Fluence Map Optimization Problem Research Articles

A conjugate gradient-assisted multi-objective evolutionary algorithm for fluence map optimization in radiotherapy treatment

Intensity-modulated radiotherapy (IMRT) is one of the most applied techniques for cancer radiotherapy treatment. The fluence map optimization is an essential part of IMRT plan designing, which has a significant impact on the radiotherapy treatment effect. In fact, the treatment planing of IMRT is an inverse multi-objective optimization problem. Existing approaches of solving the fluence map optimization problem (FMOP) obtain a satisfied treatment plan via trying different coupling weights, the optimization process needs to be conducted many times and the coupling weight setting is completely based on the experience of a radiation physicist. For fast obtaining diverse high-quality radiotherapy plans, this paper formulates the FMOP into a three-objective optimization problem, and proposes a conjugate gradient-assisted multi-objective evolutionary algorithm (CG-MOEA) to solve it. The proposed algorithm does not need to set the coupling weights and can produce the diverse radiotherapy plans within a single run. Moreover, the convergence speed is further accelerated by an adaptive local search strategy based on the conjugate-gradient method. Compared with five state-of-the-art multi-objective evolutionary algorithms (MOEAs), the proposed CG-MOEA can obtain the best hypervolume (HV) values and dose–volume histogram (DVH) performance on five clinical cases in cancer radiotherapy. Moreover, the proposed algorithm not only obtains the more optimal solution than traditional method used to solve the FMOP, but also can find diverse Pareto solution set which can be provided to radiation physicist to select the best treatment plan. The proposed algorithm outperforms dose-volume histogram state-of-the-art multi-objective evolutionary algorithms and traditional method for FMOP on five clinical cases in cancer radiotherapy.

A unified approach for optimal dose delivery and trajectory optimization for the treatment of prostate cancer

For a Fluence map optimization problem, we present a unique iterative reweighting algorithm-based method to achieve optimality. The purpose of planning is to determine a dose distribution in such a manner that it should cover the maximum part of the target without affecting the functionality of organ at risk. We suggest a unique methodology to solve dose-volume bounds while maintaining their non-convexity, as compared to earlier approaches that dedicated to convex estimation. The suggested approach is cooperative to competent procedures centered on partial minimization and certainly adjusts to tackle maximum-dose bounds. Fluence mapping is used for the inverse planning which determines the intensity profile of each beam. For the analysis, 09 equally separated beams with dose-volume limits on the organ at risk have used to treat malignancies in the prostate. Cumulative dose-volume histogram is used for the treatment plans quality measurement. We considered 4-different samples from CORT dataset named prostate, Liver, TG119 phantom, and Head & Neck to make a judgment about the generalizability of the performance of the proposed algorithm, and after getting this we finally compared our result with other method using prostate sample to validate our approach. After comparing with other methods centered on dose volume limits we found that in our approach maximum dose can deliver within minimum time across the planned target. For the optimization of trajectory we kept, the radiation delivery rate fixed to its extreme level and employed the independence property of each single leaf pair equally. By making this provision for all total T delivery times, trade-off between distribution time and fluency mapping quality is produced.

Beam angle optimization in IMRT: are we really optimizing what matters?

AbstractIntensity‐modulated radiation therapy (IMRT) is a modern radiotherapy modality that uses a multileaf collimator to enable the irradiation of the patient with nonuniform maps of radiation from a set of distinct beam irradiation directions. The aim of IMRT is to eradicate all cancerous cells by irradiating the tumor with a prescribed dose while simultaneously sparing, as much as possible, the neighboring tissues and organs. The optimal choice of beam irradiation directions—beam angle optimization (BAO)—can play an important role in IMRT treatment planning by improving organ sparing and tumor coverage, increasing the treatment plan quality. Typically, the BAO search is guided by the optimal value of the fluence map optimization (FMO)—the problem of obtaining the most appropriate radiation intensities for each beam direction. In this paper, a new score to guide the BAO search is introduced and embedded in a parallel multistart derivative‐free optimization framework that is detailed for the extremely challenging continuous BAO problem. For the set of 10 clinical nasopharyngeal tumor cases considered, treatment plans obtained for optimized beam directions clearly outperform the benchmark treatment plans obtained considering equidistant beam directions typically used in clinical practice. Furthermore, treatment plans obtained considering the proposed score clearly improve the quality of the plans resulting from the use of the optimal value of the FMO problem to guide the BAO search.

Fraction-variant beam orientation optimization for non-coplanar IMRT

Conventional beam orientation optimization (BOO) algorithms for IMRT assume that the same set of beam angles is used for all treatment fractions. In this paper we present a BOO formulation based on group sparsity that simultaneously optimizes non-coplanar beam angles for all fractions, yielding a fraction-variant (FV) treatment plan. Beam angles are selected by solving a multi-fraction fluence map optimization problem involving 500-700 candidate beams per fraction, with an additional group sparsity term that encourages most candidate beams to be inactive. The optimization problem is solved using the fast iterative shrinkage-thresholding algorithm. Our FV BOO algorithm is used to create five-fraction treatment plans for digital phantom, prostate, and lung cases as well as a 30-fraction plan for a head and neck case. A homogeneous PTV dose coverage is maintained in all fractions. The treatment plans are compared with fraction-invariant plans that use a fixed set of beam angles for all fractions. The FV plans reduced OAR mean dose and D2 values on average by 3.3% and 3.8% of the prescription dose, respectively. Notably, mean OAR dose was reduced by 14.3% of prescription dose (rectum), 11.6% (penile bulb), 10.7% (seminal vesicle), 5.5% (right femur), 3.5% (bladder), 4.0% (normal left lung), 15.5% (cochleas), and 5.2% (chiasm). D2 was reduced by 14.9% of prescription dose (right femur), 8.2% (penile bulb), 12.7% (proximal bronchus), 4.1% (normal left lung), 15.2% (cochleas), 10.1% (orbits), 9.1% (chiasm), 8.7% (brainstem), and 7.1% (parotids). Meanwhile, PTV homogeneity defined as D95/D5 improved from .92 to .95 (digital phantom), from .95 to .98 (prostate case), and from .94 to .97 (lung case), and remained constant for the head and neck case. Moreover, the FV plans are dosimetrically similar to conventional plans that use twice as many beams per fraction. Thus, FV BOO offers the potential to reduce delivery time for non-coplanar IMRT.

Computerized triplet beam orientation optimization for MRI-guided Co-60 radiotherapy.

Magnetic resonance imaging (MRI)-guided Co-60 provides daily and intrafractional MRI soft tissue imaging for improved target and critical organ tracking. To increase delivery efficiency, the system uses three Co-60 sources at 120° apart, allowing up to 600 cGy combined dose rate at isocenter. Despite the potential tripling in output, creating a delivery plan that uses all three sources is considerably unintuitive. Here, the authors computerize the triplet orientation optimization using column generation, an approach that was demonstrated effective in integrated beam orientation and fluence optimization for noncoplanar therapies. To achieve a better plan quality without increasing the treatment time, the authors then solve a fluence map optimization (FMO) problem while regularizing the fluence maps to reduce the number of deliverable MLC segments. Three patients-one prostate, one lung, and one head and neck boost plan (H&NBoost)-were evaluated in this study. For each patient, the beamlet doses were calculated using Monte Carlo, under a 0.35 T magnetic field, for 180 equally spaced coplanar beams grouped into 60 triplets. The beamlet size is 1.05 × 0.5 cm determined by the MLC leaf thickness and step size. The triplets were selected using the column generation algorithm. The FMO problem was formulated using an L2-norm dose fidelity term and an L1-norm anisotropic total variation regularization term, which allows controlling the number of MLC segments, and hence the treatment time, with minimal degradation to the dose. The authors' Fluence Regularization and Optimized Selection of Triplets (FROST) plans were compared against the clinical treatment plans (CLNs) produced by an experienced dosimetrist. PTV homogeneity, max dose, mean dose, D95, D98, and D99 were evaluated. OAR max and mean doses, as well as R50, defined as the ratio of the 50% isodose volume over the planning target volume were investigated. The mean PTV D95, D98, and D99 differ by +0.04%, +0.07%, and +0.25% of the prescription dose between planning methods. The mean PTV homogeneity was virtually same with values at 0.8788 (FROST) and 0.8812 (CLN). R50 decreased by 0.67 comparing FROST to CLN. On average, FROST reduced Dmax and Dmean of OARs by 7.30% and 6.08% of the prescription dose, respectively. The manual CLN planning processes required numerous trial and error runs. The FROST plans on the other hand required minimal human intervention. Efficient delivery of MRI-guided Co-60 therapy needs the output of multiple sources yet suffers from unintuitive and laborious manual beam selection processes. Computerized triplet orientation optimization improves both planning efficiency and plan dosimetry. The novel fluence map regularization provides additional controls over the number of MLC segments and treatment time.

TH‐AB‐BRA‐02: Automated Triplet Beam Orientation Optimization for MRI‐Guided Co‐60 Radiotherapy

Purpose:MRI guided Co‐60 provides daily and intrafractional MRI soft tissue imaging for improved target tracking and adaptive radiotherapy. To remedy the low output limitation, the system uses three Co‐60 sources at 120° apart, but using all three sources in planning is considerably unintuitive. We automate the beam orientation optimization using column generation, and then solve a novel fluence map optimization (FMO) problem while regularizing the number of MLC segments.Methods:Three patients—1 prostate (PRT), 1 lung (LNG), and 1 head‐and‐neck boost plan (H&NBoost)—were evaluated. The beamlet dose for 180 equally spaced coplanar beams under 0.35 T magnetic field was calculated using Monte Carlo. The 60 triplets were selected utilizing the column generation algorithm. The FMO problem was formulated using an L2‐norm minimization with anisotropic total variation (TV) regularization term, which allows for control over the number of MLC segments. Our Fluence Regularized and Optimized Selection of Triplets (FROST) plans were compared against the clinical treatment plans (CLN) produced by an experienced dosimetrist.Results:The mean PTV D95, D98, and D99 differ by −0.02%, +0.12%, and +0.44% of the prescription dose between planning methods, showing same PTV dose coverage. The mean PTV homogeneity (D95/D5) was at 0.9360 (FROST) and 0.9356 (CLN). R50 decreased by 0.07 with FROST. On average, FROST reduced Dmax and Dmean of OARs by 6.56% and 5.86% of the prescription dose. The manual CLN planning required iterative trial and error runs which is very time consuming, while FROST required minimal human intervention.Conclusions:MRI guided Co‐60 therapy needs the output of all sources yet suffers from unintuitive and laborious manual beam selection processes. Automated triplet orientation optimization is shown essential to overcome the difficulty and improves the dosimetry. A novel FMO with regularization provides additional controls over the number of MLC segments and treatment time.Varian Medical Systems; NIH grant R01CA188300; NIH grant R43CA183390.

TH‐EF‐BRB‐05: 4pi Non‐Coplanar IMRT Beam Angle Selection by Convex Optimization with Group Sparsity Penalty

Purpose:Integrated beam orientation and fluence map optimization is expected to be the foundation of robust automated planning but existing heuristic methods do not promise global optimality. We aim to develop a new method for beam angle selection in 4π non‐coplanar IMRT systems based on solving (globally) a single convex optimization problem, and to demonstrate the effectiveness of the method by comparison with a state of the art column generation method for 4π beam angle selection.Methods:The beam angle selection problem is formulated as a large scale convex fluence map optimization problem with an additional group sparsity term that encourages most candidate beams to be inactive. The optimization problem is solved using an accelerated first‐order method, the Fast Iterative Shrinkage‐Thresholding Algorithm (FISTA). The beam angle selection and fluence map optimization algorithm is used to create non‐coplanar 4π treatment plans for several cases (including head and neck, lung, and prostate cases) and the resulting treatment plans are compared with 4π treatment plans created using the column generation algorithm.Results:In our experiments the treatment plans created using the group sparsity method meet or exceed the dosimetric quality of plans created using the column generation algorithm, which was shown superior to clinical plans. Moreover, the group sparsity approach converges in about 3 minutes in these cases, as compared with runtimes of a few hours for the column generation method.Conclusion:This work demonstrates the first non‐greedy approach to non‐coplanar beam angle selection, based on convex optimization, for 4π IMRT systems. The method given here improves both treatment plan quality and runtime as compared with a state of the art column generation algorithm. When the group sparsity term is set to zero, we obtain an excellent method for fluence map optimization, useful when beam angles have already been selected.NIH R43CA183390, NIH R01CA188300, Varian Medical Systems; Part of this research took place while D. O'Connor was a summer intern at RefleXion Medical.

Accelerated iterative beam angle selection in IMRT.

Iterative methods for beam angle selection (BAS) for intensity-modulated radiation therapy (IMRT) planning sequentially construct a beneficial ensemble of beam directions. In a naïve implementation, the nth beam is selected by adding beam orientations one-by-one from a discrete set of candidates to an existing ensemble of (n - 1) beams. The best beam orientation is identified in a time consuming process by solving the fluence map optimization (FMO) problem for every candidate beam and selecting the beam that yields the largest improvement to the objective function value. This paper evaluates two alternative methods to accelerate iterative BAS based on surrogates for the FMO objective function value. We suggest to select candidate beams not based on the FMO objective function value after convergence but (1) based on the objective function value after five FMO iterations of a gradient based algorithm and (2) based on a projected gradient of the FMO problem in the first iteration. The performance of the objective function surrogates is evaluated based on the resulting objective function values and dose statistics in a treatment planning study comprising three intracranial, three pancreas, and three prostate cases. Furthermore, iterative BAS is evaluated for an application in which a small number of noncoplanar beams complement a set of coplanar beam orientations. This scenario is of practical interest as noncoplanar setups may require additional attention of the treatment personnel for every couch rotation. Iterative BAS relying on objective function surrogates yields similar results compared to naïve BAS with regard to the objective function values and dose statistics. At the same time, early stopping of the FMO and using the projected gradient during the first iteration enable reductions in computation time by approximately one to two orders of magnitude. With regard to the clinical delivery of noncoplanar IMRT treatments, we could show that optimized beam ensembles using only a few noncoplanar beam orientations often approach the plan quality of fully noncoplanar ensembles. We conclude that iterative BAS in combination with objective function surrogates can be a viable option to implement automated BAS at clinically acceptable computation times.

A fluence map optimization model for restoring traditional fractionation in IMRT treatment planning

One of the core problems in intensity modulated radiation therapy (IMRT) treatment planning is the fluence map optimization (FMO) problem, which determines a fluence map (or profile) for each beam used in the delivery of treatment. Radiation therapy is administered in multiple so-called daily fractions to allow for healthy tissue to recover from damage caused by the treatment. Before the advent of IMRT, the treatment was designed to ensure a constant dose to cells in the target (the areas in the patient where cancerous cells are present or suspected). In the presence of multiple targets with different prescribed doses, this design meant that treatment had to be delivered in a sequence of unequal fractions, one per prescription dose level. For example, in case of two targets treatment would consist of an initial plan aimed at treating both targets to a lower total dose, followed by a so-called boost plan aimed at delivering the additional dose at the target with higher prescribed dose. In contrast, IMRT treatment plans are often delivered with equal treatment plan for each fraction, which means that the dose per fraction cannot be the same for all targets. The important problem of restoring traditional fractionation to IMRT treatments has not yet received much attention in the literature. In this paper we propose a new optimization model that explicitly restores fractionation into the FMO problem, yielding an optimal set of fluence maps for each fraction. We illustrate the capabilities of our approach on clinical head-and-neck cancer cases.

Dose optimization with first‐order total‐variation minimization for dense angularly sampled and sparse intensity modulated radiation therapy (DASSIM‐RT)

A new treatment scheme coined as dense angularly sampled and sparse intensity modulated radiation therapy (DASSIM-RT) has recently been proposed to bridge the gap between IMRT and VMAT. By increasing the angular sampling of radiation beams while eliminating dispensable segments of the incident fields, DASSIM-RT is capable of providing improved conformity in dose distributions while maintaining high delivery efficiency. The fact that DASSIM-RT utilizes a large number of incident beams represents a major computational challenge for the clinical applications of this powerful treatment scheme. The purpose of this work is to provide a practical solution to the DASSIM-RT inverse planning problem. The inverse planning problem is formulated as a fluence-map optimization problem with total-variation (TV) minimization. A newly released L1-solver, template for first-order conic solver (TFOCS), was adopted in this work. TFOCS achieves faster convergence with less memory usage as compared with conventional quadratic programming (QP) for the TV form through the effective use of conic forms, dual-variable updates, and optimal first-order approaches. As such, it is tailored to specifically address the computational challenges of large-scale optimization in DASSIM-RT inverse planning. Two clinical cases (a prostate and a head and neck case) are used to evaluate the effectiveness and efficiency of the proposed planning technique. DASSIM-RT plans with 15 and 30 beams are compared with conventional IMRT plans with 7 beams in terms of plan quality and delivery efficiency, which are quantified by conformation number (CN), the total number of segments and modulation index, respectively. For optimization efficiency, the QP-based approach was compared with the proposed algorithm for the DASSIM-RT plans with 15 beams for both cases. Plan quality improves with an increasing number of incident beams, while the total number of segments is maintained to be about the same in both cases. For the prostate patient, the conformation number to the target was 0.7509, 0.7565, and 0.7611 with 80 segments for IMRT with 7 beams, and DASSIM-RT with 15 and 30 beams, respectively. For the head and neck (HN) patient with a complicated target shape, conformation numbers of the three treatment plans were 0.7554, 0.7758, and 0.7819 with 75 segments for all beam configurations. With respect to the dose sparing to the critical structures, the organs such as the femoral heads in the prostate case and the brainstem and spinal cord in the HN case were better protected with DASSIM-RT. For both cases, the delivery efficiency has been greatly improved as the beam angular sampling increases with the similar or better conformal dose distribution. Compared with conventional quadratic programming approaches, first-order TFOCS-based optimization achieves far faster convergence and smaller memory requirements in DASSIM-RT. The new optimization algorithm TFOCS provides a practical and timely solution to the DASSIM-RT or other inverse planning problem requiring large memory space. The new treatment scheme is shown to outperform conventional IMRT in terms of dose conformity to both the targetand the critical structures, while maintaining high delivery efficiency.

Efficient IMRT inverse planning with a new L1-solver: template for first-order conic solver

Intensity modulated radiation therapy (IMRT) inverse planning using total-variation (TV) regularization has been proposed to reduce the complexity of fluence maps and facilitate dose delivery. Conventionally, the optimization problem with L-1 norm is solved with quadratic programming (QP), which is time consuming and memory expensive due to the second-order Newton update. This study proposes to use a new algorithm, template for first-order conic solver (TFOCS), for fast and memory-efficient optimization in IMRT inverse planning. The TFOCS utilizes dual-variable updates and first-order approaches for TV minimization without the need to compute and store the enlarged Hessian matrix required for Newton update in the QP technique. To evaluate the effectiveness and efficiency of the proposed method, two clinical cases were used for IMRT inverse planning: a head and neck case and a prostate case. For comparison, the conventional QP-based method for the TV form was adopted to solve the fluence map optimization problem in the above two cases. The convergence criteria and algorithm parameters were selected to achieve similar dose conformity for a fair comparison between the two methods. Compared with conventional QP-based approach, the proposed TFOCS-based method shows a remarkable improvement in computational efficiency for fluence map optimization, while maintaining the conformal dose distribution. Compared with QP-based algorithms, the computational speed using TFOCS for fluence optimization is increased by a factor of 4 to 6, and at the same time the memory requirement is reduced by a factor of 3 to 4. Therefore, TFOCS provides an effective, fast and memory-efficient method for IMRT inverse planning. The unique features of the approach should be particularly important in inverse planning involving a large number of beams, such as in VMAT and dense angularly sampled and sparse intensity modulated radiation therapy (DASSIM-RT).

A two-phase method for selecting IMRT treatment beam angles: Branch-and-Prune and local neighborhood search

This paper presents a new two-phase solution approach to the beam angle and fluence map optimization problem in Intensity Modulated Radiation Therapy (IMRT) planning. We introduce Branch- and- Prune (B&P) to generate a robust feasible solution in the first phase. A local neighborhood search algorithm is developed to find a local optimal solution from the Phase I starting point in the second phase. The goal of the first phase is to generate a clinically acceptable feasible solution in a fast manner based on a Branch-and-Bound tree. In this approach, a substantially reduced search tree is iteratively constructed. In each iteration, a merit score based branching rule is used to select a pool of promising child nodes. Then pruning rules are applied to select one child node as the branching node for the next iteration. The algorithm terminates when we obtain a desired number of angles in the current node. Although Phase I generates quality feasible solutions, it does not guarantee optimality. Therefore, the second phase is designed to converge Phase I starting solutions to local optimality. Our methods are tested on two sets of real patient data. Results show that not only can B&P alone generate clinically acceptable solutions, but the two-phase method consistently generates local optimal solutions, some of which are shown to be globally optimal.

Computational enhancements to fluence map optimization for total marrow irradiation using IMRT

The fluence map optimization (FMO) problem is a core problem in intensity modulated radiation therapy (IMRT) treatment planning. Although it has been studied extensively for site-specific treatment planning, few studies have examined efficient computational methods for solving it for intensity modulated total marrow irradiation (IM-TMI) planning; few studies have also looked at exploiting prior beamlet information to solve the FMO problem in a beam orientation optimization context. In this study, we consider different types of line search strategies and different types of warm-start techniques to improve the speed with which the FMO problem for IM-TMI is solved and the quality of the end solution. We also consider a parallelism-enhanced algorithm to solve the FMO problem for IM-TMI treatment planning with a large number of beams (36 equispaced beams at each of 11 isocenters, for a total of 396 beams). We show that the backtracking line search strategy with step reduction exhibits the best performance and that using either of the two types of warm-start techniques which we consider leads to significant improvements in both solution time and quality. We also provide results for the aforementioned 396-beam plan and show that 30-beam solutions obtained using beam orientation optimization attain a comparable level of quality as this larger solution.

Interior point algorithms: guaranteed optimality for fluence map optimization in IMRT

One of the most widely studied problems of the intensity-modulated radiation therapy (IMRT) treatment planning problem is the fluence map optimization (FMO) problem, the problem of determining the amount of radiation intensity, or fluence, of each beamlet in each beam. For a given set of beams, the fluences of the beamlets can drastically affect the quality of the treatment plan, and thus it is critical to obtain good fluence maps for radiation delivery. Although several approaches have been shown to yield good solutions to the FMO problem, these solutions are not guaranteed to be optimal. This shortcoming can be attributed to either optimization model complexity or properties of the algorithms used to solve the optimization model. We present a convex FMO formulation and an interior point algorithm that yields an optimal treatment plan in seconds, making it a viable option for clinical applications.

Comparative analysis of 60Co intensity-modulated radiation therapy**This work was supported by the National Science Foundation under grant no. DMI-0457394 and the National Cancer Institute under grant no. R01 CA100636.

In this study, we perform a scientific comparative analysis of using 60Co beams in intensity-modulated radiation therapy (IMRT). In particular, we evaluate the treatment plan quality obtained with (i) 6 MV, 18 MV and 60Co IMRT; (ii) different numbers of static multileaf collimator (MLC) delivered 60Co beams and (iii) a helical tomotherapy 60Co beam geometry. We employ a convex fluence map optimization (FMO) model, which allows for the comparison of plan quality between different beam energies and configurations for a given case. A total of 25 clinical patient cases that each contain volumetric CT studies, primary and secondary delineated targets, and contoured structures were studied: 5 head-and-neck (H&N), 5 prostate, 5 central nervous system (CNS), 5 breast and 5 lung cases. The DICOM plan data were anonymized and exported to the University of Florida optimized radiation therapy (UFORT) treatment planning system. The FMO problem was solved for each case for 5–71 equidistant beams as well as a helical geometry for H&N, prostate, CNS and lung cases, and for 3–7 equidistant beams in the upper hemisphere for breast cases, all with 6 MV, 18 MV and 60Co dose models. In all cases, 95% of the target volumes received at least the prescribed dose with clinical sparing criteria for critical organs being met for all structures that were not wholly or partially contained within the target volume. Improvements in critical organ sparing were found with an increasing number of equidistant 60Co beams, yet were marginal above 9 beams for H&N, prostate, CNS and lung. Breast cases produced similar plans for 3–7 beams. A helical 60Co beam geometry achieved similar plan quality as static plans with 11 equidistant 60Co beams. Furthermore, 18 MV plans were initially found not to provide the same target coverage as 6 MV and 60Co plans; however, adjusting the trade-offs in the optimization model allowed equivalent target coverage for 18 MV. For plans with comparable target coverage, critical structure sparing was best achieved with 6 MV beams followed closely by 60Co beams, with 18 MV beams requiring significantly increased dose to critical structures. In this paper, we report in detail on a representative set of results from these experiments. The results of the investigation demonstrate the potential for IMRT radiotherapy employing commercially available 60Co sources and a double-focused MLC. Increasing the number of equidistant beams beyond 9 was not observed to significantly improve target coverage or critical organ sparing and static plans were found to produce comparable plans to those obtained using a helical tomotherapy treatment delivery when optimized using the same well-tuned convex FMO model. While previous studies have shown that 18 MV plans are equivalent to 6 MV for prostate IMRT, we found that the 18 MV beams actually required more fluence to provide similar quality target coverage.

Neighborhood search approaches to beam orientation optimization in intensity modulated radiation therapy treatment planning

The intensity modulated radiation therapy (IMRT) treatment planning problem consists of several subproblems which are typically solved sequentially. We seek to combine two of the subproblems: the beam orientation optimization (BOO) problem and the fluence map optimization (FMO) problem. The BOO problem is the problem of selecting the beam orientations to deliver radiation to the patient. The FMO problem is the problem of determining the amount of radiation intensity, or fluence, of each beamlet in each beam. The solution to the FMO problem measures the quality of a beam set, but the majority of previous BOO studies rely on heuristics and approximations to gauge the quality of the beam set. In contrast with these studies, we use an exact measure of the treatment plan quality attainable using a given beam set, which ensures convergence to a global optimum in the case of our simulated annealing algorithm and a local optimum in the case of our local search algorithm. We have also developed a new neighborhood structure that allows for faster convergence using our simulated annealing and local search algorithms, thus reducing the amount of time required to obtain a good solution. Finally, we show empirically that we can generate clinically acceptable treatment plans that require fewer beams than in current practice. This may reduce the length of treatment time, which is an important clinical consideration in IMRT.

An exact approach to direct aperture optimization in IMRT treatment planning**This work was supported by the National Science Foundation under grant no. DMI-0457394.

We consider the problem of intensity-modulated radiation therapy (IMRT) treatment planning using direct aperture optimization. While this problem has been relatively well studied in recent years, most approaches employ a heuristic approach to the generation of apertures. In contrast, we use an exact approach that explicitly formulates the fluence map optimization (FMO) problem as a convex optimization problem in terms of all multileaf collimator (MLC) deliverable apertures and their associated intensities. However, the number of deliverable apertures, and therefore the number of decision variables and constraints in the new problem formulation, is typically enormous. To overcome this, we use an iterative approach that employs a subproblem whose optimal solution either provides a suitable aperture to add to a given pool of allowable apertures or concludes that the current solution is optimal. We are able to handle standard consecutiveness, interdigitation and connectedness constraints that may be imposed by the particular MLC system used, as well as jaws-only delivery. Our approach has the additional advantage that it can explicitly account for transmission of dose through the part of an aperture that is blocked by the MLC system, yielding a more precise assessment of the treatment plan than what is possible using a traditional beamlet-based FMO problem. Finally, we develop and test two stopping rules that can be used to identify treatment plans of high clinical quality that are deliverable very efficiently. Tests on clinical head-and-neck cancer cases showed the efficacy of our approach, yielding treatment plans comparable in quality to plans obtained by the traditional method with a reduction of more than 75% in the number of apertures and a reduction of more than 50% in beam-on time, with only a modest increase in computational effort. The results also show that delivery efficiency is very insensitive to the addition of traditional MLC constraints; however, jaws-only treatment requires about a doubling in beam-on time and number of apertures used. Finally, we showed the importance of accounting for transmission effects when assessing or, preferably, optimizing treatment plan quality.

SU‐FF‐T‐111: Fast Nonlinear Optimization with Simple Bounds for IMRT Planning

Purpose: The advent of image‐guided adaptive conformal radiotherapy demands the development of very efficient methods of intensity modulated radiation therapy (IMRT) fluence map optimization (FMO). To enable adaptive image guidance with daily IMRT optimization the ability to generate high quality treatment plans “on‐the‐fly” to correct for changes in patient anatomy during the course of radiotherapy is needed. To this end, we have tested the feasibility of employing analytic convex nonlinear objective functions in a FMO model with simple nonnegative bounds on beamlet fluence. Method and Materials: The nonlinear FMO model is based on previously studied voxel‐based dose‐penalty functions that were approximated in linear and quadratic programming models. Well‐known gradient and quasi‐Newton algorithms were employed along with an Armijo line search that requires sufficient objective function decrease. A simple projected‐direction interior‐point method was used to enforce fluence non‐negativity. The algorithms were implemented in an in‐house treatment‐planning system. We tested the following methods: steepest‐descent, BFGS, symmetric‐rank‐1, and combinations thereof. The algorithms were applied to 10 head‐and‐neck cases. The numbers of beamlets were between 900 and 2,100 with a 3‐mm isotropic dose grid. Results: The tested methods were all found to be at least as fast as a commercial linear and quadratic programming solver (CPLEX, ILog, Inc.). While no single method was best for all cases, the steepest‐descent method typically converged the fastest. Conclusion: Nonlinear optimization methods can efficiently solve the IMRT FMO problem in well under a minute with high quality for clinical cases. A simple steepest‐descent algorithm was highly successful when combined with methods to enforce simple bounds and to perform a robust line search. Quasi‐Newton methods have been successful in solving many large‐scale nonlinear programming problems, but provide little advantage for the IMRT FMO problem. In the future, we will test the algorithm's robustness with a larger population of cases.

The equivalence of “biological” and “physical” fluence map optimization models for IMRT

We present and explain some recent theoretical finding regarding the equivalence of many intensity modulated radiation therapy (IMRT) fluence map optimization (FMO) models based on both “biological” and “physical” treatment plan criteria. We will present and discuss the clinical and algorithmic significance of this otherwise highly abstract and mathematical work. Our work is best described as a mathematical framework for studying treatment plan evaluation criteria. The framework consists of many mathematical theorems and proofs relating to multi-criteria optimization. We concisely capture the essence of our work with a central theorem that states: applying an increasing function to treatment plan evaluation criteria does not affect the corresponding set of Pareto efficient treatment plans: Theorem 1. Let G1(x),...,GL(x) denote a set of L convex treatment plan evaluation criteria (e.g., structure based EUD criteria), viewed as functions of the vector of beamlet intensities x, for FMO. Moreover, let h1,...,hL be increasing functions. Then the multi-criteria optimization formulations of FMO based on G1(x),...,GL(x) has the same set of efficient treatment plans as the multi-criteria optimization formulations of FMO based on h1(G1(x)),...,hL(GL(x)). We applied the framework to many convex and non-convex structure-based treatment plan evaluation criteria which have previously been published and are under development, including TCP, NTCP, P+, EUD, gEUD, and sigmoidal transformations of EUD and gEUD. In particular, we demonstrated that models employing these criteria are equivalent to two models: dose power functions and exponential dose functions formulated in terms of voxel-based criteria that penalize dose in individual voxels, i.e. traditional voxel-based penalty functions. By equivalent we mean to say that the set of efficient solutions in a multi-critera model are identical. Theorem 1 also implies that the potential additional complexity introduced by applying the increasing transformations, say from penalty function to gEUD to NTCP, does not change the trade-offs that are being considered in the FMO problem. Put differently, this theorem states that if a suitable decomposition of a given nonconvex criterion into a convex criterion and an increasing function can be found, the nonconvex criterion may be replaced by an equivalent convex criterion. As is well know, convex models are efficiently solvable while nonconvex models must be dealt with using heuristic approaches to optimization. Casting FMO models as multi-criteria models has shown that many sets of treatment plan evaluation criteria that have been proposed in the literature are in fact equivalent with respect to the corresponding set of efficient treatment plans, i.e. any efficient plan found with one approach is accessible by any other. In addition, we have established conditions under which nonconvex criteria can be transformed into convex criteria without changing the set of efficient treatment plans. Thus, the presented framework identifies which of the proposed treatment plan evaluation criteria are truly different, which enables researchers to focus their efforts on comparing nonequivalent models. In particular, we have shown that among the “biological” criteria of EUD, gEUD, TCP, and NTCP, as well as two sigmoidal transformations of EUD, only two truly distinct criteria exist. This result should alleviate concerns on the pros and cons of using biological vs. physical criteria that was recently debated in the literature, as they are in essence equivalent. To avoid duplication of efforts, in the pursuit of new treatment plan evaluation criteria one should consider the unifying framework to determine whether potentially new criteria indeed yield different sets of efficient treatment plans and therefore truly new models

A novel linear programming approach to fluence map optimization for intensity modulated radiation therapy treatment planning

We present a novel linear programming (LP) based approach for efficiently solving the intensity modulated radiation therapy (IMRT) fluence-map optimization (FMO) problem to global optimality. Our model overcomes the apparent limitations of a linear-programming approach by approximating any convex objective function by a piecewise linear convex function. This approach allows us to retain the flexibility offered by general convex objective functions, while allowing us to formulate the FMO problem as a LP problem. In addition, a novel type of partial-volume constraint that bounds the tail averages of the differential dose–volume histograms of structures is imposed while retaining linearity as an alternative approach to improve dose homogeneity in the target volumes, and to attempt to spare as many critical structures as possible. The goal of this work is to develop a very rapid global optimization approach that finds high quality dose distributions. Implementation of this model has demonstrated excellent results. We found globally optimal solutions for eight 7-beam head-and-neck cases in less than 3 min of computational time on a single processor personal computer without the use of partial-volume constraints. Adding such constraints increased the running times by a factor of 2–3, but improved the sparing of critical structures. All cases demonstrated excellent target coverage (>95%), target homogeneity (<10% overdosing and <7% underdosing) and organ sparing using at least one of the two models.