A Mimetic Algorithm for Simultaneous Multileaf Collimator Aperture Shape and Dosimetric Optimization in CyberKnife Robotic Radiosurgery

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1 A Mimetic Algorithm for Simultaneous Multileaf Collimator Aperture Shape and Dosimetric Optimization in CyberKnife Robotic Radiosurgery Matthew R. Witten, Owen C. Clancey, Division of Radiation Oncology, Winthrop-University Hospital, Mineola, NY Abstract A mimetic algorithm for the optimization of multileaf collimator aperture shape and beam weighting is proposed for the CyberKnife M6 delivery platform (Accuray Inc., Sunnyvale, CA). The problem of optimizing a patient-specific treatment plan necessarily requires the determination of the beam geometry, beam-defining aperture shape, and beam weighting. The robotic arm of the CyberKnife moves the linear accelerator through approximately half the solid angle, the multileaf collimator in the head of the accelerator provides for conformal aperture shaping of the radiation, and the linear accelerator generates megavoltage x-rays to irradiate the target. The multileaf collimator leaf positions are optimized, as well as beam weights, by an algorithm consisting of an EA combined with a CMA-ES intense local search. The generation of apertures uses a rule-based approach, based upon dosimetric feedback regarding the radiation dose distribution acquired during each iteration. The purpose of the optimization is to find a Pareto-optimal solution which takes into account the clinical decision-making calculus of the radiation oncologist. The algorithm was tested on CT data set for a patient with carcinoma of the prostate. Twenty algorithm runs were performed; the mean fitness value was 2.48 x 10 5, and the standard deviation was 9.40 x In all instances, the algorithm produced clinically acceptable treatment plans, meeting the user-specified dose-volume objectives for the target and critical structures. Note that there is no commercial algorithm currently available for generating these types of treatment plans, so the present work demonstrates feasibility of a novel treatment planning approach. T I. INTRODUCTION HE CyberKnife (Accuray, Inc., Sunnyvale, CA) robotic radiosurgery platform consists of a lightweight X-band linear accelerator mounted as the end effector on a six-jointed robotic arm (KUKA Roboter GmbH, Augsburg, DE), a treatment couch capable of motion in five degrees of freedom (three translational, plus roll and pitch), two ceiling-mounted kilovoltage x-ray sources, and two amorphous silicon detector panels. The x-ray sources and detector panels constitute an image-guidance system, which provide real-time images of the patient on the treatment couch; such images provide information about the motion of the tumor during treatment, and the information is processed by the treatment delivery software, which in turn moves the robotic arm to displace the radiation beam to compensate for any spatial deviation between planned and measured treatment. Patients are treated with the CyberKnife for a great variety of malignancies, which include cancers of the prostate, brain, spine, lung, liver, pancreas, kidney, and breast, as well as benign tumors such as meningiomas and acoustic neuromas, and functional conditions like trigeminal neuralgia. Patients may receive treatments in a single fraction, or the treatment may be delivered over multiple treatment fractions. Accuray, Inc. recently introduced the newest generation CyberKnife, called the M6 [1]. A major improvement in functionality is provided by the addition of the InCise multileaf collimator (MLC), which consists of 41 pairs of tungsten leaves, arranged into two banks, with each leaf having a width width of 2.5 mm, when projected to the nominal 800 mm field-size definition distance. Whereas in previous generations of the CyberKnife delivery platform, aperture shapes were limited to circles, the MLC permits the conformal shaping of the apertures into totally arbitrary openings to define the limits of the radiation beam, thus allowing healthy tissue to be spared incidental irradiation to the maximum possible extent while simultaneously conforming the dose distribution closely to the generally irregular shape of the tumor. The CyberKnife M6 is shown in Fig. 1, and the InCise MLC, with the leaves shaped into a particular aperture shape, is shown in Fig. 2. Fig. 1 The CyberKnife M6 Series treatment delivery platform. Photo courtesy of Accuray, Inc /15/$ IEEE

2 work, an algorithm was developed for MLC aperture shape optimization and beam weight optimization for the CyberKnife M6 delivery platform. The optimization algorithm was used to generate a treatment plan for a patient with carcinoma of the prostate. The results definitively demonstrate the feasibility of the approach. Fig. 2 The InCise multileaf collimator, with the individual leaves taking specified positions to create an irregular aperture shape. Photo courtesy of Accuray, Inc. At present, neither the MLC nor treatment planning software capable of generating patient treatment plans have been released for clinical use. The robotic arm of the CyberKnife moves the linear accelerator through a number of fixed points in space, called nodes. At each node, the head of the linac can be aimed in almost any direction, provided that there are no joint configuration violations or robot singularities. There may be several distinct beam directions, and several distinct apertures at each node utilized in a patient treatment plan. The nodes are arranged into path sets, which are different for extracranial and intracranial targets. While the body path is approximately hemi-ellipsoidal and the head path is approximately hemi-spherical, both paths include node points such that about one-half of the solid angle is sampled; thus the space above the treatment couch is well-covered by the nodes in the path set. A treatment plan for a particular patient is generated via inverse planning, a process in which the user specifies the desired dose distribution, and the optimization algorithm selects the machine parameters required to deliver the specified radiation dose distribution. In general, there will be one or more targets, to which a tumoricidal dose of radiation must be delivered, as well as one or more organs-at-risk (OARs), which must spared from irradiation as much as possible, such that the radiation tolerances of these critical structures are not exceeded. The user will enter dose-volume histogram (DVH) objectives, which specify how much of a structure, expressed either as a percentage or absolute volume, may receive a particular radiation dose. For the CyberKnife M6, the problem is two-fold: a set of apertures must be generated and optimized, such that the position of the MLC leaves are specified for all radiation beams, and the beam weights, or monitor units (MU), which determine how much radiation each beam delivers, must be optimized. Both the aperture shape optimization and the beam weight optimization serve to produce a radiation dose distribution which is as close as possible, given the hardware constraints of the machine, to the dose distribution specified by the user. The current study builds on the authors previous work [2]-[3], which concerned dosimetric optimization applied to strictly circular collimators of various diameters, as well as a prior paper in which a model was developed to calculate the radiation absorbed dose from irregularly shaped apertures defined by the CyberKnife M6 InCise MLC. In the present A. Patient CT Data Set II. MATERIALS AND METHODS An anonymized CT data set for a patient with adenocarcinoma of the prostate was supplied by Accuray, Inc. All algorithm runs used this data set. The gross target volume (GTV) was considered to be the entire prostate, and organs-at-risk were the bladder, rectum, and skin. All contouring of the target volume and the OARs was performed by a radiation oncologist. An isotropic treatment margin of 3 mm was placed around the GTV to create the planning target volume (PTV). In addition, two optimization annular ring structures of 1 mm thickness were used to finely control the lower isodoses of the treatment plan, and to provide for desirable steep dose gradients outside the target. They were created at 10 mm and 30 mm distance from the PTV. The dose-volume objectives for the treatment plan are listed in Table I. In the table, note that the abbreviation Vxx is meant to specify the volume of a structure that receives a radiation absorbed dose of xx cgy. The objectives were adapted from RTOG-0938, a protocol for prostate radiosurgery from the Radiation Therapy Oncology Group. TABLE I. DOSE-VOLUME OBJECTIVES FOR PLANNING Organ Objectives PTV V %,V3850 < 0% Rectum V3800 <0%, V3400<7% Rectum V3200<10%,V1750<30% Bladder V3800 <0%, V3200<10% Bladder V1300<30% Ring 10 mm V2800<0% Ring 30 mm V1800<0% Skin V1500<0% The patient-specific data, CT and contours, were imported into the MATLAB (The Mathworks, Natick, MA) version (R2009a) 64-bit environment, installed on an HP Pavilion d7 notebook PC, with dual 2.53 GHz processors and 8 GB RAM. The optimization algorithm was coded in MATLAB, and the optimization runs were performed using this hardware and software configuration. B. Optimization Points Structures, including the PTV and OARs, are represented by optimization points, at which the radiation dose is calculated. Implementing the suggestion of Spirou and Chui [4], the density of points in a particular structure is approximately 50 points per cubic centimeter. As in the authors previous work [2], the strategy of dynamic point sampling of a subset of the totality of

3 optimization points is adopted. As the authors have previously observed, not all optimization points add useful information during the algorithm run, so that subsets of the totality of points may be sampled. The limited number of optimization points at which the radiation dose is calculated provides sufficient dosimetric feedback to the algorithm. It is again recognized that inter-iteration random sampling of the optimization points adds some noise to the fitness function, but the resulting decreased computation time is a highly desirable feature of the optimization, particularly from the perspective of the end-user in a busy clinic. The small amount of noise in fitness function values is a small price to pay for the increased efficiency [2]. C. Representation An individual is represented by an n-tuple of real numbers, as in (1) below: x 1, x2,, x n. (1) The number n, the length of the genome, is initially equal to the number of open MLC apertures, i.e. those apertures for which there were some non-zero gaps between the leaves on each bank of the MLC. The length of the genome is not constant, as will be subsequently discussed. The {x n } represent the set of beam weights or MU values for the individual, which is a particular candidate treatment plan. It should be noted that the possible MU values are not continuous. Generally, the linac radiation output does not become linear until approximately 7 MU, so that no beam should have an MU value lower than 7 per fraction. In addition, it is important that no particular beam should be weighted such that unacceptably high doses of radiation are delivered to the normal tissues in the path of the beam; therefore, a user-specified maximum MU value is admitted as a hard constraint. In summary: any beam will have an MU value of 0 if it is not used in the treatment plan, or, if it is used in the treatment plan, the value per fraction will fall somewhere between 7 and the user-specified maximum per beam MU value [2]. D. MLC Aperture Representation The MLC apertures are represented using 41 x 4 matrices. Each row of the matrix is associated with a leaf pair, with the positions of the top, bottom, left bank leaf, and right bank leaf carried in the columns. Each matrix describes a particular aperture shape associated with a particular beam s eye view (BEV) geometry. E. Search Space Dimensionality Reduction As was previously noted, the length of the genome is reduced as the optimization proceeds. In this way, the dimensionality of the search space is reduced until the beam set achieves what may be termed a maximal efficiency, 2 n t b b F x,w,d,v Θ v v w Θ δ DDC x DDC x d fit p, min t, min t, min 1 thres, min 1 ij j 1 ij j p, min n 2 max max t b b Θ v t, v p, w t, max Θ DDC x δ DDC x d max 1 1 ij j thres, max p, 1 ij j orgc n 2 lm l b b Θ v v w Θ DDC x δ DDC x d l 1 m 1 lm p,lm lm 1 1 kj j thres,lm 1 kj j p,lm MU MU MU 2 Θ MU max max which may be defined as that time in the optimization when, for all individuals in the population, all beams have a non-zero beam weight. Prior to this time, if a beam had an identically zero beam weight for all individuals in the population, it was deleted from the genotype. As the length of the genotype decreases, the number of optimization points is increased, until the totality of points representing all the structures is considered when evaluating the fitness function. As the authors observed previously [2], early on in the optimization, an efficient global search may be performed without careful consideration of tiny valleys or local minima in the search space. As the optimization proceeds, the dimensionality of the search space is greatly reduced by beam reduction, with a concomitant increase in the quality of candidate solutions, and a noiseless fitness function is necessary so that careful refinement of the solutions can be performed. F. Radiation Dose Calculation The calculation of radiation dose at the optimization points is performed using the method of Khan [5]. The dose deposition coefficients (DDCs), which represent the dose deposited (in cgy) by a given beam, at a particular point, per MU, are calculated as in (2) below: SAD DDC OF TPROCR, (2) 800 where DDC is the dose deposition coefficient (cgy/mu), OF is the output factor for the effective MLC aperture size (cgy/mu, the radiation dose per MU relative to that of the calibration field size), SAD is the source-to-axis distance (mm, the distance from the x-ray source to the point of calculation), TPR is the tissue-phantom ratio (the dose at the central axis of the beam, for the given effective MLC aperture size, and depth, relative to the dose at the central axis of the beam, at a depth of 15 mm), and OCR is the off-center ratio (the dose at a particular radial distance from the central axis of the beam, relative to the dose at the central axis of the beam, for the given MLC aperture shape, and depth) [2]. Note that OF, and TPR, and OCR are all dimensionless ratios used to calculate the radiation dose; these factors account for changes in radiation dose with changes in scatter, depth, and beam profile. The OCR is of particular importance for irregularly shaped MLC apertures, since the beam profile can be quite complicated. The authors have devoted considerable time to optimizing the model parameters used to represent the beam profile for such complicated apertures, following the work of Lorenz et al. [3]. Accurate calculation of the OCR requires the superposition of the profile for each open leaf gap in the MLC aperture. G. Fitness Function The fitness function is identical to that used in the authors previous work [2]. It is a simple quadratic, as shown below in (3): 2, (3)

4 where x is a vector of beam weights, w is a vector of objective weights, d is a vector of doses associated with user-selected DVH objectives, v is a vector of volumes associated with user-selected DVH objectives, Θ is the Heaviside function, subscript t refers to target, subscript p refers to target prescription dose, n t is the number of points calculated within the target, b is the number of beam weights, org is the number of critical organs, c lm is the mth constraint for the lth critical organ, n l is the number of points in the lth critical organ. In addition, δ thres,min is the dose to that point in the target such that when all points in the target are sorted, points for which the dose is less than or equal to δ thres,min cause the minimum dose volume constraint for the target to be violated, δ thres,max is the dose to that point in the target such that when all points in the target are sorted, points for which the dose is greater than or equal to δ thres,max cause the maximum dose volume constraint for the target to be violated, and δ thres,lm is the dose to that point in the lth critical organ such that when all points in the organ are sorted, points for which the dose is greater than or equal to δ thres,lm cause the mth dose volume constraint for the organ to be violated [2]. The relative magnitudes of the user-specified weights represent the clinical trade-offs that the radiation oncologist judges appropriate for a given patient. Often this means the choice between radiation dose to the tumor and sparing the surrounding organs-at-risk, so that the process of treatment planning optimization consists in finding a Pareto-optimal solution which closely represents the trade-offs in the clinical calculus of the radiation oncologist. Essentially, the fitness function includes a penalty for underdosing or overdosing the targets, a penalty for overdosing any critical structures, and a penalty for exceeding the total user-specified MU [2]. H. Algorithmic Structure 1) Brief Overview of the Optimization Process User-specified optimization parameters are entered at the beginning of the algorithm run; they may also be subsequently revised at any point by pausing the optimization, modifying the parameters as desired, and then commencing the optimization from the point in the run at which the user interrupted it. User-specified parameters are dose-volume objectives for the target and organs-at-risk, minimum and maximum per beam MU values, total MU value for the treatment plan, a desired number of nodes to be used in the treatment plan, and the maximum number of allowable nodes. After the user enters the data, a set of apertures is initialized. This set of apertures is used to create the initial population, which is then used in the main optimization loop, which consists of the following steps: (1) calculate the DDC for each aperture, (2) employ an EA to select out a subset of apertures, (3) use CMA-ES as an intense local search to fine-tune MU values, (4) generate new apertures using a rule-based approach taking into account DVH objective violations. The steps are then repeated after a parameter update based on the results of the CMA-ES. The various components of the algorithm are reviewed below; however, it was felt that an exhaustive discussion of all algorithm parameters would be of little value, and highlights of only the more important points are included. 2) Principal EC Global Search Initially, the apertures are created to treat the target with a 3 mm margin in beam s eye view. There are four cardinal hardware constraints regarding MLC apertures: (1) no islands, i.e. there cannot be a blocked area surrounded by an area which is exposed to the beam, (2) the minimum size of the gap between any pair of leaves is 7.5 mm, (3) the minimum allowable field size is 7.5 x 7.5 mm 2, and (4) there must be a minimum of three non-zero consecutive leaf openings. These hard constraints are applicable to MLC aperture creation at any point during the optimization. The size of the population is variable, with a minimum size of 300 and a maximum size of The size of the population is determined by the length of the genome (i.e. the number of beams in a patient treatment plan); the smaller the length of the genome, the larger the permissible size of the population. A uniform background mutation with a rate of 0.01 is imposed, as suggested in the literature [6]. Parents are determined via tournament selection, with a tournament size of 10. A problem-specific BLX-α operator is used for recombination, with α set to 0.5, as recommended in the literature [7]. Only non-zero MU values of an offspring can be randomly mutated to another value between the user-specified minimum and maximum MU bounds. Any beam for which the MU value is identically zero for all individuals in the population is excised from the genome, thus reducing the dimensionality of the search space. In this way, beams that are not useful are eliminated, and the total number of beams is reduced. This allows more computation time to be devoted to dose calculation, so that a greater number of optimization points in the structures may be used. In addition, the reduction in the dimensionality of the search space allows for an increase in the size of the population, providing for greater diversity. At the beginning of each iteration, all the individuals in the population are replaced. The fraction of beams retained which may be assigned to the new individuals is randomly selected uniformly from the interval [0.5, 1]. The best individual from the CMA-ES intense local search is placed in the population to seed it with good quality apertures. This best individual is retained unless there is an improvement in fitness after the EA, in which case the best individual from the last CMA-ES local search is replaced by the best individual from the current EA global search. 3) CMA-ES Intense Local Search A CMA-ES intense local search is performed using the best individual from the last iteration through the EA. The active CMA-ES algorithm is employed, using the default settings in Jasktrebski and Arnold [8]. The CMA-ES runs until the default termination conditions are met [8], or the algorithm iterates for 250 sec, whichever comes first. The beam weights are refined through this local search. The initial starting point uses random values between the user-selected minimum and maximum MU bounds. The initial

5 population size is 30, and the maximum population size is 150 (the population size is increased in subsequent iterations through the CMA-ES local search). After the local search terminates, there is a parameter update, during which point sampling is increased to include a larger subset of optimization points and the population size is increased. At this point, new beams involving new apertures are created, and admitted into the set of possible candidate beams. 4) New MLC Aperture Generation New MLC apertures are created after the CMA-ES local search is terminated. As was stated previously, the generation of new MLC apertures is rule-based. There are three methods for generating new MLC apertures. The three distinct methods are described below: a) New MLC Aperture Generation Method 1 Randomly select an aperture from the current set of apertures and block selected overdosed target and OAR points. b) New MLC Aperture Generation Method 2 Randomly select an aperture from current set of apertures, modify the MLC leaf positions to conform to all cold target points plus some margin, and block selected overdosed target and OAR points. c) New MLC Aperture Generation Method 3 Randomly select an entirely new beam geometry and block selected overdosed target and OAR points. The methods are selected stochastically. The probabilities for the first two methods are based on the components of the fitness value of the best individual. Let T be the contribution of the target points to the total fitness value, and let O be the contribution of the organs-at-risk to the total fitness value. The probabilities of the three methods are shown in (4) below: O P(1) T O O P(2) (4) T O P(3) 0.05 In this way, new MLC apertures which block overdosed points in the target and OARs, as well as new apertures which treat underdosed target points are introduced before the next iteration of the EA. III. RESULTS AND DISCUSSION For reference, a single sagittal CT reconstruction demonstrating the geometry of the organ contours is shown in Fig. 3. Fig. 3 A sagittal reconstruction demonstrating the geometry of the relevant structures: bladder (anterior to the prostate), rectum (posterior to the prostate), and prostate. It should be noted that as of the writing of this paper, Accuray has yet to release the MLC to CyberKnife sites. The latest release of the commercially available treatment planning system does not contain any functionality for planning treatments using the MLC; therefore, no comparison could made between the results generated by the authors algorithm and an alternative method of treatment planning. It was thus decided to look at the behavior of the algorithm and the results it generated with an eye solely toward feasibility of implementation in the clinic. A. Fitness Values and Function Evaluations Twenty runs of the algorithm were performed on the single patient data set. The results are summarized in Table II, which lists the fitness value of the best individual for each algorithm run, the number of function evaluations for each algorithm run, and the mean and standard deviation for all runs. TABLE II. SUMMARY OF ALGORITHM RUNS Run Number Fitness Value # Function Eval E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+06 Mean 2.48E E+05 SD 9.40E E+05

6 In all cases, at the termination of the algorithm, a clinically useful plan was produced. Thus, even though one observes a 38% standard deviation in the fitness value of the best individual, this is not significant in terms of the optimized final treatment plan. B. Behavior Of Fitness Value During Algorithm Run Fig. 6 The rectum is shielded by the MLC, as only the parts of the target inside the opening are irradiated by the primary beam. Note that it is not necessary for each aperture to treat the entire target, only that the sum of the dose distributions from all the individual beams in the treatment plan treat the target to the dose desired. Fig. 4 Best individual fitness value versus function evaluations. Note that initially the global EC leads to improvements, but these improvements are diminished during subsequent loops through the global EC. The graph in Fig. 4 illustrates the typical history of the fitness value of the best individual throughout the algorithm run. Initially, the global EC leads to improvement in fitness, but this improvement becomes less pronounced in subsequent iterations. The CMA-ES does continue to fine-tune the treatment plan, showing improvements throughout the optimization. D. Dose-Volume Histogram Analysis Ultimately, the treatment plan can be considered clinically useful if, and only if, the user-specified dose-volume objectives are achieved. If not, the dose distribution is unacceptable. In all twenty algorithm runs, a clinically useful plan was obtained. An example is shown in Fig. 6, which shows a cumulative dose-volume histogram (DVH). The horizontal axis plots the dose (in cgy), and the vertical axis is that percentage of the volume of a structure that receives at least that dose. C. MLC Apertures Of particular concern was whether or not the algorithm would produce optimized apertures which treated the target and blocked, to the extent possible, the critical organs-at-risk. The final apertures were examined to test if this was the case, and indeed it was. As shown in Fig. 4, the MLC pattern clearly conforms to the target. Fig. 7 A representative cumulative DVH for the PTV, rectum, bladder, skin, and optimization ring structrures. User-specified dose-volume objectives are plotted as black diamonds. The curves demonstrate that the dose-volume objectives are achieved in the treatment plan. Fig. 5 One of the final apertures from an algorithm run. The optimized MLC leaf positions create an aperture shape that conforms the target, which is the prostate plus planning margin. The area outside the opening is shielded by the leaves. Additionally, an investigation of the final optimized MLC aperture shapes demonstrated that there were apertures produced which shielded the organs-at-risk while treating parts of the target, as demonstrated in Fig. 5. IV. CONCLUSION In this study, a mimetic algorithm for optimizing MLC aperture shapes and beam weights for the CyberKnife M6 radiosurgery delivery platform was proposed, and the feasibility for clinical use was tested. The algorithm consisted of an evolutionary algorithm as a global search, combined with CMA-ES as an intense local search, and a rule-based MLC aperture shape generator. The algorithm was tested using a CT data set and organ contours for a patient with adenocarcinoma of the prostate. Twenty algorithm runs were performed, and the results were analyzed in terms of fitness value statistics, as well as in terms of the clinical utility of the optimized final treatment

7 plan. The algorithm performed quite well in meeting all user-specified clinical goals. The optimized MLC apertures treat the target, and shield the critical OARs, as expected. While the initial results are encouraging, it is imperative in further research to test the algorithm on other disease sites, including brain, spine, liver, lung, and pancreas. In addition, the authors wish to test the algorithm on cases with multiple targets, and cases with targets of more complex geometries. Of particular current interest in treatment planning optimization is the possibility of multiobjective optimization. An MCO approach would be of benefit, since a set of Pareto-optimal solutions, representing many different types of clinical trade-offs, could be presented to the radiation oncologist, who is the decision-maker (DM). The DM would then be able to choose amongst the alternatives, which may present options that did not occur to the DM a priori. The authors recognize that this is a value proposition and wish to pursue an MCO approach in the next generation of the optimizer. ACKNOWLEDGMENT The authors gratefully acknowledge the invaluable aid of Colin Sims, of Accuray, Inc., who provided the patient data set, and who provided feedback about the quality of the optimized treatment plans, and John Dooley, also of Accuray Inc., who clarified many issues regarding the path set geometry and machine constraints. REFERENCES [1] Accuray, Inc. (2012, November 1). Accuray Receives FDA 510(k) Clearance for New CyberKnife M6 Series, Merging Multileaf Collimation with Non-Isocentric Robotic Delivery. Press Release. Available: a-510k-clearance-new-cyberknife%c2%ae-m6%e2%84%a2-seriesmerging [2] O. Clancey and M. Witten, A Memetic Algorithm for Dosimetric Optimization in CyberKnife Robotic Radiosurgical Treatment Planning, IEEE Congress Evol Comp., [3] F. Lorenz, J. H. Killoran, F. Wenz, and P. Zygmanski, An independent dose calculation algorithm for MLC-based stereotactic radiotherapy, Med. Phys. 34, 1605 (2007). [4] Spirou, S.V. and C.S. Chui. A Gradient Inverse Planning Algorithm with Dose-Volume Constraints, Med. Phys (1998): [5] Khan, Faiz M. The Physics of Radiation Therapy, 3 rd ed. Philadelphia: Williams & Wilkins, [6] Eiben, A.E., and J.E. Smith. Introduction to Evolutionary Computing, 1 st ed. Heidelberg:Spring-Verlag Berlin Heidelberg, [7] T. Nomura and K. Shimohara. An Analysis of Two-Parent Recombinations for Real-Values Chromosomes in an Infinite Population, Evol. Comp. 9.3, 283 (2001). [8] G. A. Jasktrebski and D.V. Arnold, Improving Evolution Strategies through Active Covariance Matrix Adaptation, IEEE Congress Evol Comp, 2006.