Automated Volumetric Modulated Arc Therapy (VMAT) using Expedited Constrained Hierarchical Optimization (ECHO)
VMAT is a widely used form of radiotherapy that delivers radiation continuously as the gantry rotates around the patient. Unlike Intensity-Modulated Radiation Therapy (IMRT), which uses a limited number of fixed beams (typically 7-9), VMAT dynamically adjusts both the radiation intensity and the beam shape during the rotation (Short IMRT YouTube Video, Short VMAT YouTube Video). VMAT allows for shorter treatment times, making VMAT a popular choice in clinical settings. However, planning VMAT treatments is more complex due to the large and non-convex optimization problems involved.
In radiotherapy, machine settings must be customized for each cancer patient, requiring extensive iterative manual tuning of parameters. This process is: 1) time-consuming, 2) labor-intensive, and 3) the plan quality is heavily dependent on the skill and experience of the planner. Our team has developed and clinically deployed an in-house automated treatment planning system called ECHO (Expedited Constrained Hierarchical Optimization). ECHO (for both IMRT and VMAT) is seamlessly integrated into the FDA-approved commercial treatment planning system and is used in MSK daily clinical routine, with more than 10,000 patients treated to date (as of May 2024). ECHO was nominated as a finalist for the prestigious INFORMS Franz Edelman award in 2021 (YouTube | Slides | Paper | Podcast).
Since VMAT continuously delivers radiation as the gantry rotates around the patient, it requires the direct optimization of machine parameters (e.g., gantry and leaf motions). This is in contrast to the IMRT technique, where an effective convex optimization approach exists (i.e., a two-phase heuristic where Phase 1 optimizes the beamlet intensities and Phase 2 optimizes leaf motions). While there is a simple linear relationship between beamlet intensities and the radiation dose delivered to the patient (
With some assumptions, one can formulate the VMAT problem as a Mixed Integer Programming (MIP) problem and solve it to global optimality (see VMAT Global Jupyter Notebook). Although this approach can provide a benchmark for developing more efficient algorithms, it is computationally extensive and might take days to solve for each patient, making it impractical.
In our papers (PMB’2021, PMB’2023), we proposed an algorithm to solve the non-convex VMAT problem by breaking it down into a sequence of convex problems, hence the name sequential convex programming (see ECHO_VMAT_Jupyter_Notebook). The key idea is to derive a convex approximation at each iteration by limiting the leaf motions, defining the search space, and merging neighboring beamlets within this search space to create the approximation (see the figure below).
(Top-left figure) The grey and white areas represent closed and open beamlets, respectively. The regions framed by red dashed lines indicate the search space within which the leaves can move (the search space consists of two beamlets in this example). (Top-right figure) The white beamlets outside the search space will remain open in the next iteration with the same intensity. These can be merged into a single beamlet called an interior beamlet (green area), whose intensity (η) will be optimized as a decision variable. For each pair of neighboring beamlets within the search space (i.e., inside the red frame), we assume they contribute equally (this is our approximation). These pairs can be merged into a boundary beamlet (blue area), whose intensity will be optimized as a decision variable while ensuring it is less than η. If the optimal intensity of a boundary beamlet becomes zero, we will close that beamlet in the next iteration. If it becomes η, we will open it, and if the intensity is in between, we will partially open it. The algorithm also iteratively changes the search space (see our paper for more details).
One of the challenges in radiotherapy is the inherent conflict between tumor irradiation and the sparing of nearby healthy tissue. This results in optimization problems with multiple objectives (also known as multi-objective or multiple-criteria optimization). This means, in the following general form of a radiotherapy optimization problem, the objective function
Subject to
Multi-objective optimization is a well-studied subfield within operations research and mathematical optimization communities, and many techniques exist in the literature to address these problems. One such technique is the weighted-sum method, which assigns weights to the objective functions and optimizes their weighted sum: (i.e.,
• Step 1: Focus on delivering the radiation dose as close as possible to the prescription dose in the tumor voxels while ensuring all critical clinical criteria are met by expressing them as hard constraints.
• Step 2: Minimize radiation to the nearby healthy tissues while preserving the results of the first step.
For more details see ECHO_VMAT_Jupyter_Notebook and our papers (PMB’2021, PMB’2023).
Important Note: Multi criteria and non-convexity are two separate challenges in radiotherapy planning. As demonstrated in our VMAT_Deep_Learning_Jupyter_Notebook, deep learning can be used to address the multi-criteria challenge by learning the mapping between patient geometry and dose distribution. This approach allows for predicting the 3D dose distribution for a new patient. However, deep learning does not address the non-convexity challenge and the sequential convex programming can be used to convert the predicted dose into a deliverable treatment plan.
To evaluate the performance of the algorithm, we compared its results to the global optimal solution obtained by solving the computationally intensive Mixed Integer Programming (MIP) problem on a small prostate case.
Figure Explanation: The above figure illustrates that our algorithm can identify optimal solutions that are close to the global optimal solution.
We have also retrospectively compared the automated VMAT plans with the manually generated VMAT plans used for patient treatment for 60 patients across three different disease sites (paraspinal, oligometastatic, lung). Below are the results for 20 lung patients.
Figure Explanation: The above figure demonstrates that the automatically generated VMAT plans are superior to or comparable with the manually generated plans in terms of tumor coverage and healthy tissue sparing.
@article{dursun2023automated,
title={Automated VMAT treatment planning using sequential convex programming: algorithm development and clinical implementation},
author={Dursun, P{\i}nar and Hong, Linda and Jhanwar, Gourav and Huang, Qijie and Zhou, Ying and Yang, Jie and Pham, Hai and Cervino, Laura and Moran, Jean M and Deasy, Joseph O and others},
journal={Physics in Medicine \& Biology},
volume={68},
number={15},
pages={155006},
year={2023},
publisher={IOP Publishing}
}
@article{dursun2021solving,
title={Solving the volumetric modulated arc therapy (VMAT) problem using a sequential convex programming method},
author={Dursun, P{\i}nar and Zarepisheh, Masoud and Jhanwar, Gourav and Deasy, Joseph O},
journal={Physics in Medicine \& Biology},
volume={66},
number={8},
pages={085004},
year={2021},
publisher={IOP Publishing}
}
