Multiobjective evolutionary optimization of number of ... - CiteSeerX

Multiobjective evolutionary optimization of number of beams, their orientations and weights for IMRT

Eduard Schreibmann1,2, Michael Lahanas1, Lei Xing2 and Dimos Baltas1,3

1. Department of Medical Physics & Engineering, Strahlenklinik, Klinikum Offenbach, 63069 Offenbach, Germany. 2. Department of Radiation Oncology, Stanford University School of Medicine, CA 94305-5304, USA. 3. Institute of Communication & Computer Systems, National Technical University of Athens, 15773 Zografou, Athens, Greece. Corresponding author: Eduard Schreibmann Department of Radiation Oncology, Stanford University School of Medicine, CA 94305-5304, USA. Tel.: +650– 724 – 3086 Fax: +650– 498 – 4015 E-mail: [email protected]

Abstract We propose a hybrid multiobjective (MO) evolutionary optimization algorithm (MOEA) for intensity modulated radiotherapy inverse planning and apply it to optimize the number of incident beams, their orientations and intensity profiles. The algorithm produces a set of efficient solutions, which represent different clinical tradeoffs and contains information such as variety of dose distributions and dosevolume histograms (DVH). No importance factors are required and solutions can be obtained in regions not accessible by conventional weighted sum approaches. The application of the algorithm using a test case, a prostate and a head and neck tumor case is shown. The results are compared with MO inverse planning using a gradient-based optimization algorithm.

E. Schreibmann et al: Multiobjective evolutionary optimization

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1. Introduction Most of the currently used methods for inverse planning in intensity-modulated radiation therapy (IMRT) consider only the determination of the beam intensity profiles given a specific configuration of beams and a specific set of importance factors. A much larger search space has to be considered if also the optimal numbers of beams and their orientations have to be found, (see appendix A for more detail discussion). Furthermore, results depend on the importance factors used (Xing et al 1999). Many possible combinations can be formed in the presence of multiple objectives, leading to a huge number of possible solutions among which the best is to be found. We proposed a deterministic MO optimization method (Lahanas et al 2003) that, given a specific configuration of beams, produces a representative set of efficient solutions. By using gradient-based optimization algorithms, such as the limited memory Broyden-Fletcher-Goldberg-Shanno algorithm (L-BFGS), we obtained global optimal solutions for variance-based dosimetric objectives. This work is extended here to include the angular degree of freedom of the incident beams. The main difficulty of beam orientation optimization lies in the coupling between the beam configuration and the intensity distribution of the beams. One approach is to divide the problem in two separate decoupled stages. A beam configuration is obtained using geometric-based criteria for a specific number of beams (Haas et al 1998, Schreibmann et al 2003). Geometric-based methods that work for conformal radiotherapy do not necessarily provide optimal directions for IMRT where the intensity distribution of a beam could be as important as its direction for the quality of the obtained solution. Simple geometrical rules such that beams should not pass through organs at risk (OAR) may not be valid for intensity modulated multi-beam configuration. Pugachev and Xing (2001) considered not only geometric but also dosimetric properties of the system. Another approach is to fix the number of beams and to perform an optimization of the intensity profiles. A small trial modification of the beam configuration is introduced and after a new optimization of the intensity profile the results are compared with the results of the previous beam configuration. The comparison uses some aggregate function of the individual objective functions. Along this line, Stein et al (1997) studied the dependence of optimization results on the number and orientation of beams. Pugachev et al (2000, 2001) and Rowbottom et al (2001) considered the role of the beam orientation in IMRT using a combination of a simulated annealing algorithm for the selection of the beam orientation with an iterative algorithm for the determination of the intensity profiles. It has been shown that the use of prior geometric and dosimetric knowledge can significantly improve the convergence behavior of the optimization process (Pugachev and Xing 2002). Recently Hou et al (2003) used a GA in combination with a deterministic algorithm for the solution of the beam orientation problem using constraints, as Bedford et al (2003) use for conformal radiotherapy. These methods are used to eliminate the need of importance factors. Also Meedt et al (2003) proposed recently an algorithm that optimizes beam configurations by adding and deleting beams that due to dose constraints oppose the delivery of the prescribed dose. All these optimization methods are single-objective based and the decision-making is made in terms of a scalar function. This approach requires knowledge of optimal importance factors or constraints that have to be satisfied. Often such knowledge does not exist or constraints are violated due to physical limitations and the optimization procedure has to be repeated, by trial and errors with different set of weights, or constraints, until a satisfactory solution is found. All methods presented for inverse planning for the beam orientation problem are a priori in nature except the work of Haas et al (1998) who used geometric based methods, without an optimization of the intensity profile. The aim of MOEA based optimization algorithms is to provide a representative set of non-dominated solutions for problems where many conflicting objectives need to be considered simultaneously instead of a single solution. Decision-making is applied at the end of the optimization either manually or using a decision engine. This method decouples the optimization from the decision making process. A new decision is possible without having to repeat the optimization. Wu and Zhu (2000, 2001) used a single objective GA to optimize the beam direction for standard conformal radiotherapy. This algorithm provides only a single solution. Even if the final result is a population of solutions the algorithm uses as a score function the weighted sum of the


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individual objective functions, therefore the solutions are more or less distributed around the plan with the optimal fitness, it is unknown if these solutions are non-dominated and representative of the true Pareto front. Multiobjective planning produces a large number of solutions and can be ideally dealt by evolutionary algorithms due to their parallel algorithmic character. We used for the IMRT inverse planning problem for a fixed beam configuration the non-dominated sorted genetic algorithm with controlled elitism NSGA-IIc, Deb and Goel (2001). This algorithm belongs to the class of Pareto ranking algorithms, which are designed to produce a representative set of non-dominated solutions. L-BFGS was used to initialize population members of NSGA-IIc, Lahanas et al (2003). This method is used to full scale inverse planning where the optimal number of beams, their orientations and the intensity profiles simultaneously are found. By including as optimization objective the number of beams, the non-dominance check will remove plans that produce the same dose distribution but use a larger number of beams. The method is applied on a phantom test case and two clinical cases; a prostate tumor case and a head and neck tumor case. Algorithm’s convergence is checked by comparing Pareto fronts obtained with NSGA-IIc and L-BFGS. We discuss the information obtained from the MO inverse planning algorithm that can be used to select an optimal solution.

2. Methods 2.1. Multiobjective Optimization In inverse planning we have to determine a vector x = ( x1 ,..., xN ) of bixel weights

x j , j = 1,..., N ,

the

so-called

decision

variables,

which

optimize

a

vector

function

f (x) = ( f1 (x), f 2 (x),..., f M (x)) of M objective functions f i (x), i = 1,..., M . The MO problem can be generally defined as a problem to find a vector x that optimizes the vector function f (x) . Normally we don’t have a situation in which all f i (x) reach their optimums at a common point x . The most commonly adopted definition of “optimality” is the Pareto optimum based on the dominance relation. A solution x1 dominates a solution x2 if the two following conditions are satisfied: • x1 is no worse than x2 in all objectives, i.e. f j ( x1 ) ≤ f j ( x 2 ) ∀j = 1,..., M •

x1 is strictly better than x2 in at least one objective, i.e. f j ( x1 ) < f j ( x 2 ) for at least one

j ∈ {1,2,..., M }

We assume, without loss of generality, that this is a minimization problem. x1 is said to be non-dominated by x2 or x1 is non-inferior to x2 and x2 is dominated by x1. Among a set of solutions P, the non-dominated set of solutions P’ are those that are not dominated by any other member of the set P. When the set P is the entire feasible search space then the set P’ is called the global Pareto optimal set. If for every member x of a set P there exists no solution in the neighborhood of x then the solutions of P form a local Pareto optimal set. The image of the Pareto optimal set is called the Pareto front. Three strategies can be used for solving MO optimization problems. •

•

An a priori method. The decision making (DCM) is specified in terms of a scalar function and an optimization engine is used to obtain the corresponding solution. This approach requires knowledge of the optimal weights (importance factors). Often such knowledge does not exist and the optimization procedure has to be repeated, by trial and error methods with different set of weights, until a satisfactory solution is found. An a posteriori method. An optimization engine exists which finds all solutions. Decision making is applied at the end of the optimization manually, or using a decision engine. This method decouples the optimization from the decision making process. A new decision is possible without having to repeat the optimization.


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•

Mixture of a priori and a posteriori methods. During the optimization periodically information obtained may be used to reformulate the goals as some of these can not be achieved. We consider only the a posteriori method. The main tasks of this approach are: Obtaining a representative set of non-dominated solutions, i.e. the optimization process. Using the trade-off information to select a solution from this set, i.e. the decision making process. As the inverse planning problem in radiotherapy is a MO problem we have to find the set of non-dominated solution. In conventional inverse planning non-dominated solutions are produced by repeating the optimization using an aggregate single objective function, the weighted sum fTot(x) of M

M

j =1

j =1

fTot (x) = ∑ w j f j (x) , ∑ w j = 1 .This approach

the individual objectives functions f j (x) , j=1,2,…,M:

provides only non-dominated solutions situated on the convex parts of the Pareto front. The optimization is repeated with different weights wj until a satisfactory solution is obtained. Constraints can be applied in MO optimization by using the constrained dominance relation. A solution x1 constraint dominates a solution x2 if: (x1 is feasible and x2 is not) or (x1 and x2 are both feasible but x1 has a smaller constraint violation than x2) or (x1 and x2 are feasible and x1 dominates x2). For single objective optimization only one solution exists. For MO optimization we have to keep all non-dominated solutions that at least satisfy some criteria. For M objectives M(M-1)/2 pair correlations exist and we need enough solutions to sample the range of objectives values that we consider clinical interesting with a sufficient resolution. Optimizations with fixed beam configurations show that 50-100 solutions are required to have a representative set of all possibilities. The number of solutions for the MO optimization of beam orientation is therefore proportional to the range of beams that we want to consider. For configurations with 3-9 beams we need 350-700 solutions.

2.2. Optimization objectives As objective functions we use the PTV dose variance f PTV around the prescription dose Dref , for the normal tissue (NT) and the OARs the sum of the squared dose values f NT and f OAR respectively and for the number of beams f B .

f PTV =

∑ (d

1

N PTV

N PTV

j =1

PTV j

− Dref

)

2

(1)

∑ (d ) N NT

f NT =

1 N NT

f OAR =

1

N OAR

N OAR

j =1

j =1

NT 2 j

(2)

∑ (d )

OAR 2 j

(3)

fB = B d

PTV j

, d

NT j

and d

OAR j

(4) th

are the calculated dose values at the j sampling point for the PTV, the NT and

each OAR respectively. N PTV , N NT and N OAR are the corresponding number of sampling points and B is the number of beams. The objectives values are calculated using dose values expressed as percentage of the prescription dose. A Clarkson dose computation model was used.

2.3. Inverse planning with the evolutionary NSGA-IIc algorithm. We use for inverse planning the MO non-dominated sorting genetic algorithm with constrained elitism NSGA-IIc that is a special version of NSGA-II (Deb et al 2000, 2001) which is one of the most effective evolutionary optimization algorithms. The role of the genetic algorithm is a search in the beam configuration space using the so-called implicit parallelism (Goldberg, 1989). The GA population keeps a population of beam configurations. Important beam directions will more likely


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survive if they provide good dose distributions. Mutation and crossover search other possible configurations but keep important information that has been accumulated through the evolutionary process. As the intensity profiles are coupled with the beam configuration we use a high performance deterministic algorithm to obtain the optimal intensity profile for each individual. Initially the intensity profiles are optimized using only a few iterations in order to find directions that depend less on the accuracy of the intensity profiles. 2.3.1. Representation We represent a solution as a two-component chromosome. The first part W contains the bixel weights of all beams with a double precision floating-point representation. The second part B is a binary string with a bit, 0 or 1, for each beam. A value 1 represents a beam that is selected (the socalled active beam). 2.3.2. Genetic operators We use arithmetic crossover with a random mixing parameter for the chromosome W. For the chromosome B we use a two-point crossover operator and a flip mutation. A repair mechanism is used to keep the number of selected beams inside a user specific range. 2.3.3. Parameters The population in the NSGA-IIc algorithm is sorted according to the dominance. Nondominated members are assigned rank 1. The remaining population is then again checked and the non-dominated are assigned rank 2. This process continues until all members are classified. What is different to the situation with standard algorithms is that the fitness of each individual is determined by its rank and not on individual or a weighted sum of objective values. Members with small rank have the largest probability of surviving to the next generation. In NSGA-IIc a parameter, the geometric ratio, set in this work to 0.65, determines to what extent individuals of higher ranks may also included in the reproduction of new solutions. This helps to preserve the so-called lateral diversity avoiding premature convergence. For optimal performance the crossover and mutation probability must be limited in the respective ranges 0.7-1.0 and 0.001-0.1. For the large (N+B)-dimensional problem with M objectives we use a population size of 100-200 solutions. Tests show that the optimization results do not change significantly after 200 generations. 2.3.4. Application of constraints The aim of Pareto based MOEAs is to provide a representative set of the entire Pareto front. A large fraction of the objective space in IMRT represents solutions which are clinically not acceptable. We use constraints that allow restricting the population into parts of the objective space which are of particular interest. One such constraint is the dose coverage for the PTV. We require that 95% receive at least 95% of the prescription dose. The constraint is implemented by defining a constrained non-domination relation which is used for the ranking of the individuals of the population. 2.3.5. Archiving of non-dominated solutions The calculation of the objective values requires a significant fraction of the optimization time due to the large number of sampling points where the dose has to be calculated. It is practically impossible to allow the algorithm to evolve for thousands of generations. In clinical practice time is important and a representative set has to be found, if possible, in a few minutes. In order to obtain a sufficient representative set of solutions we archive all non-dominated solutions found. Nondominated solutions found in the archive are removed.


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2.3.6. The inverse planning algorithm The evolutionary MO IMRT inverse planning is described by the following algorithm 1. Generate initial population. All bixel weights are initialized to 1. The beams are initialized with random or equispaced directions. 2. Modify weights and beam directions by crossover and mutation. For newly generated solutions optimize the weights with L-BFGS using random scaled importance factors. We scan therefore not only the beam configuration space but also the importance factors space. Increase periodically the number of optimization iterations. 3. Select population according to NSGA-IIc algorithm. 4. Add all non-dominated solutions to the archive and remove possible dominated archived solutions. 5. If termination conditions satisfied stop else go to 2. Random importance factors are used where the importance factor is scaled to obtain a sufficient coverage as only small fPTV values provide clinical acceptable solutions (Lahanas et al, 2003). The intensity profiles are optimized with L-BFGS initially using 30 iterations. This number increases periodically up to 300 iterations at the end of the evolutionary process. This number can be reduced but was kept large for this study in order to obtain highly optimized intensity profiles. For the initialization of the beam directions we have chosen 70% of the solutions using random selected beam configurations with 3-9 beams. The remaining beam configurations are beams that are equispaced with the first beam direction randomly selected.

2.4. Inverse planning with L-BFGS We compare the results from NSGA-IIc with results using the limited memory algorithm LBFGS (Liu et al 1989) which is globally convergent (Lahanas et al 2003). We cannot perform direct inverse planning with L-BFGS as this would require the optimization of all possible beams, including as an additional objective the number of beams. The algorithm would probably not converge due to multiple local minima as the search space is not necessary convex. We perform “inverse planning” by accumulating non-dominated solutions with L-BFGS. We repeat the optimization for a large number of cases with different number of beams, beam orientations and different importance factors using only the dosimetric objectives, fPTV, fNT and fOAR. For each of these cases the algorithm was found to produce global optimal solutions (Lahanas et al 2003). Some solutions of this set could be dominated solutions. We include the number of beams for each solution as an additional objective and through a non-dominated selection filter remove all dominated solutions. This set of non-dominated solutions provides an approximation of the Pareto front. We compare the set of non-dominated solutions produced by L-BFGS and NSGA-IIc using all two-dimensional projections of the Pareto front formed by the combinations of two objectives.

2.5. Test cases Three cases are used to test the algorithm, a phantom test case and two clinical cases a prostate and a head and neck case. For both cases a collimator width of 10x10 is used for obtaining the initial dose values, and a bixel size of 2.5 mm. 2.5.1.Phantom case The geometry for the phantom case consists of a cube shaped PTV, with all dimensions of 50 mm, situated in the middle of a bigger cube, of dimensions 200 mm, simulating the NT, see figure 1. The most preferred beam directions in this case would be the ones situated at 0°, 90°, 180° and 270°.


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Figure 1. Geometry of the test case, the PTV being the red cube is inside a larger cube representing the surrounding normal tissue. The beam directions are represented as black lines.

For the beam irradiation direction, the gantry was allowed to move from 0° to 360°, using a step size 5°, while the table was fixed at 0°. The bixels where optimized by considering a planar geometry in the isocenter’s plane. This setup generates a genome consisting of 1288 bixels for the 72 possible beams. The number of sampling points to each structure used in the optimization is shown in table 1. Object

Number of Sampling Points

PTV

3000

NT

10000

Table 1. Statistics for the 13000 sampling points used for the phantom patient case.

As optimization objectives using L-BFGS we consider the objectives fPTV and fNT, see equations 1 and 2. For NSGA-IIc we include the number of beams fB as an additional optimization objective. 2.5.2. Prostate cancer case The first clinical case is represented by a prostate tumor, see figure 2, with a PTV volume of 149 cm3. The optimization geometry consists of a coplanar plan (at a fixed table angle of 0 degrees), and a planar geometry for the bixel intensities, in the isocenter plane. The resulting genome contains 1140 bixel intensities and 72 beam orientations. The number of sampling points contained in each structure is shown in table 2.

Figure 2. Geometry of the prostate cancer case. Object PTV NT Left Femur

Number of Sampling Points 2519 5000 3000

Right Femur

3000

Rectum

2512

Table 2. Statistics for the 16031 sampling points used for the prostate case.


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As optimization objectives using L-BFGS we consider for the PTV and NT the objective functions fPTV and fNT. For the rectum, the left and right femoral heads the corresponding objectives are denoted as fRect, fFemL and fFemR respectively. For NSGA-IIc we include the number of beams fB as an additional optimization objective. 2.5.3. Head and neck cancer case The second clinical case is represented by a head and neck tumor, see figure 3. The optimization geometry consists of a coplanar plan (at a fixed table angle of 0 degrees), and a planar geometry for the bixel intensities, in the isocenter plane. The resulting genome contains 823 bixel intensities and 72 beam orientations. The number of sampling points contained in each structure is shown in table 3.

Figure 3. Geometry of the head and neck cancer case. Object Number of Sampling Points PTV 3000 NT 3000 Spinal Cord 500 Left Parotid

500

Right Parotid

500

Table 3. Statistics for the 7500 sampling points used for the head and neck case.

As optimization objectives using L-BFGS we consider for the PTV and NT the objective functions fPTV and fNT. For the spinal cord, the left and right parotid the corresponding objectives are denoted as fSC, fParL and fParR respectively. For NSGA-IIc we include the number of beams fB as an additional optimization objective.

3. Results The optimization tests were undertaken with a 2.53 GHz Intel Pentium IV PC with 512 MB RAM memory. We used the NSGA-IIc algorithm of the MO optimization library MOMHLIB++ by A. Jaszkiewicz.

3.1. Inverse planning Pareto fronts 3.1.1. Phantom case The NSGA-IIc algorithm was applied with 200 generations and a population size of 300. The optimization time was 70 minutes. The only constraint applied was 95% coverage at 95% of the prescription dose. A total of 5019 non-dominated solutions were accumulated. We applied L-BFGS to produce 1500 solutions using 3-9 beams, various random and equispaced orientations and randomly distributed importance factors. Only 43 non-dominated solutions were accumulated. The directions selected by NSGA-IIc are shown in figure 4 separately for solutions with 3 to 9 beams. For three and four beams preferred are only beam gantry angles at 0°, 90°, 180° and 270°. For a larger number of beams some few additional directions are selected that are approximately symmetric distributed around 0° and 90°.


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Figure 4. Angular distribution of beams selected by NSGA-IIc for the test case for different number of beams.

The frequency distribution of the number of beams of the solutions archived by NSGA-IIc for the test case is shown in figure 5. The majority of non-dominated solutions are found for three beams. This is remarkably as from combinatorial perspective combinations of 8 or 9 beams, see appendix A, cover more than 90% of all possible configurations and only 1 in 100000 possible combinations includes three beams.

Figure 5. Percent of non-dominated solutions as a function of the number of beams of archived non-dominated solutions obtained by NSGA-IIc for the test case.

The three two-dimension projections of the Pareto front (fPTV, fNT), (fNT, fB) and (fPTV, fB) are shown in figure 6. The results of L-BFGS are included.


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Figure 6. The three two-dimensional projections (fPTV, fNT), (fNT, fB) and (fPTV, fB) of the Pareto front obtained by NSGA-IIc and L-BFGS for the test case.

For (fPTV, fNT) we see in figure 6(a) that the dose variance in the NT increases rapidly as we approach the minimum value found with fPTV = 0.0018 which is a very small value due to the symmetry of the case. The (fNT , fB) plot, figure 6(b), shows that the NT dose variance can be reduced with increasing number of beams. The most significant decrease is if the number increases from 3 to 4 beams. For more than 4 beams the rate at which the minimum fNT value is reduced with increasing number of beams is smaller. The (fPTV – fB) plot, figure 6(c), shows that more than three beams do not improve the dose distribution inside the PTV. For (fPTV, fNT) the dependence on the number of beams is shown in figure 7. For configurations with more than three beams there is strong trade-off between fPTV and fNT as the PTV dose variance approaches its minimum value.

Figure 7. Trade-off (fPTV , fNT) obtained by NSGA-IIc for the test case. The dependence on the number of beams is shown.

3.1.2. Prostate patient case The NSGA-IIc algorithm was applied with 200 generations and a population size of 100. The optimization time was two hours. For the PTV a clinical constraint is to have at least 99% coverage at 97% of the prescription dose. We used a less strict constraint with 95% coverage at 95% of the prescription dose in order to have a larger freedom of selecting an optimal solution. A total of 5370 non-dominated solutions were accumulated. For L-BFGS we obtained 1001 non-dominated solutions out of 1500 optimization runs. The directions selected by the NSGA-IIc for the prostate case are shown in figure 8 for solutions with configurations of 3 to 9 beams separately. In contrast to the cube case more directions are selected but still a few directions are important.


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Figure 8. Angular distribution of beams selected by NSGA-IIc for the prostate case for different number of beams.

For 3-6 beams gantry angles beams close to θ = 325°, 295° and θ = 40° are preferred. For more beams a gantry angle θ = 105° is more likely accepted.

Figure 9. Gantry angles that are more likely selected by NSGA-IIc for the prostate case.

These directions have been selected as the passage of beams through the rectum or the femoral heads is then avoided, see figure 9. The frequency distribution of the number of beams of the archived solutions for the prostate case is shown in figure 10. In contrast to the test case the entire range of number of beams is selected with a larger preference for solutions with increasing number of beams.


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(%) of Solutions

25 20 15 10 5 0

3

4

5

6

7

8

9

Number of Beams Figure 10. Percent of non-dominated solutions as a function of the number of beams obtained by NSGA-IIc for the prostate case.

The two-dimensional projections of the Pareto front from the NSGA-IIc and L-BFGS algorithm are shown in figure 11. Only the dosimetric objectives are considered. The dose variance in the normal tissue and the rectum are mostly sensitive on the beam direction and NSGA-IIc finds superior solutions than L-BFGS. The reason is that NSGA-IIc performs a not completely random search in beam configuration space. For the rectum and NT we have a minimum value of 500 and 330 respectively to accept. For the NT and the rectum a minimum dose and dose variance is at least required to have clinical acceptable solutions. There is a strong trade-off between the dose variance fNT in the NT and the rectum fRect, see figure 11. Decreasing the dose variance in the PTV increases rapidly fNT and fRect. The left and right femoral head are less problematic as beam directions can be found for which the dose and the corresponding dose variance is very small or 0.


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Figure 11. Two-dimensional projections of the Pareto front of the dosimetric objectives fPTV, fNT, fRec, fFemL and fFemR obtained by NSGA-IIc and L-BFGS for the prostate case.

The dependence of the dosimetric objective values on the number of beams is shown in figure 12. Included are the results of the optimization using the algorithm L-BFGS. The strongest dependence on the number of beams is observed for the normal tissue dose variance. With increasing number of beams the dose variance in the NT can be reduced. For the PTV the dose variance can be reduced with increasing number of beams. NSGA-IIc obtained slightly better results than L-BFGS who does a random search in the beam configuration space.


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Figure 12. Dependence of the dosimetric objectives fPTV, fNT, fFemL, fRect and fFemR on the number of beams fB obtained by NSGA-IIc and L-BFGS.

3.2. Analysis of the trade-offs of the non-dominated solutions We do not consider the final decision making process, i.e. the selection of a single solution from the set of non-dominated solutions. In contrast to single objective optimization in MO decision making is another process decoupled from the optimization. We discuss only information that is provided by the efficient set of solutions that can be used for the selection of a optimal solution. 3.2.1. Phantom case The selection of a solution for the cube case depends on the dose in the NT and the dose variance in the PTV that we want to accept. The results suggest that the solution with four beams is optimal. All solutions contain some of the optimal 4 directions. Even if with more than four beams we can reduce the dose in the NT if we decrease the dose variance in the PTV we finally approach the fNT value for solutions with the optimal oriented four beams. Increasing the number of beams above 6 does not significantly improve fNT. For three beams we have to accept the largest dose variances in the NT even if we can accept a small dose homogeneity. 3.2.2. Prostate cancer case For the prostate case we apply a filter to the solutions by using constraints for the OARs. For each structure we define Dmin the minimum and Dmax the maximum dose allowed. We define upper and lower limit DVH-based constraints specified by the pair of parameters (DU, VU) and (DL, VL) respectively. The constraint (DU, VU) for a structure is satisfied only if a volume smaller than VU receives a dose larger than DU. For the constraint (DL, VL) the volume that receives at least a dose DL must be larger than VL. The target prescription dose is 74 cGy. Structure DMin DMax DU VU (%) DL VL (%) (cGy) (cGy) (cGy) (cGy) Left or Right Femur 0 45 40 20 Rectum 48 25 PTV 73 83 72 99 Table 4. Constraints imposed for the prostate case.


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For the NT it is difficult to give specific constraints (Jones and Hoban 2002) and we avoid giving a high dose value constraint for the rectum. After application of the filter 524 out of the 5370 solutions satisfy the selection criteria as defined in table 4. Among these filtered solutions 50 have 3 beams, 74 6 beams and 106 are configurations with 9 beams. We obtain a range of values for the maximum and average dose in the structures such as the PTV, NT and rectum. These ranges are shown for the solutions with 3 or 9 beams in table 5. The dose values are expressed in percent of the prescription dose. Beams 3 9

Max DPTV 100.62 - 101.96 100.45 - 102.78

Mean DPTV 99.82 -100.45 99.94 -100.12

Max DNT 100.12 – 102.24 100.11 – 101.43

Mean DNT 13.31 - 18.35 13.07 - 17.58

Max DRectum 99.14 - 101.21 98.99 - 100.72

Mean DRectum 22.70 - 54.10 19.24 - 56.68

Table 5. Range of the maximum and average dose in the PTV, NT and rectum for the filtered solutions using 3 or 9 beams.

The results in terms of the dose variances are shown in table 6. For the PTV the dose variance using 9 beams is approximately three times smaller than solutions with 3 beams. The large variances for 9 beams can be explained by the trade-off between fNT and fPTV. It is possible to improve the dose homogeneity in the PTV but only by accepting a large dose variance in the NT. Beams 3 9

fPTV 0.043 - 0.433 0.016 - 0.161

fNT 577.58 - 1109.48 331.89 - 859.23

fRect 1167.87 - 3434.34 894.16 - 3459.00

Table 6. Range of dose variances fPTV, fNT and fRect for the filtered solutions using 3 or 9 beams.

The trade offs (fNT, fRect) and (fNT, fFemL) as a function of the number of beams can be seen in figure 13. For 3 beams solutions have large fNT and fRect values. For 6 and 9 beams both fNT and fRect can be reduced but this depends on what directions and importance factors are used. For 9 beams only a slightly smaller fNT can be obtained than with 6 beams. For three beams the femoral heads are not a problem but we have to accept a large dose variance in the NT. Increasing the number of beams reduces the variance fNT and the dose homogeneity in the PTV but not simultaneously.


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Figure 13. Dependence of the trade-off (fNT , fRect ) and (fNT , fFemL) on the number of beams (3,6 or 9) for the filtered solutions. The trade-off of the set not filtered set of efficient solutions is also shown.

The DVHs of the PTV, NT and OARs of the filtered solutions with 3, 6 or 9 beams are shown in figure 14. The constraints for the PTV produce almost identical DVHs, whereas a larger spectrum for the OARs exists.


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Figure 14. DVHs of filtered solutions generated by NSGA-IIc for the PTV, NT and the OARs for the prostate case. The dependence on the number of beams (3, 6 and 9 beams) is shown.

3.2.3. Head and neck cancer case For the head and neck case the constraints for the OARs are given in table 7. The target prescription dose is 52 cGy. For this case the OAR constraints are strict since the brain and more


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sensitive OARs are considered. We use therefore less strict constraints for the PTV coverage probing thus a larger region of the Pareto front. Structure Left or Right Parotid Spinal Cord PTV

DMin (cGy) 0 48

DMax (cGy) 18 54 55

DU (cGy)

48

VU (%)

DL (cGy) 12 40

VL (%) 33.33 33.33

85

Table 7. Constraints imposed for the head and neck case.

For this case we want to consider the limits of the optimization considering the OARs. The treatment planner considered very strict constraints for the spinal cord. Using the CORVUS treatment planning system it was not possible to satisfy the constraints and the final solution accepted had a small coverage for the PTV. We limit the study to 2D coplanar distributed beams for computational reasons. The optimization was performed with a limit of 3-9 beams. A population of 200 solutions was used for the hybrid genetic algorithm. The frequency distribution of the number of beams of the 15327 accumulated solutions after 300 generations is shown in figure 15.

Figure 15. Percent of non-dominated solutions as a function of the number of beams obtained by NSGA-IIc for the head and neck case. The distribution shows that in the main population with 200 members solutions with 7-9 beams are more easily accumulated than solutions with a smaller number of beams. Therefore we performed two additional optimization runs, one with a range of 3-4 beams and one with a fixed number of 6 beams. We analyzed the Pareto front and dose distributions to see the spectrum of possible solutions. We applied a filter and removed solutions with less than 85% coverage for the PTV at 95% of the prescription dose. The number of solutions with 3-9 beams was reduced to 10105 solutions and for the other two runs with 3-4 beams and 6 beams to 2437 and 2066 solutions respectively.


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Figure 16. Trade-off between a) (fPTV , fNT ), b) (fPTV , fSC) c) (fPTV , fParL) and d) (fSC , fParL) on the number of beams (3-9, 3-4 or 6) for the filtered solutions.

The (fPTV, fNT) trade-off, figure 16(a), shows that for more than 6 beams the decrease of the dose variance in the PTV is not significant. Also the variance in the NT is not reduced strong as in the two previous cases where the NT can be protected better the more beams are used. Similar the (fPTV, fSC) and (fPTV, fParL) trade-offs, figure 16(b) and 16(c), show that more than 6 beams are not necessary. The (fSC, fParL) trade-off, figure 16(d), shows that the main problem is the simultaneous protection of the spinal cord and the parotids. A better protection can be achieved the more beams are used. We consider some dosimetric parameters that show us the limits of the dose distributions that can be obtained. We consider the average and minimum dose in the spinal cord and min(DSC) respectively and the minimum dose in left parotid min(DParL), expressed as a percent of the prescription dose, and the coverage of the PTV at 95% of Dref (PTV95).


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Figure 17. Dependence of trade-offs between dosimetric parameters for the head and neck case. a) (PTV95, min(DSC)) b) (PTV95, ) c) (PTV95, min(DParL)) and d) (min(DSC) - min(DParL)) on the number of beams (3-9, 3-4 or 6) for the filtered solutions.

The minimum and average dose in the spinal cord depends on the number of beams, see figure 17(a) and 17(b). This dependence decreases as the PTV coverage increases and approaches a limit of 98% for more than 6 beams and a limit of 95-96% for 3-4 beams. The protection of the spinal cord improves with the number of beams but we have to accept small coverage for the PTV. For a coverage PTV95 =85% we have to accept a minimum dose value in the spinal cord as large as 60% of the prescription dose. Figure 17(c) shows that above a PTV coverage of 95% for PTV95 there is a rapid increase of the dose in the left parotid. Figure 17(d) shows the minimum dose for the spinal cord to be 60% of the prescription dose but only using 9 beams at least and that in this case the minimum parotid dose can be kept approximately below 30% of the prescription dose.

4. Discussion and conclusions We have presented a MO hybrid evolutionary algorithm for IMRT inverse planning that provides a representative set of efficient solutions for the problem of how to obtain a optimal number of beams, their orientation and intensity profiles. The advantage of the a posteriori MOEA is that it provides a representative set of all possible solutions, whereas conventional single objective optimization requires to repeat the optimization if the result is not satisfactory. Another advantage is that it provides the correlation and trade-off between the mathematical objectives and the corresponding dosimetric and dose-volume histogram related values. This information can be used to reduce the number of objectives to be considered and to concentrate on the objectives that are difficult to satisfy


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and is therefore essential in the decision making process. The optimization time is approximately one hour. Similar to the work by Hou et al (2003) we use a deterministic algorithm to optimize the intensity profiles. Tests show that GAs are not efficient and produce solutions of poor quality. We use L-BFGS that produces very fast global optimal solutions for a fixed beam configuration. The role of the MO evolutionary algorithm is to search in the beam configuration space and to accumulate a representative set of non-dominated solutions. The role of L-BFGS is the optimization of the intensity profiles for a fixed beam configuration. Conventional optimization algorithms search an optimal solution by comparing the fitness values between two solutions. In the Pareto ranking based algorithm the dominance relation or if constraints are used the constrained dominance relation is used for the search. For different beam configurations the fitness is not a guarantee that only non-dominated solutions will be selected. Also optimal importance factors may depend on the number of beams and their orientation. The set of solutions contains information such as the possibilities of the dose distributions which are attainable for a range of number of beams. The planner is not required to specify unknown information such as importance factors, number of beams and their orientation, except the prescription dose and the range of number of beams to be considered. A zero critical dose value is specified for the OARs i.e. the aim is to minimize the dose in the OARs considering dose values even below what would be considered as clinical acceptable. The main constraint is a sufficient large coverage for the PTV. Other constraints can be included if desired, such as DVH-based constraints for the OARs, which can increase the number of clinical acceptable solutions. Inverse planning was performed with L-BFGS by producing a large number of solutions with different number of beams, beam orientations and importance factors and by applying a filter to remove dominated solutions. The Pareto fronts provided by L-BFGS and NSGA-IIc have been compared. NSGA-IIc with the support by L-BFGS is able to produce a representative set of nondominated solutions. The results show that more likely stable beam configurations (orientation and number of beams) are selected by this algorithm. For the phantom case we have found that the majority of solutions by the conventional approach are dominated solutions. As all single objective algorithms use a weighted objective sum it cannot be excluded that a majority of solutions found by these algorithms will be also dominated solutions. The archiving of all non-dominated solutions which are found is necessary in order to have a sufficient large set of representative solutions for the large number of objectives. It additionally allows the population size to be reasonably small and therefore reduces the optimization time. The population-based search allows the algorithm to escape from possible local minima. We have limited this study to dosimetric objectives as the variance-based objectives produce more uniform dose distributions and biological based objectives produce more non-uniform dose distributions and not necessary superior solutions (Jones L and Hoban P 2002) Methods were proposed by Xing et al (1999) or Wu and Zhu (2001) to obtain “optimal” importance factors that provide solution with DVHs very similar to specified “ideal” DVHs. These methods assume some a priori knowledge such as the optimal DVHs for the surrounding normal tissue NT and OARs. The problem is that the required dose distribution in radiotherapy cannot always be obtained, due to physical limitations and to the existence of trade-offs between the various conflicting optimization objectives. Without MO optimization the treatment planner does not know if the solution found is truly the best possible solution. As we have a combinatorial problem we do not expect to obtain a global Pareto optimal set. The comparison of the Pareto front obtained by L-BFGS and NSGA-IIc and the beam orientations found for the test case suggests that the non-dominated set which is obtained is close to the global optimal Pareto front even if only a very small subset of beam orientations can be considered. In this work all non-dominated solutions were archived. The number of non-dominated solutions increases rapidly especially for high dimensional problems. No reduction was applied to this number so that many solutions are very similar. This is not a problem as cluster reduction


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techniques can be applied that reduce the number of similar solutions and preserve the characteristics of the set. Another method is the use of the ε-non-domination relation (Laumanns et al 2002) that given a desired accuracy for each objective reduces the number of possible solutions to a maximum possible number. The algorithm is not limited to any specific set of objective functions as long as sufficiently efficient algorithms exist that can be used for the optimization of the intensity profile. This is a problem common for all other methods. The role of the MOEA algorithm is to provide the mechanism to select optimal orientations such that a representative set of non-dominated solutions is obtained that is not limited to convex functions and that is not influenced by local minima. We have chosen dose variance based objectives in order to use the very efficient L-BFGS algorithm. For the prostate case we applied soft constraints only on the PTV coverage and the used an off-line filter to select solutions that if possible satisfy all constraints. We analyzed the dependence of the solutions that passed the filter on the basis of the number of beams. The results showed that the only significant benefit of using larger number of beams is the reduction of the dose in the NT. Especially for this case only a few beam directions are significant to protect the OARs. Only for the NT a larger number of beams is necessary but for more than 6 beams the gain in NT protection seems to be less significant. The results show that the optimum number depends on which objectives are important. For the test case four beams seem to be optimal. For the head and neck case we observed that the PTV coverage and the OAR constraints require a large number of beams. The PTV coverage does not increase significant and the Normal tissue is not protected significant better if more than 6 beams are used. For this case the main problem is that the spinal cord sets a very strict problem. The larger the number of beams is the better the spinal cord can be protected but only if we accept a reduced coverage for the PTV. If the optimization is performed with a few iterations then more “stable” directions are found where the accuracy with which the intensity profile is determined is not very critical. By increasing the accuracy more configurations with a larger number of beams are obtained but these require a more complicated and accurate profile to provide non-dominated solutions. We expect that the representative set of all possible solutions obtained by the algorithm to provide valuable practical information for the beam orientation problem applied to other difficult clinical cases. The aim of the method presented is the decoupling of the optimization from the decision making process. Conventional inverse planning assumes some a priori knowledge of the possibilities and the treatment planner express this in terms of importance factors, constraints and number of beams. The reality is that planners spend sometimes hours in trying to obtain an acceptable solution by a trial and error change of the importance factors and the constraints. Our approach is to use not too strict constraints in order to obtain a representative set of all possible solutions. The planner only has to study the trade-off, the limits and the DVHs of the solutions. This can be studied as a function of the number of beams from which the planner can choose a solution that requires a small as possible number of beams.


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Appendix A. The combinatorial complexity of beam configurations The number of possible combinations if we have to choose M beams out of N possible is

N N!   = .  M  M !( N − M )!

Beam Combinations

The number of combinations for M = 3 - 9 and N = 72 is shown in figure 18, which depends approximately exponential on M, whereas for 3 beams we have 59640 possible combinations for 9 beams 85113005120 combinations exist. The number of all combinations for M in the range 3 - 9 is 98726450051.

1E10

1E7

10000

3

4

5

6

7

8

9M

Figure 18. Number of combinations of M beams from 72 possible beams as a function of for M.

An increase of the number of optimizations by a factor 100 or more is necessary if for each possible configuration a sufficiently large set of importance factors has to be scanned. Much more combinations exist if non-coplanar beam configurations are considered. We used the original NGSA-IIC algorithm including an archiving option. The results show that especially for the head and neck case solutions with a small number of beams are suppressed as it is more difficult to find optimal directions with a small number of beams for difficult cases and because the main population contains only 200 members. Increasing the population seems not to be a solution if we consider that the archived population contains thousand of solutions but the problem could be solved by a modification that includes also archived members in the selection process. For the present study we used a more simple solution in performing additional calculations where we reduced the range of number of beams. Even if many beam configurations obviously can be ignored still the problem is combinatorial complex and only a very small subset of configurations is finally examined.

Acknowledgements This research was supported through a European Community Marie-Curie Fellowship (HPTM2000-00011)


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References Bedford J L and Webb S 2003 Elimination of importance factors for clinically accurate selection of beam orientations, beam weights and wedge angles in conformal radiation therapy Med. Phys. 30 1788-804 Bortfeld T, Niemierko A, Rosenthal S, Thieke C and Wolfgang S What’s next in IMRT? Optimising the Optimisation ICRO/Ögro conference, Salzburg, May 2002 Deb K, Agrawal S, Pratap A and Meyarivan 2000 T A fast and elitist multiobjective genetic algorithm: NSGA-II. Technical Report 2000001, Indian Institute of Technology, Kanpur, Kanpur Genetic Algorithms Laboratory (KanGAL) Deb K and Jain S 2001 Running performance metrics for evolutionary multi-objective optimization, Kangal Report N0. 2002004 Deb K and Goel T 2001 Controlled elitist non-dominated sorting genetic algorithms for better convergence, In E. Zitzler et al. (eds.): Evolutionary Multi-criterion Optimization (EMO 2001), First International Conference, EMO 2001, Zurich, Switzerland, March 7-9 2001, Proceedings. Lecture Notes in Computer Science Vol. 1993, Springer. (pp. 67--81). Goldberg D E 1989 Genetic Algorithms in Search, Optimisation and Machine Learning (Massachusetts: Addison Wesley, Reading) Haas O C L, Burnham K J and Mills J A 1998 Optimization of beam orientation in radiotherapy using planar geometry Phys. Med. Biol. 43 2179–93 Hou Q, Wang J, Chen Y and Galvin M 2003 Beam orientation optimization for IMRT by a hybrid method of the genetic algorithm and the simulated dynamics Med. Phys. 30 2360-2367 Jones L and Hoban P 2002 A method for physically based radiotherapy optimization with intelligent tissue weight determination Med. Phys. 29 26-37 Jones L and Hoban P 2002 A comparison of physically and radiobiologically based optimization for IMRT Med. Phys. 29 1447-55 Lahanas M, Schreibmann E and Baltas D 2003 Multiobjective inverse planning for intensity modulated radiotherapy with constraint-free gradient-based optimization algorithms Phys. Med. Biol. 48 2843-71 Lahanas M, Schreibmann E, Milickovic N and Baltas D 2003 Intensity Modulated Beam Radiation Therapy Dose Optimization with Multiobjective Evolutionary Algorithms, (C. M. Fonseca et al Eds.): Evolutionary Multi-Criterion Optimization, Second International Conference, EMO 2003, Faro, Portugal, April 8-11, 2003, Proceedings. Lecture Notes in Computer Science 2632 Springer 2003, pp 648-661 Laumanns M, Thiele L, Deb K, and Zitzler E Archiving with Guaranteed Convergence And Diversity in Multi-objective Optimization. In GECCO 2002: Proceedings of the Genetic and Evolutionary Computation Conference, Morgan Kaufmann Publishers, New York, NY, USA, pages 439-447, July, 2002. Liu D C and Nocedal J 1989 On the limited memory BFGS method for large scale optimization Mathematical Programming 45 503-28 Meedt G, Alber M and Nüsslin F 2003 Non-coplanar beam direction optimization for intensitymodulated radiotherapy Phys. Med. Biol. 48 2999–3019 Pugachev A B, Boyer A L and Xing L 2000 Beam orientation optimization in intensity-modulated radiation treatment planning, Med. Phys. 27 1238-45


25/25

Pugachev A, Li J G, Boyer A L, Hancock S L, Le Q-T, Donaldson S S and Xing L 2001 Role of beam orientation optimization in intensity modulated radiation therapy Int. J. Radiation Oncology Biol. Phys. 50 551-60 Pugachev A and Xing L 2001 Computer-assisted selection of coplanar beam orientations in intensitymodulated radiation therapy Phys. Med. Biol. 46 2467–76 Pugachev A. and Xing, L. 2002 Incorporating prior knowledge into beam orientation optimization in IMRT Int. J. Radiat. Oncol. Biol. Phys. 54 1565-74 Rowbottom C G, Khoo V S and Webb S 2001 Simultaneous optimization of beam orientations and beam weights in conformal radiotherapy Med. Phys. 28 1696-702 Rowbottom C G, Webb S and Oldham M 1998 Improvements in prostate radiotherapy from the customization of beam directions Med. Phys. 25 1171-9 Schreibmann E, Lahanas M, Uricchio R, Theodorou K, Kappas C and Baltas D 2003 A geometry based optimization algorithm for conformal external beam radiotherapy Phys. Med. Biol. 48 1825–41 Stein J, Mohan R, Wang X-H, Bortfeld T, Wu Q, Preiser K, Ling C C and Schlegel W 1997 Number and orientation of beams in intensity modulated radiation treatments Med. Phys. 24 149-60 Xing L, Li J G, Donaldson S, Le Q T and Boyer A L 1999 Optimization of importance factors in inverse planning Phys Med. Biol. 44 2525-36 Wu X, Zhu Y, Dai J and Wang Z 2000 Selection and determination of beam weights based on genetic algorithms for conformal radiotherapy treatment planning Phys. Med. Biol. 45 2457-58 Wu X and Zhu Y 2001 An optimization method for importance factors and beam weights based on genetic algorithms for radiotherapy treatment planning Phys. Med. Biol. 46 1085-99 Jaszkiewicz A, MOMHLIB++ Open Source Multiple-Objective Metaheuristics Library in C++, http://www-idss.cs.put.poznan.pl/˜jaskiewicz/MOMHLIB/