Robust Optimization: Concepts and Applications

Robust optimization is an emerging area in research that allows addressing different optimization problems and specifically industrial optimization problems where there is a degree of uncertainty in some of the variables involved. There are several ways to apply robust optimization and the choice of form is typical of the problem that is being solved. In this paper, the basic concepts of robust optimization are developed, the different types of robustness are defined in detail, the main areas in which it has been applied are described and finally, the future lines of research that appear in this area are included

Intensity Modulated Proton Therapy Optimization Under Uncertainty: Field Misalignment and Internal Organ Motion

Intensity modulated proton therapy (IMPT) is one of the most advanced forms of radiation therapy, which can deliver a highly conformal dose to the tumor while sparing the dose in healthy tissues. Compared to conventional photon-based radiation therapy, IMPT is more flexible in delivering radiation dose according to different tumor shapes. However, this flexibility also makes the optimization problems in IMPT harder to solve, e.g., it requires larger memory to store data and longer computational time. Furthermore, proton beams are very sensitive to different uncertainties, such as setup uncertainty, range uncertainty and internal organ motion. These uncertainties can greatly impact the quality of clinical treatment. Therefore, this dissertation aims to investigate different optimization methods for treatment planning and to handle a variety of uncertainties in IMPT. First, to solve the fluence map optimization (FMO) problem in IMPT, we propose a method to formulate the FMO problem into a molecular dynamics model. So that, the FMO problem can be optimized according classical dynamics system. This method combines the advantages of gradient-based algorithms and heuristic search algorithms. Next, we develop and validate a robust optimization method for IMPT treatment plans with multi-isocenter large fields to overcome the dose inhomogeneity problem caused by the setup misalignment in field junctions. Numerical results show that the robust optimized IMPT plans create a low gradient field radiation dose in the junction regions, which can minimize the impact from misalignment uncertainty. Compare to conventional techniques, the robust optimization method leads the whole treatment much more efficient. Lastly, we focus on a two-stage method to solve the beam angle optimization (BAO) problem in IMPT with internal organ motion uncertainty. In the first stage, a $p$-median algorithm is developed for beam angle clustering. In the second stage, a bi-level search algorithm is used to find the final beam angle set for the treatment. Furthermore, Support vector machine (SVM) is used for beam angle classification to reduce the search space and the 4D-CT information is incorporated to handle the internal organ motion uncertainty. Results show that the two-stage BAO method consistently finds a high-quality solution in a short time.Industrial Engineering, Department o

Modelling small block aperture in an in-house developed GPU-accelerated Monte Carlo-based dose engine for pencil beam scanning proton therapy

Purpose: To enhance an in-house graphic-processing-unit (GPU) accelerated virtual particle (VP)-based Monte Carlo (MC) proton dose engine (VPMC) to model aperture blocks in both dose calculation and optimization for pencil beam scanning proton therapy (PBSPT)-based stereotactic radiosurgery (SRS). Methods and Materials: A block aperture module was integrated into VPMC. VPMC was validated by an opensource code, MCsquare, in eight water phantom simulations with 3cm thick brass apertures: four were with aperture openings of 1, 2, 3, and 4cm without a range shifter, while the other four were with same aperture opening configurations with a range shifter of 45mm water equivalent thickness. VPMC was benchmarked with MCsquare and RayStation MC for 10 patients with small targets (average volume 8.4 cc). Finally, 3 patients were selected for robust optimization with aperture blocks using VPMC. Results: In the water phantoms, 3D gamma passing rate (2%/2mm/10%) between VPMC and MCsquare were 99.71$\pm$0.23%. In the patient geometries, 3D gamma passing rates (3%/2mm/10%) between VPMC/MCsquare and RayStation MC were 97.79$\pm$2.21%/97.78$\pm$1.97%, respectively. The calculation time was greatly decreased from 112.45$\pm$114.08 seconds (MCsquare) to 8.20$\pm$6.42 seconds (VPMC), both having statistical uncertainties of about 0.5%. The robustly optimized plans met all the dose-volume-constraints (DVCs) for the targets and OARs per our institutional protocols. The mean calculation time for 13 influence matrices in robust optimization by VPMC was 41.6 seconds. Conclusion: VPMC has been successfully enhanced to model aperture blocks in dose calculation and optimization for the PBSPT-based SRS.Comment: 3 tables, 3 figure

Solving trajectory optimization problems in the presence of probabilistic constraints

The objective of this paper is to present an approximation-based strategy for solving the problem of nonlinear trajectory optimization with the consideration of probabilistic constraints. The proposed method defines a smooth and differentiable function to replace probabilistic constraints by the deterministic ones, thereby converting the chance-constrained trajectory optimization model into a parametric nonlinear programming model. In addition, it is proved that the approximation function and the corresponding approximation set will converge to that of the original problem. Furthermore, the optimal solution of the approximated model is ensured to converge to the optimal solution of the original problem. Numerical results, obtained from a new chance-constrained space vehicle trajectory optimization model and a 3-D unmanned vehicle trajectory smoothing problem, verify the feasibility and effectiveness of the proposed approach. Comparative studies were also carried out to show the proposed design can yield good performance and outperform other typical chance-constrained optimization techniques investigated in this paper

Nature-inspired Methods for Stochastic, Robust and Dynamic Optimization

Nature-inspired algorithms have a great popularity in the current scientific community, being the focused scope of many research contributions in the literature year by year. The rationale behind the acquired momentum by this broad family of methods lies on their outstanding performance evinced in hundreds of research fields and problem instances. This book gravitates on the development of nature-inspired methods and their application to stochastic, dynamic and robust optimization. Topics covered by this book include the design and development of evolutionary algorithms, bio-inspired metaheuristics, or memetic methods, with empirical, innovative findings when used in different subfields of mathematical optimization, such as stochastic, dynamic, multimodal and robust optimization, as well as noisy optimization and dynamic and constraint satisfaction problems

Optimization Approaches for Intensity Modulated Proton Therapy Treatment Planning

Radiation therapy is a critical modality in the field of oncology. The primary goal of radiation therapy is to destroy or control the growth of cancerous cells while minimizing damage to healthy tissues. Intensity Modulated Proton Therapy (IMPT) is a type of radiation therapy that utilizes protons to irradiate the tumor. The unique physical properties of protons enable precise control over the radiation dose distribution within the tumor and more effective sparing of healthy tissues. Typically, radiation therapy treatment planning is posed as a multi-criteria optimization problem, whereby the challenge is finding the best possible treatment plan. In this study, we formulate and compare two optimization approaches for IMPT treatment planning. We first explore a linear programming (LP) approach, followed by a moment-based approach where we incorporate the dose-volume requirements into the fluence map optimization (FMO) problem. The evaluation of these models is conducted using anonymized patient data corresponding to a lung cancer case, with a focus on generating a good-quality initial plan that is amenable to further refinement. The moment-based approach has a drawback in terms of its high memory usage. To mitigate this limitation, we explore several sparsification strategies aimed at reducing memory requirements. Employing an aggressive sparsification method, we demonstrate that the moment-based approach outperforms the LP model in dosimetric outcomes and computational run-time. We highlight a trade-off between the quality of the treatment plan and computational run-time when utilizing different sparcification strategies for the moment-based approach. By adopting a less strict sparsification method, we anticipate achieving higher-quality treatment plans at the expense of increased computational run-time

Risk-adapted Optimization in Intensity Modulated Proton Therapy (IMPT)

Due to the pronounced dose gradients generated by proton beams, proton treatment plans can be very sensitive to treatment uncertainties. However in IMPT many different solutions of the inverse problem exist which result in dose distributions of comparable quality. This thesis investigates methods to exploit this degeneracy of solutions to generate treatment plans which are robust to uncertainties. An investigation of the optimization algorithm in the used IMPT software KonRad revealed that the standard optimization algorithm is not capable to find the optimal treatment plan in a reasonable time. Thus several additional optimization algorithms were implemented and tested in KonRad. The best results were achieved using the L-BFGS algorithm. To rate the sensitivity to uncertainties of individual beamlet dose distributions the heterogeneity number H was developed. It was shown that H correlates with the dose calculation error introduced by the commonly employed pencil beam algorithm as well as with the sensitivity to setup errors of individual beamlets. Finally, the "worst case optimization" was developed to account for uncertainties during the inverse treatment planning. This technique was applied to account for range uncertainties, setup errors and a combination of both uncertainties. The treatment plans generated with this new method are much more robust to the respective uncertainties as conventional IMPT and even as conventional single-field proton plans