A fast dose calculation method based on table lookup ...

INSTITUTE OF PHYSICS PUBLISHING

PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 48 (2003) N159–N166

PII: S0031-9155(03)62200-1

NOTE

A fast dose calculation method based on table lookup for IMRT optimization Qiuwen Wu1, David Djajaputra1, Marc Lauterbach2, Yan Wu1 and Radhe Mohan3 1

Department of Radiation Oncology, Virginia Commonwealth University Health System, Box 980058, Richmond, VA 23298, USA 2 MRC Systems, Hans-Bunte-Strasse 10, Heidelberg, Germany 3 Department of Radiation Physics, Unit 94, University of Texas, M D Anderson Cancer Center, Houston, TX 77030, USA E-mail: [email protected]

Received 14 April 2003 Published 3 June 2003 Online at stacks.iop.org/PMB/48/N159 Abstract This note describes a fast dose calculation method that can be used to speed up the optimization process in intensity-modulated radiotherapy (IMRT). Most iterative optimization algorithms in IMRT require a large number of dose calculations to achieve convergence and therefore the total amount of time needed for the IMRT planning can be substantially reduced by using a faster dose calculation method. The method that is described in this note relies on an accurate dose calculation engine that is used to calculate an approximate dose kernel for each beam used in the treatment plan. Once the kernel is computed and saved, subsequent dose calculations can be done rapidly by looking up this kernel. Inaccuracies due to the approximate nature of the kernel in this method can be reduced by performing scheduled kernel updates. This fast dose calculation method can be performed more than two orders of magnitude faster than the typical superposition/convolution methods and therefore is suitable for applications in which speed is critical, e.g., in an IMRT optimization that requires a simulated annealing optimization algorithm or in a practical IMRT beam-angle optimization system. (Some figures in this article are in colour only in the electronic version)

1. Introduction Optimal dose distributions in intensity-modulated radiotherapy (IMRT) are commonly obtained through an iterative optimization method that typically requires a large number of dose calculations to converge. Depending on the complexity of the case and the number of beams used, calculations of the dose matrices using an accurate superposition/convolution (SC) 0031-9155/03/120159+08$30.00

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algorithm, which is available in many commercial treatment planning systems, may take up to a few minutes to complete. A local gradient search optimization algorithm may need several dozens of iterations while a global optimizer (e.g., the simulated annealing algorithm) may require several thousands of iterations to converge. While the use of accurate dose calculation algorithms is justified in a 3D treatment planning where only a few dose calculations are necessary, unmodified use of these algorithms in IMRT planning could lead to an unacceptably long treatment planning time. It is therefore important to have a dose calculation method or optimization strategy that can yield accurate results relatively quickly. This speed issue is even more important for other types of optimizations that employ normal IMRT optimization as an inner loop, such as the IMRT beam-angle optimization (Stein et al 1997, Pugachev et al 2001) which loops over different sets of beam angles, and the optimization of the importance factors which loops over different sets of weights for the volumes of interest (Xing et al 1999). Several techniques have been proposed previously, or implemented in commercial systems, to speed up the dose calculation, some of them tailored for the IMRT optimization. Some treatment planning systems are equipped with several dose calculation algorithms: a less accurate but fast dose calculation algorithm can be chosen to perform the initial phase of the optimization to reach an approximate solution, then a more accurate but slower algorithm is used to fine-tune the solution (Chen et al 1995, Mohan et al 1996, Siebers et al 2001). Since the approximate solution produced by the fast algorithm is already close to optimal, only a few iterations are needed with the slow algorithm. Variations of this method have also been proposed (Laub et al 2000, Siebers et al 2002). The difficulty with this approach is that several different dose calculation algorithms are involved and an interface between them needs to be developed. Another possible speed-up method is to use a pre-computed kernel (Spirou and Chui 1998, Shepard et al 1999). Prior to the optimization, the dose to each voxel from each beamlet with unit intensity in the absence of other beamlets, i.e. the kernel, is computed and stored for later use. During the optimization, the dose to each voxel is obtained by looking up these kernels and summing them up with proper weights. This is probably the fastest dose calculation that can be achieved since it involves only memory access and simple arithmetical operations. The drawbacks of this approach, however, include the storage requirement for the kernel, the special dose calculation method that needs to be developed to produce the kernel and the large amount of computer memory that may be required to efficiently carry out the large array operations involved. In this note we propose a fast dose calculation method that can be used effectively in an IMRT optimization. The technique is simple, easy to implement and should be particularly useful in situations where the accurate dose kernel used by the treatment planning system is not directly accessible to the optimizer. An initial dose is calculated with an accurate algorithm available in the treatment planning system, and an approximate kernel is then constructed and saved. In subsequent iterations, the dose is calculated directly, and quickly, by the optimizer by looking up the saved kernel. This method may be especially useful for a treatment planning system that does not provide several dose calculation algorithms. 2. Methods and materials The dose to the ith voxel, Di, is written as M Kij · wj . Di =

(1)

j =1

Here i is the voxel index (i = 1, . . . , N); j is the beamlet index ( j = 1, . . . , M) and wj is the intensity weight for the jth beamlet. Note that in a multiple-beam setting the beamlet index j

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Initialize intensity matrices Compute dose Compute score

Converged ?

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Compute final dose distributions End Figure 1. Flowchart of a normal IMRT optimization process. Nonessential components not related to this note are not shown. The initial intensity for each beam is uniform for those beamlets that pass through the target volumes, as described in Wu and Mohan (2000). The dose is computed using the accurate superposition/convolution method available in the treatment planning system (Pinnacle3 , Philips Radiation Oncology Systems, Milpitas, CA).

comprises both the beam index and the beamlet index in each beam. The dose kernel Kij is a sparse matrix because dose contributions to each voxel from one beam come mostly from only a few beamlets in that beam. In the system that we use, the dose is computed with an SC algorithm from a Pinnacle3 treatment planning system. While the dose Di is readily available from many treatment planning systems, in most cases the kernel Kij is not accessible to an optimizer that is interfaced with the system. It is therefore necessary for the optimizer to call the accurate but slow dose engine of the treatment planning system to perform a new dose calculation each time the intensity matrices are updated. Figure 1 shows the flowchart of this normal IMRT optimization process. An exact kernel Kij should contain not only the primary but also the scatter component of the dose information. Since the kernel matrix is sparse, a first-order approximation to the kernel Kij can be obtained by ignoring the scatter component. This can be implemented using a simple raytracing algorithm, i.e. assuming that the beamlet deposits dose only along its path. We therefore use the following formula to calculate the kernel: D  i if voxel i is on the path of beamlet j, Kij = wj (2)  0 otherwise. This approximate form of Kij can be computed easily with a single function call to the accurate dose calculation engine of the treatment planning system. Once stored in a table, subsequent dose calculations can be performed by looking up this kernel matrix according to equation (1). Figure 2 shows the modified optimization flowchart that uses this table-lookup (TLP) method. Inaccuracies that are introduced by the TLP approximation can be minimized by updating the kernel several times during the optimization, as represented by the outer loop in figure 2.

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Initialize intensity matrices Compute dose Compute kernel Compute dose using TLP

Compute score

Converged ?

No

Update intensity matrices

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More dose calculation ? No

Compute final dose distributions End Figure 2. Flowchart of the IMRT optimization using the kernel that is based on the table lookup (TLP). Shaded boxes indicate the additional steps for TLP.

The main source of inaccuracies in the TLP method is the fact that the kernel is calculated for a specific set of beam intensity profiles. For this set of beam profiles, the dose computed using the TLP method, by construction, is exactly the same as the SC algorithm for each voxel that is intersected by a beamlet raytrace. As the beam profiles are modified during the optimization, however, the discrepancy between the two methods increases because the beam profiles for which the kernel is used are increasingly different from those for which the kernel is calculated. By performing several updates of the kernel during the optimization we ensure that the dose is always calculated with a sufficiently accurate kernel. It is also worth pointing out that the final dose distributions are always calculated with the accurate SC dose algorithm in both figures 1 and 2. Some other steps, which are not shown in the flowcharts, such as intensity filtering and prescription updating, are also done prior to the final dose calculation. In the TLP method, the dose to each voxel (from each beam) is assumed to come from one, and only one, beamlet. In principle two approaches can be used to assign this beamlet–voxel correspondence. The first approach is to loop over the intensity matrix elements (beamlets). A ray from the virtual source to the centre of the beamlet pixel is extended to the dose grid and voxels that are intersected by this ray are assigned to this beamlet. Although this approach can be implemented easily concomitant with the ray tracing, it may give rise to a sampling problem which is illustrated in figure 3. For simplicity, the rays in figure 3 have been assumed to be parallel, they are actually divergent in the dose calculations by SC and TLP methods. Beam B has the proper resolution so that all voxels are intersected by each beamlet. Beam A,


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Figure 3. Illustrations of the relationships between the beamlet resolution and the voxel resolution. A simple 4 × 4 dose grid is shown. Beam A comes from top to bottom with beamlets A1 to A4 represented by the dotted lines, beam B comes from left to right with beamlets B1 to B6. For simplicity, the beams have been assumed to be parallel in the drawing.

on the other hand, has a coarse resolution, and some voxels are not intersected by any beamlet. Voxel V32 (third row, second column), e.g., is on the path of beamlet A2, but voxel V13 is not on the path of any beamlet. In this approach it is therefore possible that the dose to some voxels from certain beams is not updated at all during the optimization since the voxels are not on the path of any beamlet in these beams. A related problem is encountered when the intensity matrix is finer than necessary to cover all the voxels. Instead of using this loop-over-beamlets approach, we use another approach to the beamlet–voxel correspondence, namely to loop over the dose voxels. For each voxel in the dose grid, a ray is constructed from the virtual source to the centre of the voxel. The beamlet whose intensity matrix pixel is intersected by this ray is then assigned to this voxel. This approach is free of the resolution problem because the ray to each voxel intersects one, and only one, intensity matrix pixel. Note that this one-to-one correspondence between the dose voxels and the beamlets in each beam greatly simplifies the lookup problem compared to using a full-scatter kernel where a substantial book-keeping effort has to be made. In addition, this approach greatly reduces the memory requirement for the kernel. The total number of kernel elements that needs to be saved in this approach is equal to the number of voxels multiplied by the number of beams. In the case of a full-scatter lookup, this number should be multiplied further by the number of pixels that lies within the cut-off range of the 2D cross section of the dose kernel. 3. Results Figure 4 illustrates the performance of the TLP method, compared with the SC method, in optimizing an IMRT plan. The displayed results have been obtained for a typical head-andneck IMRT case with nine coplanar beams. The SC method, whose scores are shown in figure 4 as white circles, required about 1 h to finish on a 750 MHz Sun workstation. In contrast, the TLP optimization for the same case, shown in figure 4 as black squares, can be finished in about 15 min. Note that the final scores produced by these two different methods are about the same. The optimizer utilizes Newton’s method to minimize an objective

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Iteration Number Figure 4. A comparison between the table lookup (TLP) and the superposition/convolution (SC) method in optimizing a typical head and neck IMRT case with nine noncoplanar beams.

function (score) which is constructed using the dose–volume criteria as described by Wu and Mohan (2000). Although the TLP method requires more (about twice as many in this case) dose calculations as the SC method, it requires much less time to finish (about one fourth in this case) than the SC method. This is because the total computing time for the IMRT optimization is dominated by the time spent on the dose calculation by the slow SC method. A TLP dose calculation is typically more than two orders of magnitude faster than a SC dose calculation and therefore it only adds a little to the total computing time. There are four SC dose calculations for the TLP method (corresponding to the initial score and the peak of each ‘score jump’) in figure 4, compared to 18 for the SC method, hence the speed-up factor of about four. In the TLP method, the fact that the score jumps slightly each time the dose is recalculated with the SC method indicates that an optimal solution considered by the TLP may not be optimal when the SC is used. In most cases in our experience, as exemplified in figure 4, successive SC peaks in the TLP method do decrease with more iteration, along with the size of the score jumps. This shows that the TLP method has sufficient accuracy to reliably guide the optimization process to an optimal, or a close-to-optimal, solution. There is no guarantee, however, that the solution produced by the TLP is identical to the one produced by the SC method, even in the case where their scores are similar. The fast TLP method described in this note may be particularly useful for the ‘super’ optimization problems in which a normal IMRT optimization process is just an inner part of a bigger loop. Included in this category are the beam-angle optimization problem and the optimization of the importance factors. To be useful in these problems, the TLP method has to be able to rank treatment plans according to their final scores when optimized with the accurate SC method. A plan that is better than another according to the SC-based optimizer has to be predicted accordingly by the TLP-based optimizer. Note that for this application, the TLP method is not required to produce an absolute score that is exactly the same as that produced by the SC method. A systematic shift in the score produced by the TLP method, e.g., relative to the score produced by the SC method, does not neutralize its speed advantage as long as the scores that it generates can still be used reliably to rank the treatment plans.

Score with TLP using 3 SC calculations


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Score with SC calculation at each iteration Figure 5. Correlation between the final scores obtained using the TLP method and those obtained using the SC method in the IMRT optimization. Two cases were studied: a head-and-neck (open triangles) and a prostate case (closed circles).

Figure 5 displays the correlation between the final score obtained using the TLP method and that obtained using the SC method in the IMRT optimization. One hundred combinations of coplanar beam angles are randomly selected for each of two IMRT cases: a head-and-neck case with nine beams (the same one as used in figure 4) and a prostate case with five beams. These two cases represent the two extreme levels of complexity typically encountered in clinical IMRT, as is evident from the clear demarcation of the two groups of scores displayed in figure 5. The scores for the head-and-neck case are higher because it is more difficult in this case to satisfy all the criteria provided by the planner due to the high complexity in shape and position of the tumours and organs-at-risk. Figure 5 shows that the scores generated by the TLP method are well correlated with those produced by the SC method both for the prostate and the head-and-neck cases. A slight difference in slope between the two groups can be discerned in figure 5 but, as stated in the previous paragraph, this does not affect the ability of the method to rank multiple plans and therefore its usefulness for this application. 4. Discussions and conclusions The TLP technique proposed in this note has several advantages: (1) it is simple and easy to implement; (2) it only uses one type of dose calculation algorithm and no internal knowledge of the dose calculation algorithm is necessary; (3) it is fast: for a normal IMRT optimization only a few dose calculations are required and a reasonable solution can be obtained in a predictable short period of time; (4) it is especially suitable for other types of optimization problem such as the IMRT beam-angle optimization in which a large number of dose calculations are needed or IMRT optimizations which require simulated annealing algorithms. Another use of the TLP technique is to tune the optimization parameters. There are many optimization parameters involved in an IMRT planning and many of them need to be adjusted a few times to reach a desired solution. With the TLP method, it is possible to probe the effects of these changes quickly. Once a favourable set of parameters is obtained, an optimization based on SC or other accurate methods can be started. Yet another use of TLP is to find

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an approximate initial solution which is then followed by other dose calculation methods, as already demonstrated by others (Chen et al 1995, Mohan et al 1996, Siebers et al 2001). The solution found by the TLP method may not be as good as the one found by the SC method, even after several SC dose calculations. The difference may or may not be significant when all other factors are considered such as the limitations of the delivery system. Improvements to the accuracy of the solution may be obtained by increasing the number of SC dose calculations or by incorporating the scatter component into the dose kernel, either through an analytical or a numerical parametrization. These modifications, however, will significantly reduce the speed advantage of the TLP method. Acknowledgments This work is supported in part by grant CA74043 from the National Cancer Institute and a research grant from Philips Radiation Oncology Systems. References Chen Z, Wang X, Bortfeld T, Mohan R and Reinstein L 1995 The influence of scatter on the design of optimized intensity modulations Med. Phys. 22 1727–33 Laub W, Alber M, Birkner M and Nusslin F 2000 Monte Carlo dose computation for IMRT optimization Phys. Med. Biol. 45 1741–54 Mohan R, Wu Q, Wang X and Stein J 1996 Intensity modulation optimization, lateral transport of radiation, and margins Med. Phys. 23 2011–21 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. Radiat. Oncol. Biol. Phys. 50 551–60 Shepard D M, Olivera G, Angelos L, Sauer O, Reckwerdt P and Mackie T R 1999 A simple model for examining issues in radiotherapy optimization Med. Phys. 26 1212–21 Siebers J V, Lauterbach M, Tong S, Wu Q and Mohan R 2002 Reducing dose calculation time for accurate iterative IMRT planning Med. Phys. 29 231–7 Siebers J V, Tong S, Lauterbach M, Wu Q and Mohan R 2001 Acceleration of dose calculations for intensitymodulated radiotherapy Med. Phys. 28 903–10 Spirou S V and Chui C-S 1998 A gradient inverse planning algorithm with dose-volume constraints Med. Phys. 25 321–33 Stein J, Mohan R, Wang X H, Bortfeld T, Wu Q, Preiser K, Ling C C and Schlegel W 1997 Number and orientations of beams in intensity-modulated radiation treatments Med. Phys. 24 149–60 Wu Q and Mohan R 2000 Algorithms and functionality of an intensity modulated radiotherapy optimization system Med. Phys. 27 701–11 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