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Sadowski B, Milewska K, Ginter J. Machine Learning Based Prediction of Gamma Passing Rate for VMAT Radiotherapy Plans. J Pers Med 2022; 12:jpm12122071. [PMID: 36556291 PMCID: PMC9781864 DOI: 10.3390/jpm12122071] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/30/2022] [Revised: 12/11/2022] [Accepted: 12/13/2022] [Indexed: 12/23/2022] Open
Abstract
The use of machine learning algorithms (ML) in radiotherapy is becoming increasingly popular. More and more groups are trying to apply ML in predicting the so-called gamma passing rate (GPR). Our team has developed a customized approach of using ML algorithms to predict global GPR for electronic portal imaging device (EPID) verification for dose different 2% and distance to agreement 2 mm criteria for VMAT dynamic plans. Plans will pass if the GPR is greater than 98%. The algorithm was learned and tested on anonymized clinical data from 13 months which resulted in more than 3000 treatment plans. The obtained results of GPR prediction are very interesting. Average specificity of the algorithm based on an ensemble of 50 decision tree regressors is 91.6% for our criteria. As a result, we can reduce the verification process by 50%. The novel approach described by our team can offer a new insight into the application of ML and neural networks in GPR prediction and dosimetry.
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Affiliation(s)
- Bartłomiej Sadowski
- Medical Physics Department, The Maria Sklodowska-Curie National Research Institute of Oncology, 02-781 Warsaw, Poland
| | - Karolina Milewska
- Biomedical Physics Division, Institute of Experimental Physics, Faculty of Physics, University of Warsaw, 02-093 Warsaw, Poland
| | - Józef Ginter
- Biomedical Physics Division, Institute of Experimental Physics, Faculty of Physics, University of Warsaw, 02-093 Warsaw, Poland
- Correspondence:
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2
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Eriksson O, Zhang T. Robust automated radiation therapy treatment planning using scenario-specific dose prediction and robust dose mimicking. Med Phys 2022; 49:3564-3573. [PMID: 35305023 PMCID: PMC9310773 DOI: 10.1002/mp.15622] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/21/2021] [Revised: 02/17/2022] [Accepted: 03/14/2022] [Indexed: 12/01/2022] Open
Abstract
Purpose We present a framework for robust automated treatment planning using machine learning, comprising scenario‐specific dose prediction and robust dose mimicking. Methods
The scenario dose prediction pipeline is divided into the prediction of nominal dose from input image and the prediction of scenario dose from nominal dose, each using a deep learning model with U‐net architecture. By using a specially developed dose–volume histogram–based loss function, the predicted scenario doses are ensured sufficient target coverage despite the possibility of the training data being non‐robust. Deliverable plans may then be created by solving a robust dose mimicking problem with the predictions as scenario‐specific reference doses. Results Numerical experiments are performed using a data set of 52 intensity‐modulated proton therapy plans for prostate patients. We show that the predicted scenario doses resemble their respective ground truth well, in particular while having target coverage comparable to that of the nominal scenario. The deliverable plans produced by the subsequent robust dose mimicking were showed to be robust against the same scenario set considered for prediction. Conclusions We demonstrate the feasibility and merits of the proposed methodology for incorporating robustness into automated treatment planning algorithms.
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Affiliation(s)
- Oskar Eriksson
- RaySearch Laboratories, Eugeniavägen 18, Solna, Stockholm, SE-171 64, Sweden
| | - Tianfang Zhang
- Department of Mathematics, KTH Royal Institute of Technology, Stockholm, SE-100 44, Sweden
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3
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Chen G, Cui J, Qian J, Zhu J, Zhao L, Luo B, Cui T, Zhong L, Yang F, Yang G, Zhao X, Zhou Y, Geng M, Sun J. Rapid Progress in Intelligent Radiotherapy and Future Implementation. Cancer Invest 2022; 40:425-436. [PMID: 35225723 DOI: 10.1080/07357907.2022.2044842] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/06/2023]
Abstract
Radiotherapy is one of the major approaches to cancer treatment. Artificial intelligence in radiotherapy (shortly, Intelligent radiotherapy) mainly involves big data, deep learning, extended reality, digital twin, radiomics, Internet plus and Internet of Things (IoT), which establish an automatic and intelligent network platform consisting of radiotherapy preparation, target volume delineation, treatment planning, radiation delivery, quality assurance (QA) and quality control (QC), prognosis judgment and post-treatment follow-up. Intelligent radiotherapy is an interdisciplinary frontier discipline in infancy. The review aims to summary the important implements of intelligent radiotherapy in various areas and put forward the future of unmanned radiotherapy center.
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Affiliation(s)
- Guangpeng Chen
- Department of Oncology, Xinqiao Hospital, Army Medical University, Chongqing 400037, P.R. China
| | - Jianxiong Cui
- Department of Oncology, Xinqiao Hospital, Army Medical University, Chongqing 400037, P.R. China.,Department of Oncology, Sichuan Provincial Crops Hospital of Chinese People's Armed Police Forces, Leshan 614000, Sichuan, P.R. China
| | - Jindong Qian
- Department of Oncology, Xinqiao Hospital, Army Medical University, Chongqing 400037, P.R. China
| | - Jianbo Zhu
- Department of Oncology, Xinqiao Hospital, Army Medical University, Chongqing 400037, P.R. China
| | - Lirong Zhao
- Department of Oncology, Xinqiao Hospital, Army Medical University, Chongqing 400037, P.R. China
| | - Bangyu Luo
- Department of Oncology, Xinqiao Hospital, Army Medical University, Chongqing 400037, P.R. China
| | - Tianxiang Cui
- Department of Oncology, Xinqiao Hospital, Army Medical University, Chongqing 400037, P.R. China
| | - Liangzhi Zhong
- Department of Oncology, Xinqiao Hospital, Army Medical University, Chongqing 400037, P.R. China
| | - Fan Yang
- Department of Oncology, Xinqiao Hospital, Army Medical University, Chongqing 400037, P.R. China
| | - Guangrong Yang
- Qijiang District People's Hospital, Chongqing 401420, P.R. China
| | - Xianlan Zhao
- Department of Oncology, Xinqiao Hospital, Army Medical University, Chongqing 400037, P.R. China
| | - Yibing Zhou
- Department of Oncology, Xinqiao Hospital, Army Medical University, Chongqing 400037, P.R. China
| | - Mingying Geng
- Department of Cancer Center, Daping Hospital, Army Medical University, Chongqing 400042, P.R. China
| | - Jianguo Sun
- Department of Oncology, Xinqiao Hospital, Army Medical University, Chongqing 400037, P.R. China
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4
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Zhang T, Bokrantz R, Olsson J. Probabilistic Pareto plan generation for semiautomated multicriteria radiation therapy treatment planning. Phys Med Biol 2022; 67. [PMID: 35061602 DOI: 10.1088/1361-6560/ac4da5] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2021] [Accepted: 01/21/2022] [Indexed: 11/12/2022]
Abstract
Objective.We propose a semiautomatic pipeline for radiation therapy treatment planning, combining ideas from machine learning-automated planning and multicriteria optimization (MCO).Approach.Using knowledge extracted from historically delivered plans, prediction models for spatial dose and dose statistics are trained and furthermore systematically modified to simulate changes in tradeoff priorities, creating a set of differently biased predictions. Based on the predictions, an MCO problem is subsequently constructed using previously developed dose mimicking functions, designed in such a way that its Pareto surface spans the range of clinically acceptable yet realistically achievable plans as exactly as possible. The result is an algorithm outputting a set of Pareto optimal plans, either fluence-based or machine parameter-based, which the user can navigate between in real time to make adjustments before a final deliverable plan is created.Main results.Numerical experiments performed on a dataset of prostate cancer patients show that one may often navigate to a better plan than one produced by a single-plan-output algorithm.Significance.We demonstrate the potential of merging MCO and a data-driven workflow to automate labor-intensive parts of the treatment planning process while maintaining a certain extent of manual control for the user.
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Affiliation(s)
- Tianfang Zhang
- Department of Mathematics, KTH Royal Institute of Technology, Stockholm SE-100 44, Sweden.,RaySearch Laboratories, Eugeniavägen 18, Solna, Stockholm SE-171 64, Sweden
| | - Rasmus Bokrantz
- RaySearch Laboratories, Eugeniavägen 18, Solna, Stockholm SE-171 64, Sweden
| | - Jimmy Olsson
- Department of Mathematics, KTH Royal Institute of Technology, Stockholm SE-100 44, Sweden
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5
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Artificial Intelligence in Radiotherapy and Patient Care. Artif Intell Med 2022. [DOI: 10.1007/978-3-030-64573-1_143] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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6
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Hybrid Grey Wolf Optimization-Based Gaussian Process Regression Model for Simulating Deterioration Behavior of Highway Tunnel Components. Processes (Basel) 2021. [DOI: 10.3390/pr10010036] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/17/2023] Open
Abstract
Highway tunnels are one of the paramount infrastructure systems that affect the welfare of communities. They are vulnerable to higher limits of deterioration, yet there are limited available funds for maintenance and rehabilitation. This state of circumstances entails the development of a deterioration model to forecast the performance condition behavior of critical tunnel elements. Accordingly, this research paper proposes an integrated deterioration prediction model for five highway tunnel elements, namely, cast-in-place tunnel liners, concrete interior walls, concrete portal, concrete ceiling slab, and concrete slab on grade. The developed deterioration model is envisioned in two fundamental components, which are model calibration and model assessment. In the first component, an integrated model of Gaussian process regression and a grey wolf optimization algorithm (GWO-GPR) is introduced for deterioration behavior prediction of highway tunnel elements. In this regard, the grey wolf optimizer is exploited to improve the prediction accuracies of the Gaussian process through optimal estimation of its hyper parameters and to automatically interpret the significant deterioration factors. The second component involves three tiers of performance evaluation comparison, statistical significance comparisons, and consolidated ranking to assess the prediction accuracies of the developed GWO-GPR model. In this regard, the developed model is validated against six widely acknowledged machine learning models, which are back-propagation artificial neural network, Elman neural network, cascade forward neural network, generalized regression neural network, support vector machines, and regression tree. Results demonstrate that the developed GWO-GPR model significantly outperformed other deterioration prediction models in the five tunnel elements. In cast-in-place tunnel liners it accomplished a mean absolute percentage error, mean absolute error, root mean square percentage error, root relative squared error, and relative absolute error of 1.65%, 0.018, 0.21%, 0.018, and 0.147, respectively. In this context, it was inferred that the developed GWO-GPR model managed to reduce the prediction errors of the back-propagation artificial neural network, Elman neural network, and support vector machines by 84.71%, 76.91%, and 69.6%, respectively. It can be concluded that the developed deterioration model can assist transportation agencies in creating timely and cost-efficient maintenance schedules of highway tunnels.
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Zhang T, Bokrantz R, Olsson J. Probabilistic feature extraction, dose statistic prediction and dose mimicking for automated radiation therapy treatment planning. Med Phys 2021; 48:4730-4742. [PMID: 34265105 DOI: 10.1002/mp.15098] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2021] [Revised: 07/07/2021] [Accepted: 07/08/2021] [Indexed: 11/10/2022] Open
Abstract
PURPOSE We propose a general framework for quantifying predictive uncertainties of dose-related quantities and leveraging this information in a dose mimicking problem in the context of automated radiation therapy treatment planning. METHODS A three-step pipeline, comprising feature extraction, dose statistic prediction and dose mimicking, is employed. In particular, the features are produced by a convolutional variational autoencoder and used as inputs in a previously developed nonparametric Bayesian statistical method, estimating the multivariate predictive distribution of a collection of predefined dose statistics. Specially developed objective functions are then used to construct a probabilistic dose mimicking problem based on the produced distributions, creating deliverable treatment plans. RESULTS The numerical experiments are performed using a dataset of 94 retrospective treatment plans of prostate cancer patients. We show that the features extracted by the variational autoencoder capture geometric information of substantial relevance to the dose statistic prediction problem and are related to dose statistics in a more regularized fashion than hand-crafted features. The estimated predictive distributions are reasonable and outperforms a non-input-dependent benchmark method, and the deliverable plans produced by the probabilistic dose mimicking agree better with their clinical counterparts than for a non-probabilistic formulation. CONCLUSIONS We demonstrate that prediction of dose-related quantities may be extended to include uncertainty estimation and that such probabilistic information may be leveraged in a dose mimicking problem. The treatment plans produced by the proposed pipeline resemble their original counterparts well, illustrating the merits of a holistic approach to automated planning based on probabilistic modeling.
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Affiliation(s)
- Tianfang Zhang
- Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden.,RaySearch Laboratories, Stockholm, Sweden
| | - Rasmus Bokrantz
- Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden
| | - Jimmy Olsson
- Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden
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Jiao SX, Wang ML, Chen LX, Liu XW. Evaluation of dose-volume histogram prediction for organ-at risk and planning target volume based on machine learning. Sci Rep 2021; 11:3117. [PMID: 33542427 PMCID: PMC7862493 DOI: 10.1038/s41598-021-82749-5] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/14/2020] [Accepted: 01/25/2021] [Indexed: 12/13/2022] Open
Abstract
The purpose of this work is to evaluate the performance of applying patient dosimetric information induced by individual uniform-intensity radiation fields in organ-at risk (OAR) dose-volume histogram (DVH) prediction, and extend to DVH prediction of planning target volume (PTV). Ninety nasopharyngeal cancer intensity-modulated radiation therapy (IMRT) plans and 60 rectal cancer volumetric modulated arc therapy (VMAT) plans were employed in this study. Of these, 20 nasopharyngeal cancer cases and 15 rectal cancer cases were randomly selected as the testing data. The DVH prediction was performed using two methods. One method applied the individual dose-volume histograms (IDVHs) induced by a series of fields with uniform-intensity irradiation and the other method applied the distance-to-target histogram and the conformal-plan-dose-volume histogram (DTH + CPDVH). The determination coefficient R2 and mean absolute error (MAE) were used to evaluate DVH prediction accuracy. The PTV DVH prediction was performed using the IDVHs. The PTV dose coverage was evaluated using D98, D95, D1 and uniformity index (UI). The OAR dose was compared using the maximum dose, V30 and V40. The significance of the results was examined with the Wilcoxon signed rank test. For PTV DVH prediction using IDVHs, the clinical plan and IDVHs prediction method achieved mean UI values of 1.07 and 1.06 for nasopharyngeal cancer, and 1.04 and 1.05 for rectal cancer, respectively. No significant difference was found between the clinical plan results and predicted results using the IDVHs method in achieving PTV dose coverage (D98,D95,D1 and UI) for both nasopharyngeal cancer and rectal cancer (p-values ≥ 0.052). For OAR DVH prediction, no significant difference was found between the IDVHs and DTH + CPDVH methods for the R2, MAE, the maximum dose, V30 and V40 (p-values ≥ 0.087 for all OARs). This work evaluates the performance of dosimetric information of several individual fields with uniform-intensity radiation for DVH prediction, and extends its application to PTV DVH prediction. The results indicated that the IDVHs method is comparable to the DTH + CPDVH method in accurately predicting the OAR DVH. The IDVHs method quantified the input features of the PTV and showed reliable PTV DVH prediction, which is helpful for plan quality evaluation and plan generation.
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Affiliation(s)
- Sheng Xiu Jiao
- School of Physics, Sun Yat-Sen University, 135 Xin Gang Road West, Guangzhou, 510275, China
| | - Ming Li Wang
- State Key Laboratory of Oncology in South China, Sun Yat-Sen University Cancer Center, 651 Dong Feng Road East, Guangzhou, 510060, China
| | - Li Xin Chen
- State Key Laboratory of Oncology in South China, Sun Yat-Sen University Cancer Center, 651 Dong Feng Road East, Guangzhou, 510060, China.
| | - Xiao-Wei Liu
- School of Physics, Sun Yat-Sen University, 135 Xin Gang Road West, Guangzhou, 510275, China.
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9
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Artificial Intelligence in Radiotherapy and Patient Care. Artif Intell Med 2021. [DOI: 10.1007/978-3-030-58080-3_143-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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10
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Chow JC. Computer method and modeling: Medical biophysics applications in cancer therapy, medical imaging and drug delivery. AIMS BIOPHYSICS 2021. [DOI: 10.3934/biophy.2021017] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
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11
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Shin DS, Kim KH, Kang SW, Kang SH, Kim JS, Kim TH, Kim DS, Cho W, Suh TS, Chung JB. Dose Super-Resolution in Prostate Volumetric Modulated Arc Therapy Using Cascaded Deep Learning Networks. Front Oncol 2020; 10:593381. [PMID: 33304852 PMCID: PMC7701297 DOI: 10.3389/fonc.2020.593381] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2020] [Accepted: 10/14/2020] [Indexed: 11/13/2022] Open
Abstract
Purpose This study proposes a cascaded network model for generating high-resolution doses (i.e., a 1 mm grid) from low-resolution doses (i.e., ≥3 mm grids) with reduced computation time. Methods Using the anisotropic analytical algorithm with three grid sizes (1, 3, and 5 mm) and the Acuros XB algorithm with two grid sizes (1 and 3 mm), dose distributions were calculated for volumetric modulated arc therapy plans for 73 prostate cancer patients. Our cascaded network model consisted of a hierarchically densely connected U-net (HD U-net) and a residual dense network (RDN), which were trained separately following a two-dimensional slice-by-slice procedure. The first network (HD U-net) predicted the downsampled high-resolution dose (generated through bicubic downsampling of the baseline high-resolution dose) using the low-resolution dose; subsequently, the second network (RDN) predicted the high-resolution dose from the output of the first network. Further, the predicted high-resolution dose was converted to its absolute value. We quantified the network performance using the spatial/dosimetric parameters (dice similarity coefficient, mean dose, maximum dose, minimum dose, homogeneity index, conformity index, and V95%, V70%, V50%, and V30%) for the low-resolution and predicted high-resolution doses relative to the baseline high-resolution dose. Gamma analysis (between the baseline dose and the low-resolution dose/predicted high-resolution dose) was performed with a 2%/2 mm criterion and 10% threshold. Results The average computation time to predict a high-resolution axial dose plane was <0.02 s. The dice similarity coefficient values for the predicted doses were closer to 1 when compared to those for the low-resolution doses. Most of the dosimetric parameters for the predicted doses agreed more closely with those for the baseline than for the low-resolution doses. In most of the parameters, no significant differences (p-value of >0.05) between the baseline and predicted doses were observed. The gamma passing rates for the predicted high-resolution does were higher than those for the low-resolution doses. Conclusion The proposed model accurately predicted high-resolution doses for the same dose calculation algorithm. Our model uses only dose data as the input without additional data, which provides advantages of convenience to user over other dose super-resolution methods.
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Affiliation(s)
- Dong-Seok Shin
- Department of Biomedical Engineering, Department of Biomedicine and Health Sciences, College of Medicine, The Catholic University of Korea, Seoul, South Korea.,Research Institute of Biomedical Engineering, College of Medicine, The Catholic University of Korea, Seoul, South Korea
| | - Kyeong-Hyeon Kim
- Department of Biomedical Engineering, Department of Biomedicine and Health Sciences, College of Medicine, The Catholic University of Korea, Seoul, South Korea.,Research Institute of Biomedical Engineering, College of Medicine, The Catholic University of Korea, Seoul, South Korea
| | - Sang-Won Kang
- Department of Biomedical Engineering, Department of Biomedicine and Health Sciences, College of Medicine, The Catholic University of Korea, Seoul, South Korea.,Research Institute of Biomedical Engineering, College of Medicine, The Catholic University of Korea, Seoul, South Korea
| | - Seong-Hee Kang
- Department of Radiation Oncology, Seoul National University Bundang Hospital, Bundang, South Korea
| | - Jae-Sung Kim
- Department of Radiation Oncology, Seoul National University Bundang Hospital, Bundang, South Korea
| | - Tae-Ho Kim
- Proton Therapy Center, National Cancer Center, Goyang, South Korea
| | - Dong-Su Kim
- Korea Atomic Energy Research Institute, Daejeon, South Korea
| | - Woong Cho
- Department of Radiation Oncology, Seoul National University Boramae Medical Center, Seoul, South Korea
| | - Tae Suk Suh
- Department of Biomedical Engineering, Department of Biomedicine and Health Sciences, College of Medicine, The Catholic University of Korea, Seoul, South Korea.,Research Institute of Biomedical Engineering, College of Medicine, The Catholic University of Korea, Seoul, South Korea
| | - Jin-Beom Chung
- Department of Radiation Oncology, Seoul National University Bundang Hospital, Bundang, South Korea
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Zhang T, Bokrantz R, Olsson J. Direct optimization of dose-volume histogram metrics in radiation therapy treatment planning. Biomed Phys Eng Express 2020; 6. [PMID: 34035188 DOI: 10.1088/2057-1976/abb5ea] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2020] [Accepted: 09/07/2020] [Indexed: 12/20/2022]
Abstract
We present a method of directly optimizing on deviations in clinical goal values in radiation therapy treatment planning. Using a new mathematical framework in which metrics derived from the dose-volume histogram are regarded as functionals of an auxiliary random variable, we are able to obtain volume-at-dose and dose-at-volume as infinitely differentiable functions of the dose distribution with easily evaluable function values and gradients. Motivated by the connection to risk measures in finance, which is formalized in this framework, we also derive closed-form formulas for mean-tail-dose and demonstrate its capability of reducing extreme dose values in tail distributions. Numerical experiments performed on a prostate and a head-and-neck patient case show that the direct optimization of dose-volume histogram metrics produced marginally better results than or outperformed conventional planning objectives in terms of clinical goal fulfilment, control of low- and high-dose tails of target distributions and general plan quality defined by a pre-specified evaluation measure. The proposed framework eliminates the disconnect between optimization functions and evaluation metrics and may thus reduce the need for repetitive user interaction associated with conventional treatment planning. The method also has the potential of enhancing plan optimization in other settings such as multicriteria optimization and automated treatment planning.
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Affiliation(s)
- Tianfang Zhang
- Mathematical Statistics, Department of Mathematics, KTH Royal Institute of Technology, Stockholm SE-100 44, Sweden.,RaySearch Laboratories, Sveavägen 44, Stockholm SE-103 65, Sweden
| | - Rasmus Bokrantz
- RaySearch Laboratories, Sveavägen 44, Stockholm SE-103 65, Sweden
| | - Jimmy Olsson
- Mathematical Statistics, Department of Mathematics, KTH Royal Institute of Technology, Stockholm SE-100 44, Sweden
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