a new mlc segmentation algorithmÕsoftware for step-and ...pages.cs.wisc.edu/~athula/luan.pdf ·...

A new MLC segmentation algorithm Õsoftware for step-and-shootIMRT delivery

Shuang Luan,a) Chao Wang, Danny Z. Chen, and Xiaobo S. HuDepartment of Computer Science and Engineering, University of Notre Dame, Notre Dame, Indiana 46556

Shahid A. Naqvi and Cedric X. YuDepartment of Radiation Oncology, University of Maryland School of Medicine, Baltimore, Maryland 21201

Chad L. LeeDepartment of Radiation Oncology, Sinai Hospital, Baltimore, Maryland 21215

~Received 23 July 2003; revised 17 December 2003; accepted for publication 17 December2003; published 10 March 2004!

We present a new MLC segmentation algorithm/software for step-and-shoot IMRT delivery. Ouraim in this work is to shorten the treatment time by minimizing the number of segments. Our newsegmentation algorithm, called SLS~an abbreviation forstatic leaf sequencing!, is based on graphalgorithmic techniques in computer science. It takes advantage of the geometry of intensity maps.In our SLS approach, intensity maps are viewed as three-dimensional~3-D! ‘‘mountains’’ made ofunit-sized ‘‘cubes.’’ Such a 3-D ‘‘mountain’’ is first partitioned into special-structured submountainsusing a new mixed partitioning scheme. Then the optimal leaf sequences for each submountain arecomputed by either a shortest-path algorithm or a maximum-flow algorithm based on graph models.The computations of SLS take only a few minutes. Our comparison studies of SLS with CORVUS~both the 4.0 and 5.0 versions! and with the Xia and Verhey segmentation methods on Elekta Linacsystems showed substantial improvements. For instance, for a pancreatic case, SLS used onlyone-fifth of the number of segments required by CORVUS 4.0 to create the same intensity maps,and the SLS sequences took only 25 min to deliver on an Elekta SL 20 Linac system in contrast tothe 72 min for the CORVUS 4.0 sequences~a three-fold improvement!. To verify the accuracy ofour new leaf sequences, we conducted film and ion-chamber measurements on phantom. The resultsshowed that both the intensity distributions as well as dose distributions of the SLS delivery matchwell with those of CORVUS delivery. SLS can also be extended to other types of Linac systems.© 2004 American Association of Physicists in Medicine.@DOI: 10.1118/1.1646471#

Key words: conformal therapy, IMRT, leaf sequencing, MLC segmentation algorithm

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I. INTRODUCTION

Intensity Modulated Radiation Therapy~IMRT! is a newtechnique for cancer treatments and aims to deliver a higconformal dose to the target volume while keeping the dto other surrounding organs at risk~OAR! below tolerancelevels.1–6,10,15,17,25,26,30,34The implementation of IMRT re-quires an ability to deliver two-dimensional nonuniform flence distributions, calledintensity mapsor intensity modu-lated beams~IMBs! @Fig. 1~a!#.

Currently, IMRT can be delivered by multiple overlappinfields shaped by a multileaf collimator16 ~MLC! @see Fig.1~b! for a schematic representation of a MLC#. Using aMLC, IMRT can be implemented either statically or dynamcally. In dynamic approaches, the MLC leaves keep movacross a treatment field while the radiation remains onstatic approaches, also referred to as ‘‘step-and-shoIMRT, the MLC leaves stay stationary during irradiation, amove to reshape the beam while the radiation is turned ofnumber of dynamic and static MLC techniques have bdescribed in Webb’s books.26,27

Due to the complexity of dynamic intensity modulatebeams, the verification of all aspects of the planning a

695 Med. Phys. 31 „4…, April 2004 0094-2405 Õ2004Õ31„4

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delivery procedures becomes difficult. Thus, ‘‘step-anshoot’’ is the currently preferred method for delivering IMRin clinical practices. Advantages of the ‘‘step-and-shootechnique include precise delivery, easy verification, ageneral availability. A key disadvantage, however, is thamay require a prolonged treatment time, since radiationto be constantly switched on and off to allow MLC leavesreshape. In some machines, such as Elekta and Siemenintersegment delay dominates the total treatment time istep-and-shoot delivery. Thus, in such cases, it is highlysirable to use a leaf sequence with the fewest segmentssible to deliver the desired intensity map.32

This paper is one of a series of papers on new segmetion algorithms for step-and-shoot IMRT delivery.7,8,18,19,31

The new MLC segmentation algorithm, called SLS~an ab-breviation for static leaf sequencing! seeks to shorten thetreatment times of IMRT plans by significantly reducing tnumber of aperture shapes used. The input to SLS is insity maps computed by current planning systems, andoutput is optimized leaf sequences.

In contrast to the known segmentation algorithms, SLSbased on graph algorithmic techniques in computer scieand is designed to take advantage of the geometric struct

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696 Luan et al. : A new MLC segmentation algorithm 696

of intensity maps. Intuitively, in SLS, intensity maps~orIMBs! are viewed as 3-D ‘‘mountains’’ made of unit-size‘‘cubes.’’ Such a 3-D ‘‘mountain’’~possibly with a complexshape! is first partitioned into a set of special-structured sumountains using a new mixed partitioning scheme. Thenoptimal leaf sequences for each resulting submountaincomputed by a shortest path algorithm or a maximum flalgorithm based on a graph model. Treating an intensity mas a three-dimensional solid has also been considereSiochi;23 however, his ‘‘rod pushing’’ algorithm is not baseon graph algorithmic techniques.

We have also developed a SLS software package usinprogramming language on Linux workstations. We coducted comparison studies of SLS with CORVUS~versions4.0 and 5.0!, a popular commercial planning system devoped by NOMOS Corp., and with our implementation of tXia and Verhey algorithm.32 Our comparisons between SLand CORVUS indicated that for the same intensity maps,numbers of aperture shapes computed by SLS are up toless than CORVUS 4.0 and 40% less than CORVUS~31% less than CORVUS 5.0 with the leakage correctturned off! on Elekta Linac systems. Analysis based on soware simulations, delivered films, and ion chamber measments showed that the 3-D doses produced by modiIMRT plans of SLS match those of CORVUS, and thus co

FIG. 1. ~a! A sample intensity-modulated beam~IMB ! to be delivered by asequence of MLC shapes.~b! The projection of a MLC aperture shape othe isocentric plane.

Medical Physics, Vol. 31, No. 4, April 2004

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firmed the clinical feasibility of SLS. The comparisons btween SLS and the Xia and Verhey algorithm also showsignificant improvements. For the same set of intensity munder the Elekta constraints, SLS outperformed the XiaVerhey algorithm by about 33% in terms of the numbersaperture shapes.

Based on these encouraging studies, we have starteapply SLS clinically in the Department of Radiation Oncoogy at the University of Maryland Medical Center. For eample, for one of the clinical cases that we have treatedan Elekta SL20 Linac system, SLS shortened the treatmtime of the CORVUS 4.0 IMRT plans from 72 to 25 min.

The rest of the paper is organized as follows. In Sec~Methods and Materials! we present our new SLS algorithmsoftware. In Sec. III~Results! we give some comparison results between SLS and CORVUS or the Xia and Verheygorithm. In Sec. IV we conclude the paper and discuss soof the possible extensions of our new segmentationproach.

II. METHODS AND MATERIALS

A. The SLS segmentation algorithm

In this section, we present our new segmentation alrithm SLS. Recall that the goal of a MLC segmentationgorithm is to generate the minimum number of apertushapes~also calledsegments! for creating a desired intensitmap. While most known approaches formulate the problas a matrix partition problem~i.e., partitioning the intensitymap matrix into a set of special matrices, each definingaperture shape!, our approach is very different in the senthat it exploits the geometric nature of the problem.

1. A geometric viewpoint of the segmentationproblem

Geometrically, one may view a MLC aperture shape apolygon on thexy plane that also has a uniform ‘‘height’’ inthe z axis; the ‘‘height’’ of the polygon represents the radition intensity to be delivered for that aperture shape. In tview, each aperture shape can be treated as a ‘‘plateau’’ in3-D space with a uniform heighth for some integerh.0

FIG. 2. ~a! A 3-D IMB mountain and the corresponding intensity map.~b!–~e! Four ‘‘plateaus’’~segments! of a unit height for building the 3-D IMB mountainin ~a!.

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FIG. 3. ~a! and ~b! illustrate two segments corresponding to two MLC apertures.~c! When laying the segmenin ~b! on top of the one in~a!, some of the unit cubes othe one in~b! drop to the first layer, where the lighcells represent the segment in~a! and the dark cellsrepresent the segment in~b!.

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@e.g., see Figs. 2~b!–2~e!#. Hence, intuitively, one may viewthe segmentation problem in ‘‘step-and-shoot’’ deliveryplaying the following game: A given IMB~intensity map! isa 3-D ‘‘mountain’’ made of unit-sized [email protected]., Fig. 2~a!#,and one seeks to ‘‘build’’ this mountain by stacking up tminimum number of plateaus~segments! @e.g., Figs. 2~b!–2~e!#. It is important to note that each plateau used, like3-D IMB mountain, is not one whole ‘‘rigid’’ object, i.e., it ismade of unit cubes as well. Thus, when two segments‘‘stacked’’ together, some of the cubes may fall to lowpositions as if they are pulled down by gravity.~See Fig. 3for illustrations.!

Geometrically, the MLC segmentation problem is to ptition a 3-D IMB ‘‘mountain’’ into the minimum number ofplateaus~segments! of uniform heights. Since these plateaare defined by a MLC aperture shape~i.e., the cross-sectionof a plateau on thexy plane corresponds to a MLC apertushape!, the mechanical constraints of the MLC preclude ctain plateaus from being used to ‘‘build’’ the IMBmountains.11,16 One such constraint is called theminimumleaf separation~e.g., on an Elekta or Varian Linac system!,which requires the distance between the two opposite leaof any MLC leaf pair to be no smaller than a given valued~e.g.,d51 cm!. Another common constraint is calledno in-terdigitation ~e.g., on an Elekta or Siemens Linac system!,which forbids the leaf tip of each MLC leaf to surpassneighboring leaves on the opposing leaf bank. These cstraints prevent opposite MLC leaves from colliding ineach other and being damaged.~Figure 4 shows these constraints on the popular clinical Linac systems.!

In this paper, we focus on dealing with the Elekta MLconstraints. However, it is possible to extend our approacother Linac collimator systems.

2. Main ideas of the SLS algorithm

In this paper, we solve three versions of the segmentaproblem. These problems are as follows.


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~1! The 2-D segmentation problem: Given an IMB with en-tries of either 0 or 1~called zero–one IMB!, find theminimum number of segments for creating the IMsuch that each segment has a height of 1. See Fig.~a!for an example of a zero–one IMB.

~2! The Bortfeld–Boyer segmentation problem: Given anar-bitrary IMB, find the minimum number of segments fothe IMB, such that each segment hasone unitof inten-sity. See Fig. 2 for an example of the Bortfeld–Boysegmentation problem.

~3! The 3-D segmentation problem: Given anarbitrary IMB,find the minimum number of segments for the IMB, suthat each segment has anarbitrary integral intensitylevel.

Observe that both the 2-D segmentation problem aBortfeld–Boyer segmentation problem are special casethe 3-D segmentation problem.

In our solution for the 3-D segmentation problem, we fioptimally solve the 2-D segmentation problem~Sec. II A 3!and the Bortfeld–Boyer segmentation problem~Sec. II A 4!.Here, the word ‘‘optimal’’ means that the optimal outpquality of the solutions for these special cases can be promathematically. Then, based on our optimal solutions for thspecial cases, we present an effective segmentation algorfor the 3-D segmentation problem~Sec. II A 5!.

3. Special case I: The 2-D segmentation problem

The first special case of the 3-D segmentation problthat we consider is the 2-D segmentation problem~e.g., seeFig. 5!: Given a zero–one IMB~with entries of either 0 or 1!,find the minimum number of segments for creating the zerone IMB, such that each segment has a height of 1. Obsthat this 2-D version can be viewed as apolygon partitionproblemthat seeks to partition a given planar domain into tminimum number of desired polygons; here, each resultpolygon corresponds to a MLC aperture shape.

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FIG. 5. ~a! An IMB with entries of either 0~blank! or 1~shaded!. ~b! An IMB, where the shaded area denoteentries>4. ~c! The output of a ‘‘sliding window’’-likemethod for the 2-D segmentation problem on thshaded area~six MLC aperture shapes, assuming thMLC leaves are moving along they axis!. Some of thecells in ~c! are stretched for the sake of clarity.~d! Theoptimal output of our 2-D segmentation algorithm~three MLC aperture shapes!.

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a. Why 2-D segmentation?The 2-D segmentation problem is important in that

optimal solution for this problem can lead to potential improvements to the segmentation algorithm by Xia aVerhey.32 To illustrate the importance of this 2-D problemwe first review the main ideas of the Xia and Verhey algrithm. The following is the major steps of their algorithbased on the ‘‘reducing level’’ technique used in the papeGalvin et al.14

~1! Choose an intensity level d52i , where i5Int„log2(Imax)…21 ~for any valueI .1, Int„log2(I)… de-notes the power of 2 closest toI! and I max is the maxi-mum intensity level of the IMB under consideration.

~2! From the IMB, find an aperture shape with the largarea on thexy plane and with an intensity leveld; re-move this aperture shape from the IMB.~Note that thealgorithm by Xia and Verhey was mainly designedaddress the Siemens constraints; hence, to adaptalgorithm to satisfying the Elekta constraints, we needfind the largest aperture shape subject to the Elekta cstraints in this step.!

~3! Repeat steps~1! and~2! on the remaining IMB, until theremaining IMB is empty.

For example, on the IMB in Fig. 5~b!, the maximum in-tensity level I max57, i 5Int„log2(Imax)…2152, and d52i

54. If one keeps applying the above algorithm to segmthe IMB in Fig. 5~b! until I max is no longer 7, then thissegmentation process produces the segments in Fig. 5~c!, allhaving a height ofd54. ~We assume in this paper that thMLC leaves always move along they axis.!

Observe that in the above segmentation process, theand Verhey algorithm essentially performs a partition ofshaded area in Fig. 5~b! ~i.e., containing all entries whosvalues are at least 4!. In Fig. 5~b!, if one views the shadedarea as containing entries that are all marked by 1 andunshaded areas as containing entries marked by zero,


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the Xia and Verhey algorithm is in fact dealing with the 2-segmentation problem@partitioning the shaded area in Fig5~a!#. Thus, the 2-D segmentation problem arises as asubproblem in the Xia and Verhey algorithm.

In the Xia and Verhey algorithm, a ‘‘sliding window’’-likemethod is used to solve the 2-D segmentation problewhich repeatedly finds the largest-area deliverable apershapes. Applying the sliding window method to the shadarea in Fig. 5~a! generates six aperture shapes, each witheight of 4 @Fig. 5~c!#, while the mathematically optimasolution produced by our algorithm has only three apertshapes@Fig. 5~d!#, cutting down the number of aperturshapes by 50% for this example. This implies that solvthe 2-D segmentation problem optimally can lead to iprovements to the Xia and Verhey algorithm.

b. The 2-D segmentation algorithm.Now we present our optimal 2-D segmentation algorith

called 2-D SLS. The key idea of the 2-D SLS is based onobservation calledpeak eliminations. For convenience, hereafter in this section, we refer to any zero–one IMB aspolygonP on thexy plane. Note that such a polygonP maycontain holes and the boundary edges ofP are either verticalor horizontal. We say that a vertical boundary edgee of apolygon P is a left peak if the two horizontal edges ofPadjacent toe are both to the left ofe @e.g., see Fig. 6~a!#.Similarly, a right peak is a vertical edge ofP whose twoadjacent horizontal edges are both to its right.

Observe that the existence of left and right peaks isreason for which the polygonP cannot be delivered by usina single MLC aperture shape. This is because when deliing a MLC aperture shape, each pair of MLC leaves defianopeningof the shape as a continuous vertical strip for texposure of radiation. Left and right peaks separate thevolved vertical strips of the IMB into disjoint pieces, anhence prevent the IMB from being delivered as a singleerture shape. Thus, the key to segmentingP into MLC aper-ture shapes is to ‘‘eliminate’’ all left and right peaks by pa

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titioning P into multiple ‘‘nice’’ polygons~i.e., MLC apertureshapes!.

Observe that, to eliminate a left~resp., right! peake, onemust divideP into two smaller polygons using a polygoncurve that connects the peake to some pointp on the bound-ary of P, such thatp is not to the left~resp., right! of e @Fig.6~b!#. We call such polygonal curves used in partitioningPthe dividing curves. Each dividing curve should be deliveable by MLC leaves, that is, it should consist of edges tare only vertical or horizontal, and the widths of the horizotal edges should fit the width of the MLC leaves. Furtherdividing curve should not introduce new left and right peahence, starting at the left end of the curve and walking alothe curve, one should always go rightward and neverleftward ~i.e., the curve is from left to right, with possiblups and downs!.

Introducing a dividing curve to eliminate a peak ofP mayincrease the number of resulting polygons~MLC apertureshapes!. Hence, to partitionP into the minimum number ofaperture shapes, we need to introduce the minimum numof dividing curves. Since each left or right peak can be elimnated by one dividing curve, and one dividing curve canbest eliminate two peaks~i.e., connecting a left peak toright peak!, it follows that we need to find a maximum setmutually disjoint dividing curves such that each curve conects a left peak and a right [email protected]., see Fig. 6~b!#. Notethat the dividing curves must be mutually disjoint to avointroducing redundant aperture shapes. In fact, findinmaximum set of such dividing curves is the key to solvithe 2-D segmentation problem. For the remaining uncnected peaks, it is relatively easy to eliminate them by cnecting each of them to some nonpeak boundary point oP.

To compute a maximum set of connections betweenleft and right peaks, we formulate the task as a maxim

FIG. 6. ~a! The left and right peaks of a polygon.~b! The three dividingcurves~dashed! yield an optimal segmentation of the polygon.~c! The di-rected grid graph modeling the 2-D segmentation problem: The thick eindicate a maximum flow.


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flow problem on a directed grid graph induced by the IMgrid @Fig. 6~c!#. ~For discussions on the maximum flow prolem and its solutions, see the book of Cormenet al.12! Spe-cifically, we assign a unit flow capacity to all vertices anedges of the grid graph~to be defined below!. It can beproved7 that a maximum flow in this directed grid grapyields a maximum set of sought connections that specifieoptimal solution for the 2-D segmentation problem.

The main steps of our 2-D SLS algorithm are as follow

~1! Construct a directed graphG induced by the underlyinggrid of the given zero–one IMB. The vertices ofG in-clude all intersection points of the IMB grid that areither part of a peak or in the interior of the polygonPcorresponding to the IMB. The edges ofG include thoseedges of the IMB grid whose end points are verticesG. Directions are assigned to the edges ofG based on thefollowing rules:~i! For an edge involving a left peak vertex, its directio

is away from the left peak.~ii ! For an edge involving a right peak vertex, its dire

tion is toward the right peak.~iii ! The direction of a horizontal edge that does not

volve any peak vertex is always from left to right~iv! The direction of a vertical edge that does not i

volve any peak vertex is always bidirectional.~2! For all vertices corresponding to the same left~resp.,

right! peak, introduce a source~resp., sink! vertex toG,and edges from the source vertex~resp., peak vertices! tothose peak vertices~resp., the sink vertex! @e.g., see Fig.6~c!#.

~3! Assign a unit flow capacity to each vertex and edge ofG.Note that this flow capacity assignment ensures thatdirected path of a unit flow from a source vertex to a sivertex inG always connects a left peak to a right peakP.

~4! Compute a maximum flow in the directed grid graphGfrom the source vertices to the sink vertices, and baon the resulting maximum flow, extract the connectiobetween the left and right peaks ofP.

~5! Eliminate the remaining unconnected peaks, and extthe resulting segments.

Observe that a unit flow capacity on each vertex and eof G guarantees that a maximum flow inG yields a maxi-mum set of mutually disjoint paths~from the sources to thesinks!, which specifies precisely a maximum set of sougconnections between the left and right peaks ofP. It can beproved that the above 2-D SLS algorithm takes a polynomtime to solve the 2-D segmentation problem.

4. Special case II: The Bortfeld-Boyer segmentationproblem.

Another key special case of the 3-D MLC segmentatproblem is theBortfeld–Boyer versionof the segmentationproblem:1–6 Given an arbitrary IMB, find a minimum set osegments for creating that IMB, such that each segmentexactly one unit of intensity. Geometrically, the Bortfeld

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Boyer version aims to partition a given 3-D IMB ‘‘mountain’’ into ‘‘plateaus’’ ~segments! of a unit height~e.g., seeFig. 2!. Note that for the general 3-D segmentation problethe uniform height of each segment can be any integer>1~e.g., in the Xia and Verhey algorithm32!. Unless otherwisespecified, in this section~Sec. II A 4!, all segments are of aunit height.

a. Why Bortfeld–Boyer segmentation?Studying the Bortfeld–Boyer segmentation problem

important because the maximum heights of the majorityIMBs used in current clinical treatments are around 5 toand because a mathematically optimal solution for this scial case on such IMBs is often very close to an optimsolution for the general 3-D segmentation problem.

b. Main ideas.Geometrically, one may view a 3-D IMB mountain a

consisting of a series of IMB ‘‘mountain slices,’’ each owhich is a 2-D object defined on a column of the IMB gr@see Fig. 7~c! for examples of IMB ‘‘mountain slices’’#. Ac-tually, such an IMB ‘‘mountain slice’’ corresponds to thconcept ofintensity profile33 or 1-D IMB.18,28

Let S5$Si u i PI % denote a set of segments that ‘‘builds’’3-D IMB mountain, whereI is an index set. For theBortfeld–Boyer segmentation problem, since the heighteach segmentSi is 1,Si , in fact, builds a continuous block oa unit height on every IMB columnCj that intersects theprojection of Si on the IMB grid. We call the continuoublock on an IMB columnCj created by a segmentSi a fieldopeningand denote it byBi j . Note that when delivering asegmentSi , each of its field openingsBi j is delivered by apair of MLC leaves that is aligned to its corresponding IMcolumnCj .

Suppose that one ‘‘collects’’ the field openings createdall segments in$Si u i PI % on an IMB columnCj . Then this‘‘collection’’ $Bi j u i PI % of field openings actually ‘‘builds’’the IMB mountain slice~intensity profile! defined onCj . Wecall a collection of field openings that builds an IMB moutain slice defined on an IMB column adelivery optionof thecorresponding intensity profile. For example, the boxesthe delivery options in Fig. 7~d! represent the field openings

From the above analysis, we can solve the BortfeBoyer segmentation problem by first selecting a ‘‘good’’ d

FIG. 7. ~a! A simple IMB. ~b! The IMB in ~a! can be partitioned into twosegments of a unit height~assuming the MLC leaves move along they axis!.~c! The intensity profiles~IP! of the three columns of the IMB in~a!. ~d! Thedelivery options~DO! of the three intensity profiles in~c! and the stitchingof their field openings~indicated by arrows!; each path following the arrowscorresponds to a segment in~b!.


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livery option for building each IMB mountain slice, and thesomehow ‘‘stitching’’ together the field openings along tconsecutive IMB columns to form segments. Here, ‘‘stitcing’’ means which of the selected field openings for columCj should follow which field opening for columnCj 11 in asegment@see Fig. 7~d! for an illustration#. Observe that, if no‘‘stitching’’ occurs, then a given 3-D IMB mountain can b‘‘built’’ by letting each selected field opening be a segmeFor example, the IMB in Fig. 7~a! can be delivered using sixsegments, with each segment being one of the field openin Fig. 7~d!. Observe that each time we ‘‘stitch’’ two fieldopenings together~without violating the MLC constraints!,the number of segments used is reduced by 1. In Fig. 7~d!,four ‘‘stitchings’’ can be done, thus reducing the numbersegments to 2@see Fig. 7~b!#.

It can be shown8 that it is sufficient tolocally ‘‘stitch’’together the field openings of the two selected deliverytions for any two consecutive IMB columns, by computingmaximum cardinality matching between the two correspoing sets of intervals, with each interval representing exacone selected field opening9—the number of segments thuresulted is mathematically minimumwith respect tothe se-lected delivery options~with one delivery option for eachIMB column!. Every segment consists of a sequence of fiopenings thus stitched together, with one field opening freach of the selected delivery options for a consecutive lisIMB columns.

The above approach for solving the Bortfeld–Boyer sementation problem, however, faces a major difficulty: Tintensity profile of an IMB column often can be createdusing any one of a large number of different delivery optioand any such delivery option may possibly be used byoptimal set of segments for creating that IMB. In fact, in tworst case, there can be up toN! different delivery optionsfor an intensity profile, whereN is the minimum number offield openings needed for creating the intensity profile.28,33

For instance, the intensity profile in Fig. 8~a! has 3!56 dif-ferent delivery options in Fig. 8~b!.

Our key idea for resolving this difficulty is hinged ongeometric observation. We find out that to compute an omal solution for the Bortfeld–Boyer segmentation probleit is sufficient to use only a few special delivery options oof all possible delivery options. Being able to use dramacally fewer delivery options enables our algorithms/softwato run very fast.

We need to introduce the notion of left and right MLleaf positions of an intensity profile as in Webb’s paper28

Geometrically, an intensity profile is a functional curvef thatis defined on an interval and consists of only vertical ahorizontal [email protected]., the thick solid curve in Fig. 8~a!#. Aswe ‘‘walk’’ along such a curvef from left to right, eachunit-length upward vertical edge segment~in our walkingdirection! is called aleft leaf position~L position!, and eachunit-lengthdownwardvertical edge segment is called arightleaf position~R position!. Note that a vertical edge of lengtL on f consists ofL leaf positions~here,L is an integer!. Forconvenience, we label the left leaf positions of an intensprofile asLi ’s, and the right leaf positions asRi ’s, in increas-

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FIG. 8. ~a! An intensity profile, with the left leaf posi-tions L1 , L2 , andL3 , and right leaf positionsR1 , R2 ,andR3 . ~b! The 3!56 delivery options for the intensityprofile in ~a!; the one enclosed by the dashed box is tonly canonical delivery option.

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ing order as we traverse its curve from left to [email protected]., seeFig. 8~a!#. Clearly, the numbers of theL positions andRpositions of each intensity profile are always the same.

It is easy to see that everyL or R position corresponds toexactly one end point of a field opening in any deliveoption for an intensity [email protected]., see Fig. 8~b!#; that is, allL andR positions of the intensity profile correspond to somendpoints of the field openings of each of its delivery otions. However, not every end point of the field openingsa delivery option necessarily corresponds to anL or R posi-tion of the intensity profile; that is, a field opening of a dlivery option can have end points that need not be anL or Rposition of the intensity profile.

We call the left~resp., right! end point of a field openingthat does not correspond to anL position ~resp.,R position!of its intensity profile aleft ~resp., right! Steiner point.Steiner points can be somehow specified on an intensityfile ~say, randomly! for defining field openings, with thenumbers of specified left and right Steiner points beingsame. A delivery option with Steiner points has more fieopenings than a delivery option without Steiner points forintensity profile. Note that the concept of Steiner points halso been described in Webb’s paper,29 in which they are alsocalled ‘‘splits.’’

The advantage of using delivery options with Steinpoints is that the number of segments may be reduced.example, the optimal solution in Figs. 9~c!–9~d! is obtainedby ‘‘breaking’’ one of the field openings for the intensitprofile of the middle IMB column of Fig. 9~a! at someSteiner points@Fig. 9~c!#, thus enabling us to connect thbottom two segments in Fig. 9~b! into one segment and reducing the number of segments used.

We call the union of all left~resp., right! Steiner points~ifany specified! andL positions~resp.,R positions! on an in-tensity profile the set ofleft ~resp.,right! field opening endpoints (FO end points)of the intensity profile, and still labethem asLi ’s ~resp.,Ri ’s! in the left-to-right order. As shownin the papers,28,33 a delivery option for an intensity profilecan be formed by a pairing between its left and right FO epoints, such that each left FO end point is paired with exaone right FO end point that is to its right.

Definition 1 (Canonical delivery option)—Le$(Ll i

,Rr i)u1< i<k% be a delivery option based on a set S

FO end points for an intensity profile, with k5uSu/2, suchthat each field opening(Ll i

,Rr i) has a unit height. Then the

delivery option is said to becanonicalif and only if for anytwo field openings(Ll i

,Rr i) and (Ll j

,Rr j), iÞ j , neither the


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[e.g., see Fig. 8(b)], where x(•) denotes the x-coordinate oan FO end point.

The following facts have been shown in the paper:7 ~1!For any set of left and right FO end points on an intensprofile, the canonical delivery option is unique and$(Li ,Ri)u1< i<k%; ~2! it is sufficient to use only the canonical delivery option~among all possible delivery options! foreach set of left and right FO end points of the intensity pfile of every IMB column to compute a mathematically otimal set of segments for the Bortfeld–Boyer segmentatproblem.

c. The algorithm.To highlight some of the key aspects, below we pres

first a preliminary version of our algorithm for solving thBortfeld–Boyer segmentation problem. The assumptionthis preliminary algorithm is that for the intensity profile oeach IMB column, only one set of left and right FO enpoints is used~this assumption will be removed later, ocourse!.

~1! For the intensity profile of every IMB column, generathe unique canonical delivery option for the given setFO end points.

FIG. 9. ~a! An IMB. ~b! An optimal solution without using Steiner points~c!–~d! An optimal solution with Steiner points.

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~2! For every two consecutive IMB columns, computemaximum matching between their canonical deliveoptions.9 Here, two field openings can be matched if thsatisfy the no interdigitation constraint.~Note that theminimum leaf separation constraint has already btaken into account when a delivery option is generate!A maximum matching here corresponds to the ‘‘stitcing’’ process discussed previously.

~3! Extract the segments for the input IMB based onmatched field openings of the canonical delivery optioin the increasing order of consecutive IMB columns.

Now we present our solution to the Bortfeld–Boyer sementation problem.

Note that, for an IMB column ofn entries~bixels!, thenumber of possible locations for Steiner points isn21. Sincemultiple Steiner points may be inserted at the same locatthe number of possible sets for insertingm Steiner pointsonto the intensity profile is bounded byO(nm). Our experi-ence based on clinical datasets actually indicated thatIMRT planning, the numbern is around 10 and the numbermof Steiner points needed for producing an optimal setsegments is usually no bigger than 5. Hence, to computoptimal set of segments for an IMB, we only need to usfew tens of thousands of different sets of Steiner pointsform the sets of FO end points. For each such set of FOpoints, we only use its unique canonical delivery option.this way, using Steiner points in our algorithm does not cate a severe problem to the running time. In fact, our pgram runs in only a few minutes using the sets of Fendpoints thus generated. Yet, the usage of Steiner poreduces the number of segments by 30% compared tousing Steiner points.

To distinguish from our algorithm for the general 3-segmentation problem, we call the following algorithm fthe Bortfeld–Boyer segmentation problem thebasic 3-DSLS.

~1! Choose the numberm of Steiner points for forming setof FO end points.

~2! Generate the canonical delivery option for each setFO end points containing at mostm Steiner points forthe intensity profile of every IMB column.

~3! Construct a directed graphG, such that each vertex ofGcorresponds to exactly one delivery option generatedstep~2!.

~4! For every two vertices ofG corresponding to deliveryoptions for the intensity profiles of any two consecutiIMB columnsCj andCj 11 , put a directed edge from thvertex for columnCj to the vertex for columnCj 11 , andassign a weight to that edge based on a maximum maing between the two sets of field openings of the tcorresponding delivery options.~Note thatG thus con-structed is acyclic.!

~5! A shortest path inG from a vertex of the first IMB col-umn to a vertex of the last IMB column then specifiesoptimal set of segments for the input IMB.~For discus-sions on the shortest path problem and its algorithms,


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the book of Cormenet al.13! Note that when Steinepoints are used, the intensity profile of each IMB columhas many sets of FO end points and thus many canondelivery options~one canonical delivery option for eacset of FO end points!. Then finding an optimal set osegments becomes a shortest path problem8 in G.

The above basic 3-D SLS algorithm solves the BortfelBoyer segmentation problem in a polynomial time ofn if mis a constant~e.g.,m<5).

5. The general 3-D segmentation algorithm

Our algorithm for the general 3-D segmentation probleis crucially based on our two algorithms in Secs. II A 3 aII A 4. To be able to make use of these two algorithms,need to partition the input 3-D IMB mountain into a set‘‘nice’’ sub-IMBs, such that each such sub-IMB can bhandled optimally by one of these two algorithms~i.e., to useour 2-D SLS algorithm for the 2-D segmentation problemresulting sub-IMB must have a uniform height; to use obasic 3-D SLS algorithm for the Bortfeld–Boyer segmention problem, the maximum height of such a sub-IMB shoube reasonably small, say, around 5!.

A beamlet~i.e., an entry! of the input IMB with a valueiis called az postof height i ~z stands for thez axis corre-sponding to intensity values!. We use the following schemeto solve the general 3-D segmentation problem.

~1! Let I max be the maximum intensity level of the IMBunder consideration. IfI max is small ~e.g., no more than5!, then go to step~5! and perform option~a! on theIMB.

~2! Choose an intensity leveld with 1<d<I max.~3! Consider only allz posts of the IMB whose heights are

leastd ~i.e., for the time being, allz posts lower thandare ignored!. ‘‘Cut’’ each ‘‘tall’’ z post ~whose height is>d) at the leveld, i.e., the part of thez post at levels.d is ignored.@See Fig. 5~b! for an illustration, in whichthe chosen intensity level isd54; the shaded portions othe IMB are the ‘‘tall’’ z posts.# Observe that what is nowleft of the 3-D IMB mountain is a ‘‘plateau’’ consistingof z posts of height exactlyd.

~4! Treat this ‘‘plateau’’~of a uniform heightd! as the inputof a 2-D segmentation problem, and partition it intominimum set of aperture shapes of heightd using the2-D SLS algorithm in Sec. II A 3.

~5! Remove the ‘‘plateau’’ from the 3-D IMB mountain, anrecursively process the remaining IMB by using exacone of the following two options, depending on thmaximum height of the remaining IMB.~a! If the maximum height of the remaining IMB is

small, then apply our basic 3-D SLS algorithm~Sec.II A 4 ! to obtain the segments for the IMB.

~b! Otherwise repeat steps~2!–~5! on the remainingIMB.

A key to the above scheme is how to choose an approate leveld to form the ‘‘plateaus’’~for producing a small se


Medical Physics, Vo

TABLE I. SLS vs CORVUS 4.0~with leakage correction!.

Numberof IMBs I max Field size~in cm!

Number of segments Number of MUs

CORVUS 4.0 SLS CORVUS 4.0 SLS

Head and neck 5 5 636 143 32 1314 660Thyroma 7 5 1039 235 83 1808 1101Pancreas 7 5 10312 381 84 2752 1491Prostate 7 5 939 281 57 2213 1416

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of segments for the input 3-D IMB mountain!. There areseveral possible choices ford: a function of the maximumintensity levelI max ~e.g.,d5@ I max/2#), a function of the vol-ume of the resulting ‘‘plateau,’’ a random choice, etc. Oexperiments on these choices ofd using various real medicadatasets showed that different choices ofd give good resultsfor different types of datasets. To obtain the ‘‘best’’ qualioutput, our general 3-D SLS algorithm combines varioways of choosingd in a ‘‘mixed’’ fashion, and we call it themixed partitioning method. Specifically, in the mixed parti-tioning method, every time when step~2! of the previousscheme is performed, we use each of the ‘‘good’’ ways

l. 31, No. 4, April 2004

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choosed, and the algorithm, based on every chosen valued, is carried out recursively on the remaining IMB. At thend, our 3-D SLS algorithm selects the best overall resThe number of different ways of choosingd that we actuallyuse is four.~Of course, based on the characteristics ofintensity maps in hand, one may develop and incluexclude any partitioning methods into/from this scheme.!

The four methods of choosingd used in our current ver-sion of the general 3-D SLS algorithm/software are as flows:

~a! The Xia and Verhey partitioning method:32 For the

cal axis ishe profil

FIG. 10. Comparisons of the SLS and CORVUS 4.0 dose profiles. These profiles were plotted using the RIT113 Film Dosimetry System. The vertifor absolute doses in cGy, and the horizontal axis is for locations in cm. The CORVUS dose profiles are in dark, and SLS dose profiles are in light. Tesin ~a! and ~b! are orthogonal to the leaf motion direction, and the profiles in~c! and ~d! are along the leaf motion direction.


FIG. 11. Illustrating how MLC leavesand theY jaws are used for blockingradiation: ~a! CORVUS 4.0, and~b!SLS.

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IMB under consideration, choose the power of 2 thaclosest toI max/2 @i.e., d52i , where i 5Int„log2(Imax)…21].

~b! Que’s half-cutting method:22 d5@ I max/2#.~c! The maximum volume method:d is chosen such tha

the volume of the resulting plateau is maximized.~d! The ‘‘slice’’ method:21,24 Let d be the smallest positive

entry of the IMB.

The program for our general 3-D SLS algorithm basedthe above mixed partitioning method usually runs in onlyfew minutes. Yet, the mixed partitioning method helps pduce 10%–20% fewer segments according to our expments.

B. Computer and clinical implementation

To further study the effectiveness and performance ofnew 3-D SLS algorithm, we implemented it using the C pgramming language on a Linux workstation. Our softwapackage is named SLS. SLS is designed to use the CORintensity maps as input, and the Elekta Precise TreatmSystem™~RT Desktop! as the target delivery machine.

A typical IMRT planning process with SLS is carried ousing the following major steps.

~1! Generate the CORVUS prescriptions and intensity ma~2! Import the CORVUS intensity maps and compute t

SLS aperture shapes~segments! for these intensity maps~3! Import the CORVUS prescription files and compute t

monitor units~MU! for the SLS aperture shapes. For tMU computation, we use two different methods: a fublown Monte Carlo based dosage calculation progra20

and a much simpler discrete intensity map guided Mcalculation program. The underlying idea of usingMonte Carlo based MU calculation is to make sure tthe two leaf sequences~SLS and CORVUS! match eachother in 3-D dose, by adjusting the amount of moniunits prescribed. The idea of an intensity map guidMU calculation is to guarantee that the SLS prescrtions match the CORVUS prescriptions in beamlet intesities. Using these two methods for MU calculation ehances the error tolerance of our SLS software.

~4! Encode and transfer the SLS prescriptions, by usinDICOM RT protocol, to the radiation delivery machineFor this step, we developed a DICOM encoding a


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transferring software using the MergeCOM-3™ Avanced Integrator’s Tool Kit for UNIX of Red Hat Linuxsystems.

III. RESULTS

After the SLS software package was developed, we cducted experiments and comparisons between SLS andpopular segmentation algorithms/software: the CORVUSverse planning system~the 4.0 and 5.0 versions! and the Xia

FIG. 12. Comparisons of dose distributions on the coronal plane.~a! TheCORVUS plan isodose lines: 90%, 75%, 60%, 40%, 20%, and 10%.~b! TheSLS plan isodose lines: 90%, 75%, 60%, 40%, 20%, and 10%.~c! Overlay-ing the isodose lines of 75% and 40% in~a! and ~b!.


Medical Physics, Vo

TABLE II. SLS vs CORVUS 5.0~with and without leakage correction! and the Xia and Verhey algorithm.

Numberof IMBs

Maximumintensity level

Field size~in cm!

Number of segments

CORVUS 5.0

Xia and Verhey SLSWith leakage

correctionWithout leakage

correction

Brain 5 5 233 22 24 23 17Head andneck

6 10 435 64 65 72 637 5 16315 262 215 216 133

Prostate 5 5 738 72 61 63 39

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and Verhey segmentation algorithm. Some of our experimtal results are presented in this section. In general, our reindicated the following.

~1! The SLS software was implemented correctly.~2! The treatment plans produced by SLS are feasible

patient treatments.~3! The SLS plans have a significantly smaller number

segments and therefore can be delivered in a mshorter treatment time.

A. SLS vs CORVUS 4.0 and the clinical feasibilityof SLS

We conducted comparisons between our SLS softwand the CORVUS 4.0 planning system used in the Depment of Radiation Oncology, University of Maryland Schoof Medicine. A main objective of this study was to verify thclinical feasibility of the SLS software package. Hence,comparisons were largely focused on whether the plans cputed by SLS match the original CORVUS plans in a 3dose.

Table I gives some of our comparison results, which shthat SLS significantly outperforms CORVUS 4.0 in termsthe number of segments computed. In Table I, the numbof segments computed by SLS are up to 80% less than CVUS 4.0. As a result, the treatment time has also beenproved substantially by the SLS plans. For example, forpancreatic case in Table I, the treatment time was shortefrom 72 min for the CORVUS 4.0 plans to 25 min for thSLS plans. Note that the treatment time was measured oElekta Linac system, and should be accelerated on Siemand Varian machines. We believe that even for machiwithout a large intersegment delay as on Elekta, the substial reduction of the number of segments used~e.g., from 381to 84 as the pancreas case in Table I! should significantlyshorten the treatment time.

Generally speaking, the dose distributions created by Splans and CORVUS 4.0 plans for the same set of intenmaps are very similar. The difference between the isodlines is within 2 mm, and the difference of doses measuby ion chamber at the isocenters is within 3%. Below we ua pancreatic case as an example to show the similaritiestween the SLS plans and the original CORVUS plansterms of their dose distributions.

Figure 10 shows the dose profiles of the CORVUS 4.0 aSLS plans for a pancreatic case.

l. 31, No. 4, April 2004

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As can be seen from the figure, these dose profilesvery similar. The bigger differences exhibited that are cloto the left and right ends of the dose profiles are due toleakage of the CORVUS 4.0 plans. These differencesoutside the beams, and are caused by the way that COR4.0 uses the MLC leaves and backup diaphragms to blradiation. Note that on the Elekta Linac system, MLC leavcannot be completely closed and must maintain a minimleaf separation distance. Hence, to achieve the effect of fblocking the radiation through a pair of MLC leaves, wneed to bring in the backup diaphragms. The backup dphragms are also referred to as theX andY jaws~theY jawshave a 10% leakage!. The way theY jaws are used for block-ing leaf openings in CORVUS 4.0 is shown in Fig. 11~a!.Basically, CORVUS 4.0 assumes that theY jaws and MLCleaves are ‘‘perfect’’ blockers. In contrast, the way theY jawsare used for blocking leaf openings in SLS is shown in F11~b!. The shaded area in Fig. 11~a! indicates the place inwhich most leakage occurs. As can be seen, CORVUShas more leakage, especially around the boundaries ofbeams.

Figure 12 shows the CORVUS 4.0 and SLS dose disbutions and the comparisons on the coronal plane forsame pancreatic case. In this figure, the isodose lines of ttwo types of plans are very similar. Based on our measument, the difference between the two 75% lines is withinmm.

Since the dose measured by films has a 5%–10% ewe also performed an ion chamber measurement at thecenters. In this case, the dose measured for the SLS pla183 cGy, while the CORVUS 4.0 prescribed dose is 178 cThe difference is within 3%.

B. SLS vs CORVUS 5.0 and the Xia and Verheyalgorithm

We also conducted comparisons between SLS and CVUS 5.0 as well as the Xia and Verhey segmentation alrithm. The focus of these comparisons was primarily onnumber of segments computed by each algorithm/softwa

CORVUS 5.0 is the latest version of the CORVUS planing system. The comparisons were carried out at NOMCorp. in Pittsburgh. We also chose to compare SLS withXia and Verhey algorithm because their algorithm has bcoded by many people and has been used as a gene

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accepted benchmark program. In our study, we implementhe Xia and Verhey algorithm with the Elekta constrainbased on the reducing level method.

Table II gives the comparison results. Since the currversion of the SLS software does not yet include the tonand groove correction, for a fair comparison, we incluCORVUS 5.0 results, both with the leakage correction awithout the leakage correction. As shown by Table II, Soutperforms CORVUS 5.0 by a factor of about 40%~31%when the leakage correction is turned off! in terms of thenumber of segments computed, and outperforms the XiaVerhey algorithm by a factor of about 33%. On all the casSLS performs better than CORVUS 5.0 and the Xia aVerhey algorithm in various degrees.

IV. CONCLUSION

In this paper, we presented a new segmentation algoritcalled SLS, for step-and-shoot IMRT delivery. Unlike prevous segmentation approaches, SLS is based on graphrithmic techniques in computer science, and exploits the gmetric nature of the problem. Our implementation aexperiments of SLS on an Elekta Linac system showedthe SLS software substantially outperforms CORVUS~ver-sions 4.0 and 5.0! and the Xia and Verhey algorithm in termof minimizing the number of MLC aperture shapes ashortening the treatment time. Film and ion chamber msurements proved that the SLS plans match well withCORVUS commercial planning system in 3-D composdose and hence paved the way for the clinical applicationSLS.

For future research, we plan to do the following.

~1! We plan to extend our SLS algorithm/software to Semens and Varian Linac systems. We believe thattreatment time should be significantly reduced due toless strict MLC constraints and shorter intersegmentlays on these two systems.

~2! The SLS algorithm as in its current state does not taccount of the ‘‘tongue and groove’’ effect.29,32 This isalso evident in Fig. 12, where the 90% isodose linescontinuous for CORVUS and broken up for SLS. Thefore, for future work, we also plan to integrate th‘‘tongue and groove’’ consideration into our algorithmWhile the effect of adding the tongue and groove corrtions to our current SLS algorithm is not completeclear at this point, we do notice that the number of sments tends to increase by a factor of 10%–20% baon the estimates by several authors23,32 on other algo-rithms.

ACKNOWLEDGMENTS

The authors are very grateful to the anonymous reviewfor their highly constructive comments and suggestions. Twork was supported in part by the National Science Fountion under Grant No. CCR-9988468. The work of the fi


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author was also supported in part by a fellowship fromCenter for Applied Mathematics at the University of NotDame.

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a new mlc segmentation algorithmÕsoftware for step-and ...pages.cs.wisc.edu/~athula/luan.pdf ·...

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