application of constrained optimization to radiotherapy planning

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Application of constrained optimization to radiotherapy planning Otto A. Sauer, David M. Shepard, and T. Rock Mackie Citation: Medical Physics 26, 2359 (1999); doi: 10.1118/1.598750 View online: http://dx.doi.org/10.1118/1.598750 View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/26/11?ver=pdfcov Published by the American Association of Physicists in Medicine Articles you may be interested in Direct leaf trajectory optimization for volumetric modulated arc therapy planning with sliding window delivery Med. Phys. 41, 011701 (2014); 10.1118/1.4835435 A role for biological optimization within the current treatment planning paradigm Med. Phys. 36, 4672 (2009); 10.1118/1.3220211 A constrained tracking algorithm to optimize plug patterns in multiple isocenter Gamma Knife radiosurgery planning Med. Phys. 32, 3132 (2005); 10.1118/1.2044430 Penalized likelihood fluence optimization with evolutionary components for intensity modulated radiation therapy treatment planning Med. Phys. 31, 2335 (2004); 10.1118/1.1773631 Algorithms and functionality of an intensity modulated radiotherapy optimization system Med. Phys. 27, 701 (2000); 10.1118/1.598932

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Application of constrained optimization to radiotherapy planningOtto A. Sauer, David M. Shepard, and T. Rock Mackie

Citation: Medical Physics 26, 2359 (1999); doi: 10.1118/1.598750 View online: http://dx.doi.org/10.1118/1.598750 View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/26/11?ver=pdfcov Published by the American Association of Physicists in Medicine Articles you may be interested in Direct leaf trajectory optimization for volumetric modulated arc therapy planning with sliding window delivery Med. Phys. 41, 011701 (2014); 10.1118/1.4835435 A role for biological optimization within the current treatment planning paradigm Med. Phys. 36, 4672 (2009); 10.1118/1.3220211 A constrained tracking algorithm to optimize plug patterns in multiple isocenter Gamma Knife radiosurgeryplanning Med. Phys. 32, 3132 (2005); 10.1118/1.2044430 Penalized likelihood fluence optimization with evolutionary components for intensity modulated radiation therapytreatment planning Med. Phys. 31, 2335 (2004); 10.1118/1.1773631 Algorithms and functionality of an intensity modulated radiotherapy optimization system Med. Phys. 27, 701 (2000); 10.1118/1.598932

Application of constrained optimization to radiotherapy planningOtto A. Sauera)

Universitat Wurzburg, Klinik fur Strahlentherapie, Josef-Schneider-Straße 4, 97080 Wu¨rzburg, Germany

David M. Shepard and T. Rock MackieUniversity of Wisconsin, Medical School, Madison, Wisconsin 53706-1532

~Received 24 November 1998; accepted for publication 18 August 1999!

Essential for the calculation of photon fluence distributions for intensity modulated radiotherapy~IMRT! is the use of a suitable objective function. The objective function should reflect the clinicalaims of tumor control and low side effect probability. Individual radiobiological parameters forpatient organs are not yet available with sufficient accuracy. Some of the major drawbacks of somecurrent optimization methods include an inability to converge to a solution for arbitrary inputparameters, and/or a need for intensive user input in order to guide the optimization. In this work,a constrained optimization method was implemented and tested. It is closely related to the de-manded clinical aims, avoiding the drawbacks mentioned above. In a prototype treatment planningsystem for IMRT, tumor control was guaranteed by setting a lower boundary for target dose. Theaim of low complication is fulfilled by minimizing the dose to organs at risk. If only one type oftissue is involved, there is no absolute need for radiobiological parameters. For different organs,threshold dose, relative seriality of the organs or an upper dose limit could be set. All parameters,however, were optional, and could be omitted. Dose–volume constraints were not used, avoidingthe possibility of local minima in the objective function. The approach was benchmarked throughthe simulation of both a head and neck and a lung case. A cylinder phantom with precalculated dosedistributions of individual pencil beams was used. The dose to regions at risk could be significantlyreduced using at least seven ports of beam incidence. Increasing the number of ports beyond sevenproduced only minor further gain. The relative seriality of organs was modeled through the use ofan added exponent to the dose. This approach however increased calculation time significantly. Thealternative of setting an upper limit is much faster and allows direct control of the maximum dose.Constrained optimization guarantees high tumor control probability, it is computationally moreefficient than adding penalty terms to the objective function, and the input parameters are doselimits known in clinical practice. ©1999 American Association of Physicists in Medicine.@S0094-2405~99!02711-X#

Key words: intensity modulated radiotherapy~IMRT!, optimization, objective function

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I. INTRODUCTION

In order to avoid local recurrence, curative radiotherapy ato eradicate all clonogenic tumor cells within a defined tarvolume.1 This has to be done without severely injuring adcent organs. Intensity modulated radiation therapy~IMRT!provides the potential to simultaneously fulfill both of therequirements.2 Due to the large number of parameters toconsidered, such as the fluence of individual pencil beaconventional interactive optimization techniques are not psible. This has led to the concept of inverse treatment pning: The aim of the treatment has to be described uscertain treatment goals, so that a mathematical optimizaalgorithm can find the parameters which lead to an optimdose distribution. For each pencil or elementary beamprimary fluence, the source position, the beam directionspace, the beam energy, and the radiation modality coultreated as variables.3 In practice, however, the solution spais usually restricted such that only the primary fluenweights vary. This obtains the most therapeutic gain whcompared to conventional treatment techniques. Optimtion of beam directions is important for a small number

2359 Med. Phys. 26 „11…, November 1999 0094-2405/99/26

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ports only.4 Different mathematical methods have been ulized in order to perform optimization. They can be divideinto two groups: stochastic methods such as simulaannealing5,6 and deterministic methods such as linear or qdratic programming, used by most groups working on tsubject. Although deterministic methods are faster, thererisk that, depending on the start up values, the algoritcould converge only to a local minimum. However, Deacould show that nonconvex objective functions occur onlydose–volume constraints are explicitly considered.7

The most important task is the definition of a sensibtreatment goal. This is accomplished through building anjective function. It is evident that this function has to includradiobiological considerations of the irradiated tissue. A smoidal dose response curve was suggested by many autstarting with Holthusen8 in 1936. Brahme and co-workerapplied Poisson statistics in order to model sigmoidal dresponse.9 Tissue may be characterized by the dose vaD50, the dose resulting in a probability of 50% for the effeand the dose response gradientg at theD50 dose level. Inaddition, a method that accounts for inhomogeneities of

2359„11…/2359/8/$15.00 © 1999 Am. Assoc. Phys. Med.

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2360 Sauer, Shepard, and Mackie: Application of constrained optimization 2360

dose distribution must be applied. Ka¨llman and co-workers,suggested defining a relative seriality, or its opposite, a rtive parallelity of an organ.10 A serial organ is damaged ione subvolume of that organ is damaged, while a totaparallel organ losses its complete functionality only if allthe subvolumes are damaged. The complex physiologyliving organs, however, makes it difficult to adequatemodel all possible radiation effects, and the clinical dare rare and insecure. Another problem is the need to aage between different patients with different radsensitivities.11,12 The applicability of both steep theoreticand flat averaged dose response functions for an individpatient is questionable. If many assumptions and parameare not well justified, it is better at present to use more splistic models that require a smaller number of parameterwhich we have more confidence. In addition, simpler modresult in faster numerical calculation and they are ableconverge to a global optimum. Various approaches haveready been used: maximize the minimum dose to the tarminimize the maximum dose to the regions at risk, lesquares or an equivalent procedure to fit dose to a prescrdistribution, total energy imparted to the target and rvolumes.3,5,13–17 All of these methods generally result idose distributions superior to those from conventional ddistribution plans. The end point, however, does not necsarily correspond to the treatment goal for an individualtient. In some methods it is necessary to use importancetors with no direct correspondence to biological parametOthers, due to the possibility of using too rigorous dose cstraints, lead to infeasibility of the optimization problem.

In this work, we investigated a strategy where dosepoints in the target is forced to lie between a lower andupper limit. With the lower limit, tumor control probability isadjustable. Different bounds may be attributed to differtarget points, if required. The dose value to any point wita region at risk is minimized. With this strategy, a weightibetween tumor control and normal tissue complication prability is avoided. If only one type of region at risk is acounted for, generally no parameters are necessary in oto specify the radiation response of that organ. A dose asas possible is automatically achieved. Certainly, the toance value where complication probability starts to riseto be known by the clinician in order to recognize plawhere no curative treatment is possible. This case is demstrated for a U-shaped target, similar to a head and ntarget volume. For an organ of parallel type, it mightdesirable to apply a seriality parameter, in order to maximthe region of low dose. Alternatively a simple dose resporelation could be used. This is proportional to dose for ldose values and almost flat beyond a specified thresdose. If more than one type of vulnerable regions arevolved, relative sensitivity and seriality parameters couldapplied. Alternatively upper limit constraints may be asigned to each region. This is demonstrated for an L-shatarget surrounded with vulnerable regions which correspto lung and spinal cord. With these approaches sophisticoptimization algorithms can be used together with clinicaknown tissue parameters.

Medical Physics, Vol. 26, No. 11, November 1999

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II. MATERIAL AND METHODS

In order to test the proposed optimization strategy, a trement planning system was written using theMATLAB5 pro-gramming language.18 It consists of easily replaceable modules for: ~1! setting up geometry, beams, objective functioand constraints,~2! calculating pencil beam weights, and~3!evaluating and documenting results. Previous versions ofcode had already been used to address a number of issuradiation beam optimization.19,20

A. Software modules and optimization algorithm

The software follows the operations as needed for trement planning. Regions of interest are entered as polygand converted to binary matrices. The number of portsvary. The gantry angles may be chosen manually or set edistantly after input of a start value. In this work, for a number of ports greater than four, the automatic option was usFor a smaller number of ports the same rules as in convtional interactive treatment planning were applied.

The dose distribution from each pencil beam wprecalculated.21 For this purpose a pencil beam of 2 Meprimary photons with width of 2 mm and length of 110 mwas scanned over a water cylinder of 200 mm diameter.dose distribution in the water cylinder was calculated for 1positions of incidence, i.e., every 2 mm and stored in ohundred 1003100 matrices. For the dose calculation, tsuperposition method of Monte Carlo pregenerated enedeposition kernels was employed. Beam divergence wasincluded. An example of a pencil beam dose distributionshown in Fig. 1. Due to cylindrical symmetry, the pencbeam dose kernels are invariant with respect to rotati

FIG. 1. Dose distribution from a pencil beam calculated with the convolutmethod. The cylinder phantom is 20 cm in diameter. The pixel size 2 mTo demonstrate the energy spread, the gray-scale is logarithmic.

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2361 Sauer, Shepard, and Mackie: Application of constrained optimization 2361

around the central axis of the cylinder. Therefore, theybe used at any gantry angle. Only pencil beams potenticontributing to the target dose with a significant amount wused. It was set to 0.2% of the maximum dose of a pebeam. The dose was calculated in one transversal planeFor the purpose of evaluating different methods, this stdardized phantom is of advantage. However, the systemwith any dose calculation engine.

The density of dose points used for the optimization mbe chosen randomly. Our experience shows that a highsity is necessary in regions where a high gradient of the ddistribution is desired. For the examples shown here a dsity of 80% for all regions of interest and 8% for the restthe phantom was used.

For determination of the optimum fluence weights tMATLAB routine CONSTR is invoked.18 CONSTR is based onthe method of Lagrange multipliers. At a minimum pointx*of the objective functionO(x) the Kuhn–Tucker equation

¹O~x* !1¹lg~x* !50

with

lg~x* !50 ~1!

is fulfilled. g~x* ! is the vector of constraint functions,l theset of Langrange parameters. The Kuhn–Tucker equatiosolved using sequential quadratic programming. This methe problem is approximated by a quadratic form at eiteration point. The quadratic subproblem is solved by usthe active set strategy: only constraints not fulfilled atparticular iteration point are considered. A similar stratewas published by Hristov and Fallone.22

B. Objective function and optimization parameters

The objective functionO was constructed to minimize thdose to all points at risk. If there is only one kind of tissuespecially if it is of serial type, the objective function maysimply the sum of dose to all points at risk. However, toable to account for differences in sensitivityw and serialitySthe following function was implemented:

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Di is the dose, the exponentSi accounts for the relativeseriality, wi is the sensitivity at risk pointi, and RP is thenumber of risk points considered. To organs where evesmall volume receiving a relatively high dose causes sevdamage, a highShas to be attributed. This can be seen asapproximation of the lower part of the sigmoidal dosesponse relation. Instead ofw, it is more convenient to usethreshold valuesD th for the occurrence of complications aorgans at risk.D th andw are connected by setting the produD th

S equal to the same value for different organs. By dothis, all dose points receiving just the accordant threshdose are penalized by the same amount, while dose vaabove the threshold are penalized according to their seriexponent. Setting the value of thewDth

S to 1,

Medical Physics, Vol. 26, No. 11, November 1999

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For a parallel type of tissue, if regions to save shall not,cannot be identified, it might be desirable to have the dbelow a certain level for as much volume as possible. Tcan be achieved by increasing the burden with dose togions being anyway above the threshold for complicatioThis behavior may be modeled by setting the exponenS,1 in Eq.~2!. Alternatively the dose response may be simlated by settingS51 and substituting the doseD in Eq. ~2!with the effectE, where

E5D for D,D th

E5a~D2D th! for D>D th. ~4!

a is a little greater than 0, in order to guide the optimizatiprocedure into the right direction. It was set to 0.1 in thwork. Equation~4! approximates the upper part of the simoidal dose response relation.

For the target volume, only a minimum desired valueset. This is done by defining a lower limit matrix~LLM ! forthe entire phantom, where we set

LLM 50.95 for target points, LLM50 elsewhere. ~5!

By scaling the dose distribution with the prescribed dothis setting guarantees a high tumor control probability, wiout detailed knowledge of the dose response curve. Additally an upper limit matrix~ULM ! is defined. Although anupper limit in the target region is desirable for certain disesites, this matrix is generally not necessary, because duthe minimization of dose to normal tissue, the occurrencehot spots is unlikely. The main reason for setting an uplimit was the reduction of the solution space, thus speedup the calculation. The setting used in this work, if not oerwise stated was

ULM51.1 for target points, ULM50.95 elsewhere. ~6!

The introduction of upper limits bears the risk that a slution might not be feasible, if the limits are too restrictivThis is due to physical laws, especially the depth dose cuHowever, with very little experience, appropriate values cbe set to avoid infeasibility. The upper limit matrix alsopens the possibility to set upper boundaries for the dosregions at risk. This boundary can be set to the toleravalue of serial organs.

C. Documentation and evaluation of results

In order to compare different results, besides dose disbutions and dose–volume histograms~DVH!, basic param-eters of the dose distributions, like minimum, maximumean value, and standard deviation of the dose to a speregion of interest were calculated. The importance of arameter is linked to the type of tissue under consideratiFor an entirely serial organ, the maximum dose is the liming factor, while for a parallel structured organ the radioblogical response is linked closer to the mean dose andstandard deviation.

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2362 Sauer, Shepard, and Mackie: Application of constrained optimization 2362

Considering two dose distributions having similar medose values, the one with the higher variance could havevulnerable effect. This subject is controversial in the liteture. The result of a recent survey from Kwaet al.23 is thatmean dose predicts radiation pneumonitis, while for Zaiand Amols24 empirical data indicate that irradiating a partivolume of lung results in a lower complication rate thirradiating the whole organ uniformly. For eradication oftumor, generally the minimum target dose is decisive. Adtionally the quantity ‘‘equivalent uniform dose’’~EUD! asproposed by Niemjerko was calculated.25 EUD is the doseresulting in the same probability of cell survival as the inhmogeneous dose distribution within the target volume,

EUD52D0 lnF 1

N (i

eDi /D0G , ~7!

where D051/a522 Gy/ln~SF2) and SF2 is the survivalfraction at 2 Gy.Di is the dose at voxeli andN the numberof voxels. In this worka50.347 Gy21, equivalent to SF250.5 were used. For the mean dose, 70 Gy was assuFor easier comparison EUD was normalized to 1 by dividEUD with 70 Gy again. When the dose distribution is highinhomogeneous the EUD tends to the minimum dosewhen it is highly homogeneous EUD tends to the mean do

III. RESULTS AND DISCUSSION

To evaluate the proposed methods, they were applietwo cases related to clinical practice; a head and neck alung tumor treatment.

A. Head and Neck case

Treatment plans were calculated for a U-shaped tawith a region at risk within its concavity~see Fig. 2!. For thevulnerable tissue no radiobiological parameters were set.dose was minimized for all points at risk. The number

FIG. 2. Optimized dose distribution for a U-shaped target with a regionrisk within its concavity. Seven incident ports with equidistant gantry angwere set. The lower boundary for target dose was 0.95. Dose to the poinrisk were minimized. Dash-dotted lines are the 0.2 and 0.5 isodose linessolid lines denote the 0.9 and 0.99 isodoses, respectively.

Medical Physics, Vol. 26, No. 11, November 1999

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incident ports with different gantry angles is varied froonly one up to thirteen ports. Except for the case withsingle port, the dose to the target is constrained betw0.95 and 1.1, Due to physical reasons, this is not possibonly one port is applied. Relaxing the upper limit, howevachieves a clinical meaningful solution even for this caThe dose distributions conform well with the target. Thisshown for seven ports in Fig. 2 as an example. The flueprofiles for this dose distribution are plotted in Fig. 3. Onlyfew target voxels receive a dose less than 0.95. The amis controlled by an accuracy parameter for the limits. Hereaccuracy of 0.01 is set, small enough not to compromisetumor control probability. Additionally points not used fooptimization may receive dose outside the specified limThis is controlled by the point density used. For all poinwithin the region at risk, a very low dose is achieved. Figshows the corresponding dose volume histograms. Withknowledge of the dose necessary for the tumor and theof side effects, a safe treatment can be delivered.

Figure 5 shows the dependence of different parameterthe number of ports. For the number of ports equal or grethan two the target dose parameters are constant. The mdose is a little above one, in the middle between the low

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FIG. 3. Fluence profile corresponding to the dose distribution in Fig. 2. Tgantry angles are indicated.~0° is from left to right.!

FIG. 4. Cumulated dose–volume histogram of the dose distribution in FigDue to an accuracy parameter, a very small fraction of the target volreceives a dose less than the lower limit of 0.95. The curve labeled ‘‘Phtom’’ is for the entire phantom.

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2363 Sauer, Shepard, and Mackie: Application of constrained optimization 2363

and upper limits. This guarantees a EUD very close todesired value of 1.0. All dose parameters for the regionrisk are significantly decreasing with increasing numberports. However, for a number of ports greater than sevenbenefit is rather small. This is in agreement with findinfrom other authors.4,26

B. Lung case

In a more general case, the relative difference in the dresponse properties of different tissues has to be takenaccount. Therefore the optimization approach was testedan L-shaped target with a region at risk of a more senature within its elbow. This setup mimics the spinal coand is surrounded by a lung, an organ of parallel physiolocal structure~Fig. 6!. The threshold dose for the lung was sto 0.3, its seriality to 1. For the spinal cordD th50.5 wasused, and the seriality exponent S was varied. Figures 67 show a dose distribution and a corresponding dose volhistogram for an example with seven ports andS58 for thespinal cord. The behavior of the dose distribution with vaing seriality exponent is summarized in Fig. 8. IncreasinSup to a value of 16 leads to a maximum dose for the orgarisk ~risk 1, spinal cord! almost equal to the desired threshovalue of 0.5. As seen in Fig.6, only very small regions atcorners of risk 1 receive a dose exceeding 0.5. All doserameters for risk 2~lung! are almost independent ofS for thespinal cord and the mean dose is well below the limit of 0

FIG. 5. Dependence of basic parameters of the dose distribution forU-target case on the number of ports. Dashed and dotted lines neapoints indicating maximum and minimum dose represent mean valuesthe 8 pixels with the highest and lowest dose, respectively.

Medical Physics, Vol. 26, No. 11, November 1999

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Due to its position, around the target, the maximum dosethe lung is of the same order as target dose. The dose totarget itself is as enforced by the dose limit matrices. Tapproach eliminates the risk of ‘‘overcharging’’ the systewhich would lead to infeasibility of the problem. Certainlthe end point of the optimization is determined not onlythe radiobiological parametersw andSused here, but also bythe geometry of the problem, and the technical and physconstraints and the degrees of freedom.

In order to investigate its gain, the optimization was pformed for different numbers of ports set between one athirteen ~Fig. 9!. Threshold dose values were set as preously, andS58 was used as the seriality exponent for rivolume 1. Again, up to the number of four ports, the ganangles were set manually. As expected, the main effecincreasing the number of ports is on risk 1 volume. Dose

etheor

FIG. 6. Optimized dose distribution for the L-shaped target, surroundedspinal cord and lung. Optimization was performed using Eqs.~1! and ~2!,with S58 for spinal cord andS51 for lung. D th values were 0.5 and 0.3respectively. Isodose lines as in Fig. 2.

FIG. 7. Cumulated dose–volume histogram of the dose distribution in FigBecause of the high seriality set for the spinal cord, only a small volureceives a higher dose than the threshold set to 0.5. The low seriality enent for lung allows much of the volume to receive a dose higher thanthreshold which was set to 0.3.

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2364 Sauer, Shepard, and Mackie: Application of constrained optimization 2364

continuously decreasing, but for more than seven ports,gain is very small. The decrease for the dose in risk 2 volu~lung! is small. While the standard deviation for three poand more is rather constant, the mean dose is reduced0.3 for one port to 0.2 for thirteen ports.

Both results shown in Fig. 8 and in Fig. 9 indicate that tmaximum dose, being determinative for the risk of injurithe spinal cord, is controllable by the seriality parameHowever, the control is not direct and increasing the exnent S in Eq. ~2!, increases the computational burden. Itpossible to control the maximum dose directly by settupper limit values for each dose point within the regioninterest. Having upper and lower limits with the lower limgreater than the upper limit at adjacent volumes, as for taand risk 1 in this example, involves the danger of not beable to reach a feasible solution. A strategy has to be appin order to avoid such a situation. As shown in Fig. 10,our test case, this problem occurs only if less than seports are applied. For less than seven ports, the upper

FIG. 8. Dependence of basic parameters of the dose distribution forL-target case on the seriality exponent for spinal cord. The maximum dfor the spinal cord decreases with increasing exponent, while the valuethe target and the lung remain nearly constant.

Medical Physics, Vol. 26, No. 11, November 1999

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dose for risk 1 was increased, until a feasible solutionreached. The calculation time for this approach was an oof magnitude less than using an exponent of 8. This confithe advantage of using dose limit constraints directly. Toverall CPU time needed for 100 pencil beams and 104 dosepoints was about 4 min on a DECa4100 computer. Howevera big potential is left in order to speed up the calculation

The capability to increase the low dose region within tlung is demonstrated in Fig. 11. According to Eq.~4!, theincrease of effect was reduced for dose values greaterCompared to the dose distribution from Fig. 6, the regionlow dose is increased. This is paid with a higher dose tohigh dose region. If a dose greater than 0.5 destroystissue anyway, this more inhomogeneous dose distribushould lead to a lower complication probability for the lunA similar dose distribution was achieved using Eq.~2! withan exponentS,1.

IV. CONCLUSIONS

We have implemented an optimization strategy whguarantees a high tumor control probability by simply sett

esefor

FIG. 9. Dependence of the basic parameters of the dose distribution foL-target case on the number of ports. Input parameters are as in Fig. 5

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2365 Sauer, Shepard, and Mackie: Application of constrained optimization 2365

a lower limit constraint on the target volume. Complicatiprobability for organs at risk is controlled by minimizing thdose to that organ. It was shown that if only a serial typevulnerable tissue has to be considered no radiobiologicaput parameter is necessary. If the parameters of the dresponse curve for an individual patient’s organ areknown with sufficient certainty, it is safer to keep the doselow as technically achievable. For a number of tissue tybeing involved, radiobiological parameters generally havebe set in order to guide the optimization process. The numof parameters should be as small as possible, simpleclosely linked to clinical experience. Our method setsthreshold dose causing damage to an organ and its relseriality controlling the mean, the variance, and the mamum dose. These parameters are closely related to tolerdose and volume effect conventionally used in clinical prtice. An explicit dose–volume constraint, e.g., as usedBortfeld, Stein, and Preiser is not necessary in our metho27

Dose and volume of too high dose for lung are minimiz

FIG. 10. Same as Fig. 9, but nowS51 for both lung and spinal cord. Anupper limit was set for the spinal cord. This was 0.5 when the numbeports was greater than or equal to seven and as low as feasible for lnumbers.

Medical Physics, Vol. 26, No. 11, November 1999

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automatically, avoiding possible infeasibility and computional problems due to the occurrence of multiple locminima if such constraints are set.7 To take advantage of thecomputational efficiency of modern optimization algorithmit is desirable to use upper limit constraints for vulnerabregions. As demonstrated, this significantly improvestreatment plan and does not lead to infeasibility if usedsmall volumes of serial structure only. A similar tactic wused by Powliset al.15 However, they were satisfied by finding a feasible solution by interactively changing the dolimits and did not attempt to further optimize the dose dtribution.

The controversial issue of how many ports are requiredorder to improve uncomplicated tumor control~UTC! is dis-cussed in the literature4,26 Our results confirm that in generamore than five ports should be used. However the gachieved with more than seven ports is small. CalculatUTC with very steep dose response curves may not resua significant difference in UTC for three and seven porHowever, knowing the uncertainty in dose response of invidual patients, the achievable reserve in tolerance shonot be given away.

We have written a prototype of an ‘‘inverse treatmeplanning system’’ with the following features: The systeminimizes dose to regions at risk, maintaining a minimudose to target. The only parameter needed is the lower bofor target dose. The number of input parameters to spevulnerable regions is optional. Corresponding to the coplexity of the problem, no parameter or up to two parametmay be set, i.e., a threshold dose, a seriality exponentalternatively an upper bound. This makes the system flexdependent on the knowledge of an individual case. Caretaken to avoid infeasibility. The system does not break doeven if only a single port is used, which seriously shrinks

fer

FIG. 11. Dose distribution achieved using Eq.~4!. The increase of effect forthe ‘‘lung’’ was set low forD.0.5. The upper limit for the ‘‘spinal cord’’was 0.5. Isodose lines as in Fig. 2.

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2366 Sauer, Shepard, and Mackie: Application of constrained optimization 2366

ability to optimize the dose distribution. Optimization runautomatically, user interaction is not necessary. Only ifper limit constraints cannot be fulfilled, may the user wishalter these settings. With an appropriate dose calculationgine the system developed may be used with real patieThen, additionally technical constraints of the dose delivdevice ~such as a tomotherapy unit or a linear accelerawith multileaf-collimation! must be taken into account.

ACKNOWLEDGMENTS

O. S. was partly supported by Deutsche ForschunGemeinschaft~Sa 481/2-1!. D. S. was supported by NIHGrant No. CA48902.

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