Algorithms and functionality of an intensity modulated radiotherapy optimizationsystemQiuwen Wu and Radhe Mohan

Citation: Medical Physics 27, 701 (2000); doi: 10.1118/1.598932 View online: http://dx.doi.org/10.1118/1.598932 View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/27/4?ver=pdfcov Published by the American Association of Physicists in Medicine Articles you may be interested in Improving intensity-modulated radiation therapy using the anatomic beam orientation optimization algorithm Med. Phys. 35, 2170 (2008); 10.1118/1.2905026 A new method of incorporating systematic uncertainties in intensity-modulated radiotherapy optimization Med. Phys. 32, 2567 (2005); 10.1118/1.1954161 Clinical applications of IMRT to adenocarcinoma of the prostate: Portal dose verification and intensity modulatedneutron radiotherapy Med. Phys. 32, 302 (2005); 10.1118/1.1827731 Penalized likelihood fluence optimization with evolutionary components for intensity modulated radiation therapytreatment planning Med. Phys. 31, 2335 (2004); 10.1118/1.1773631 Application of constrained optimization to radiotherapy planning Med. Phys. 26, 2359 (1999); 10.1118/1.598750

Algorithms and functionality of an intensity modulated radiotherapyoptimization system

Qiuwen Wua) and Radhe MohanDepartment of Radiation Oncology, Medical College of Virginia, Virginia Commonwealth University andMcGuire VA Hospital, Richmond, Virginia 23298-0058

~Received 31 August 1999; accepted for publication 12 January 2000!

The main purpose of this paper is to describe formalisms, algorithms, and certain unique features ofa system for optimization of intensity modulated radiotherapy~IMRT!. The system is coupled to acommercial treatment planning system with an accurate dose calculation engine based on the kernelsuperposition algorithm. The system was designed for use for research as well as for routine clinicalpractice. It employs dose– and dose–volume-based objective functions. The system can optimizeIMRT plans with multiple target volumes simultaneously. Each target volume may be assigned adifferent prescription dose with constraints on either underdosing, or overdosing, or both. Fororgans at risk more than one constraint may be applied. This feature allows simultaneous treatmentof primary, regional disease and electively treated nodes. The system allows specification of con-straints on logical combinations of anatomic structures, such as a region of overlap between theprostate planning target volume and rectum or the volume of lung excluding the tumor. Theoptimization may also be performed on plans which, in addition to intensity-modulated beams,include other modalities such as non-IMRT photon and electron beams and brachytherapy sources.The various features of the system are illustrated with one phantom example and two clinicalexamples: a brain stereotactic radiosurgery case and a nasopharynx case. In the cylindrical phantomexample, the use of the system for overlap regions is demonstrated. The brain stereotactic radio-surgery example shows the improvement of IMRT plans over the conventional arcs based plan andthe three-dimensional conformal plan with multiple fixed gantry angles and demonstrates the ap-plication of our system to cases where small grid sizes are important. The nasopharynx exampleshows the potential of IMRT to simultaneously treat large and boost fields. It also illustrates thepower of IMRT to protect normal anatomic structures for highly complex situations and the effi-ciency in planning and delivery achievable with IMRT. The overall IMRT planning time is typi-cally less than 2 h on a SunUltrasparc workstation, most of which is spent in repeated computationof dose distributions. ©2000 American Association of Physicists in Medicine.@S0094-2405~00!01904-0#

Key words: intensity modulated radiotherapy, optimization, 3D treatment planning

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I. INTRODUCTION

Intensity modulated radiotherapy~IMRT! has received considerable attention recently. It has already been implemeinto routine clinical practice at several centers,1–6 althoughits full potential is yet to be realized. Clinical implementtions include both commercial systems as well as systdeveloped in-house. Commercial treatment planning stems, such as Peacock™ and Corvus~NOMOS Corporation,Sewickley, PA!7 and Helios~Varian Oncology Systems, PalAlto, CA!, provide integrated IMRT planning, whereas tin-house systems, such as the one described in this papedeveloped as independent, stand-alone systems which cacoupled to a commercial treatment planning system. Mosthe development effort in this field is focused on two areasIMRT methodology: algorithms for optimizing intensitieand algorithms to verify and deliver such treatments.

The optimization algorithms basically fall into two caegories: gradient based and stochastic. Gradient method8–14

tend to be fast in reaching the optimum. The numberiterations is typically less than 100. However, they genera

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require that the objective function be convex, which limthe choice of objective functions. Random search algorithsuch as simulated annealing methods,15–20 are not subject tothis limitation. Therefore, arbitrary objective functions cabe employed. However, these algorithms are slow in reaing the optimum, typically needing tens of thousands oferations. Clinical objectives of IMRT optimization may bexpressed in terms of limits on dose, limits on dose–volucombinations, or in terms of dose-response indices. Opinvary as to which kind of objective function is best suited fclinical use.9,21–26Ideally, the objective functions should incorporate dose-response information. Unfortunately,present such information is sparse, unreliable, and podocumented, and most IMRT optimization systems usejective functions in terms of dose and dose–volume comnations.

In this paper, we present an IMRT optimization systewhich utilizes dose- and dose–volume-based objective futions. Instead of developing a stand-alone inverse plannsystem, we interface our optimization program to a comm

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cial three-dimensional~3D! radiation treatment planning system~Pinnacle3 from ADAC, Milpitas, CA!, thereby eliminat-ing many redundant data entries. More importantly, wetake advantage of the accurate dose calculation engine otreatment planning system, thus overcoming the problemuncertainty in accuracy of dose present in other systems.system is designed to optimize IMRT plans with multipintensity-modulated beams incident from stationary ganand couch positions. In Sec. II, we describe the optimizatalgorithms, objective functions, program flow, and tschemes for interfacing with the 3DRTP system. In Sec.we illustrate some of the capabilities of the program usphantom and clinical examples. In the cylindrical phantoexample, we demonstrate that the overlap region betweentarget and the organ at risk can be handled in three diffeways depending on the treatment objectives. In the bradiosurgery example, the IMRT plan using micro-MLCcompared with the conventional arc plan and fixed field cformal plan. Improvement in both conformity and dose gdient is shown. In the nasopharynx case, a single IMRT pis applied to give multiple prescription dose levels to targand nodes with many neighboring critical organs and syields acceptable dose distribution. In Sec. IV we discstrengths and weaknesses of our choices of algorithmsobjective functions.

II. METHODS AND MATERIALS

As mentioned previously, we define our clinical objetives in terms of dose limits or in terms of limits on volumreceiving certain specified dose. We use a variant of Nton’s methods for optimizing the objective function. To alow for a margin for penumbra or to provide a safety margaround a critical normal structure, we create transient innal volumes by expanding the volumes of interest definwithin the 3DRTP system by specified margins. The folloing sections describe the methodology in detail and theportant features of the system.

A. Newton’s methods and objective functions

There are many variations of Newton’s methods. The flowing describes the one we use in our system.27 Given aconvex objective functionf 5 f (x), wherex is a vector rep-resenting many variables, which for the case of IMRT opmization are the ray weights. The minimum off can be foundusing the following iterative process.

~1! Compute the first and second derivatives,f x and f xx ,respectively.

~2! Let the step size atith iteration be denoted bydxi anddefined byxi 115xi1dxi . Set

dxi52@ f xx~x!#21• f x~x!. ~1!

Here @ f xx(x)#21 is the inverse matrix of@ f xx(x)#.~3! Terminate the iteration when the predefined conv

gence criteria are met, otherwise repeat steps~1! and ~2!.The advantage of this method is that it converges rap

to the optimum compared with other methods. Disadvtages are that~1! the computation of the inverse matrix of th

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second derivatives can be slow or impractical when the nuber of variables is large, and~2! in common with other gra-dient methods, if the objective function has multiple locminima, the program may become trapped in one of theTherefore, the choice of objective functions is importawhen such algorithms are used. The introduction of dosvolume constraints may introduce local minima,28 though wedid not observe any in our present work.

The goal of optimization in treatment planning isachieve a balance between adequate target coverage ansparing of critical structures. When the problem is translainto a mathematical form, there are three possibleproaches:~1! to optimize a function of target dose with constraints on the organs at risk~OAR!, ~2! to minimize the doseto the OARs with constraints on the target dose, or~3! tooptimize a function of target dose and dose to OARs. Whave chosen the third approach for which the general formthe objective function may be written as

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and pnT is the weight~i.e., penalty for deviating from the

specified constraints! for that target. Similarly,f mOAR is the

mth OAR andpmOAR is the corresponding penalty factor. Th

penalties are adjusted to reflect the overall treatment obtives and depend on the treatment site and the locationsize of the target volume. The value of the objective functfor a particular set of parameters and for a given dose disbution is also called ‘‘the plan score.’’~In this case, a lowerscore means a better plan.!

B. Dose-based objective function

The fundamental form of the objective function we haselected is as follows:

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H~Di2D0OAR!~Di2D0

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Here NT and NOAR are the number of voxels in a targevolume and in an organ at risk, respectively. We hadropped the subscriptsn andm for clarity. Di is the dose inthe ith voxel, D0

T is the prescription dose, andD0OAR is the

specified tolerance dose for the OAR.H(Di2D0OAR) is the

step function defined as

H~Di2D0OAR!5H 1, Di.D0

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In other words, only the OAR points with dose greater ththe tolerance dose will contribute to its objective functicomponent.

Dose at a point is the sum of dose contributions fromthe rays from all the beams,

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703 Q. Wu and R. Mohan: Algorithms and functionality 703

Di5( Ki j v j , ~6!

wherev j is the weight forjth ray and is the optimizationvariable.Ki j is the contribution of thejth ray to theith point.Ki j depends on the patient geometry and incident phobeam characteristics. For the sake of simplicity and claritypresentation, we assume that the objective function canwritten as

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Each ray contributes to only a small number of poinlaterally in its neighborhood. If scatter is ignored, the secoderivative matrix is highly sparse. Each diagonal elementhe second derivative matrix contains contributions ofpoints the ray passes through. Nondiagonal elementsnonzero if two rays pass through a common point, in whcase the matrix element value contains the contribution frthat one point only. It is, therefore, relatively small in manitude, and it is a reasonable approximation to assumethe second derivative matrix is diagonal, especially whenmatrix dimension is very large. With this approximation, tinverse can easily be calculated. The scatter contributioincorporated in the forward computation of doseDi itera-tively. The change in ray weight in each iteration is thcomputed using the following expression:

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with the constraint thatv j>0, i.e., the ray weight cannot bnegative. This condition is checked and corrected at theof each iteration.Ki j is computed as follows: dose to a poiDi in Eq. ~6! is the sum over all beams. If scatter is ignorethen only one ray in each beam contributes toDi . ThenKi j

can be approximated to be the ratio of dose from that beto the ray weightv j . Alternative ways of obtainingKi j arepresented in Sec. IV. The ellipses~•••! in Eq. ~10! denotesadditional similar terms for other targets and OARs.

The dose-based objective function can be made mflexible by splitting the target terms into two parts,

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whereD low<D0<Dhigh, D0 is the prescription dose for thtarget, D low is the low threshold below which the dosepenalized. SimilarlyDhigh is the high threshold above whicdose is be penalized.H is the step function defined prevously. Dose values betweenD low andDhigh are not penalized.plow andphigh are the weights. This feature provides the fleibility of assigning different penalties for cold and hot spo

C. Dose–volume-based objectives

In the current state-of-the-art of IMRT optimization, thdose–volume-based objective functions are perhaps ccally the most useful type. We use a method similar toone suggested by Bortfeld9 to incorporate the dose–volumebased objectives. Figure 1 is a simple schematic exampleone OAR explaining how the optimization based on dosvolume objectives works. The dose–volume constraintspecified as:V(.D1),V1 . In other words, the volume receiving dose greater thanD1 should be less thanV1 . Toimplement the constraint into the objective function, we seanother dose valueD2 so that in the current dose volumhistogramV(D2)5V1 . The objective function may then bwritten as

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H~D22Di !•H~Di2D1!~Di2D1!21••• D .

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That is, only the points with dose values betweenD1 andD2

contribute to the score. Therefore, they are the only owhich are penalized. The ellipsis means that additional cstraints for the same OAR may be specified.

For the target volumes, two types of dose–volume critemay be specified to limit hot and cold spots. For instance,

FIG. 1. Optimization technique for dose–volume-based objectives.dose–volume criteria are specified to constrain the volume receiving dgreater thanD1 to be less thanV1 . This means that only the points withdose betweenD1 andD2 are considered, and points with dose aboveD2 areignored.

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704 Q. Wu and R. Mohan: Algorithms and functionality 704

the desired target dose of 80 Gy, we may specifyV~.82 Gy!,5% andV~.79 Gy! .95%. In other words, the volume othe target receiving dose greater than 82 Gy should bethan 5%, and the volume of target receiving 79 Gy or higshould be at least 95%. Dose-based criteria can be conered as a subset of the dose–volume criteria in whichvolume is set to an extreme value~0 or 100% as appropriate!. Dose–volume criteria provide more flexibility for thoptimization process and greater control over the dose dibutions. For example, referring to Fig. 1, for an OAR, tdose–volume-based optimization process attempts to bonly the points betweenD1 andD2 into compliance with theconstraint. Therefore the intensity of rays passing throuthese dose points needs to be adjusted. In contrast, thebased optimization process attempts to constrain all ofpoints aboveD1 .

D. Target volume expansion

The prescription dose is normally specified for the planing target volume~PTV!. To allow for the beam penumbrathis volume must be expanded by specified margins to fothe ‘‘extended PTV’’~or EPTV!. Figure 2 shows the targevolume expansion scheme. The margins chosen may deon the number of beams and beam energies, but masignificantly smaller than those used in 3D conformtherapy~3DCRT!.29 To expand the volume, it is first represented in terms of a three-dimensional bit matrix~see thefollowing for more details!. Then a ‘‘rolling ball’’ scheme,similar to standard expansion algorithms available incomputer science literature, is used to enlarge the bit maThe goal of optimization is to reduce the dose outsideextended planning target volume as much as possible. Hever, dose cannot be forced to very low levels immediatoutside the EPTV without compromising the target doTherefore, a region called the ‘‘transition zone’’~TZ! is de-fined. This region includes all points within user specifi

FIG. 2. Planning target volume and its expansion to create the unifor‘‘expanded PTV’’ or EPTV, the ‘‘transition zone’’ and the ‘‘normal tissueregion. Dashed lines shown are the 3D dose calculation box in the plansystem. Outer~skin! contour, not shown, generally encloses the box.

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TZ margins with respect to the EPTV. The points in TZ afree from any constraints and do not contribute to the objtive function.

Outside the TZ, and within a margin specified with rspect to TZ, another region, called the ‘‘normal tissue’’~NT!is defined. The NT region encompasses generic normal tisthat does not belong to any particular OAR. However, door dose–volume constraints, similar to the ones for an OAare applied. The importance of defining this region is toduce the dose outside the transition zone in a controlled mner by appropriately selecting the width and constraintsthe NT region. One could, in principle, use all the normtissues bounded by the surface contours as NT. This isdesirable for most cases since it necessitates dose caltions in the entire body even when only a smaller regionrelevant for IMRT optimization. Repeated calculationslarge regions, especially using kernel superposition alrithms, require considerable computer time. For this reasdose calculations are performed in a 3D box enclosing othe regions of interest. One could consider the portion of3D dose matrix outside the TZ as the generic normal tisvolume. However, as shown in Fig. 2, this could lead toincorrect assignment of ray weights. Consider, for examptwo rays A and B. If the calculation dose matrix bounda~the dashed box in Fig. 2! is used as the boundary of thgeneric normal tissue, ray B will pass through more normtissue points than ray A, i.e., the radiological path lengthNT is different for rays A and B. Therefore, the weight of raB will be incorrectly set to a lower value than the weightray A, although, because of the axial symmetry of theample, they are equivalent.

E. Interface with the treatment planning system

The Pinnacle3 3D treatment planning system is usedthe platform for the IMRT system. The system has a scring utility which allows a group of actions to be recorded aplayed later to automate some of the treatment plannsteps. To facilitate interfacing with external programs, tvendor has provided an application programming interfa~API! called ‘‘PinnCom.’’ The API consists of a dozen routines, allowing the external programs to query or set thetributes of Pinnacle’s3 internal objects, such as contourbeams, and prescriptions. The API is based on the intercess communication protocol. The IMRT program is not tito the treatment planning system, but rather is an indepdently compiled and linked program. The major advantageintegration in this manner is to provide the encapsulationprotection needed for both systems. To accommodateneeds of the IMRT program, a special type of compensain the form of a transmission matrix has been introduced ithe 3D planning system.

F. IMRT program flow

The interaction between the IMRT optimization systeand the 3D treatment planning system is shown in Fig.The IMRT optimization system is designed with the emphsis on the integration, i.e., to use as much of the informat

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705 Q. Wu and R. Mohan: Algorithms and functionality 705

as already available in the 3DRTP system, and to minimthe duplication of data entry. Boxes on the top row reflectusual 3D planning procedure. The IMRT program fetchcontour and beam data from the 3DRTP system and innally calls the planning system’s dose computation engrepeatedly for iterative optimization. The resulting planevaluated with the tools available in the planning system

The internal IMRT optimization program flow is shownFig. 4. The program first performs some initialization funtions, which, among other tasks, involves acquiring~from astored file or from the user! relevant optimization data sucas the choice of objective function and its parameters sucgrid size and convergence criteria. It then reads the contof all relevant anatomic structures from the planning systeIt uses this information to create volumes of interest~VOIs!,which are logical combinations of anatomic structures. Infmation on how to create VOIs and the treatment objectiveobtained from the user or a stored file.

Routines to produce ‘‘bit and bitmap matrices’’~see de-tails in Sec. II G! are then invoked to determine the scopeeach VOI. Beam information is acquired from the plannisystem through the interface routines. The intensity matriinitialized to be uniform within the beam aperture and fback to the planning system for dose computation for ebeam. The dose computation engine used in the treatmplanning system uses the so-called ‘‘collapsed cone conlution’’ algorithm.30–32 After dose calculation is completeddose–volume histograms and the treatment plan scorecomputed. Program convergence is evaluated based on

FIG. 3. Interaction of the IMRT optimization system with the 3D RTP sytem.

FIG. 4. IMRT optimization program flowchart.

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criteria specified at the beginning of the program. The pgram terminates if~1! the maximum number of iterations iexceeded, or~2! score increases instead of decreases, or~3!the change of score value is very small. The latter ispressed as

d5Af i2Af i 11

Af i

<«,

where f i is the score function at theith iteration and« is apredefined small positive value, e.g., 0.5%. Eitherf or itssquare root can be used for defining the convergence crion. The reason for using the square root is thatAf i is re-lated to the average dose difference, which is a more meingful quantity to observe as the optimization proceprogresses. It can be easily shown thatd is just a factor of 2larger in magnitude near the optimum iff i is used instead ofAf i . The program terminates when the specified criteriamet. Otherwise, the intensity of each ray is adjusted accoing to Eq.~10!, the updated intensity matrices are convertinto the appropriate format of the planning system andloaded back into the planning system, and the program ctinues with the next iteration.

G. Volumes of interest „VOI…, bit matrices, bitmapmatrix, and intensity matrices

As mentioned previously, a VOI is a logical combinatioof anatomic structures. For example, the VOI of lungs canthe expressed as the logical OR of ‘‘left lung’’ and ‘‘righlung’’; VOI of the transition zone is ‘‘EPTV1TZ margin’’NOT ‘‘EPTV’’; an ‘‘overlap’’ VOI can be defined as ‘‘tar-get’’ AND ‘‘rectum.’’ Here ‘‘OR,’’ ‘‘AND,’’ and ‘‘NOT’’are all logical operators. Treatment objectives are specifor the VOIs. The need for introducing the concept of VOwill be demonstrated in the example in Sec. III.

To determine the scope of each anatomic structure anefficiently implement the algorithm, the concepts of ‘‘bit mtrix’’ and ‘‘bitmap matrix’’ are introduced. A ‘‘bit matrix’’ isa three-dimensional matrix of 1’s and 0’s and is spatiacoincident with the CT image matrix. If a CT voxel lieinside the structure, the corresponding bit is set to 1, othwise to 0. One bit matrix is constructed for each anatomstructure. Bit matrices are used to construct a ‘‘bitmap mtrix.’’ The points on a bitmap matrix coincide with the poinon the dose matrix, which, in turn coincides with the Cmatrix. ~To accelerate dose computations, a coarser dosetrix, coincident with every other, or every third pixel, maalternatively be chosen.! Each point on the bitmap matrix irepresented by a long integer, with each VOI representeda bit in the integer. The bit is set if the dose matrix pobelongs to the corresponding VOI. One point may belongmore than one VOI, therefore more than one bit may befor that point. Points with no bit set do not belong to aVOI and are not relevant to the optimization process and ctherefore, be discarded to conserve space. A bitmap mastores information of all anatomic structures and their ext

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706 Q. Wu and R. Mohan: Algorithms and functionality 706

sions relevant to the optimization process. To consememory, the bit matrices can be discarded once the bitmmatrix is constructed.

One two-dimensional ‘‘intensity matrix’’ for each beamalso created. Each pixel of the intensity matrix representscross section of a single ray, or beamlet. Rather than beifixed value as in most other systems, the intensity grid sare user-specified variables for each plan, this facilitatesuse of the system with different type of MLCs as illustratin Sec. III. The ray weight is the variable to be determinby the optimization system. In each iteration, a ray issumed to affect only those dose points which lie alongclose to its path. To identify which points on the bitmamatrix lie along the path, a 3D ray tracing routine, basedBresenham’s algorithm,33 is used. The information about thpoints along each ray is obtained once and used repeatedthe iterative process.

H. Regions of overlap

According to ICRU Report 50,34 the planning target vol-ume ~PTV! includes a margin for setup uncertainty and ogan motion. Therefore, it is possible that an overlap betwthe PTV and a nearby OAR may exist. For example, in3D conformal treatment of prostate patients, due to the ormotion and setup errors, the PTV typically overlaps wrectum volume, and the overlap region is treated togewith PTV as target in the initial stage. However, in the bophase, the overlap region is blocked to maintain the tolerato rectum. The region of overlap may be considered tolong to the PTV, to the OAR, or neither depending on tspecific clinical requirements. For instance, if the injurythe normal organ cannot be tolerated regardless of the tudose, the region of overlap would be considered as parthe OAR. Flexible handling of the overlap region is providin the optimization system. A schematic phantom exampleSec. III illustrates this capability.

I. Optimization based on previously delivered doseand beam weight optimization

An important feature of the optimization system is thacan be applied to design plans which, in addition to intensmodulated beams, include other modalities such as nIMRT photon and electron beams and brachytherasources. In addition, IMRT optimization can be performon a patient who has been treated previously. The syscompensates for the previous dose distribution in an atteto achieve the specified overall objectives. To handle scases, the objective function requires a minor modificatiFor example, the target component of the objective functEq. ~3!, becomes

f T51

NT(i 51

NT

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T!2, ~13!

where Di0 is the value of the previous dose at that poi

Similar modifications may be made to OAR terms as we

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The optimization system can also be used for beweight optimization, or a combination of beam weight opmization and an IMRT boost. For such situations, each ofnon-IMRT beams is treated as if it was a single ray.

III. RESULTS

We use three examples to demonstrate some of thetures of the IMRT system. In the first example, using a scmatic cylindrical phantom shown in Fig. 5~a!, we illustratevarious methods to account for overlap regions. The phtom is 30 cm in diameter. The target is a cylinder of 6 cmdiameter at the center. The OAR is of the same size butcm below the target. Five coplanar, equispaced 18 Mbeams, with one incident from the top, are used for plannIMRT. Field sizes of each beam are designed to enclosetarget in the beam’s-eye-view window. The overlap regiontreated in three different ways: as a part of the target, asof the OAR, or as a special independent region. Dose-baobjectives are used in this example. The prescription dosethe target and tolerance dose for the OAR are set to beand 50 Gy, with penalties of 1 and 3, respectively. Whenoverlap region is treated as the special independent typeprescription dose to this region is set to be 80 Gy withpenalty of 5. The results are shown in Figs. 5~b!–5~d!. Threeisodose levels of value 90, 70, and 50 Gy are plotted. In F5~b!, where overlap is assigned to be a part of the target,isodose line of 90 Gy encloses the target with a slight redtion in target dose at points near the OAR. In Fig. 5~c!,where the overlap region is assigned to be a part of the O

FIG. 5. A schematic example illustrating the use of overlap region.~a! Lay-out of the target, organ-at-risk~OAR!, and overlap regions.~b! Overlap istreated as a part of the target.~c! Overlap is treated as a part of the OAR.~d!Overlap is treated as a special region with dose range between the targeOAR prescriptions.

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707 Q. Wu and R. Mohan: Algorithms and functionality 707

the 90 Gy isodose line encloses the target except in andthe overlap region. In Fig. 5~d!, where the overlap region iconsidered as an independent entity, the prescription forregion is specified to be between 70 and 90 Gy. This leada dose distribution in which the 90 Gy isodose line is alothe upper border of the overlap region, and the 70 Gy linalong its lower border. This simple phantom study illustrahow the optimization program can be applied to produdifferent dose distributions to meet a range of clinicalquirements.

In the second example, we applied the program to a bstereotactic radiosurgery case. The patient is a 62-yearfemale with recurrent meningioma on the left occipital lobThe tumor volume was 5.8 cc, and the prescription wasdeliver 13.5 Gy in a single fraction with 6 MV radiation. Thpatient was actually treated with the conventional arc-bastereotactic radiosurgery. To evaluate the performance oIMRT optimization program, a 3D conformal treatment plwas also generated. For the arc treatment@Fig. 6~b!#, twoisocenters were utilized to compensate for the irregular shof the tumor, with 5 arcs for each isocenter. The 3D confmal plan used seven nonopposing, noncoplanar beams, mmally avoiding each other within the physical limits of thtreatment machine. The block margin for penumbras armm for each beam. The IMRT plan employed the saseven beams, assuming the use of a 3 mm leaf wmicroMLC ~Brainlab, Germany!. A dose-based objectivefunction is used in this example. Unlike other examples,intensity matrix grid sizes for this one are set to be 1.5 mby 3.0 mm, where the latter dimension corresponds to

FIG. 6. The brain stereotactic radiosurgery example.~a! Anatomic descrip-tion. ~b! Conventional plan involving two isocenters with five arcs each.~c!The 3D conformal plan.~d! The IMRT plan.

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leaf width of the microMLC. The brain stem is the oncritical structure in the field. For all three plans, the treatmobjectives require that 95% of the target volume receive 1Gy, with as low a dose to the brain stem and surroundnormal tissues as possible. The isodose distributions forthree plans are shown in Figs. 6~b!–6~d!. Two end points areused for the comparison: conformality and dose gradieThese are especially important to stereotactic radiosurgdue to the high dose given in a single fraction. In the aplan, a sharp dose fall-off in the region of interface betwethe target and the brain stem is observed. On the other hthe use of the circular collimator makes the plan less conmal than desired. In the 3D plan, each beam conforms totumor shape and, therefore, the conformality of the plangood, but the dose gradient is poor mainly due to the liminumber of beams used. In the IMRT plan, both high confmality of the 3D plan and a high dose gradient are achievThe latter is due to the fact that IMRT has the abilityraising the intensities at the tumor edges,29 effectively sharp-ening penumbras. Other plan statistics are shown in Tab

The third example, shown in Fig. 7, is that of head aneck cancer. The patient is a 67-year-old female withsopharynx cancer, staged at T2N3M0, with extensive bieral neck adenopathy. The patient was treated with convtional 3D conformal 6 MV photon beams and 9 Meelectron beams to the posterior neck. Two different corosections are shown in Figs. 7~a! and 7~b!. Primary tumor~labeled Tumor1 in the figures! has a volume of 25.4 cc, anthe gross disease volumes in the involved left and rilymph nodes have volumes of 79.9 and 5.3 cc, respectivThe prescription is to deliver 70 Gy to the primary tumor ainvolved lymph nodes~gross target volume, or GTV!, 60 Gyto the microscopic extensions~clinical target volume, orCTV, which, in this case, is the volume within a margin ofcm in all directions around the GTV!, and 50 Gy to elec-tively treated volume~ETV! including bilateral retropharyn-geal nodes, spinal accessory nodes, and level I–VI noThe critical structures to be considered include the spcord, brain stem, larynx, and parotids. The IMRT plan eploys nine fixed gantry, coplanar, equispaced noncollin

TABLE I. Comparisons of arcs, 3D, and IMRT plans for stereotactic radsurgery case. PITV is defined as the ratio of prescription dose volumtarget volume.

Arcs 3D IMRTPITV 2.8 1.5 1.8

Percent ofprescription dose Percent of brainstem

100 1.4 2.4 1.090 2.1 7.7 2.580 2.9 12.3 4.6

Percent ofprescription dose

Volume ~cc! ofnontarget brain

100 9.8 2.4 4.190 12.0 8.1 6.780 15.1 13.1 9.750 34.5 36.6 27.0

708 Q. Wu and R. Mohan: Algorithms and functionality 708

FIG. 7. The nasopharynx example.~a! and~b! Two different coronal sections showing the areas to be treated and relevant critical structures.~c! and~d! Theisodose distributions for the IMRT plan.

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beams with one beam incident from the anterior directiThe dose–volume-based optimization criteria were usedthis case. The primary tumor and the gross disease innodes were logically combined into a single volume of intest, as were all electively treated nodes, and the left and rparotids.

Isodose distributions are shown in Figs. 7~c! and 7~d!, andthe dose–volume histograms for the relevant structuresshown in Fig. 8. It is clear that the IMRT plan produc

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highly conformal dose distributions. Primary tumor annodal gross disease are covered with 70 Gy, and the 50line encloses all the electively treated nodes. The brain sand spinal cord are within their tolerance limits of 55 andGy, respectively. The larynx receives less than 45 Gy.parotids, more than 60% of the volume receives less thanGy. The intensity profile for the anterior beam is shownFig. 9. Intensity profiles may contain a significant amountfluctuations~‘‘noise’’ !. The degree of fluctuations depend

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709 Q. Wu and R. Mohan: Algorithms and functionality 709

on many factors such as anatomy, tolerances, grid sizesthe choice of optimization algorithms. For efficient delivewith DMLC, minimum leakage, and improved accuracy, itdesirable to smooth out fluctuations if they exist, but onlythe extent that this does not significantly affect the dosetributions. This, however, is a nontrivial task. The effect

FIG. 8. Dose–volume histograms of IMRT plan for the nasopharynx ca

FIG. 9. Intensity profile for the anterior beam in the nasopharynx exam~a! 3D intensity profile.~b! 2D simulated gray scale image display, thdarker area represents higher intensity, the white cross is at the isocen

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fluctuations of intensity profiles on DMLC delivery is beyond the scope of this paper and will appear in anotherport.

This example illustrates one of the unique featureIMRT, which is that the high dose to GTV, intermediadose to CTV, and low dose to ETV are delivered at the satime, in effect a simultaneous integrated boost~SIB!, whichis undoubtedly more efficient to plan and deliver than mtiple plans needed to deliver standard 3D radiotheraIMRT dose distributions for SIB are superior comparedwhen IMRT is used only in the boost phase following tinitial large field treatment.35

This example illustrates another advantage of SIB IMRConventional head and neck treatments for upper necklower neck are usually separated, with fields matched atinterface. The lower neck is usually treated with anterior aposterior fields for the elective bilateral supraclavicunodes. Such a complicated setup is not required for SIMRT in which both the upper the and lower neck regioare treated simultaneously. We should also note thatusual electron beam boost fields are also eliminated, whfurther simplifies the planning and treatment process.

IV. DISCUSSION

A. Optimization algorithms

As indicated previously, the advantage of the optimizatalgorithm we have adopted, namely Newton’s method, isspeed of convergence. This is due to its reliance on secderivatives to compute the step size for each iteration. Inwork so far with hundreds of optimization runs, we hafound that the program typically reaches its optimum intotal run time of about 2 h on the SunUltrasparc-2200 work-station, most of which is spent in repeated dose comptions. For a typical case employing ten beams and 15315315 cm3 calculation box with a 0.430.430.4 cm3 dosegrid, each beam requires 30 s to compute the dose, theretotal dose computation time for the optimization of 20 itertions is about 100 min.

We also tested other gradient methods, such as the cogate gradient13 and steepest descent methods, and fouthem to require more iterations. If dose calculation is pformed in each iteration, the program would take a conserably longer time and may be impractical clinically, espcially if the dose spread kernel superposition methodutilized. One way to solve this problem may be to compudose distributions for each single ray once, and storecontributionKi j of each ray to each neighboring point in thcomputer memory or files which can be accessed later.ing so also eliminates the approximation ofKi j used in thispaper, which assumes that a ray deposits dose at pointsalong its path. However, this method requires that the trement planning system be capable of computing dose cobutions for each single ray. Methods of this type are usedsystems which require a large number of iterations.7,12,13Theadvantage is the speed improvement of optimization sidose computation at each iteration becomes a table loooperation. The drawbacks include:~1! the demand for

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710 Q. Wu and R. Mohan: Algorithms and functionality 710

memory and disk storage is high,~2! initial dose computa-tion for each ray takes a long time, and~3! any changes inthe beam configuration will necessitate a new dose comtation, thus optimization of beam angles with this methodlikely impractical.

B. Dose- and dose–volume-based objectives

In the process of applying the IMRT optimization systewe found that optimization with dose-based objectivesquires fewer iterations and converges faster than with dovolume-based objectives. However, the resulting plantained using dose-based criteria often deviates to a gredegree from the desired plan. We also found that the dovolume-based objectives require less trial-and-error to adoptimization parameters to achieve a suitable plan. Theson, as explained in Sec. II, is that dose-based optimizapenalizes all the points above the dose limit. On the othand, dose–volume-based optimization penalizes onlysubset of points within the lower end of the range of dovalues above the dose limit. The following simple examfurther clarifies this point. Let us consider an organ-at-radjacent to a target. Let us assume that the prescribed tdose is 100 Gy and that the OAR tolerance is 50 Gy. Whdose-based objectives are used, all points within the Owith dose values higher than 50 Gy are penalized. Powith higher doses, which usually happen to be the pointsregions close to the target boundary, contribute the greaamount to the total score due to the quadratic nature ofobjective function, and thus will be penalized more thpoints with lower doses. Attempting to reduce the dosepoints close to the target may compromise the target dSuch a compromise may not be clinically necessary sincis usually acceptable to allow a small portion of the OAnear the target to be exposed to high levels of dose. Wdose–volume-based objectives are used in this case,those points with dose values between 50 Gy and some ovalue, for instance 70 Gy, will contribute to the total scoand will be penalized. Those points with dose values higthan 70 Gy do not contribute to the score at all, thus theytolerated by the optimization program. Such ‘‘tolerancgives the program the ability to reach the optimum inareas including other OARs and normal tissues, and thfore the overall plan may be superior to the plan generaby the dose-based objective functions. This has been demstrated in another paper.36 Similar arguments can be madeexplain the need for the ‘‘transition zone’’ in which poindo not contribute to the total score. Since the dose outthe target cannot drop immediately to low values withoaffecting the target dose, omitting points in a narrow baoutside the target volume from the optimization constraimakes it feasible to find a suitable solution.

V. SUMMARY

In this paper we have described a general purpose IMoptimization system developed for research and clinicaland interfaced to a commercial treatment planning systThe system in its current form employs a gradient optimi

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tion algorithm and dose- and dose–volume-based objecfunctions. The structure of the program is such that it caneasily expanded to include other objective functions andtimization algorithms. The interface between the IMRT otimization system and the treatment planning system is wdefined. Therefore, the program, with minor modificationcan be adapted to other treatment planning systems asprovided similar interface API is available. The program hbeen applied to an array of clinical sites including head aneck, brain, gynecological malignancies, and prostateachieve considerable improvement in dose distributions.

ACKNOWLEDGMENTS

The authors would like to thank Mark Gehring and ToMcNutt of ADAC for providing technical support on interfacing with the treatment planning system. We also thankYang for his work on the bitmap and ray tracing routineThis work is supported in part by Grant No. CA74043 frothe National Cancer Institute.

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