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The use of mixed-integer programming for inverse treatment planning with pre-defined field

segments

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2002 Phys. Med. Biol. 47 2235

(http://iopscience.iop.org/0031-9155/47/13/304)

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INSTITUTE OF PHYSICS PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 47 (2002) 2235–2245 PII: S0031-9155(02)36001-9

The use of mixed-integer programming for inversetreatment planning with pre-defined field segments

Greg Bednarz, Darek Michalski, Chris Houser, M Saiful Huq,Ying Xiao, Pramila Rani Anne and James M Galvin

Department of Radiation Oncology, Kimmel Cancer Center of the Jefferson Medical College,Thomas Jefferson University, Philadelphia, PA 19107, USA

E-mail: Greg.Bednarz@mail.tju.edu

Received 19 April 2002Published 20 June 2002Online at stacks.iop.org/PMB/47/2235

AbstractComplex intensity patterns generated by traditional beamlet-based inversetreatment plans are often very difficult to deliver. In the approach presentedin this work the intensity maps are controlled by pre-defining field segmentsto be used for dose optimization. A set of simple rules was used to definea pool of allowable delivery segments and the mixed-integer programming(MIP) method was used to optimize segment weights. The optimizationproblem was formulated by combining real variables describing segmentweights with a set of binary variables, used to enumerate voxels in targetsand critical structures. The MIP method was compared to the previouslyused Cimmino projection algorithm. The field segmentation approach wascompared to an inverse planning system with a traditional beamlet-basedbeam intensity optimization. In four complex cases of oropharyngeal cancerthe segmental inverse planning produced treatment plans, which competedwith traditional beamlet-based IMRT plans. The mixed-integer programmingprovided mechanism for imposition of dose–volume constraints and allowedfor identification of the optimal solution for feasible problems. Additionaladvantages of the segmental technique presented here are: simplified dosimetry,quality assurance and treatment delivery.

1. Introduction

Three-dimensional (3D) radiation treatment planning techniques are routinely used to conformthe high radiation dose to the target volume and to reduce the radiation dose to non-involvedtissue. However, conventional 3D planning techniques can be inadequate in some cases, forinstance, if there is a need to design a conformal treatment plan for targets with invaginationsthat surround critical structures. Inverse planning techniques (combined with an appropriate

0031-9155/02/132235+11$30.00 © 2002 IOP Publishing Ltd Printed in the UK 2235

2236 G Bednarz et al

dose delivery system) may offer a solution in such cases by solving for a distribution of beamintensities that obey set dose constraints (Sternick 1997, Boyer et al 1997, Wu et al 2000).The beam intensities are usually optimized over a fine grid of elementary beamlets, e.g.,using a 1 × 1 cm2 grid. The modulated beams can be delivered using a variety of methods,e.g., by superimposing a number of static fields generated with a multileaf collimator (MLC)(Galvin et al 1993). The optimized beamlet-based intensity maps tend to be very complex(Sternick 1997, Webb et al 1998) and often require a large number of delivery segments. Anumber of beam-smoothing techniques have been proposed to control the complexity of theintensity maps. Some of these techniques rely on averaging intensity patterns or on introducingadditional constraints in the beamlet optimization to control intensity variations (Spirou et al2001, Alber et al 2001). Both methods can potentially degrade the final dose distribution.In the approach investigated in this work the intensity maps are controlled by pre-definingfield segments to be used for dose optimization. These segments are created in addition tobeam’s-eye-view (BEV) conformal fields in order to make it possible to create invaginations inthe dose distribution. The idea of adding additional field segments within the BEV conformalfields has been investigated in the literature (Webb 1991, De Neve et al 1996, Derycke et al1997, Eisbruch et al 1998, Frass et al 1999, Xiao et al 2000, De Gersem et al 2001, Xia et al2001). These segments could be introduced, e.g., for the purpose of boosting a low dose regionand/or shielding critical structures. The selection and placement of the segments depend onplanning objectives and planner experience (Eisbruch et al 1998, Xia et al 2001) and alsoon techniques employed to optimize their relative weights (De Neve et al 1996, Derycke et al1997, Frass et al 1999, Xiao et al 2000, De Gersem et al 2001).

This paper is concerned with the optimization aspect of the inverse treatment planningwith pre-defined segments. Using these segments as the only allowable fields, the computer-based optimization was employed to determine the segment weights. The mixed-integerprogramming with dose–volume constraints was used to solve for the desired dose distribution.The MIP method was compared to the previously used Cimmino projection algorithm(Xiao et al 2000). These two algorithms represented two different classes of optimizationmethods. The use of the mixed-integer programming was investigated because this methodallowed for straightforward implementation of dose–volume constraints and for introductionof the objective function in the optimization. The MIP could yield the optimal solution forfeasible (e.g., solvable) optimization problems. In contrast, the Cimmino method convergedon a solution satisfying a given set of constraints. The inverse treatment planning withthe pre-defined delivery segments and using these two optimization methods was comparedto the NOMOS CORVUS inverse planning system with a traditional beamlet-based beamintensity optimization. Four complex cases of patients treated for oropharyngeal cancer wereused as test cases. These cases were selected for their complexity due to multiple boost levelsrequired to treat the disease and the need to create multiple invaginations in dose distribution toachieve critical organs sparing. One aim of this investigation was to determine if this methodof simplifying IMRT delivery could give clinically acceptable solutions even for complextreatment planning scenarios and produce dose distributions similar to those obtained fromthe beamlet-based optimization.

2. Materials and methods

2.1. Field segmentation

The segmental planning technique described here (implemented on the CMS/FOCUS planningsystem) started with conformal fields at nine equally spaced, co-planar gantry angles.

Mixed-integer programming for IMRT with pre-defined field segments 2237

Figure 1. Field segmentation example. This MLC segment irradiates the PTV but excludes thespinal cord. Two spinal cord MLC segments are superimposed on the conformal field, if the spinalcord casts the shadow through the middle of the target volume.

For these gantry angles, additional fields were added that only partially irradiated the targetso that invaginations were created within the dose distribution. The rules and rationalefor field segmentation were described previously (Bednarz et al 1999, Galvin et al 2000).They stem from our earlier work (Bednarz et al 1999), where we investigated the use ofpartial transmission blocks to create invaginations in dose distribution. We found that thedose distributions we were able to obtain for a number of complex head and neck caseswere comparable to those obtained using beamlet-based inverse planning. In practice, partialtransmission blocks can be replaced with multiple MLC segments. The number of the segmentsdepends on the number of fixed gantry positions used for the planning, the number of criticalstructures to be shielded and the number of target volumes. As an example, for the spinalcord shadowing one of target volumes we would create 18 segments to irradiate the targetbut exclude the spinal cord (two segments superimposed on each of the conformal fields atthe nine gantry angle, assuming that the spinal cord casts the shadow through the middle ofthe target volume), figure 1. This procedure would be repeated for other critical structuresand target volumes. The margins around the segments were selected to account for the beampenumbra. In the present work, the segmentation process produced from 70 to 111 segmentsper case. After the segmentation process was complete, the segment weights were optimizedto satisfy prescribed dose constraints.

2.2. Segment weight optimization

For dose calculation, the volume was divided into a grid of voxels and the dose matriceswere extracted from the CMS FOCUS system for the optimization purposes. Volumes ofinterest were modelled by subsets of voxels enumerated by two indices: S referring to a givenstructure, i referring to a given voxel. The dose calculation and dose limits were written as aset of linear inequalities

LS < DSi =J∑j

wjdij < US (1)

where DSi is the dose delivered to the ith voxel of Sth structure, dij is the element ofdose calculation matrix, wj is the component of the segment’s intensity (weight) vector

2238 G Bednarz et al

W = {w1, w2, . . . , wJ }, and LS and US are the lower and upper dose limits for structure S.At this point a projection algorithm, e.g., the Cimmino algorithm (Censor et al 1997) could beapplied to solve for a weight vector satisfying these inequalities. That segment optimizationapproach has been reported on previously by Xiao et al (2000) for segmented treatment ofprostate cancer. The Cimmino algorithm is based on relaxed projections that provide iterativelydeterministic correction of weights, starting from some initial values. It is related to a knownfixed-point iteration technique of Jacobi for solving linear equalities. The iterative update ofweights in the Cimmino algorithm takes the form

wk+1j = wk

j + λ∑

i

zici

(wk

j

)dij (2)

where

ci(wk) = 0 if L < Di < U

ci(wk) = (U − Di)

/ ∑j

d2ij if U � Di

ci(wk) = (L − Di)

/∑j

d2ij if Di � L

and ∑i

zi = 1 and zi > 0.

k enumerates the iterations and λ is the relaxation parameter between 0 and 2. The value of zj

can be interpreted in terms of relative importances of organs in a plan, e.g., if every voxel isequally important then all zj should be equal to 1/N, where N is the total number of voxels. Animportant feature of the Cimmino algorithm is its ability to deal with infeasible problems, e.g.when the intersection of all the constraint sets is empty. In the infeasible case the algorithmconverges to the point that minimizes the weighted sum of squares of distances to all violatedsets. This sum is called the proximity function and it is used as a criterion to terminate theiteration process.

The set of inequalities in equation (1) can be also treated in the context of the mixed-integerprogramming that adds an objective function to the optimization problem (Kolman et al 1995).For example, a solution can be sought, which minimizes doses to critical organs or maximizesdoses to the targets, if the initial optimization problem is feasible. If the problem is infeasible,the strict dose limits can be relaxed through incorporation of the dose–volume constraints. TheMIP allows for a straightforward and rigorous implementation of the dose–volume constraintsby combining real variables describing segment weights with a set of binary variables, used toenumerate voxels in targets and critical structures. The binary variables could be set to zero orone depending on the current dose level at a given voxel, providing a mechanism of imposingmultiple dose–volume constraints. The single voxel dose constraints from equation (1)are relaxed by introducing underdose (fd) and overdose (Fd) fractions:(

1 − bS

i fd

)L < Di <

(1 + BS

i Fd

)U (3)

where bi and Bi are binary variables. The amount of underdosed and/or overdosed volume iscontrolled by requiring that

V∑i

bSi < n1 (4a)

andV∑i

BSi < N1 (4b)

Mixed-integer programming for IMRT with pre-defined field segments 2239

where n1 = fv∗ V and N1 = Fv

∗ V are the numbers of the underdosed and overdosed voxels,respectively and V is the volume of the Sth structure. The fractions fv and Fv are the fractionsof the Sth structure volume which can be underdosed or overdosed to (1−fd)L and (1+Fd)U,

respectively. The optimization is over the segment weight vector W = {w1, w2, . . . , wJ } andthe binary variables.

For the present work the Cimmino optimization was performed using in-house softwarerunning on a PC Linux workstation with two Pentium III-800 MHz processors. The MIPoptimization was carried out using commercially available optimization software (IBMMIP Optimization Solutions, IBM Corp., Armonk, NY), running either on the same PCLinux workstation or on a PC Windows 2000 workstation with Pentium IV, 1.4 GHzprocessor. The IBM mixed integer solver used a branch and bound method with the simplexalgorithm (Kolman et al 1995). The input files for the solver were written in a standardmathematical programming system (MPS). Two linear objective functions were used: (1)minimization/maximization of the dose to the targets and (2) minimization of the numberof non-compliant voxels, i.e., the number of under- or overdosed voxels. The voxel volumeused for calculations was either 2 or 3 mm3, resulting in optimization problems with a fewthousand binary variables. In general, MIP problems with thousands of integer variables mayrequire many hours of computational time, which would make this optimization approachimpractical. Therefore, we introduced a number of techniques reducing computational burdenand optimization times. One of these techniques relied on sparsing of the initial dose matrices.That is, instead of investigating the dose compliance at each voxel, the constraints wereimposed on selected voxels, e.g., on every second voxel. This approximation was based on theassumption that, if two voxels, which were 2 mm apart were either compliant or non-compliantthen the voxel between them was likely to follow the pattern. Note that this procedure wasnot equivalent to the dose averaging over 4 mm3 voxels, since the dose was still sampledwith 2 or 3 mm3 resolution. The optimization times were on average reduced by a factor offive for the 50% reduction in the number of integer variables. We also found that the furtherreduction in the computational burden could be achieved by using an appropriate optimizationtechnique. One approach was to set the optimization goal to minimize the dose to targetssubject to lower dose volume constraints and lower dose limits, and also subject to the doseconstraints to critical structures. The upper dose volume constraints were removed and theupper dose limits were imposed according to the dose limit for the final boost. This approachalso significantly reduced the search time for an integer solution and it was usually used inthe first optimization pass. In the second pass the results could be refined using, e.g., theminimization of the number of the non-compliant voxels with full DVH constraints. Anotherreason to use the above approach was to test the feasibility of the optimization problem. Theresults of the first pass would indicate which target dose inhomogeneity was likely to exceedthe test limits.

The optimized weights were transferred back to the FOCUS planning system. Forthe examples discussed here the doses were calculated using the scatter integration witha heterogeneity correction. The comparison of the different planning and optimizationapproaches was based on examination of dose distributions on cross-sectional CT imagesand analysis of the dose–volume histograms.

2.3. Case selection and dose constraint specifications

Four complex cases of patients treated for oropharyngeal cancer were analysed. In each caseeither one or two boost levels were required to treat the disease and complex dose distributionswere needed to achieve critical organs sparing. All dose–volume constraints were set according

2240 G Bednarz et al

Figure 2. Dose–volume histograms for case 1. This figure compares the MIP plan (solid lines)with the CORVUS simulated annealing plan for the MIMiC collimator (dashed lines). DVHs forthe spinal cord, the right parotid, the PTV (54 Gy level) and the GTV (66 Gy level) are shown.

to the recommendations of the RTOG protocol H0022, designed to test IMRT for this diseasesite. Two of the cases involved three dose levels to treat the gross tumour (GTV) to 66 Gy,subclinical PTV at high risk to 60 Gy and bilateral lymph nodes (initial PTV) to 54 Gy. Thedose was delivered in 30 fractions and all the targets were treated concurrently. The other twocases involved two dose levels (66 Gy to the gross tumour and 54 Gy to the bilateral lymphnodes). Critical normal structures included brain stem (<54 Gy), spinal cord (<45 Gy),mandible (<70 Gy) and parotid glands (50% of at least one gland below 30 Gy). The coveragerequirement was to have at least 95% of each target volume receive the prescribed dose. Theinhomegenity limits were such that not more than 20% of the volume of targets could receivea dose above 110% of the prescribed dose and not more than 1% of the volume could receivea dose less than 93% of the prescribed dose.

3. Results and discussion

The mixed-integer programming generated solutions which compared favourably with theCORVUS solutions. It is noteworthy that this result was obtained despite restricting intensitymaps by pre-defining the delivery segments. Figures 2 and 3 show dose–volume histogramsfor one of the one-boost level cases. Figures 4–6 show dose–volume histograms for one ofthe more complex two-boost level cases and figure 7 shows various target volumes and thedose distribution for the MIP plan in a transverse CT slice for this case. Similar results wereobtained for the remaining two cases. Tables 1 and 2 give more quantitative comparisonof the results by listing percentage coverages and dose inhomogeneity for selected targetvolumes for all the cases and planning methods. The MIP method and the Cimmino algorithmproduced comparable results; however, the mixed-integer solutions held slight advantage overthe Cimmino solutions in limiting the high dose tails and yielding better target coverages.This was consistent with inherent features of these optimization methods, i.e., for feasibleproblems MIP yields optimal solution as opposed to the Cimmino algorithm, which convergeson a feasible solution. The MIP-optimized plans satisfied all the dose–volume constraintsof the RTOG IMRT protocol, figures 2–6 and tables 1 and 2. The Cimmino plans andthe CORVUS with MIMiC collimator plans deviated in a minor way from the protocol by

Mixed-integer programming for IMRT with pre-defined field segments 2241

Figure 3. Dose–volume histograms for case 1. The figure compares the MIP plan (solid lines)with the CORVUS simulated annealing plan for a standard MLC (dashed lines). DVHs for thespinal cord, the right parotid, the PTV (54 Gy level) and the GTV (66 Gy level) are shown.

Figure 4. Dose–volume histograms for case 4. This figure compares the MIP plan (solid lines)with the CORVUS simulated annealing plan for the MIMiC collimator (dashed lines). DVHs forthe spinal cord, the right parotid, the initial PTV (54 Gy level), the PTV for the region of subclinicaldisease at high risk (60 Gy level) and the GTV (66 Gy level) are shown.

exceeding the inhomogeneity limit in case 2 at the 54 Gy dose level. The CORVUS with astandard collimator plan exceeded the inhomogeneity limit at the 54 Gy dose level for cases 1and 2, and at the 54 and 60 Gy dose levels for case 4.

The MIP optimization times were in the range from 20 min to over 120 min. These timeswere longer than the optimization times for runs with the Cimmino algorithm, which took onaverage 20 min. However, multiple runs with small adjustments of importance parameterswere usually required for the optimization with the Cimmino algorithm to obtain a satisfactorysolution. Moreover, the final result from this type of optimization could also depend on theuser experience in adjusting the importance parameters and other factors. In the case of the

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Figure 5. Dose–volume histograms for case 4. This figure compares the MIP plan (solid lines)with the CORVUS simulated annealing plan for a standard MLC (dashed lines). DVHs for thespinal cord, the right parotid, the initial PTV (54 Gy level), the PTV for the region of subclinicaldisease at high risk (60 Gy level) and the GTV (66 Gy level) are shown.

Figure 6. Dose–volume histograms for case 4. The figure compares the MIP plan (solid lines)with the Cimmino plan (dashed lines). DVHs for the spinal cord, the right parotid, the initial PTV(54 Gy level), the PTV for the region of subclinical disease at high risk (60 Gy level) and the GTV(66 Gy level) are shown.

MIP-based optimization results are largely independent of user experience. There is no needto guess proper values of importance parameters driving the optimization. In such context theMIP computational times do not seem excessive and they can easily be reduced in the future,taking into account rapid increase in the computer speed. The majority of the MIP solutionswere obtained using selective voxels sampling as described in the previous section. Usuallyevery second voxel was sampled in a given structure. The final statistics were computed usingall the voxels. We found that the solutions obtained with voxel sampling differed little from

Mixed-integer programming for IMRT with pre-defined field segments 2243

Figure 7. The dose distribution in a transverse CT slice for the MIP plan for case 4. The picturealso shows contours of various target volumes—these contours are drawn using solid white lineand labelled by the prescribed dose levels. Solid black lines represent the isodose lines at 54, 60and 66 Gy levels, as indicated by the labelled arrows.

Table 1. The performance of the MIP, the Cimmino algorithm and the CORVUS beamlet-basedsimulated annealing with either the MIMiC collimator or a standard MLC (SMLC) for the twoone-boost level cases. The PDC is the percentage coverage of the target with the prescription doseand the Inh is the dose inhomogeneity, expressed as the percentage of the target volume, whichreceived the dose above 110% of the prescribed dose.

Case 1 (54 Gy) Case 1 (66 Gy) Case 2 (54 Gy) Case 2 (66 Gy)

Method PDC% Inh% Inh% PDC% PDC% Inh% PDC% Inh%

MIP 96.5 15.3 95.7 0 95.1 17.5 96.6 0Cimm 95.7 19.8 97.9 0 95.0 21.4 95.9 0MIMiC 95.6 18.8 98.2 0 95.1 23.4 100 0SMLC 96.2 29.7 95.1 0 95.1 31.8 95.0 0

the solutions obtained using all the voxels—the observed changes in target coverage were onthe order of a fraction of the per cent, if we sampled every second voxel. The difference wouldbecome more pronounced when even fewer voxels were sampled.

The segmentation process described here is not yet automated. The rules for creatingsegments are simple and intuitive, but the segment creation is time consuming even usingcontouring tools offered by a modern radiation treatment planning system. The contouringtools that would speed up the segments creation should allow the user to automatically outlinesegments that conform to a given target but exclude structures shadowing this target. Itseems that this new capability could be introduced in a relatively simple manner by extendingcapabilities currently offered by a number of 3D RTP systems for drawing conformal blocks.The segment creation procedure described in this work also opens a question whether editingthe existing segments, e.g., the amount of margins around the target volumes or adding more

2244 G Bednarz et al

Table 2. The performance of the MIP, the Cimmino algorithm and the CORVUS beamlet-basedsimulated annealing with either the MIMiC collimator or a standard MLC (SMLC) for the twotwo-boost level cases. The PDC is the percentage coverage of the target with the prescription doseand the Inh is the dose inhomogeneity expressed as the percentage of the target volume, whichreceived the dose above 110% of the prescribed dose. Only 54 Gy and 60 Gy levels are shown.For all the planning methods except the SMLC the coverage at the 66 Gy level was approaching100% (95.1% for the SMLC plan) and the Inh was zero.

Case 3 (54 Gy) Case 3 (60 Gy) Case 4 (54 Gy) Case 4 (60 Gy)

Method PDC% Inh% PDC% Inh% PDC% Inh% PDC% Inh%

MIP 95.1 13.8 95.7 16.5 98.1 10.1 99.6 4.4Cimm 95.6 18.8 95.2 18.7 95.1 19.8 98.0 14.5MIMiC 95.6 15.2 95.6 10.1 99.4 9.5 98.2 5.3SMLC 96.7 13.4 97.1 8.1 98.5 25.6 99.3 21.9

segments would significantly improve the results. Initially, we allowed for sufficient marginsaround all the target volumes to account for the beam penumbra. However, we found that suchmargins were not necessary around the GTVs. The presence of these margins increased thedose inhomogeneity within the initial PTVs and better solutions were obtained with the muchtighter margins. Based on our initial results, we also introduced segments that accountedfor the radiation dose deposited within the initial PTV by the smaller segments boosting theGTV. These additional segments conformed to the PTV volume but shielded the GTV andtheir introduction led to significant reduction of the dose inhomogeneity within the PTV. Wedid not introduce segments accounting for the patient curvature, although these segmentscould further improve the results. These segments were not necessary to meet dose volumeconstraints in the four cases investigated here. Fewer than 50 segments were needed to deliverthese complex treatments, which in each case constituted approximately 50% of the segmentsavailable for the optimization. The time needed to deliver 50 MLC segments using step andshoot method was estimated to be in the range from 15 to 20 min for a linear acceleratorwith a standard magnetron and it could be significantly shorter with the use of a fast-tuningmagnetron (Budgell et al 2001).

4. Conclusions

The mixed-integer programming (MIP) method was used to solve for the prescribed dosedistribution in the inverse treatment planning technique, which employs pre-defined segmentsin order to limit the complexity of the IMRT dose delivery. The MIP method was comparedto the Cimmino projection method, which was tested in the earlier work. Both methods wereused to optimize the weights of allowable segments in four complex cases of oropharyngealcancer. The dose–volume histograms from the segmental planning were similar to thoseobtained through beamlet-based optimization, proving the feasibility of the segmentalapproach. The mixed-integer programming method offered a straightforward way ofimplementing dose–volume constraints. It allowed for identification of the optimal solution, ifthe problem was feasible. The advantage of the Cimmino algorithm was its robustness, i.e., itcould yield a solution closely approaching the acceptable solution for the infeasible set of doseconstraints. The segmental IMRT technique presented here could offer several advantagesover a traditional beamlet-based IMRT in terms of simplified dosimetry, quality assurance andtreatment delivery. It avoids very small MLC segments often resulting from segmentation ofthe beamlet-based fluence patterns and aims at minimizing the number of segments needed todeliver complex treatments.

Mixed-integer programming for IMRT with pre-defined field segments 2245

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