Patient feature based dosimetric Pareto front prediction in esophageal cancer radiotherapy Jiazhou Wang, Xiance Jin, Kuaike Zhao, Jiayuan Peng, Jiang Xie, Junchao Chen, Zhen Zhang, Matthew Studenski, and Weigang Hu Citation: Medical Physics 42, 1005 (2015); doi: 10.1118/1.4906252 View online: http://dx.doi.org/10.1118/1.4906252 View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/42/2?ver=pdfcov Published by the American Association of Physicists in Medicine Articles you may be interested in The dosimetric impact of inversely optimized arc radiotherapy plan modulation for real-time dynamic MLC tracking delivery Med. Phys. 39, 1588 (2012); 10.1118/1.3685583 A support vector machine (SVM) for predicting preferred treatment position in radiotherapy of patients with breast cancer Med. Phys. 37, 5341 (2010); 10.1118/1.3483264 Using Bayesian logistic regression to evaluate a new type of dosimetric constraint for prostate radiotherapy treatment planning Med. Phys. 37, 1768 (2010); 10.1118/1.3367013 A novel flexible framework with automatic feature correspondence optimization for nonrigid registration in radiotherapy Med. Phys. 36, 2848 (2009); 10.1118/1.3134242 Accuracy of patient dose calculation for lung IMRT: A comparison of Monte Carlo, convolution/superposition, and pencil beam computations Med. Phys. 33, 3149 (2006); 10.1118/1.2241992
Patient feature based dosimetric Pareto front prediction in esophageal cancer radiotherapy Jiazhou Wang Department of Radiation Oncology, Fudan University Shanghai Cancer Center, Shanghai 200032, China and Department of Oncology, Shanghai Medical College, Fudan University, Shanghai 200032, China
Xiance Jin The 1st Affiliated Hospital of Wenzhou Medical College, Wenzhou, Zhejiang 325000, China
Kuaike Zhao, Jiayuan Peng, Jiang Xie, Junchao Chen, and Zhen Zhang Department of Radiation Oncology, Fudan University Shanghai Cancer Center, Shanghai 200032, China and Department of Oncology, Shanghai Medical College, Fudan University, Shanghai 200032, China
Matthew Studenski Department of Radiation Oncology, University of Miami-Miller School of Medicine, Miami, Florida 33136
Weigang Hua) Department of Radiation Oncology, Fudan University Shanghai Cancer Center, Shanghai 200032, China and Department of Oncology, Shanghai Medical College, Fudan University, Shanghai 200032, China
(Received 11 August 2014; revised 17 November 2014; accepted for publication 6 January 2015; published 29 January 2015) Purpose: To investigate the feasibility of the dosimetric Pareto front (PF) prediction based on patient’s anatomic and dosimetric parameters for esophageal cancer patients. Methods: Eighty esophagus patients in the authors’ institution were enrolled in this study. A total of 2928 intensity-modulated radiotherapy plans were obtained and used to generate PF for each patient. On average, each patient had 36.6 plans. The anatomic and dosimetric features were extracted from these plans. The mean lung dose (MLD), mean heart dose (MHD), spinal cord max dose, and PTV homogeneity index were recorded for each plan. Principal component analysis was used to extract overlap volume histogram (OVH) features between PTV and other organs at risk. The full dataset was separated into two parts; a training dataset and a validation dataset. The prediction outcomes were the MHD and MLD. The spearman’s rank correlation coefficient was used to evaluate the correlation between the anatomical features and dosimetric features. The stepwise multiple regression method was used to fit the PF. The cross validation method was used to evaluate the model. Results: With 1000 repetitions, the mean prediction error of the MHD was 469 cGy. The most correlated factor was the first principal components of the OVH between heart and PTV and the overlap between heart and PTV in Z-axis. The mean prediction error of the MLD was 284 cGy. The most correlated factors were the first principal components of the OVH between heart and PTV and the overlap between lung and PTV in Z-axis. Conclusions: It is feasible to use patients’ anatomic and dosimetric features to generate a predicted Pareto front. Additional samples and further studies are required improve the prediction model. C 2015 American Association of Physicists in Medicine. [http://dx.doi.org/10.1118/1.4906252] Key words: Pareto front, esophagus, dosimetric prediction, model training
1. INTRODUCTION Intensity-modulated radiotherapy (IMRT) has a much greater power to shape dose distribution than conventional radiotherapy with uniform beams. However, the task of making “the best plan” is quite challenging and time-consuming, even impossible. Moreover, it is difficult to decide whether the plan is already optimal. The quantitative metrics applied to treatment plan parameters vary based on personal experience, preference, and treatment facility expectations. Therefore, a reasonable dosimetric outcome prediction before planning will be of great help. Such a prediction can be used as a reference either for plan quality control and automatic planning in the future.1 1005
Med. Phys. 42 (2), February 2015
For the plan prediction, Wu et al.2 provided an effective quality control mechanism for evaluating the DVHs of the organs at risk (OARs) for head and neck (H&N) patients using the overlap volume histogram (OVH) to describe the geometry between treatment target and OAR. This method was extended to automatic optimization objectives generation in IMRT and volume arc modulated radiotherapy (VMAT)3–5 and showed the feasibility to accelerate the planning process and improve planning efficiency. Zhu et al.6 developed a plan quality quantitative evaluation tool based on a machine learning approach to generate the DVHs of the OARs in prostate radiotherapy. The principal component analysis (PCA) was used to characterize features of DVH and the distance-to-target histogram (DTH) and support vector regression was used to
0094-2405/2015/42(2)/1005/7/$30.00
© 2015 Am. Assoc. Phys. Med.
1005
1006
Wang et al.: Patient feature based Pareto front prediction
1006
F. 1. The whole workflow.
model their correlation. After that, more anatomical features (such as volume of PTV and OAR) and more sophisticated features selection methods (stepwise multiple regression) were introduced to generate the prediction model for prostate and H&N radiotherapy.7,8 Moore et al.9,10 developed an innovative statistical formula between the patient anatomy information and the DVH for plan evaluation. These methods do not consider the trade-off between different objectives, which means each one patient’s anatomical structure will only generate only one best plan. As a multicriteria optimization problem, this may not be the case for inverse planning optimization in radiotherapy. For example, the physicist or oncologist often need to compromise between delivering high dose to tumor target volume and sparing the surrounding healthy structures during treatment planning. To solve this problem, an improved mathematical model for achievable dose sparing prediction in parotid glands in H&N IMRT which incorporate single-side sparing was developed.11 This model can predict the parotid dose that takes single-sided or double-sided parotid sparing into account. However, this solution was only suitable for parotid sparing and a more general solution was required. The multicriteria optimization (MCO)12–14 is one of the general methods to handle the trade-off between tumor coverage and healthy tissue sparing for treatment planning. It is based on the Pareto front (PF), which is composed of a series of Pareto optimized plans. The Pareto optimized plan represents the plan that is the best in every measurable dimension. A commercial treatment planning system, Raystation (RaySearch Laboratories, Stockholm, Sweden), has incorporated the MCO and showed its superiority in terms of both planning efficiency and dose distribution quality.15,16 By incorporating MCO into a prediction model, we may have a general solution for the dosimetry prediction with compromise function. We want to design a dosimetry prediction solution with the MCO concept. As a complex solution which includes model development, plan navigation, and automatic planning, the first step should be to establish a prediction model with the MCO concept included. In this study, we propose a dosimetry prediction algorithm with the MCO concept and validate the Medical Physics, Vol. 42, No. 2, February 2015
feasibility of using this method in esophageal radiotherapy. Specifically, we try to predict the mean heart dose (MHD) and mean lung dose (MLD) based on patient anatomical features before planning. The prediction algorithm considers the trade-off between different objectives.
2. METHODS AND MATERIALS Patients’ data were collected and stored into a database for generating and validating the prediction model. The database includes the OVH, DVH, and some other features such as volumes of the PTVs and OARs. A machine learning method was used for model development and validation. The whole workflow was shown in Fig. 1. 2.A. Feature database generation 2.A.1. Patient selection
Eighty esophageal cancer patients previously treated in our institution were enrolled in this study. The prescription dose was 6120 cGy (180 cGy/fraction) and the oncologist delineated the heart, normal lung (lung minus PTV), and spinal cord. All images and contours were exported to the RayStation treatment planning system. The base characteristics of these contours are listed in Table I. 2.A.2. Treatment plans generation
To minimize the beam number and beam angle variation and influence, all plans were reoptimized with equally T I. Base characteristics of the contours.
PTV volume Heart volume Lung-PTV volume Beam numbers
Average
Range
420 587 3406 9
119∼925 365∼989 1345∼6650 9
1007
Wang et al.: Patient feature based Pareto front prediction
distributed, nine field plans; the gantry angle were 0◦, 40◦, 80◦, 120◦, 160◦, 200◦, 240◦, and 280◦. We used Raystation’s MCO module to generate a set of step-and-shoot IMRT plans. All plans were generated using 6 MV on a Trilogy Linac with a millennium MLC (Varian Medical Systems, Palo Alto, CA). The optimization constraints, trade-off objectives, and optimization settings were shown in Table II. For each patient, 55 plans were generated based on these constraints and trade-off objectives. The DVHs of these plans were exported using the self-developed script. The PTV homogeneity index (PTVHI), MHD, MLD, and the max dose of the spinal cord (SCmax) were calculated from these DVHs. We calculated convex hull of those points to remove non-Pareto front plans. The convex hull algorithm was based on Qhull.17 This process is similar to the Holdsworth’s non-Pareto plan elimination method,18 as this method is more suitable for MCO optimization. 2.A.3. OVH calculation and principal component analysis
A script was created to calculate the OVHs of these patients. For each patient, three OVHs (heart with PTV, lung-PTV with PTV, and spinal cord with PTV) were calculated and exported to . PCA reduced OVHs’ dimensions and selected the most significant features. The first and second principle component scores (HeartPCS1, LungPCS1, SCPCS1, and HeartPCS2, LungPCS2 SCPCS2, respectively) were implemented for the feature selection. 2.A.4. Z -axis overlap index
As most patients were treated with coplanar beams, the jaws in the head-foot direction can block most of the radiation. Therefore, the relative position between PTV and normal tissue in the z-axis (head-foot direction) would have the
1007
greatest impact in dose distribution for normal tissue. This ideal is similar to the “out of field” feature which was used by Lulin, and we just focus on the z-axis.7 Therefore, a z-axis overlap index was calculated to account for this effect. The overlap in z-axis (OVZ) was shown in Fig. 2. The z-axis length of heart (Zheart), lung (Zlung), PTV with heart (ZPTV and heart), and PTV with lung (ZPTV and lung) was extracted from the TPS. The OVZs were calculated as follows: ZPTV and heart , Heartovz = Zheart ZPTV and lung Lungovz = . Zlung
2.B. Feature selection and modeling 2.B.1. Preliminarily feature selection
To preliminarily exclude irrelevant variables, Spearman’s rank correlation coefficient (ρ2) was used to evaluate the correlation between the anatomical features and dosimetric features. If ρ2 = 0, there is no correlation between the two variables while ρ2 = 1 means that each of the variables is a perfect monotone function of the other. Features with ρ2 < 0.01 were excluded. 2.B.2. Model generation
Anatomic features and dosimetric parameters were fitted by a linear regression model. A stepwise method was used for further feature selection. This method added the most significant factor to the model and eliminated the least significant factor. The significances of the factors were assessed using R-square. Factors would be considered as significant and enrolled into fitting if it could increase the R-square by >0.01. In order to account for the nonlinear
T II. The constraints, trade-off objectives, and optimization settings. Constraints Spinal cord max dose PTV D95
4500 cGy 6120 cGy Trade-off objectives
Lung max gEUD Heart max gEUD Spinal cord max dose PTV uniform dose
500 cGy 500 cGy 4000 cGy 6120 cGy Optimization setting
Target priority Optimization tolerance Max number of iterations Iteration before conversion Max number of segments Min segment area Min segment per fraction
30 1E-5 40 7 50 4 cm2 2 F. 2. OVZ demonstration.
Medical Physics, Vol. 42, No. 2, February 2015
1008
Wang et al.: Patient feature based Pareto front prediction
effect such as interaction between two factors, the product term (e.g., MLD * PTVHI) was generated and added to the model to test its correlation with dosimetric outcome. 2.C. Validation 2.C.1. Cross validation
A cross validation method was used for model validation after feature selection. This method can determine how the results of statistical analysis will generalize to an independent data set. The whole dataset was randomly separated into training dataset (90% of patients) and validation dataset (10% of patients). The regression model was developed using the training dataset and applied to the validation dataset. To evaluate the prediction accuracy, the standard deviation of the prediction error was calculated. The workflow can be seen in Fig. 3. 2.C.2. External validation
Five additional esophageal patients were selected for external validation for a more intuitive evaluation of the prediction. The actual value of the Pareto front of these five patients was manually collected. Raystation’s MCO module was used to collect these data. First, we generated the Pareto front by this module, we fixed the PTVHI and SCmax values and adjusted the MHD and MLD weight, and the value of the MHD and MLD were manually recorded. Since the Pareto front is a high-dimension hyperplane, we fixed the other parameters and created a 2D Pareto front chart for easy comparison and visualization. A predicted Pareto front was calculated using a model, which was based on the 80 patient database.
1008
3. RESULTS 3.A. Features database generation
A total of 2928 IMRT plans were selected after the convex hull calculation. On average, each patient had 36.6 plans. The algorithm for contour expansion and shrinkage in TPS was used in the OVH calculation. An example of an OVH is shown in Fig. 4(a). Figure 4(b) shows the first two PCSs of the OVH. Obviously, most variables of the OVH can be explained by a small number of principle components. Based on the first two PCSs, 99%, 98%, and 92% OVH variances of heart, lung-PTV, and spinal cord can be explained, respectively. 3.B. Feature selection
The results of the preliminarily feature selection are listed in Table III. The LungPCS2 and heart volume were excluded in the MLD prediction. Heart volume and lung volume were excluded in the MHD prediction. After a further feature selection using the stepwise method, the final enrolled features for the MHD prediction were MLD, PTVHI, LungPCS1, heart volume, HeartOVZ, LungOVZ MLD * PTVHI, and MLD * HeartOVZ. The final enrolled features for the MLD prediction were MHD, PTVHI, HeartPCS1, HeartPCS2, LungPCS1, SCPCS2, HeartOVZ, LungOVZ, and MHD * PTVHI. 3.C. Model training and validation
The mean standard deviation for MHD prediction error was 469 cGy for the validation dataset with 1000 times repetition of cross validation. The mean standard deviation for MLD prediction error was 284 cGy. Figure 5 shows the result of cross validation. The external validation is shown in Fig. 6. For the additional five patients, patent 1 is completely consistent
F. 3. The workflow of the model training and validation. The whole dataset was randomly separated into training dataset and validation dataset. The regression model was developed using the training dataset and applied to the validation dataset. Medical Physics, Vol. 42, No. 2, February 2015
1009
Wang et al.: Patient feature based Pareto front prediction
1009
F. 4. An example OVH for esophageal radiotherapy.
with the actual Pareto front, where others still have deviations between the predicted front and the actual Pareto front.
4. DISCUSSION This study was the first attempt to predict the dosimetric outcome by anatomic information and incorporation of the MCO concept. The relationship between the different treatment planning objectives can be predicted and quantified by implementing the MCO concept. Furthermore, when compared to the previous head and neck or prostate cases, this study was the first attempt to predict the dosimetric outcome for the esophageal radiotherapy. Although there were fewer involved organs in esophageal radiotherapy than head and neck radiotherapy, the dosimetric relation (e.g., lung dose vs heart dose and left parotid dose vs right parotid dose) between different organs was similar. By using multicriteria optimization and the Pareto front, we can make sure the plans we used for training were the best plans. Traditionally, physicist and physician need to confirm and review each plan, to make sure it is the best plan and then use it for model T III. Feature selection. Factors HeartPCS1 HeartPCS2 LungPCS1 LungPCS2 SCPCS1 SCPCS2 PTV volume Heart volume Lung volume HeartOVZ LungOVZ a
ρ 2 with MLD
ρ 2 with MHD
0.809 0.137 0.456 0.101 0.052 0.419 0.179 0.001 0.011 0.813 0.532
0.509 0.095 0.492 0.004 0.077 0.309 0.176 0.000a 0.027 0.474 0.477
This feature was excluded for further analysis.
Medical Physics, Vol. 42, No. 2, February 2015
training. By incorporating the MCO concept, we can avoid the tedious and subjective best plan selection process for training dataset. The prediction accuracy in this study is far from perfect. We could not capture all relative features to predict the final result. Some features were quite difficult to quantitate clearly. For example, the PTV with irregular shape was more difficult for beam orientation design and could have higher normal tissue dose. Finally, the algorithm we used for outcome prediction might not be optimized. Different learning algorithms might improve the prediction accuracy. In this study, we directly used the MCO algorithm of the RaySearch TPS. This algorithm cannot make sure better plans do not exist although according to Craft’s research,19 36 plans can control the error to within 5% for a 10 objective brain case and 55 plans can control the error to within 5% for a 6 objective prostate case. Here, we used 55 plans to generate our Pareto surface and we used four trade-off objectives. This should make the error from the MCO algorithm acceptable. Four dosimetric object functions and nine anatomic features were involved in model development. Incorporating more plan features and using more advanced nonlinear models could increase the model accuracy. However, these methods would increase the complexity of the model. The study of Craft et al. showed that N+ 1 plans sufficiently represented a high-dimensional Pareto front13 for N dosimetric object functions but the number predictors should be less than m/10 or m/20 (where m is the sample size) to get a reliable fitted regression model in the model training process. This requires huge number plans for model training. Additionally, different objects can have an inverse relationship or one organ might have two objective functions, which require more interaction items as the model input and many more samples for model training. We did not introduce all the treatment plan parameters (such as beam angles or treatment target complexity) into model because of the difficulty in quantifying and to keep the model simple, although these parameters would have an impact on the plan quality (demonstrated by Mittauer et al.20
1010
Wang et al.: Patient feature based Pareto front prediction
1010
F. 5. An evaluation for the model in training and validation datasets.
in head and neck radiotherapy). Linear regression was selected in this study as reported in the study of Yuan et al.,7,11 because it is easy to implement and flexible by introducing interaction between features.
One of the problems in using MCO for radiotherapy is that some objectives in the optimization were not convex. In this study, we select the MHD and MLD as trade-off objectives. For MHD, Martel et al. and Wei et al. implicated
F. 6. External validation. The first row displays the OVHs of the patients in external validation. The second row displays the Pareto front deviation between the prediction and manual collection. Medical Physics, Vol. 42, No. 2, February 2015
1011
Wang et al.: Patient feature based Pareto front prediction
that MHD was one predictor of pericarditis in esophageal cancer radiotherapy21,22 and the MHD was highly correlated with other dosimetric parameters, making comparisons of their predictive abilities difficult. The MLD was widely used as a predictor of radiation pneumonitis owing to its simplicity and effectiveness.23 Although some other predictors may be more clinically related, the mean dose was a convex objective functions and dose–volume-based criteria have proven to be nonconvex.24 As we use MCO in this study, we prefer using convex objective functions more. Some points in the Pareto front were not attainable using both nonconvex objective (e.g., dose–volume constraints) in treatment planning and convex hull in Pareto front generating.25 The practical consequences of the nonconvex objects should be studied further. In esophageal cancer treatments, the doses to heart and lung could not be precisely predicted solely by anatomical structures. This phenomenon may not be as significant in other sites. In other words, an accurate prediction model can be established in some sites just by using anatomic information. For example, in the prostate radiotherapy, the doses to the bladder and rectum would be decided just using their anatomic features. In the future, we need to determine which sites need this technique for dose prediction.
5. CONCLUSIONS It is feasible to use patients’ anatomic and dosimetric features to generate a predicted Pareto front. Additional samples and further studies are required to improve the prediction model.
ACKNOWLEDGMENTS The authors thank Matthew Studenski for reviewing and scientific editing of this paper. This work is partially supported by research funding from the Chinese National Natural Science Foundation (11205033). The authors report no conflicts of interest in conducting the research. a)Author
to whom correspondence should be addressed. Electronic mail: [email protected]; Fax: 86-21-6417 4774; Telephone: 8615921292224. 1K. L. Moore, G. C. Kagadis, T. R. McNutt, V. Moiseenko, and S. Mutic, “Vision 20/20: Automation and advanced computing in clinical radiation oncology,” Med. Phys. 41, 010901 (13pp.) (2014). 2B. Wu, F. Ricchetti, G. Sanguineti, M. Kazhdan, P. Simari, M. Chuang, R. Taylor, R. Jacques, and T. McNutt, “Patient geometry-driven information retrieval for IMRT treatment plan quality control,” Med. Phys. 36, 5497–5505 (2009). 3Y. Yang, E. C. Ford, B. Wu, M. Pinkawa, B. van Triest, P. Campbell, D. Y. Song, and T. R. McNutt, “An overlap-volume-histogram based method for rectal dose prediction and automated treatment planning in the external beam prostate radiotherapy following hydrogel injection,” Med. Phys. 40, 011709 (10pp.) (2013). 4B. Wu, D. Pang, P. Simari, R. Taylor, G. Sanguineti, and T. McNutt, “Using overlap volume histogram and IMRT plan data to guide and automate VMAT
Medical Physics, Vol. 42, No. 2, February 2015
1011 planning: A head-and-neck case study,” Med. Phys. 40, 021714 (7pp.) (2013). 5S. F. Petit, B. Wu, M. Kazhdan, A. Dekker, P. Simari, R. Kumar, R. Taylor, J. M. Herman, and T. McNutt, “Increased organ sparing using shape-based treatment plan optimization for intensity modulated radiation therapy of pancreatic adenocarcinoma,” Radiother. Oncol.: J. Eur. Soc. Ther. Radiol. Oncol. 102, 38–44 (2012). 6X. Zhu, Y. Ge, T. Li, D. Thongphiew, F. F. Yin, and Q. J. Wu, “A planning quality evaluation tool for prostate adaptive IMRT based on machine learning,” Med. Phys. 38, 719–726 (2011). 7L. Yuan, Y. Ge, W. R. Lee, F. F. Yin, J. P. Kirkpatrick, and Q. J. Wu, “Quantitative analysis of the factors which affect the interpatient organat-risk dose sparing variation in IMRT plans,” Med. Phys. 39, 6868–6878 (2012). 8J. Lian, L. Yuan, Y. Ge, B. S. Chera, D. P. Yoo, S. Chang, F. Yin, and Q. J. Wu, “Modeling the dosimetry of organ-at-risk in head and neck IMRT planning: An intertechnique and interinstitutional study,” Med. Phys. 40, 121704 (9pp.) (2013). 9L. M. Appenzoller, J. M. Michalski, W. L. Thorstad, S. Mutic, and K. L. Moore, “Predicting dose-volume histograms for organs-at-risk in IMRT planning,” Med. Phys. 39, 7446–7461 (2012). 10K. L. Moore, R. S. Brame, D. A. Low, and S. Mutic, “Experience-based quality control of clinical intensity-modulated radiotherapy planning,” Int. J. Radiat. Oncol., Biol., Phys. 81, 545–551 (2011). 11L. Yuan, Q. J. Wu, F. F. Yin, Y. Jiang, D. Yoo, and Y. Ge, “Incorporating single-side sparing in models for predicting parotid dose sparing in head and neck IMRT,” Med. Phys. 41, 021728 (8pp.) (2014). 12D. L. Craft, T. F. Halabi, H. A. Shih, and T. R. Bortfeld, “Approximating convex pareto surfaces in multiobjective radiotherapy planning,” Med. Phys. 33, 3399–3407 (2006). 13D. Craft, T. Halabi, and T. Bortfeld, “Exploration of tradeoffs in intensitymodulated radiotherapy,” Phys. Med. Biol. 50, 5857–5868 (2005). 14M. Monz, K. H. Kufer, T. R. Bortfeld, and C. Thieke, “Pareto navigation: Algorithmic foundation of interactive multi-criteria IMRT planning,” Phys. Med. Biol. 53, 985–998 (2008). 15D. L. Craft, T. S. Hong, H. A. Shih, and T. R. Bortfeld, “Improved planning time and plan quality through multicriteria optimization for intensitymodulated radiotherapy,” Int. J. Radiat. Oncol., Biol., Phys. 82, e83–e90 (2012). 16W. Hu, J. Wang, G. Li, J. Peng, S. Lu, and Z. Zhang, “Investigation of plan quality between RapidArc and IMRT for gastric cancer based on a novel beam angle and multicriteria optimization technique,” Radiother. Oncol.: J. Eur. Soc. Ther. Radiol. Oncol. 111, 144–147 (2014). 17C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Software 22, 469–483 (1996). 18C. Holdsworth, M. Kim, J. Liao, and M. Phillips, “The use of a multiobjective evolutionary algorithm to increase flexibility in the search for better IMRT plans,” Med. Phys. 39, 2261–2274 (2012). 19D. Craft and T. Bortfeld, “How many plans are needed in an IMRT multiobjective plan database?,” Phys. Med. Biol. 53, 2785–2796 (2008). 20K. Mittauer, B. Lu, G. Yan, D. Kahler, A. Gopal, R. Amdur, and C. Liu, “A study of IMRT planning parameters on planning efficiency, delivery efficiency, and plan quality,” Med. Phys. 40, 061704 (13pp.) (2013). 21M. K. Martel, W. M. Sahijdak, R. K. Ten Haken, M. L. Kessler, and A. T. Turrisi, “Fraction size and dose parameters related to the incidence of pericardial effusions,” Int. J. Radiat. Oncol., Biol., Phys. 40, 155–161 (1998). 22X. Wei, H. H. Liu, S. L. Tucker, S. Wang, R. Mohan, J. D. Cox, R. Komaki, and Z. Liao, “Risk factors for pericardial effusion in inoperable esophageal cancer patients treated with definitive chemoradiation therapy,” Int. J. Radiat. Oncol., Biol., Phys. 70, 707–714 (2008). 23L. B. Marks, S. M. Bentzen, J. O. Deasy, F. M. Kong, J. D. Bradley, I. S. Vogelius, I. El Naqa, J. L. Hubbs, J. V. Lebesque, R. D. Timmerman, M. K. Martel, and A. Jackson, “Radiation dose-volume effects in the lung,” Int. J. Radiat. Oncol., Biol., Phys. 76, S70–S76 (2010). 24J. O. Deasy, “Multiple local minima in radiotherapy optimization problems with dose-volume constraints,” Med. Phys. 24, 1157–1161 (1997). 25T. Halabi, D. Craft, and T. Bortfeld, “Dose-volume objectives in multicriteria optimization,” Phys. Med. Biol. 51, 3809–3818 (2006).
Patient feature based dosimetric Pareto front prediction in esophageal cancer radiotherapy Jiazhou Wang Department of Radiation Oncology, Fudan University Shanghai Cancer Center, Shanghai 200032, China and Department of Oncology, Shanghai Medical College, Fudan University, Shanghai 200032, China
Xiance Jin The 1st Affiliated Hospital of Wenzhou Medical College, Wenzhou, Zhejiang 325000, China
Kuaike Zhao, Jiayuan Peng, Jiang Xie, Junchao Chen, and Zhen Zhang Department of Radiation Oncology, Fudan University Shanghai Cancer Center, Shanghai 200032, China and Department of Oncology, Shanghai Medical College, Fudan University, Shanghai 200032, China
Matthew Studenski Department of Radiation Oncology, University of Miami-Miller School of Medicine, Miami, Florida 33136
Weigang Hua) Department of Radiation Oncology, Fudan University Shanghai Cancer Center, Shanghai 200032, China and Department of Oncology, Shanghai Medical College, Fudan University, Shanghai 200032, China
(Received 11 August 2014; revised 17 November 2014; accepted for publication 6 January 2015; published 29 January 2015) Purpose: To investigate the feasibility of the dosimetric Pareto front (PF) prediction based on patient’s anatomic and dosimetric parameters for esophageal cancer patients. Methods: Eighty esophagus patients in the authors’ institution were enrolled in this study. A total of 2928 intensity-modulated radiotherapy plans were obtained and used to generate PF for each patient. On average, each patient had 36.6 plans. The anatomic and dosimetric features were extracted from these plans. The mean lung dose (MLD), mean heart dose (MHD), spinal cord max dose, and PTV homogeneity index were recorded for each plan. Principal component analysis was used to extract overlap volume histogram (OVH) features between PTV and other organs at risk. The full dataset was separated into two parts; a training dataset and a validation dataset. The prediction outcomes were the MHD and MLD. The spearman’s rank correlation coefficient was used to evaluate the correlation between the anatomical features and dosimetric features. The stepwise multiple regression method was used to fit the PF. The cross validation method was used to evaluate the model. Results: With 1000 repetitions, the mean prediction error of the MHD was 469 cGy. The most correlated factor was the first principal components of the OVH between heart and PTV and the overlap between heart and PTV in Z-axis. The mean prediction error of the MLD was 284 cGy. The most correlated factors were the first principal components of the OVH between heart and PTV and the overlap between lung and PTV in Z-axis. Conclusions: It is feasible to use patients’ anatomic and dosimetric features to generate a predicted Pareto front. Additional samples and further studies are required improve the prediction model. C 2015 American Association of Physicists in Medicine. [http://dx.doi.org/10.1118/1.4906252] Key words: Pareto front, esophagus, dosimetric prediction, model training
1. INTRODUCTION Intensity-modulated radiotherapy (IMRT) has a much greater power to shape dose distribution than conventional radiotherapy with uniform beams. However, the task of making “the best plan” is quite challenging and time-consuming, even impossible. Moreover, it is difficult to decide whether the plan is already optimal. The quantitative metrics applied to treatment plan parameters vary based on personal experience, preference, and treatment facility expectations. Therefore, a reasonable dosimetric outcome prediction before planning will be of great help. Such a prediction can be used as a reference either for plan quality control and automatic planning in the future.1 1005
Med. Phys. 42 (2), February 2015
For the plan prediction, Wu et al.2 provided an effective quality control mechanism for evaluating the DVHs of the organs at risk (OARs) for head and neck (H&N) patients using the overlap volume histogram (OVH) to describe the geometry between treatment target and OAR. This method was extended to automatic optimization objectives generation in IMRT and volume arc modulated radiotherapy (VMAT)3–5 and showed the feasibility to accelerate the planning process and improve planning efficiency. Zhu et al.6 developed a plan quality quantitative evaluation tool based on a machine learning approach to generate the DVHs of the OARs in prostate radiotherapy. The principal component analysis (PCA) was used to characterize features of DVH and the distance-to-target histogram (DTH) and support vector regression was used to
0094-2405/2015/42(2)/1005/7/$30.00
© 2015 Am. Assoc. Phys. Med.
1005
1006
Wang et al.: Patient feature based Pareto front prediction
1006
F. 1. The whole workflow.
model their correlation. After that, more anatomical features (such as volume of PTV and OAR) and more sophisticated features selection methods (stepwise multiple regression) were introduced to generate the prediction model for prostate and H&N radiotherapy.7,8 Moore et al.9,10 developed an innovative statistical formula between the patient anatomy information and the DVH for plan evaluation. These methods do not consider the trade-off between different objectives, which means each one patient’s anatomical structure will only generate only one best plan. As a multicriteria optimization problem, this may not be the case for inverse planning optimization in radiotherapy. For example, the physicist or oncologist often need to compromise between delivering high dose to tumor target volume and sparing the surrounding healthy structures during treatment planning. To solve this problem, an improved mathematical model for achievable dose sparing prediction in parotid glands in H&N IMRT which incorporate single-side sparing was developed.11 This model can predict the parotid dose that takes single-sided or double-sided parotid sparing into account. However, this solution was only suitable for parotid sparing and a more general solution was required. The multicriteria optimization (MCO)12–14 is one of the general methods to handle the trade-off between tumor coverage and healthy tissue sparing for treatment planning. It is based on the Pareto front (PF), which is composed of a series of Pareto optimized plans. The Pareto optimized plan represents the plan that is the best in every measurable dimension. A commercial treatment planning system, Raystation (RaySearch Laboratories, Stockholm, Sweden), has incorporated the MCO and showed its superiority in terms of both planning efficiency and dose distribution quality.15,16 By incorporating MCO into a prediction model, we may have a general solution for the dosimetry prediction with compromise function. We want to design a dosimetry prediction solution with the MCO concept. As a complex solution which includes model development, plan navigation, and automatic planning, the first step should be to establish a prediction model with the MCO concept included. In this study, we propose a dosimetry prediction algorithm with the MCO concept and validate the Medical Physics, Vol. 42, No. 2, February 2015
feasibility of using this method in esophageal radiotherapy. Specifically, we try to predict the mean heart dose (MHD) and mean lung dose (MLD) based on patient anatomical features before planning. The prediction algorithm considers the trade-off between different objectives.
2. METHODS AND MATERIALS Patients’ data were collected and stored into a database for generating and validating the prediction model. The database includes the OVH, DVH, and some other features such as volumes of the PTVs and OARs. A machine learning method was used for model development and validation. The whole workflow was shown in Fig. 1. 2.A. Feature database generation 2.A.1. Patient selection
Eighty esophageal cancer patients previously treated in our institution were enrolled in this study. The prescription dose was 6120 cGy (180 cGy/fraction) and the oncologist delineated the heart, normal lung (lung minus PTV), and spinal cord. All images and contours were exported to the RayStation treatment planning system. The base characteristics of these contours are listed in Table I. 2.A.2. Treatment plans generation
To minimize the beam number and beam angle variation and influence, all plans were reoptimized with equally T I. Base characteristics of the contours.
PTV volume Heart volume Lung-PTV volume Beam numbers
Average
Range
420 587 3406 9
119∼925 365∼989 1345∼6650 9
1007
Wang et al.: Patient feature based Pareto front prediction
distributed, nine field plans; the gantry angle were 0◦, 40◦, 80◦, 120◦, 160◦, 200◦, 240◦, and 280◦. We used Raystation’s MCO module to generate a set of step-and-shoot IMRT plans. All plans were generated using 6 MV on a Trilogy Linac with a millennium MLC (Varian Medical Systems, Palo Alto, CA). The optimization constraints, trade-off objectives, and optimization settings were shown in Table II. For each patient, 55 plans were generated based on these constraints and trade-off objectives. The DVHs of these plans were exported using the self-developed script. The PTV homogeneity index (PTVHI), MHD, MLD, and the max dose of the spinal cord (SCmax) were calculated from these DVHs. We calculated convex hull of those points to remove non-Pareto front plans. The convex hull algorithm was based on Qhull.17 This process is similar to the Holdsworth’s non-Pareto plan elimination method,18 as this method is more suitable for MCO optimization. 2.A.3. OVH calculation and principal component analysis
A script was created to calculate the OVHs of these patients. For each patient, three OVHs (heart with PTV, lung-PTV with PTV, and spinal cord with PTV) were calculated and exported to . PCA reduced OVHs’ dimensions and selected the most significant features. The first and second principle component scores (HeartPCS1, LungPCS1, SCPCS1, and HeartPCS2, LungPCS2 SCPCS2, respectively) were implemented for the feature selection. 2.A.4. Z -axis overlap index
As most patients were treated with coplanar beams, the jaws in the head-foot direction can block most of the radiation. Therefore, the relative position between PTV and normal tissue in the z-axis (head-foot direction) would have the
1007
greatest impact in dose distribution for normal tissue. This ideal is similar to the “out of field” feature which was used by Lulin, and we just focus on the z-axis.7 Therefore, a z-axis overlap index was calculated to account for this effect. The overlap in z-axis (OVZ) was shown in Fig. 2. The z-axis length of heart (Zheart), lung (Zlung), PTV with heart (ZPTV and heart), and PTV with lung (ZPTV and lung) was extracted from the TPS. The OVZs were calculated as follows: ZPTV and heart , Heartovz = Zheart ZPTV and lung Lungovz = . Zlung
2.B. Feature selection and modeling 2.B.1. Preliminarily feature selection
To preliminarily exclude irrelevant variables, Spearman’s rank correlation coefficient (ρ2) was used to evaluate the correlation between the anatomical features and dosimetric features. If ρ2 = 0, there is no correlation between the two variables while ρ2 = 1 means that each of the variables is a perfect monotone function of the other. Features with ρ2 < 0.01 were excluded. 2.B.2. Model generation
Anatomic features and dosimetric parameters were fitted by a linear regression model. A stepwise method was used for further feature selection. This method added the most significant factor to the model and eliminated the least significant factor. The significances of the factors were assessed using R-square. Factors would be considered as significant and enrolled into fitting if it could increase the R-square by >0.01. In order to account for the nonlinear
T II. The constraints, trade-off objectives, and optimization settings. Constraints Spinal cord max dose PTV D95
4500 cGy 6120 cGy Trade-off objectives
Lung max gEUD Heart max gEUD Spinal cord max dose PTV uniform dose
500 cGy 500 cGy 4000 cGy 6120 cGy Optimization setting
Target priority Optimization tolerance Max number of iterations Iteration before conversion Max number of segments Min segment area Min segment per fraction
30 1E-5 40 7 50 4 cm2 2 F. 2. OVZ demonstration.
Medical Physics, Vol. 42, No. 2, February 2015
1008
Wang et al.: Patient feature based Pareto front prediction
effect such as interaction between two factors, the product term (e.g., MLD * PTVHI) was generated and added to the model to test its correlation with dosimetric outcome. 2.C. Validation 2.C.1. Cross validation
A cross validation method was used for model validation after feature selection. This method can determine how the results of statistical analysis will generalize to an independent data set. The whole dataset was randomly separated into training dataset (90% of patients) and validation dataset (10% of patients). The regression model was developed using the training dataset and applied to the validation dataset. To evaluate the prediction accuracy, the standard deviation of the prediction error was calculated. The workflow can be seen in Fig. 3. 2.C.2. External validation
Five additional esophageal patients were selected for external validation for a more intuitive evaluation of the prediction. The actual value of the Pareto front of these five patients was manually collected. Raystation’s MCO module was used to collect these data. First, we generated the Pareto front by this module, we fixed the PTVHI and SCmax values and adjusted the MHD and MLD weight, and the value of the MHD and MLD were manually recorded. Since the Pareto front is a high-dimension hyperplane, we fixed the other parameters and created a 2D Pareto front chart for easy comparison and visualization. A predicted Pareto front was calculated using a model, which was based on the 80 patient database.
1008
3. RESULTS 3.A. Features database generation
A total of 2928 IMRT plans were selected after the convex hull calculation. On average, each patient had 36.6 plans. The algorithm for contour expansion and shrinkage in TPS was used in the OVH calculation. An example of an OVH is shown in Fig. 4(a). Figure 4(b) shows the first two PCSs of the OVH. Obviously, most variables of the OVH can be explained by a small number of principle components. Based on the first two PCSs, 99%, 98%, and 92% OVH variances of heart, lung-PTV, and spinal cord can be explained, respectively. 3.B. Feature selection
The results of the preliminarily feature selection are listed in Table III. The LungPCS2 and heart volume were excluded in the MLD prediction. Heart volume and lung volume were excluded in the MHD prediction. After a further feature selection using the stepwise method, the final enrolled features for the MHD prediction were MLD, PTVHI, LungPCS1, heart volume, HeartOVZ, LungOVZ MLD * PTVHI, and MLD * HeartOVZ. The final enrolled features for the MLD prediction were MHD, PTVHI, HeartPCS1, HeartPCS2, LungPCS1, SCPCS2, HeartOVZ, LungOVZ, and MHD * PTVHI. 3.C. Model training and validation
The mean standard deviation for MHD prediction error was 469 cGy for the validation dataset with 1000 times repetition of cross validation. The mean standard deviation for MLD prediction error was 284 cGy. Figure 5 shows the result of cross validation. The external validation is shown in Fig. 6. For the additional five patients, patent 1 is completely consistent
F. 3. The workflow of the model training and validation. The whole dataset was randomly separated into training dataset and validation dataset. The regression model was developed using the training dataset and applied to the validation dataset. Medical Physics, Vol. 42, No. 2, February 2015
1009
Wang et al.: Patient feature based Pareto front prediction
1009
F. 4. An example OVH for esophageal radiotherapy.
with the actual Pareto front, where others still have deviations between the predicted front and the actual Pareto front.
4. DISCUSSION This study was the first attempt to predict the dosimetric outcome by anatomic information and incorporation of the MCO concept. The relationship between the different treatment planning objectives can be predicted and quantified by implementing the MCO concept. Furthermore, when compared to the previous head and neck or prostate cases, this study was the first attempt to predict the dosimetric outcome for the esophageal radiotherapy. Although there were fewer involved organs in esophageal radiotherapy than head and neck radiotherapy, the dosimetric relation (e.g., lung dose vs heart dose and left parotid dose vs right parotid dose) between different organs was similar. By using multicriteria optimization and the Pareto front, we can make sure the plans we used for training were the best plans. Traditionally, physicist and physician need to confirm and review each plan, to make sure it is the best plan and then use it for model T III. Feature selection. Factors HeartPCS1 HeartPCS2 LungPCS1 LungPCS2 SCPCS1 SCPCS2 PTV volume Heart volume Lung volume HeartOVZ LungOVZ a
ρ 2 with MLD
ρ 2 with MHD
0.809 0.137 0.456 0.101 0.052 0.419 0.179 0.001 0.011 0.813 0.532
0.509 0.095 0.492 0.004 0.077 0.309 0.176 0.000a 0.027 0.474 0.477
This feature was excluded for further analysis.
Medical Physics, Vol. 42, No. 2, February 2015
training. By incorporating the MCO concept, we can avoid the tedious and subjective best plan selection process for training dataset. The prediction accuracy in this study is far from perfect. We could not capture all relative features to predict the final result. Some features were quite difficult to quantitate clearly. For example, the PTV with irregular shape was more difficult for beam orientation design and could have higher normal tissue dose. Finally, the algorithm we used for outcome prediction might not be optimized. Different learning algorithms might improve the prediction accuracy. In this study, we directly used the MCO algorithm of the RaySearch TPS. This algorithm cannot make sure better plans do not exist although according to Craft’s research,19 36 plans can control the error to within 5% for a 10 objective brain case and 55 plans can control the error to within 5% for a 6 objective prostate case. Here, we used 55 plans to generate our Pareto surface and we used four trade-off objectives. This should make the error from the MCO algorithm acceptable. Four dosimetric object functions and nine anatomic features were involved in model development. Incorporating more plan features and using more advanced nonlinear models could increase the model accuracy. However, these methods would increase the complexity of the model. The study of Craft et al. showed that N+ 1 plans sufficiently represented a high-dimensional Pareto front13 for N dosimetric object functions but the number predictors should be less than m/10 or m/20 (where m is the sample size) to get a reliable fitted regression model in the model training process. This requires huge number plans for model training. Additionally, different objects can have an inverse relationship or one organ might have two objective functions, which require more interaction items as the model input and many more samples for model training. We did not introduce all the treatment plan parameters (such as beam angles or treatment target complexity) into model because of the difficulty in quantifying and to keep the model simple, although these parameters would have an impact on the plan quality (demonstrated by Mittauer et al.20
1010
Wang et al.: Patient feature based Pareto front prediction
1010
F. 5. An evaluation for the model in training and validation datasets.
in head and neck radiotherapy). Linear regression was selected in this study as reported in the study of Yuan et al.,7,11 because it is easy to implement and flexible by introducing interaction between features.
One of the problems in using MCO for radiotherapy is that some objectives in the optimization were not convex. In this study, we select the MHD and MLD as trade-off objectives. For MHD, Martel et al. and Wei et al. implicated
F. 6. External validation. The first row displays the OVHs of the patients in external validation. The second row displays the Pareto front deviation between the prediction and manual collection. Medical Physics, Vol. 42, No. 2, February 2015
1011
Wang et al.: Patient feature based Pareto front prediction
that MHD was one predictor of pericarditis in esophageal cancer radiotherapy21,22 and the MHD was highly correlated with other dosimetric parameters, making comparisons of their predictive abilities difficult. The MLD was widely used as a predictor of radiation pneumonitis owing to its simplicity and effectiveness.23 Although some other predictors may be more clinically related, the mean dose was a convex objective functions and dose–volume-based criteria have proven to be nonconvex.24 As we use MCO in this study, we prefer using convex objective functions more. Some points in the Pareto front were not attainable using both nonconvex objective (e.g., dose–volume constraints) in treatment planning and convex hull in Pareto front generating.25 The practical consequences of the nonconvex objects should be studied further. In esophageal cancer treatments, the doses to heart and lung could not be precisely predicted solely by anatomical structures. This phenomenon may not be as significant in other sites. In other words, an accurate prediction model can be established in some sites just by using anatomic information. For example, in the prostate radiotherapy, the doses to the bladder and rectum would be decided just using their anatomic features. In the future, we need to determine which sites need this technique for dose prediction.
5. CONCLUSIONS It is feasible to use patients’ anatomic and dosimetric features to generate a predicted Pareto front. Additional samples and further studies are required to improve the prediction model.
ACKNOWLEDGMENTS The authors thank Matthew Studenski for reviewing and scientific editing of this paper. This work is partially supported by research funding from the Chinese National Natural Science Foundation (11205033). The authors report no conflicts of interest in conducting the research. a)Author
to whom correspondence should be addressed. Electronic mail: [email protected]; Fax: 86-21-6417 4774; Telephone: 8615921292224. 1K. L. Moore, G. C. Kagadis, T. R. McNutt, V. Moiseenko, and S. Mutic, “Vision 20/20: Automation and advanced computing in clinical radiation oncology,” Med. Phys. 41, 010901 (13pp.) (2014). 2B. Wu, F. Ricchetti, G. Sanguineti, M. Kazhdan, P. Simari, M. Chuang, R. Taylor, R. Jacques, and T. McNutt, “Patient geometry-driven information retrieval for IMRT treatment plan quality control,” Med. Phys. 36, 5497–5505 (2009). 3Y. Yang, E. C. Ford, B. Wu, M. Pinkawa, B. van Triest, P. Campbell, D. Y. Song, and T. R. McNutt, “An overlap-volume-histogram based method for rectal dose prediction and automated treatment planning in the external beam prostate radiotherapy following hydrogel injection,” Med. Phys. 40, 011709 (10pp.) (2013). 4B. Wu, D. Pang, P. Simari, R. Taylor, G. Sanguineti, and T. McNutt, “Using overlap volume histogram and IMRT plan data to guide and automate VMAT
Medical Physics, Vol. 42, No. 2, February 2015
1011 planning: A head-and-neck case study,” Med. Phys. 40, 021714 (7pp.) (2013). 5S. F. Petit, B. Wu, M. Kazhdan, A. Dekker, P. Simari, R. Kumar, R. Taylor, J. M. Herman, and T. McNutt, “Increased organ sparing using shape-based treatment plan optimization for intensity modulated radiation therapy of pancreatic adenocarcinoma,” Radiother. Oncol.: J. Eur. Soc. Ther. Radiol. Oncol. 102, 38–44 (2012). 6X. Zhu, Y. Ge, T. Li, D. Thongphiew, F. F. Yin, and Q. J. Wu, “A planning quality evaluation tool for prostate adaptive IMRT based on machine learning,” Med. Phys. 38, 719–726 (2011). 7L. Yuan, Y. Ge, W. R. Lee, F. F. Yin, J. P. Kirkpatrick, and Q. J. Wu, “Quantitative analysis of the factors which affect the interpatient organat-risk dose sparing variation in IMRT plans,” Med. Phys. 39, 6868–6878 (2012). 8J. Lian, L. Yuan, Y. Ge, B. S. Chera, D. P. Yoo, S. Chang, F. Yin, and Q. J. Wu, “Modeling the dosimetry of organ-at-risk in head and neck IMRT planning: An intertechnique and interinstitutional study,” Med. Phys. 40, 121704 (9pp.) (2013). 9L. M. Appenzoller, J. M. Michalski, W. L. Thorstad, S. Mutic, and K. L. Moore, “Predicting dose-volume histograms for organs-at-risk in IMRT planning,” Med. Phys. 39, 7446–7461 (2012). 10K. L. Moore, R. S. Brame, D. A. Low, and S. Mutic, “Experience-based quality control of clinical intensity-modulated radiotherapy planning,” Int. J. Radiat. Oncol., Biol., Phys. 81, 545–551 (2011). 11L. Yuan, Q. J. Wu, F. F. Yin, Y. Jiang, D. Yoo, and Y. Ge, “Incorporating single-side sparing in models for predicting parotid dose sparing in head and neck IMRT,” Med. Phys. 41, 021728 (8pp.) (2014). 12D. L. Craft, T. F. Halabi, H. A. Shih, and T. R. Bortfeld, “Approximating convex pareto surfaces in multiobjective radiotherapy planning,” Med. Phys. 33, 3399–3407 (2006). 13D. Craft, T. Halabi, and T. Bortfeld, “Exploration of tradeoffs in intensitymodulated radiotherapy,” Phys. Med. Biol. 50, 5857–5868 (2005). 14M. Monz, K. H. Kufer, T. R. Bortfeld, and C. Thieke, “Pareto navigation: Algorithmic foundation of interactive multi-criteria IMRT planning,” Phys. Med. Biol. 53, 985–998 (2008). 15D. L. Craft, T. S. Hong, H. A. Shih, and T. R. Bortfeld, “Improved planning time and plan quality through multicriteria optimization for intensitymodulated radiotherapy,” Int. J. Radiat. Oncol., Biol., Phys. 82, e83–e90 (2012). 16W. Hu, J. Wang, G. Li, J. Peng, S. Lu, and Z. Zhang, “Investigation of plan quality between RapidArc and IMRT for gastric cancer based on a novel beam angle and multicriteria optimization technique,” Radiother. Oncol.: J. Eur. Soc. Ther. Radiol. Oncol. 111, 144–147 (2014). 17C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Software 22, 469–483 (1996). 18C. Holdsworth, M. Kim, J. Liao, and M. Phillips, “The use of a multiobjective evolutionary algorithm to increase flexibility in the search for better IMRT plans,” Med. Phys. 39, 2261–2274 (2012). 19D. Craft and T. Bortfeld, “How many plans are needed in an IMRT multiobjective plan database?,” Phys. Med. Biol. 53, 2785–2796 (2008). 20K. Mittauer, B. Lu, G. Yan, D. Kahler, A. Gopal, R. Amdur, and C. Liu, “A study of IMRT planning parameters on planning efficiency, delivery efficiency, and plan quality,” Med. Phys. 40, 061704 (13pp.) (2013). 21M. K. Martel, W. M. Sahijdak, R. K. Ten Haken, M. L. Kessler, and A. T. Turrisi, “Fraction size and dose parameters related to the incidence of pericardial effusions,” Int. J. Radiat. Oncol., Biol., Phys. 40, 155–161 (1998). 22X. Wei, H. H. Liu, S. L. Tucker, S. Wang, R. Mohan, J. D. Cox, R. Komaki, and Z. Liao, “Risk factors for pericardial effusion in inoperable esophageal cancer patients treated with definitive chemoradiation therapy,” Int. J. Radiat. Oncol., Biol., Phys. 70, 707–714 (2008). 23L. B. Marks, S. M. Bentzen, J. O. Deasy, F. M. Kong, J. D. Bradley, I. S. Vogelius, I. El Naqa, J. L. Hubbs, J. V. Lebesque, R. D. Timmerman, M. K. Martel, and A. Jackson, “Radiation dose-volume effects in the lung,” Int. J. Radiat. Oncol., Biol., Phys. 76, S70–S76 (2010). 24J. O. Deasy, “Multiple local minima in radiotherapy optimization problems with dose-volume constraints,” Med. Phys. 24, 1157–1161 (1997). 25T. Halabi, D. Craft, and T. Bortfeld, “Dose-volume objectives in multicriteria optimization,” Phys. Med. Biol. 51, 3809–3818 (2006).
