sof 918. Corresponding weights for Plan 4 are 137 and 1respectively, and the score is 1353. For comparison, the ODVHs of Plans 1 and 2 of Fig. 2~b! are also plotted. TargeDVHs for all four plans were essentially identical and are nshown. Note that Plan 4~the plan trapped in the higher scolocal minimum! DVH is similar to that of Plan 1, the planwith equally weighted POPs, whereas Plan 3 DVH is in btween the Plan 1 and Plan 2 DVHs.

We would like to digress a little to point out a flaw in thtype of methods used here for implementing the dosvolume-based objective function. The volume above 20for Plan 3 is 72% as compared with 40% for Plan 2. Thbased on the dose–volume constraint that the volume recing 20 Gy or higher be less than 30%, Plan 2 would bebest among the DVHs shown. However, based on sco

FIG. 5. Score as a function of the POP 1 and 3 weights~MU! for manuallydesigned plans delivering a total of 265 MU from all four beams. The blline corresponds to the beam 1 weight set equal to the beam 3 weight, wthe beam 2 weight set is equal to the beam 4 weight. The gray~red! linecorresponds to plans with the ratios of beam 1 to beam 3 weights and b2 to beam 4 weights set at 1/3. For the dark gray~green! line, the beam 3and beam 4 weights are set to 0.

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calculated according to the method described previouPlan 3 is the best~Plan 3 score of 918 versus Plan 2 score1540!. A close scrutiny of the Plan 2 and 3 DVHs reveathat this is due to the fact that, while the volume exceed20 Gy for Plan 3 is much higher than for Plan 2, the averadose of points in these subvolumes exceeding the speclimit is much smaller for Plan 3 than for Plan 2, producingcalculated score that is lower for the former plan. This flawinherent in the manner in which the dose–volume-basedjective function is implemented but is not an obstaclefinding satisfactory solutions because one can set thestraints to some artificial values to meet the desired dovolume constraints. Furthermore, this flaw is not an issuethe results presented in this paper, since the objective fution, as we have demonstrated, does have multiple miniand our aim is to study the characteristics of multiple miniand to test our software system’s ability to locate theminima. We should note that this flaw is not unique to osystem but exists in all commercial and research systknown to us employing the same objective functions.

To study the effect of the choice of objective functioparameters, we modified the OAR constraint and set theume receiving 20 Gy or higher to be less than 40%, i(volume.20 Gy),40%. The POP 2 and 4 and POP 1 andscatter plot is shown in Fig. 7~a!. There are still three clusterbut the positions of the clusters at the ends have chanfrom ~75, 190! to ~35, 230! for the upper left cluster and threverse for the lower right cluster, indicating that the nconstraint further increases the difference betweenweight of one POP relative to the other and that one paiPOP beams is more favorable than the previous constrawhich is expected from the constraint parameters.

Changing the objective function to require that no partthe OAR receives a dose greater than 39 Gy i.e., volu~.39 Gy!50, we obtained the POP weight distributio

FIG. 6. Organ-at-risk DVHs for representative optimized plans. TarDVHs are not plotted since they are nearly identical. Plan 3 has a sco918, and Plan 4 has a score of 1353. For comparison, DVHs for the mally designed Plans 1 and 2 of Fig. 2 are also plotted. The center oftriangle is the OAR dose–volume constraint.


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shown in Fig. 7~b!. All the plans have merged into a singcluster, indicating that there is a single minimum and thboth POPs have the same weight in the optimized plans. Tshould not be a surprise since, for this value of dose–voluparameter, there is no volume effect, and the objective fution is reduced to a pure quadratic form that can be shoanalytically to have just a single minimum that is similardose-based criteria.

We also examined the effect of the number of variablesthe probability of getting trapped in local minima. We dvided each of the four beams into left and right halvesdepicted in the inset of Fig. 8~a!, thus changing the numbeof variables from four to eight. Using the same dose–voluconstraints as for the data shown in Fig. 4, we obtainedresults shown in Figs. 8~a! and 8~b!. The peaks in Fig. 8~a!are now lower and wider than the clusters in Fig. 8~b!, andalthough they are still decipherable, they are beginningblur. We should point out that the POP 1 and 3 and PO

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FIG. 7. POP weight scatter plots with modified objective function paraeters. Plot~a! is for OAR volume (.20 Gy),40% and plot~b! is for OARVolume (.39 Gy),50%.

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FIG. 8. Impact of increasing the number of variables on multiple minima. Frequency distributions@~a! and~c!# and POP weight scatter plots@~b! and~d!# forcases when each beam was divided into two@~a! and ~b!# or four @~c! and ~d!# equal parts as shown in the insets. POP 1 and 3 and POP 2 and 4 werepresent the sum of MU for all their respective components.

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and 4 weights represent MU sums for the four halves oftwo beams of each pair, doubling the number of MU requifrom all the components of the POP.

Splitting the beams further into quarters as shown in F8~c! inset, which made the number of variables 16, led tofrequency distribution histogram and scatter plots shownFigs. 8~c! and 8~d!. Now, there is only one broad peak in thfrequency distribution, and the POP weight scatter plot iscontinuous cluster. To increase the probability of findinglocal minima, we ran the 16-beam case 2000 times. Fthese exercises, one could infer that as the number of vables increases, it may become more difficult to detect mtiple local minima. To explain this observation qualitativewe should note that when the number of variables increathe number of solutions increases exponentially~the numberof variables being the exponent!. When the number of vari-ables is large, it is practically impossible to test every initcondition to ensure that all minima have been found. Theven when no local minima are located, they may stillthere. By repeating the search process a large numbetimes starting with a randomly selected initial numbervariables, we simply reduce the probability of missing the

Regardless of whether or not local minima were locatit should be observed that the solutions did not converg


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exactly the same score. There may be several reasonthis. For one, very large number of undetected multiminima may exist. For another, this effect can also be cauby the discreteness of quantities in numerical calculaticarried out by computers, or by the nonzero magnitude ofconvergence criteria. Furthermore, approximations madethe implementation of optimization algorithms can also leto this effect. A comforting note, however, may be that trange of final scores is quite narrow~900–1500! as com-pared to initial scores which, as mentioned previously, was high as 19000.

The next question is whether multiple minima canfound in clinical IMRT problems and whether we needresort to stochastic optimization to find a satisfactory sotion considering the fact that the number of variables~beam-lets! in IMRT problems is in the thousands. To study thmultiple minima issue for clinical IMRT cases, we appliethe above-described methodology to one head-and-neckand one lung case. Considering that many hundreds of IMplans need to be generated for each case, a more thorstudy involving a number of patients for each site wouldvery time consuming and is deferred to future investigatioNevertheless, useful inferences can be drawn from theited work presented here.

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B. Head-and-neck example

For the head-and-neck case, we studied the final scoDVHs, dose distributions, and intensities. The frequency dtribution of scores for 500 plans with randomly set initiintensity distributions is shown in Fig. 9~a!, is in Gaussianform, and has a mean value of about 9600. The frequedistribution of final scores is plotted in Fig. 9~b!. There is abroad peak centered on the score of 570. It is not cleathere are multiple peaks in Fig. 9~b! or simply random fluc-tuations. This process was repeated with a different seedthe random number generator for another 500 runs; thescore distributions were similar and centered aroundsame scores, but without the noticeable fine structures.gardless, the finals scores were all near each other and scompared with the starting values. These results implyno Type 1 multiple local minima could be located~i.e., planscores are nearly the same!.

FIG. 9. Frequency distribution of head-and-neck IMRT plans generated500 different randomly selected initial intensity distributions.~a! Scorescorresponding to initial intensity distributions.~b! Final scores corresponding to optimized intensity distributions.


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Figure 10 shows DVHs for PTV3~corresponding toGTV!, PTV1, spinal cord and left parotid for four randomselected optimized IMRT plans. The final scores for thefour plans are 562, 539, 537, and 569, respectively. Tplans are not identical but similar. The only significant dferences are for the left parotid volumes below 20 Gy. Dopoints below 20 Gy in parotids contribute negligibly to thplan score. Thus, no Type 2 minima could be found. We aexamined dose distributions of the same four plans in F11. While there are some differences in low-dose regionsthere is a small hot spot in one of the plans, by and large,isodose patterns are very similar in terms of target volucoverage and sparing of critical normal structures, indicatno evidence of Type 3 local minima.

1. Intensity distributions

Figure 12 shows intensity distributions in gray scalethe AP beam for two randomly selected optimized plaFigures 12~a! and 12~b! show initial, randomly assigned intensities and appear to be uncorrelated. Note that the insities are assigned nonzero values only for rays that pthrough the target. Figures 12~c! and 12~d! show optimizedintensities. While there appears to be no correlation infine detail, rough similarities in dark and light regions haemerged. Smoothing these with a standard 3 pixel33 pixel‘‘box-car’’ filter leads to the intensity distributions shown iFigs. 12~e! and 12~f!. Now, the broad similarities betweethe two patterns are clear. As explained earlier, the highquency noise inherent in the intensity distribution optimiztion process obscures the broad features of intensity distrtion. Thus, while intensity distributions do not appear tothe same in fine detail, they are similar in broad terms alead to similar dose distributions. This leads us to infer twe could not find Type 4 minima either.

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FIG. 10. PTV3, PTV1, cord and left parotid DVHs for four randomly slected head-and-neck optimized IMRT plans.~See the definitions of PTV1and PTV3 in the text.!


FIG. 11. Isodose distributions for four head-and-neck optimized IMRT plans that correspond to the DVHs shown in Fig. 10.

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Observations about intensity distributions can be plaon a quantitative basis by calculating correlation coefficie~CCs! between pairs of intensity distributions. We used tfollowing two types of CCs:

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total number of the elements in the matrix, and avg(A) is theaverage value of the elements of the matrix. The CCsnormalized so that they are unitless. CC1 can be thoughas relative and CC2 as absolute. A higher value of Cwould mean a greater degree of correlation of the matricwhereas a lower value of CC2 would mean the same. Sboth matrices are positive definite, the maximum of CC11.0, and the minimum of CC2 is 0.0.

We calculated CC1 for pairs of intensity matrices for eaof the beams for all unique permutations of the 500 IMRplans. The distribution of CC1’s is plotted in Fig. 13~a!. Thisplot also shows CC1 distributions for initial intensity matrces and for final matrices smoothed using box-car averagThe CC1 for initial random intensity distributions is;0.75,as expected from the definition given in Eq.~3!. The CC1 forfinal ~optimized! intensity distributions is;0.86 and in-

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creases to a value of;0.976 for smoothed optimized distrbutions. Corresponding CC2 distributions are shown in F13~b!. The CC2 for initial random intensity distributions;1.35. The CC2 for optimized intensity distributions is;1.0and for smoothed optimized distributions is;0.36. The sig-nificant improvement in correlation with smoothing indicatthe presence of high frequency noise in optimized intendistributions.

2. Dependence of local minima on objectivefunction parameters

In the phantom case, we saw that the existence of lominima depended on the parameters of the objective fution. To see whether our conclusions about local minimathe head-and-neck case would change with the changparameters, we repeated the above-mentioned experiwith a modified set of parameters. We called the new separameters the ‘‘strong criteria’’ since the constraints forcritical structures are increased significantly, as listedTable I, column 3. The DVHs of four plans selected radomly from the set of 500 plans are given in Fig. 14 ashow that, while there is some expected degradation in P

FIG. 12. Intensity patterns for the AP beam of two randomly selected plInitial random intensity patterns show no correlation between the pshown in~a! and ~b!. Correlation between the intensity-modulated plans~c! and~d! exists, but is obscured by the high frequency noise inherent tooptimization. Smoothing of optimized intensity patterns makes correlamore apparent.


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doses, the DVHs for the four plans are almost identicThus, changing criteria do not appear to have had any efon the local minima.

C. Lung example

As demonstrated previously for the test phantom casesuggested in the literature,1 local minima may exist for caseinvolving structures with large volume effect, e.g., lung. Thprompted us to apply our methods for a lung case as wellfor the head-and-neck example, the final score distribut~not shown here! did not exhibit multiple peaks. DVHs forfour plans, selected randomly from the set of 500, are shoin Fig. 15. The DVHs are all similar except for the cord. Fthe cord, based on objective function parameters showTable II, a dose below 42 Gy to any point does not contribto the plan score. Thus, the different dose distributioshown in Fig. 15 cannot be distinguished by the optimizatsystem.

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FIG. 13. Correlation coefficient for the intensity distributions.~a! CC1 and~b! CC2 for initial randomly selected intensities, optimized intensities, asmoothed optimized intensities.

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IV. SUMMARY AND CONCLUSIONS

We have demonstrated that multiple local minima do exin dose–volume-based objective functions. Starting the omization process from a large number of different initial sof beam weights, our optimization system was able to locmultiple minima for a simple schematic phantom andbeam weight optimization. We found that whether or notlocal minima exist depends on parameter choices for thejective function. Strong constraints on the volume of an atomic structure receiving a low dose increase the chancemultiple minima will exist. The probability of the optimization system to get trapped in a local minimum may depeon the number of variables~beams or beamlets!. Therefore,for intensity-modulated beams, where one encounters thsands of beamlets, it is likely that optimization will not gtrapped in a poor solution far from the global minimum. Thwas borne out in our limited study of two realistic cases. Fboth the head-and-neck case and the lung case, we fothat, starting with random initial intensities, the optimizatiprocess converged to nearly the same~though not identical!solutions. Quantitative analyses showed that the optimi

FIG. 14. DVHs for four randomly selected head-and-neck IMRT planssigned with stronger constraints on the critical structures.~see Table I!.

FIG. 15. DVHs for four randomly selected lung IMRT plans.


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intensity patterns are well correlated with each other. Tsmall differences in solutions appeared to be due to thegeneracy of objective function~i.e., different dose distribu-tions leading to the same score!, the discreteness inherent icomputer processes, and the nonzero value of the congence criteria.

It is also clear that optimized intensity distributions anoisy and that smoothing these distributions significantly iproves correlation among intensity distributions. Our owexperience, which is consistent with that of other investigtors, is that high frequency noise is inherent in optimizintensity distributions and that smoothing, carried out juciously, has negligible effect on dose distributions.

While we found no clear evidence of multiple minimpresenting impediments to finding a suitable IMRT solutiin our very limited study, this does not necessarily mean tmultiple minima do not exist. The existence of local minimdepends on many factors, only a few of which have beexplored in this study. To be more definitive, one would neto apply the methodology described in this paper to mamore cases for different sites and perhaps repeat optimizafor much more than 500 times, which would be extremetime consuming. It would also be necessary to studydependence of multiple minima on the nature and parameof the objective function.

Even though this limited study cannot be considered cclusive, the results do increase our confidence in the valiof gradient methods in IMRT optimization. At least for thtime being, we can continue to assume that multiple miniare not of major concern in IMRT planning carried out wigradients methods as long as the treatment objectivesmet. However, at the same time, it may be useful to continsmall experiments of the type described in this paper. If mtiple minima were found to be a major concern in such futustudies, then stochastic methods—such as simulated aning and random search—or, more likely, hybrid methods tcombine stochastic and gradient methods would need toemployed.

We did not address the complex issue of beam angletimization, a process that is known to have multiple minimand does require stochastic approaches. Investigatorsreported the use of the simulated annealing approach totimize beam angles, combined with the gradient approachoptimizing intensity distributions to address thproblem.10,12

ACKNOWLEDGMENT

This work is supported by Grant Nos. CA74043 aCA84430 from the National Cancer Institute.

a!Author to whom all correspondence should be addressed; electronic [email protected]. O. Deasy, ‘‘Multiple local minima in radiotherapy optimization prolems with dose-volume constraints,’’ Med. Phys.24, 1157–1161~1997!.

2C.-S. Chui and S. V. Spirou, ‘‘Inverse planning algorithms for externbeam radiation therapy,’’ Med. Dosim26, 189–197~2001!.

3S. M. Morrill, K. S. Lam, R. G. Lane, M. Langer, and I. I. Rosen, ‘‘Verfast simulated reannealing in radiation therapy treatment plan optimtion,’’ Int. J. Radiat. Oncol., Biol., Phys.31, 179–188~1995!.

-

nys

ll,py

by

byys

, St

t

C.ity

S.inys.

-

y

u-e

icm,’’

n

ivey,’’


4S. M. Morrill, R. G. Lane, G. Jacobson, and I. I. Rosen, ‘‘Treatmeplanning optimization using constrained simulated annealing,’’ PhMed. Biol. 36, 1341–1361~1991!.

5I. I. Rosen, K. S. Lam, R. G. Lane, M. Langer, and S. M. Morri‘‘Comparison of simulated annealing algorithms for conformal theratreatment planning,’’ Int. J. Radiat. Oncol., Biol., Phys.33, 1091–1099~1995!.

6S. Webb, ‘‘Optimisation of conformal radiotherapy dose distributionssimulated annealing,’’ Phys. Med. Biol.34, 1349–1370~1989!; erratum35, 297 ~1990!.

7S. Webb, ‘‘Optimization of conformal radiotherapy dose distributionssimulated annealing. II. Inclusion of scatter in the 2D technique,’’ PhMed. Biol. 36, 1227–1237~1991!.

8R. Mohan, G. S. Mageras, B. Baldwin, L. J. Brewster, G. J. KutcherLeibel, C. M. Burman, C. C. Ling, and Z. Fuks, ‘‘Clinically relevanoptimization of 3D conformal treatments,’’ Med. Phys.19, 933–944~1992!.

9G. S. Mageras and R. Mohan, ‘‘Application of fast simulated annealingoptimization of conformal radiation treatments,’’ Med. Phys.20, 639–647 ~1993!.

10J. Stein, R. Mohan, X. H. Wang, T. Bortfeld, Q. Wu, K. Preiser, C.Ling, and W. Schlegel, ‘‘Number and orientations of beams in intensmodulated radiation treatments,’’ Med. Phys.24, 149–160~1997!.


t.

.

.

o

-

11A. Pugachev, J. G. Li, A. L. Boyer, S. L. Hancock, Q. T. Le, S.Donaldson, and L. Xing, ‘‘Role of beam orientation optimizationintensity-modulated radiation therapy,’’ Int. J. Radiat. Oncol., Biol., Ph50, 551–560~2001!.

12A. B. Pugachev, A. L. Boyer, and L. Xing, ‘‘Beam orientation optimization in intensity-modulated radiation treatment planning,’’ Med. Phys.27,1238–1245~2000!.

13Q. Wu and R. Mohan, ‘‘Algorithms and functionality of an intensitmodulated radiotherapy optimization system,’’ Med. Phys.27, 701–711~2000!.

14T. Bortfeld, J. Stein, and K. Preiser, ‘‘Clinically relevant intensity modlation optimization using physical criteria,’’ XII International Conferencon the Use of Computers in Radiation Therapy, 1997, pp. 1–4.

15L. Ma, P. B. Geis, and A. L. Boyer, ‘‘Quality assurance for dynammultileaf collimator modulated fields using a fast beam imaging systeMed. Phys.24, 1213–1220~1997!.

16Y. Chen, A. L. Boyer, and C. M. Ma, ‘‘Calculation of x-ray transmissiothrough a multileaf collimator in process citation,’’ Med. Phys.27, 1717–1726 ~2000!.

17J. Chang, G. S. Mageras, C. S. Chui, C. C. Ling, and W. Lutz, ‘‘Relatprofile and dose verification of intensity-modulated radiation therapInt. J. Radiat. Oncol., Biol., Phys.47, 231–240~2000!.

Multiple local minima in IMRT optimization based on dose–volume criteria

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