Fluence Map Optimization Model Research Articles

A predicted three-dimensional dose sequence based treatment planning optimization method for gynecologic IMRT.

In knowledge-based treatment planning (KBTP) for intensity-modulated radiation therapy (IMRT), the quality of the plan is dependent on the sophistication of the predicted dosimetric information and its application. In this paper, we propose a KBTP method that based on the effective and reasonable utilization of a three-dimensional (3D) dose prediction on planning optimization. We used an organs-at-risk (OARs) dose distribution prediction model to create a voxel-based dose sequence based optimization objective for OARs doses. This objective was used to reformulate a traditional fluence map optimization model, which involves a tolerable spatial re-assignment of the predicted dose distribution to the OAR voxels based on their current doses' positions at a sorted dose sequencing. The feasibility of this method was evaluated with ten gynecology (GYN) cancer IMRT cases by comparing its generated plan quality with the original clinical plan. Results showed feasible plan by proposed method, with comparable planning target volume (PTV) dose coverage and greater dose sparing of the OARs. Among ten GYN cases, the average V30 and V45 of rectum were decreased by 4%±4% (p=0.02) and 4%±3% (p<0.01), respectively. V30 and V45 of bladder were decreased by 8%±2% (p<0.01) and 3%±2% (p<0.01), respectively. Our predicted dose sequence-based planning optimization method for GYN IMRT offered a flexible use of predicted 3D doses while ensuring the output plan consistency.

A Novel Definition of Equivalent Uniform Dose Based on Volume Dose Curve

With the improvement of mobile device performance, the requirement of equivalent dose description in intensity-modulated radiation therapy is increasing in mobile multimedia for healthcare. The emergence of mobile cloud computing will provide cloud servers and storage for intensity-modulated radiotherapy (IMRT) mobile applications, thus realizing visualized radiotherapy in a real sense. Equivalent uniform dose (EUD) is a biomedical indicator based on the dose measure. In this paper, the dose volume histogram is used to describe the dose distribution of different tissues in target and nontarget regions. The traditional definition of EUD, such as the exponential form and the linear form, has only a few parameters in the model for fast calculation. However, there is no close relationship between this traditional definition and the dose volume histogram. In order to establish the consistency between the EUD and the dose volume histogram, this paper proposes a novel definition of EUD based on the volume dose curve, called VD-EUD. By using a unique organic volume weight curve, it is easy to calculate VD-EUD for different dose distributions. In definition, different weight curves are used to represent the biological effects of different organs. For the target area, we should be more careful about those voxels with a low dose (cold point); thus, the weight curve is monotonically decreasing. While for the nontarget area, the curve is monotonically increasing. Furthermore, we present the curves for parallel, serial, and mixed organs of nontarget areas separately, and we define the weight curve form with only two parameters. Medical doctors can adjust the curve interactively according to different patients and organs. We also propose a fluence map optimization model with the VD-EUD constraint, which means that the proposed EUD constraint will lead to a large feasible solution space. We compare the generalized EUD (gEUD) and the proposed VD-EUD by experiments, which show that the VD-EUD has a closer relationship with the dose volume histogram. If the biological survival probability is equivalent to the VD-EUD, the feasible solution space would be large, and the target areas can be covered. By establishing a personalized organic weight curve, medical doctors can have a unique VD-EUD for each patient. By using the flexible and adjustable EUD definition, we can establish the VD-EUD-based fluence map optimization model, which will lead to a larger solution space than the traditional dose volume constraint-based model. The VD-EUD is a new definition; thus, we need more clinical testing and verification.

Spatiotemporally Optimal Fractionation in Radiotherapy

We present a spatiotemporally integrated formulation of the optimal fractionation problem using the standard log-linear-quadratic survival model. Our objective is to choose a fluence map and a number of fractions to maximize the biological effect of tumor dose averaged over its voxels subject to maximum dose, mean dose, and dose-volume constraints for various normal tissues. Constraints are expressed in biologically effective dose equivalents. We propose an efficient convex programming method to approximately solve the resulting computationally difficult model. Through extensive computer simulations on 10 head-and-neck and prostate cancer test cases with a broad range of radiobiological parameters, we compare the biological effect on tumors obtained by our integrated approach relative to that from two other models. The first is a traditional intensity modulated radiation therapy (IMRT) fluence map optimization model that does not optimize the number of fractions. The second assumes that a fluence map is available a priori from a traditional IMRT optimization model and then optimizes the number of fractions, thus separating the spatial and temporal components. The improvements in tumor biological effect over IMRT were 9%–52%, with an average of 22% for head-and-neck, and 53%–108%, with an average of 69% for prostate. The improvements in tumor biological effect over the spatiotemporally separated model were 15%–45%, with an average of 27%, and 17%–23%, with an average of 21%, for head-and-neck and prostate, respectively. This suggests that integrated optimization of the fluence map and the number of fractions could improve treatment efficacy, as measured within the linear-quadratic framework.

WE‐AB‐209‐07: Explicit and Convex Optimization of Plan Quality Metrics in Intensity‐Modulated Radiation Therapy Treatment Planning

Purpose:To formulate objective functions of a multicriteria fluence map optimization model that correlate well with plan quality metrics, and to solve this multicriteria model by convex approximation.Methods:In this study, objectives of a multicriteria model are formulated to explicitly either minimize or maximize a dose‐at‐volume measure. Given the widespread agreement that dose‐at‐volume levels play important roles in plan quality assessment, these objectives correlate well with plan quality metrics. This is in contrast to the conventional objectives, which are to maximize clinical goal achievement by relating to deviations from given dose‐at‐volume thresholds: while balancing the new objectives means explicitly balancing dose‐at‐volume levels, balancing the conventional objectives effectively means balancing deviations. Constituted by the inherently non‐convex dose‐at‐volume measure, the new objectives are approximated by the convex mean‐tail‐dose measure (CVaR measure), yielding a convex approximation of the multicriteria model.Results:Advantages of using the convex approximation are investigated through juxtaposition with the conventional objectives in a computational study of two patient cases. Clinical goals of each case respectively point out three ROI dose‐at‐volume measures to be considered for plan quality assessment. This is translated in the convex approximation into minimizing three mean‐tail‐dose measures. Evaluations of the three ROI dose‐at‐volume measures on Pareto optimal plans are used to represent plan quality of the Pareto sets. Besides providing increased accuracy in terms of feasibility of solutions, the convex approximation generates Pareto sets with overall improved plan quality. In one case, the Pareto set generated by the convex approximation entirely dominates that generated with the conventional objectives.Conclusion:The initial computational study indicates that the convex approximation outperforms the conventional objectives in aspects of accuracy and plan quality.

An Automatic Approach for Satisfying Dose-Volume Constraints in Linear Fluence Map Optimization for IMPT.

Prescriptions for radiation therapy are given in terms of dose-volume constraints (DVCs). Solving the fluence map optimization (FMO) problem while satisfying DVCs often requires a tedious trial-and-error for selecting appropriate dose control parameters on various organs. In this paper, we propose an iterative approach to satisfy DVCs using a multi-objective linear programming (LP) model for solving beamlet intensities. This algorithm, starting from arbitrary initial parameter values, gradually updates the values through an iterative solution process toward optimal solution. This method finds appropriate parameter values through the trade-off between OAR sparing and target coverage to improve the solution. We compared the plan quality and the satisfaction of the DVCs by the proposed algorithm with two nonlinear approaches: a nonlinear FMO model solved by using the L-BFGS algorithm and another approach solved by a commercial treatment planning system (Eclipse 8.9). We retrospectively selected from our institutional database five patients with lung cancer and one patient with prostate cancer for this study. Numerical results show that our approach successfully improved target coverage to meet the DVCs, while trying to keep corresponding OAR DVCs satisfied. The LBFGS algorithm for solving the nonlinear FMO model successfully satisfied the DVCs in three out of five test cases. However, there is no recourse in the nonlinear FMO model for correcting unsatisfied DVCs other than manually changing some parameter values through trial and error to derive a solution that more closely meets the DVC requirements. The LP-based heuristic algorithm outperformed the current treatment planning system in terms of DVC satisfaction. A major strength of the LP-based heuristic approach is that it is not sensitive to the starting condition.

Inverse Planning Optimization Method for Intensity Modulated Radiation Therapy

In order to facilitate the leaf sequencing process in intensity modulated radiation therapy (IMRT), and design of a practical leaf sequencing algorithm, it is an important issue to smooth the planned fluence maps. The objective is to achieve both high-efficiency and high-precision dose delivering by considering characteristics of leaf sequencing process. The key factor which affects total number of monitor units for the leaf sequencing optimization process is the max flow value of the digraph which formulated from the fluence maps. Therefore, we believe that one strategy for compromising dose conformity and total number of monitor units in dose delivery is to balance the dose distribution function and the max flow value mentioned above. However, there are too many paths in the digraph, and we don't know the flow value of which path is the maximum. The maximum flow value among the horizontal paths was selected and used in the objective function of the fluence map optimization to formulate the model. The model is a traditional linear constrained quadratic optimization model which can be solved by interior point method easily. We believe that the smoothed maps from this model are more suitable for leaf sequencing optimization process than other smoothing models. A clinical head-neck case and a prostate case were tested and compared using our proposed model and the smoothing model which is based on the minimization of total variance. The optimization results with the same level of total number of monitor units (TNMU) show that the fluence maps obtained from our model have much better dose performance for the target/non-target region than the maps from total variance based on the smoothing model. This indicates that our model achieves better dose distribution when the algorithm suppresses the TNMU at the same level. Although we have just used the max flow value of the horizontal paths in the diagraph in the objective function, a good balance has been achieved between the dose conformity and the total number of monitor units. This idea can be extended to other fluence map optimization model, and we believe it can also achieve good performance.

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.

Fluence map optimization (FMO) with dose–volume constraints in IMRT using the geometric distance sorting method

A new heuristic algorithm based on the so-called geometric distance sorting technique is proposed for solving the fluence map optimization with dose–volume constraints which is one of the most essential tasks for inverse planning in IMRT. The framework of the proposed method is basically an iterative process which begins with a simple linear constrained quadratic optimization model without considering any dose–volume constraints, and then the dose constraints for the voxels violating the dose–volume constraints are gradually added into the quadratic optimization model step by step until all the dose–volume constraints are satisfied. In each iteration step, an interior point method is adopted to solve each new linear constrained quadratic programming. For choosing the proper candidate voxels for the current dose constraint adding, a so-called geometric distance defined in the transformed standard quadratic form of the fluence map optimization model was used to guide the selection of the voxels. The new geometric distance sorting technique can mostly reduce the unexpected increase of the objective function value caused inevitably by the constraint adding. It can be regarded as an upgrading to the traditional dose sorting technique. The geometry explanation for the proposed method is also given and a proposition is proved to support our heuristic idea. In addition, a smart constraint adding/deleting strategy is designed to ensure a stable iteration convergence. The new algorithm is tested on four cases including head–neck, a prostate, a lung and an oropharyngeal, and compared with the algorithm based on the traditional dose sorting technique. Experimental results showed that the proposed method is more suitable for guiding the selection of new constraints than the traditional dose sorting method, especially for the cases whose target regions are in non-convex shapes. It is a more efficient optimization technique to some extent for choosing constraints than the dose sorting method. By integrating a smart constraint adding/deleting scheme within the iteration framework, the new technique builds up an improved algorithm for solving the fluence map optimization with dose–volume constraints.

SU-E-T-608: Integrated Beam Orientation and Fluence Map Optimization in Radiation Therapy Treatment Planning.

To efficiently select high-quality coplanar or non-coplanar beam orientations for IMRT treatments while formally and explicitly incorporating the effect of the selected beam orientations on the quality of the dose distribution obtained by the treatment plan optimization model. Beam orientation models consider a discrete set of potential coplanar and/or non-coplanar beam locations around the patient. A new greedy algorithm is proposed to solve a model that integrates beam orientation optimization (BOO) and fluence map optimization (FMO). The algorithm iteratively adds beams to a FMO model. In each iteration, an attractiveness measure is associated with each remaining candidate beam orientation. This attractiveness measure is based explicitly on an optimal dose distribution that allows only the currently selected set of beams to be used. Several alternate attractiveness measures are considered which use either first-order information or both first and second-order information. Performance of the algorithm was assessed on a clinical lung cancer case. The developed beam selection algorithm was applied to a lung cancer case using either coplanar beams or both coplanar and non-coplanar beams. In the coplanar case, beam orientations were found that produce a superior dose distribution to that using an equal number of equi-spaced beams. In the non-coplanar case it was found that fewer beams were needed to produce a dose distribution of comparable quality to that found in the coplanar case. The developed solution approach showcases the potential benefits of integrating different steps in the treatment plan optimization process. By integrating the BOO and FMO models, treatment plan quality was explicitly incorporated into the beam selection process. BOO can be automated and implemented efficiently, which eliminates the guesswork involved in manually adjusting beam orientations in IMRT treatment planning.

An efficient approach to incorporating interfraction motion in IMRT treatment planning

Intensity modulated radiation therapy (IMRT) is one of the most widely used delivery modalities for radiation therapy for cancer patients. A patient is typically treated in daily fractions over a period of 5–9 weeks. In this paper, we consider the problem of accounting for changes in patient setup location and internal geometry between the treatment fractions, usually referred to as interfraction motion. The conventional method is to add a margin around the clinical tumor volume (CTV) to obtain a planning target volume (PTV). A fluence map optimization (FMO) model is then solved to determine the optimal intensity profiles to deliver to the patient. However, a margin-based method may not adequately model the changes in dose distributions due to the random nature of organ motion. Accounting for interfraction motion in the FMO model essentially transforms the deterministic optimization problem into a stochastic one. We propose a stochastic FMO model that employs convex penalty functions to control the treatment plan quality and uses a large number of scenarios to characterize interfraction motion uncertainties. Some effects of radiotherapy are impacted mainly by the dose distribution in a given treatment fraction while others tend to manifest themselves over time and depend mostly on the total dose received over the course of treatment. We will therefore formulate an optimization model that explicitly incorporates treatment plan evaluation criteria that apply to the total dose received over all treatment fractions and ones that apply to the dose per fraction. Particularly when many structures fall into the former category, this can lead to significant reductions in the dimension of the optimization model and therefore the time required to solve it. We test an example of our model on five clinical prostate cancer cases, showing the efficacy of our approach. In particular, compared to a traditional margin-based treatment plan, our plans exhibit significantly improved target dose coverage and clinically equivalent critical structure sparing at only a modest increase in computational effort.

An Automated Intensity-Modulated Radiation Therapy Planning System

We design and implement an intensity-modulated radiation therapy plan generation technology that effectively and efficiently optimizes beam geometry as well as beam intensities. Our approach is based on an existing linear programming-based fluence map optimization model that approximates dose-volume requirements using conditional value-at-risk (C-VaR) constraints. We show how the parameters of the C-VaR constraints can be used to control various metrics of treatment plan quality. Next, we develop an automated search strategy for parameter tuning. Finally, beam angle selection is integrated with fluence map optimization. The beam angle selection scheme employs a bicriteria scoring of beam angle geometries and a selection mechanism to choose from among the set of nondominated geometries. The overall technology is automated and generates several high-quality treatment plans satisfying dose prescription requirements in a single invocation and without human guidance. The technology has been tested on various real-patient cases with uniform success.

A novel fluence map optimization model incorporating leaf sequencing constraints

A novel fluence map optimization model incorporating leaf sequencing constraints is proposed to overcome the drawbacks of the current objective inside smoothing models. Instead of adding a smoothing item to the objective function, we add the total number of monitor unit (TNMU) requirement directly to the constraints which serves as an important factor to balance the fluence map optimization and leaf sequencing optimization process at the same time. Consequently, we formulate the fluence map optimization models for the trailing (left) leaf synchronized, leading (right) leaf synchronized and the interleaf motion constrained non-synchronized leaf sweeping schemes, respectively. In those schemes, the leaves are all swept unidirectionally from left to right. Each of those models is turned into a linear constrained quadratic programming model which can be solved effectively by the interior point method. Those new models are evaluated with two publicly available clinical treatment datasets including a head–neck case and a prostate case. As shown by the empirical results, our models perform much better in comparison with two recently emerged smoothing models (the total variance smoothing model and the quadratic smoothing model). For all three leaf sweeping schemes, our objective dose deviation functions increase much slower than those in the above two smoothing models with respect to the decreasing of the TNMU. While keeping plans in the similar conformity level, our new models gain much better performance on reducing TNMU.

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.

A computational implementation and comparison of several intensity modulated proton therapy treatment planning algorithms

The authors present a comparative study of intensity modulated proton therapy (IMPT) treatment planning employing algorithms of three-dimensional (3D) modulation, and 2.5-dimensional (2.5D) modulation, and intensity modulated distal edge tracking (DET) [A. Lomax, Phys. Med. Biol. 44, 185-205 (1999)] applied to the treatment of head-and-neck cancer radiotherapy. These three approaches were also compared with 6 MV photon intensity modulated radiation therapy (IMRT). All algorithms were implemented in the University of Florida Optimized Radiation Therapy system using a finite sized pencil beam dose model and a convex fluence map optimization model. The 3D IMPT and the DET algorithms showed considerable advantages over the photon IMRT in terms of dose conformity and sparing of organs at risk when the beam number was not constrained. The 2.5D algorithm did not show an advantage over the photon IMRT except in the dose reduction to the distant healthy tissues, which is inherent in proton beam delivery. The influences of proton beam number and pencil beam size on the IMPT plan quality were also studied. Out of 24 cases studied, three cases could be adequately planned with one beam and 12 cases could be adequately planned with two beams, but the dose uniformity was often marginally acceptable. Adding one or two more beams in each case dramatically improved the dose uniformity. The finite pencil beam size had more influence on the plan quality of the 2.5D and DET algorithms than that of the 3D IMPT. To obtain a satisfactory plan quality, a 0.5 cm pencil beam size was required for the 3D IMPT and a 0.3 cm size was required for the 2.5D and the DET algorithms. Delivery of the IMPT plans produced in this study would require a proton beam spot scanning technique that has yet to be developed clinically.

SU-FF-T-159: Direct Aperture Optimization in IMRT Treatment Planning

Purpose: To develop an exact approach to direct aperture optimization that can be applied to a large class of IMRT fluence map optimization (FMO) models that employ convex treatment plan evaluation criteria in the objective function and/or constraints and test its ability to obtain high-quality treatment plans using a reduced number of apertures and limited beam-on-time. Method and Materials: An exact approach was developed that iteratively adds high-quality deliverable apertures while the treatment plan quality can be improved. The most promising new apertures are found by solving a subproblem, which can account for deliverability constraints imposed by the multi-leaf collimator (MLC) system such as interdigitation, connectedness, or rectangular shapes. The approach can explicitly account for transmission through the leaves that are not shielded by backup jaws in the MLC system. It was compared to the traditional method of solving a beamlet-based FMO model followed by a leaf-sequencing step to determine apertures and intensities. Results: For a set of ten clinical head-and-neck cancer cases, treatment plans comparable in quality to plans obtained by the traditional method were obtained with a reduction of more than 75% in number of apertures and a reduction of more than 50% in beam-on-time, with only a modest increase in computational effort. The importance of incorporating transmission effects when assessing treatment plan quality was shown as well. Conclusion: This work provides a direct aperture optimization method that incorporates many physical and biological treatment plan evaluation criteria, accounts for leaf transmission effects, and significantly limits both total treatment time and beam-on-time. The iterative nature of the approach allows it to be implemented in an interactive system where a trade-off can be made between treatment plan efficiency and efficacy. The ability of the approach to account for different MLC deliverability constraints allows for a comparison of different MLC architectures.

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

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

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

A unifying framework for multi-criteria fluence map optimization models

Models for finding treatment plans for intensity modulated radiation therapy are usually based on a number of structure-based treatment plan evaluation criteria, which are often conflicting. Rather than formulating a model that a priori quantifies the trade-offs between these criteria, we consider a multi-criteria optimization approach that aims at finding the so-called undominated treatment plans. We present a unifying framework for studying multi-criteria optimization problems for treatment planning that establishes conditions under which treatment plan evaluation criteria can be transformed into convex criteria while preserving the set of undominated treatment plans. Such transformations are identified for many of the criteria that have been proposed to date, establishing equivalences between these criteria. In addition, it is shown that the use of a nonconvex criterion can often be avoided by transformation to an equivalent convex criterion. In particular, we show that models employing criteria such as tumour control probability, normal tissue complication probability, probability of uncomplicated tumour control, as well as sigmoidal transformations of (generalized) equivalent uniform dose are equivalent to models formulated in terms of separable voxel-based criteria that penalize dose in individual voxels.