Strategies and Techniques for efficient planning in IMRT Radiotherapy

Transcription

1 Strategies and echniques for efficient planning in IMR Radiotherapy I. Rodrigo Gómez 1, J.M. Artacho errer 1 1 Comunication echnologies Group (GC) Aragon Institute of Engineering Research (I3A). University of Zaragoza, Mariano Esquillor s/n, 518, Zaragoza, Spain. el , Fax , { irodrigo, jartacho }@ unizar.es Abstract Intensity modulated radiation therapy (IMR) is an important modality in radiotherapy. Due to the complexity of IMR treatment plans, optimization methods are needed to design high quality treatments. he goal of a treatment is to deliver a prescribed amount of radiation to the tumor, while limiting the amount absorved by the surrounding healthy and critical organs. Planning an IMR treatment requires determining fluence maps, each consisting of hundreds or more beamlet intensities. Since it is difficult or impossible to deliver a sufficient dose to a tumor without irradiating nearby critical organs, radiation oncologists have developed guidelines to allow tradeoffs by introducing so-called dose volume constraints, which specify a given percentage of volume for each critical organ that can be sacrified if necessary. Such constraints, however, are of combinatorial nature and pose significant challenges to the fluence map optimization problem. he purpose of this paper is two fold. In this study we show several methods for getting the best results for these beam weights. he process for their calculation is a complex mathematical procedure of constrained optimization. he function needed for minimization may tae several formulations, depending on the complexity that can be assumed, and the physical meaning behind the formula. We show here diffent formulations for the problem, and several methods for solving it. Besides, in case the results need any improvements, we propose some postprocessing techniques that can tune the weights values in order to get smoother results. Keywords: dose planning, objective function, constrained optimization, postprocessing techniques. 1 Introduction Radiation therapy is the treatment of cancer with ionizing radiation, in such a way that radiation, when passing through the tissue, damages tumor cells slowing or reversing the growth of tumors. A relevant development in the last decade is the intensity modulated radiotherapy (IMR) system, where the distribution of the dose can be controlled with spatial accuracy using multileaf collimators (MLCs). At the surface of a collimator the radiation delivered by the corresponding beam can be seen as a two-dimensional discrete pattern of beamlets.

2 Numerical values associated to these beamlets, corresponding to fluence or energy fluence, are called beamlet weights. A schematic diagram of the system is shown in Fig. 1. Figure 1: A schematic diagram for IMR (3 beams). Inverse planning of IMR systems consist of automatically determining the beamlet weights, so that a prescribed dose can be attained at discrete volume locations (voxels). Other parameters (such as the number of beams and their orientations) could additionally be considered, although this issue is not taen into account here (Artacho J.M. et al, 27). Several mathematical framewors have been proposed in order to solve this IMR inverse planning (Jelen U., 27). Most of them are particular ways of analyzing large illconditioned systems of equations, where the beamlet weights are linearly related to the data, which are the prescribed doses at voxels. Because of the nature of the problem, different types of constraints should also be considered. Physical constraints arise because of the limitations of the IMR system (nonnegativeness of the beamlet weights, for example). Medical constraints are prescribed by the oncologist (limits on the received dose for different organs, for example). he problem can be so hard that semi-automatic or interactive solutions are sometimes regarded as practical trade-offs. Fully automatic framewors include, among others: algebraic methods (constrained least squares), stochastic optimization methods (such as simulated annealing and genetic or evolutionary algorithms) or bacprojection-based methods. It is important to eep in mind that, despite the number of available theoretical framewors mentioned so far, several versions of nonstochastic traditional gradientsearch methods remain extensively applied. his basic idea can be expressed with different names in IMR planning: constrained optimization, active set algorithms or 2 AMI-5-1-AR

3 nonlinear programming. For this ind of methods, mathematic definitions must be given for the specific objective function and for the constraints of the problem (Romejin H. et al, 28). he main objectives in this problem are focused on the IMR dose planning optimization. In order to improve the results for the beamlet weights, some different ways to pose the problem are shown and some optimization methods have been considered to get an effective solution. In Sections 2, 3 and 4, the mathematical problem is studied and new strategies are shown and developped. he rest of the paper is organized as follows. In Sections 5 and 6, the goal is to introduce the techniques used in this wor developped to analyze solutions from the practical point of view. his is carried out by two different strategies; postprocessing techniques, and stopping point criteria study. In order to assess the performance of the techniques two measures are used: dose-volume histogram and fluence matrixes. Dose Volume Histograms (HDV) represent, at the same time, the percentage of dose obtained in each organ for the corresponding volume percentage. It is used to measure the treatments quality. he fluence matrixes can be obtained by ordering the values in the solution so that a beamlet representation is possible. It is desirable that, even if heterogeneity is always necessary, the fluence matrixes are as homogeneous as possible. his is a need for the quality of the treatment, because, if heterogeneity is very high, there will be a lot of segments in the following stages of the process, and the exposure time for the patient to the radiation would be increased. IMR radiotheraphy s main advantage is the use of non uniform beams for the dose administration. It is possible to distribute the dose in an intelligent way, trying not to radiate the healthy tissues (OAR: Organs at ris) and, at the same time, concentrating the dose on the damaged organs (CV: Clinical arget Volume). his objective is, most of the times, impossible to achieve, because the organs are so close that it is difficult not to damage the adjacent organs. Before this administration is carried out, a dose planning phase is needed. his planning is unique for each patient and it gives the weights for each beam as a result. Dose planning can be done by planning and solving an inverse problem that can be solved with constrained optimization methods. he main goal of this wor is the study and development of optimization methods that are suitable for this purpose, always trying to get a plan that is compatible with the dose specifications prescribed by the physician, taing into account the physical conditions for the problem and the different limitations that exist (positive weigths, dose limits on organs at ris). 3 AMI-5-1-AR

4 2 Material and Methods 2.1 IMR Planning he external beam radiation therapy with intensity modulation poses a problem which can be represented by a mathematical model to relate the dose delivered by the linear accelerator (linac) that generates the radiation beam and the dose absorbed in the anatomy of the patient. he area affected by the tumour is divided in small cubes called voxels, usually tens of thousands, in order to obtain an accurate representation of the volume. Liewise, radiation beams are discretized in beamlets. he radiation beam delivers a dose which is described in terms of a matrix of weights (or fluence matrix), where each weight determines the dose delivered by its corresponding beamlet (see Fig. 1). he dose delivered to each voxel is given by n D f w i 1... m (1) Oi ij j j1 Where n is the number of beamlet weights, w j is the number of voxels, is the value of the jth weight, and f ij is the contribution of dose from the jth beamlet to the ith voxel. he reordering of the weights of the fluence matrices in a vector allows the previous expression to be expressed as a matrix DO F w (2) where F is an mxn matrix whose elements are the coefficients f ij and D O is an m dimensional vector where the doses corresponding to all the voxels are grouped. Consequently, once the matrix F is calculated, the dose obtained for a specific intensity pattern can be found directly. he inverse problem for radiotherapy treatment planning is based on the nowledge of matrix F. he goal is to find the weighting vector that approximates the dose prescribed by the oncologist in each organ or structure. Usually, a distinction is made between clinical target volume (CV) (or affected organ), organs at ris (OARs) (or healthy organs), and unspecified tissues (Us). In IMR planning, linear or quadratic objective functions are the most usual choices. When the constraints are also linear, these special objective functions can lead to wellnown optimization problems such as linear programming (LP) or quadratic programming (QP). LP and QP have good theoretic properties, such as well-nown 4 AMI-5-1-AR

5 conditions for convexity. Moreover, an important advantage is that a great number of practical algorithmic solutions are available for these problems. he election of quadratic objective functions, compared to linear ones, has certain advantages. Least-squares emerge as a natural solution for IMR problems and, moreover, quadratic objective functions can be thought as more flexible models than linear ones (Croos S. et al, 22). he price to be paid for this choice is the increased complexity of the problem. It has been argued that, for IMR planning, this is specially critical, as it is a large-scale problem which includes a high number of constraints. For the formulation of the problem, an objective error function has to be defined. It has been built as the sum of quadratic terms, measuring the difference between prescribed and obtained doses. In the definition of the objective function, a set of organs or structures is considered (CVs, OARs, and U). he th structure comprises N voxels. is a set of indexes for the voxels belonging to this structure. N enters a normalization term in the definition of the objective function preventing that big organs dominate its value. Moreover, a weighting coefficient P is introduced, allowing the oncologist to attach more relevance to one organ or another in the treatment. Objective functions considering similar criteria than the ones described have been proposed elsewhere. he objective function used here is defined as follows: P Gw ( ) ( DOi DPi ) N S S S S i P 1 P 1 N m CV, OAR1, OAR2,... 2 (3) his function can be written as a quadratic function of the weights as 1 Gw ( ) wqwrw c (4) 2 being 2P 2P Q F F R F D N N c S S P Dp D N p S p (5) 5 AMI-5-1-AR

6 where F and parts related to the structure. D p are, respectively, submatrix/subvector of F and D taing only the As several structures of the volume (OARs) need to be particularly protected, when considering the rows of matrix which belong to the voxels or structures whose dose must be constrained by an upper bound, matrix A U is obtained, being bu a vector that contains the information of the maximum allowed dose. We apply dose constraints by means of imposing U U Aw b (6) m U being the number of rows in AU and n the number of beamlet weights One of the main concerns of this wor is the proposal of special strategies in order to reduce the computational cost when nonlinear objective functions are used. he effectiveness of the proposed techniques is shown on real cases of prostate cancer and larynx cancer. Some recent wors confirm that a wise election of a particular algorithmic solution, exploiting all the available nowledge, can have a great impact regarding practical concerns such as the computation time. For the type of QP problem solved here, significant differences have been found between algorithms. he previous formulation has only one important drawbac. In order to get the best results, it is needed to adjust the values for the priorities involved. hese values are not evident, and it is necessary a trial and error process. We propose here a new formulation to the problem that tries to reduce this dependence. he new formulation to the objective function is: w j. 1 Gw ( ) ( Di O Di P ) N S D OAR S CV D i prescribed 2 (7) his way, the terms in the objective function corresponding to OARs are removed, but they are taen into consideration in the problem through the conditions over their average doses. 2.2 Optimization Methods Once the objective function is stablished, a mathematical process is needed to solve the problem and get the best solutions to the beamlet weights. here is not a universal 6 AMI-5-1-AR

7 method to solve the problem, each of them having their special characteristics, which are a matter of study. In this wor, an exhaustive research for useful methods has been done. On the preselected methods, some important issues have been taen into consideration (computational efficiency, convergence ). One of the most important characteristics that were taen into consideration is the method s order. Methods can be first or second order. First order methods only need the gradient information, whereas second order methods also need the hessian information. Although we formulated a constrained optimization problem before, some unconstrained optimization methods were studied. We too special attention to Steepest Descent, Conjugate Gradient and Newton methods. However, after studying them, it was evident they didn t contribute to improve the results. hey were not too difficult to compute, but convergence was not guaranteed, they were not computationally efficient, and what is more important, it was impossible to condition the problem, so the results could be completely inadequate (or even negative). It was necessary to explore the constrained optimization methods. he most common and studied method in constrained optimization is Lagrange multipliers. his is not an efficient method at all, but, however, it is the base for almost all methods in constrained optimization. he algorithm is based on the idea of forming a new objective function, adding the original one, the equality conditions multiplying a β factor, named Lagrange Multiplier factor. his factor is always positive. he algorithm only wors with equality conditions, but it is possible the use of inequality conditions, by introducing slac variables (Rao S., 1995). One of the main differences of these methods with the unconstrained optimization is the ind of objective function they can use. he advantages of the quadratic formulation have already been studied in the previous section. Nevertheless, we proposed studying methods woring with linear objective functions, in case the advantages were so favourable, that the formulation could be changed. Simplex Method is one of the most used out of this group. However, it was proved that convergence was not always guaranteed. here are also methods that can wor with quadratic functions and other nonlinear functions. Several methods in this group use tables to get the solution to the problem. hese methods have no order and their woring way is similar to the solution of a linear equations system. Leme and Simplex Quadratic Programming (an extension of Simplex Method for quadratic functions) are the best characterized in this group (Rodrigo I. et al, 29) (Arora J.S., 24). 7 AMI-5-1-AR

8 here are others that can only be used with inequality constraints, which is consistent with the formulation proposed. Many algorithms in this group have good characteristics, but we only considered Zoutendij, Rosen, heil Van der Panne and Complex, which were the best ones from our point of view (Rao S., 1995) Out from these groups, methods lie Sequential Quadratic Programming, Active Set or Shor were also considered. Shor s algorithm has some special good characteristics, because it can be used with very complex formulations for the objective functions and the conditions involved. It can also be started from a point not belonging to the feasible region (Rodrigo I. et al, 29) (Ashgar M., 2). Algorithms for Simplex Quadratic Programming, Active Set or Shor s methods can be summarized as follows. A. Simplex Quadratic Programming he problem is required to be formulated in this way: Minimize 1 f x x Qx c_ Q x c (8) 2 Subject to: N x e (9) A x b (1) x (11) Introducing the corresponding slac variables, Lagrangian can be expressed: Lc x.5 x Qxu A xs b xv ( N x e) (12) hen, we apply: L cqx Au Nv x (13) A xsb (14) N x e (15) u s ; i 1 m (16) i i x ; i 1 n (17) i i s, u para i m; (18) i i i parai 1 n (19) v y z (2) Equations (13), (14) and (15) can be expressed in matrix form this way: 8 AMI-5-1-AR

9 x u Q A I N N c_ Q ( n) ( nxm) A ( mxm) ( mxn) I( m) ( mxp) ( mxp) b s N ( pxm) ( pxn) ( pxm) ( pxp) ( pxp) e y z And the problem can be defined as: B X D (22) From this point on, the way for getting the solution is just following the steps of Simplex method for linear equations. (21) B. Active Set he problem is required to be formulated in this way: 1 Minimize: f ( x) cc_ Q x x Q x 2 (23) Subject to: A x b (24) C x d (25) his method is baed on the feasible directions theory. he iterative scheme to be followed is: 1 x x d (26) he problem can be formulated in this way as well, which is more convenient in this case: 1 Minimize: c_ Q x d ( x d ) Q( x d ) (27) 2 Subject to: Ac d (28) Where Ac is the matrix that is formed by the active solutions set. Reordering these expressions we get: 1 Minimize: ( Qx c _ Q) d d Q d (29) 2 Subject to: Ac d (3) And renaming the terms as: We get: Minimize: g Q x c (31) 1 g d d Qd (32) 2 9 AMI-5-1-AR

10 Subject to: Ac d (33) Karush-Kuhn ucer conditions (KK conditions) lead to the following linear equations system: Qd g Ac v (34) Ac d (35) Which can also be expressed in matrix form: Q Ac d g Ac v Once this system is solved, the steplength is calculated as follows: (36) bi AI x Min[1, for AI d ] (37) A d I Being A I and b I the matrixes that contain the inactive solutions set. If this steplength is negative, a new inequality condition enters the active set. On the contrary, if any v element is negative, a new condition must be removed from the active set. If there are many cases, the condition that is to be removed, is the one associated to the most negative value in v. C. Shor Shor s algorithm begins with the initialization of the values in matrix B. his special matrix is very important in the process. It conatins the history of the process till the th iteration. his matrix can be restarted and history begains to be calculated again. he steps that have to be followed are: 1. Select an initial point, an initial steplength (h) and initialize matrix B. 2. Calculate the value of function f and subgradient g1 at that point 3. Compute the following operations: gt B g1 (38) r gt g1 (39) r 1 r (4) R 1 x1 1 ( x 1) 1 (41) R 1 Inv( R 1) (42) B B R ( ) (43) 4. Calculate gamma 1 1 g B 1 g1 f (44) K1= (45) f( x )) (46) () () 1 1 AMI-5-1-AR

11 5. Follow these steps 11 1 x 1 x 1h B 1 g1 (47) ( 11) ( 11) f f( x 1 ) ) (48) 1=1+1 (49) 6. If ( 1) ( 11) f f h h/5.1 (5) 1= (51) Go to step 5 7. If not so, verify ( 1) ( 11) f f 8. If it s true, recalcúlate the steplength (h) as: 9. If not, update x as: 2h h h 1.5h h 1 2 And go to step 5 x 11 1 x 1 (52) (53) verify the stopping criteria 1. If stopping criteria is not accomplished, chec if B is needed to restart, and if so, restart it and go to step 2. If not, go straight to step If stopping criteria is accomplished, the algorithm is finished. 3 Results he results we obtained can be grouped in two different categories. he first one is a comparison of the optimization methods heil Van der Panne, Complex, Simplex Quadratic Programming, Rosen, Active Set, Leme and Shor. heil Van de Panne and Complex methods wored efficiently with specific examples, but they couldn t wor with real patients information. Simplex Quadratic Programming wored efficiently with real patients information, but with certains limitations, because it couldn t deal with the large amount of values, variables and conditions that can be handled with larynx cancer treatments or prostate cancer with several beams. Rosen Method gave suitable results in many situations. However, it couldn t find a valuable solution in the most complex cases. 11 AMI-5-1-AR

12 Leme, Active Set and Shor were the only three that could wor efficiently in all cases and situations. We can see the results they give for a 5 beamlet treatment of a prostate cancer patient in the following figures. (a) (b) (c) Figure 2: HDV for (a)leme, (b)shor,(c)active Set optimization methods. 36 degrees 18 degrees 324 degrees 18 degrees 252 degrees Figure 3: Beam position, organ projection and fluence matrixes for IMR planning optimization. 12 AMI-5-1-AR

13 Comparing the formulations for the objective function, let us remember we proposed two different forms for the objective function. he first one taes into account all the CVs and OARs involved in the problem and their corresponding priorities. he second one only has information about the CVs but also cares about the OARs by including them in the conditions through the average dose. he complexity about getting the best priority values is shown in figures (4) and (5), where we can see the HDV that we get with inadequate priority values and adequate ones. his problem can be solved using the second formulation. Besides, the CV histogram improves, because the descent is steepest, as it can be seen in the figures below. Figure 4: HDV obtained with inadequate priority values Figure 5: HDV obtained with adequate priority values It is possible that the results obtained by the optimization methods are not good enough. his can be possible because, although the dose volume histogram may be very good, the heterogeneities may be too high, with the drawbacs it has. o improve these results, we propose here some postprocessing techniques and a deep study of the stopping point criteria. A. Postprocessing techniques Fluence matrixes can be treated as images and, therefore, image methods for deblurring can be applied. One of the techniques that can be used here and that is quite used in image processing is filtering. he filters that we evaluated were average filter, median filter and Gaussian filter. 13 AMI-5-1-AR

14 he results are shown over a larynx cancer patient, that was treated with 5 coplanar beams. Results are shown in figures 6 (HDV) and 7 (fluence matrixes). (a) (b) (c) (d) Figure 6: HDV obtained with (a) no filtering, (b) average filtering, (c) median filtering and (d) Gaussian filtering (a) -1-1 (b) (c) (d) Figure 7: Fluence matrixes obtained with (a) no filtering, (b) average filtering, (c) median filtering and (d) Gaussian filtering. 14 AMI-5-1-AR

15 he results show that median and Gaussian filters can improve the beamlet solutions, not worsening the histograms too much. he main disadvantage that this method has, is that the filtering is applied once the optimization is complete, so, it is liely that most of the good wor developed by the optimization process can be lost. his can be improved by another implementation, which is filtering but inside the optimization algorithm. his is, every a certain number of iterations, a filter is applied to the solution, so the filtering is done but the optimization process can continue in order to get the best solution from both points of view; the optimization s view and the processing techniques one. Let us consider that, after filtering the solution, it might not be a feasible solution, so the problem could not be restarted. his was proved using Leme and Active Set algorithms. However, Shor algorithm offered very good solutions with this technique. It can be shown in the following figure, where the results are shown for a patient with prostate cancer, being treated with a 5 beam treatment. Figure 8: HDV and fluence matrixes obtained with Shor Optimization and inside filtering. 15 AMI-5-1-AR

16 B. Stopping Criteria aing into consideration the particular characteristics of each optimization method, it is possible that, finishing the process before the best solution is reached, it could be possible to get better solutions, or, at least, more homogeneous solutions. For that reason, it is necessary to study the way each method gets the solution. It is needed that the heterogeneities appear in the last iterations of the algorithm, this way, stopping the algorithm can help getting better results. Leme s algorithm was not that ind. It could be proved that heterogeneities appeared from the beginning of the process. Stopping the algorithm before the minimum is reached could even worsen the results, because some weights could not have a value yet, so the heterogeneities could even increase. However, Active Set algorithm, wored exactly the way we wanted. It is based on the feasible directions theory and the tendency is getting the solution step by step, so heterogeneities only appear at the end of the process. Shor s algorithm was particularly different. he results are near the minimum from the very beginning of the process, and, in the last iterations, it taes a lot of time to improve the solution with very short steps. It is not useful for this new method. Active Set was used with this aim. In this example, prostate cancer 5 beams, the algorithm too 1781 iterations to get the solution. With a trial and error process, we determined that stopping the algorithm 3 iterations before it is finished, the results would be quite acceptable, as it is shown in figures 9 and 1. Figure 9: HDV obtained with Active Set Optimization (first column (a)) and incomplete Active Set Optimization (second column (b)). 16 AMI-5-1-AR

17 (a) (b) Figure 1: Fluence matrixes obtained with Active Set Optimization (first column (a)) and incomplete Active Set Optimization (second column (b)). 4 Discussion he results shown in this paper are highly satisfactory. he formulation of the objective function is a decisive fact in the optimization process and, therefore, in the IMR planning stage. It was proved that the quadratic function formulation was an advantageous option. Using this quadratic formulation, the problem can be tacled by two diffent philosophies; an objective function containing information for CVs and OARs, weighted by some priority factors, and another one considering only CVs in the objective function and stablishing conditions over the dose average for the OARs. Once the problem is formulated, an exhaustive research over the optimization methods was done. Several algorithms wored efficiently but only Active Set, Leme and Shor contributed to improve the results. hese can wor in all situations, not worrying the number of unnowns or the conditions to the problem. Some of them can even wor in hardest conditions. he results could even get better if we can mae them more homogeneous. his was proposed over two different ways; postprocessing techniques and stopping criteria development and study. 5 Acnowledgments I acnowledge the Aragon Institute of Engineering Investigation for granting me in all this investigation period. 17 AMI-5-1-AR

18 References Arora J.S. (24). Introduction to Optimum Design. Elsevier Academic Press. Artacho errer J.M. et al. (27). A Feasible Application of Constrained Optimization in the IMR System. IEEE ransactions on Biomedical Engineering, 54(3): Asghar M. (2). Practical Optimization Methods: With Mathematica Applications. Springer-Verlag Croos S. and Xing L. (22). Application of constrained least squares techniques to IMR reatment Planning. Int. J. Radiant. Interantional Journal of Radiation Oncology Bioogy. Physics, 54: Jelen U. and Alber M. (27). A finite size pencil beam algorithm for IMR Dose Optimization: density corrections. Physics in Medicine and Biology, 52: Members of the staff of Memorial Sloan-Kettering Cancer Center (23). A practical guide to Intensity-Modulated Radiation herapy. Medical Physics Publishing Rao S. (1995). Optimization heory and Applications. Wiley Eastern Limited Rodrigo Gómez I. et al. (29). Métodos Active Set y Simplex QP aplicados en planificación de radioterapia. Simposium Nacional de la Unión Científica Internacional de Radio, pp Romejin H. and Dempsey J. (28). Intensity Modulated Radiation herapy treatment plan optimization, OP, 16: Shephard D. et al. (1999), Optimizing the delivery of radiation therapy to cancer patients. SIAM review, 41(4): AMI-5-1-AR