Configuration space analysis of common cost functions in radiotherapy beam-weight optimization algorithms

Optimization in treatment planning of high dose-rate brachytherapy - Review and analysis of mathematical models

Treatment planning in high dose-rate brachytherapy has traditionally been conducted with manual forward planning, but inverse planning is today increasingly used in clinical practice. There is a large variety of proposed optimization models and algorithms to model and solve the treatment planning problem. Two major parts of inverse treatment planning for which mathematical optimization can be used are the decisions about catheter placement and dwell time distributions. Both these problems as well as integrated approaches are included in this review. The proposed models include linear penalty models, dose-volume models, mean-tail dose models, quadratic penalty models, radiobiological models, and multiobjective models. The aim of this survey is twofold: (i) to give a broad overview over mathematical optimization models used for treatment planning of brachytherapy and (ii) to provide mathematical analyses and comparisons between models. New technologies for brachytherapy treatments and methods for treatment planning are also discussed. Of particular interest for future research is a thorough comparison between optimization models and algorithms on the same dataset, and clinical validation of proposed optimization approaches with respect to patient outcome.

Three IMRT advanced planning tools: A multi-institutional side-by-side comparison

Purpose

To assess three advanced radiation therapy treatment planning tools on the intensity‐modulated radiation therapy (IMRT) quality and consistency when compared to the clinically approved plans, referred as manual plans, which were planned without using any of these advanced planning tools.

Materials and Methods

Three advanced radiation therapy treatment planning tools, including auto‐planning, knowledge‐based planning, and multiple criteria optimization, were assessed on 20 previously treated clinical cases. Three institutions participated in this study, each with expertise in one of these tools. The twenty cases were retrospectively selected from Cleveland Clinic, including five head‐and‐neck (HN) cases, five brain cases, five prostate with pelvic lymph nodes cases, and five spine cases. A set of general planning objectives and organs‐at‐risk (OAR) dose constraints for each disease site from Cleveland Clinic was shared with other two institutions. A total of 60 IMRT research plans (20 from each institution) were designed with the same beam configuration as in the respective manual plans. For each disease site, detailed isodoseline distributions and dose volume histograms for a randomly selected representative case were compared among the three research plans and manual plan. In addition, dosimetric endpoints of five cases for each site were compared.

Results

Compared to the manual plans, the research plans using advanced tools showed substantial improvement for the HN patient cases, including the maximum dose to the spinal cord and brainstem and mean dose to the parotid glands. For the brain, prostate, and spine cases, the four types of plans were comparable based on dosimetric endpoint comparisons.

Conclusion

With minimal planner interventions, advanced treatment planning tools are clinically useful, producing a plan quality similarly to or better than manual plans, improving plan consistency. For difficult cases such as HN cancer, advanced planning tools can further reduce radiation doses to numerous OARs while delivering adequate dose to the tumor targets.

Using the axis of rotation of polar navigator echoes to rapidly measure 3D rigid-body motion

An improved technique to prospectively correct three-dimensional rigid-body motion using polar spherical navigator (pNAV) echoes is presented. The technique is based on acquiring pNAVs of an object in a baseline and rotated position and determining the axis of rotation (AOR) between data sets, thereby reducing 3D rotations to a 2D, planar rotation. Finding the AOR is simplified by prerotating the baseline trajectory, which forces the axis to lie within a specific polar region of a spherical shell in k-space. Orbital navigator echoes are interpolated from the pNAV data in planes orthogonal to the AOR and cross-correlated to determine the 2D rotation. The rotation about the AOR is used in conjunction with its orientation to calculate the overall 3D rotation. The pNAV-AOR technique was tested for its precision, accuracy, and processing speed in detecting compound rotations and translations of varying magnitude. In comparison to the spherical navigator echo technique, the pNAV-AOR technique is noniterative, fast, and independent of rotation magnitude and direction. At low SNR, the technique can detect compound rotations to 0.5 degrees accuracy in an estimated 100 msec, indicating that prospective 3D rigid-body motion correction may be feasible with this technique.

Speed and convergence properties of gradient algorithms for optimization of IMRT

Gradient algorithms are the most commonly employed search methods in the routine optimization of IMRT plans. It is well known that local minima can exist for dose-volume-based and biology-based objective functions. The purpose of this paper is to compare the relative speed of different gradient algorithms, to investigate the strategies for accelerating the optimization process, to assess the validity of these strategies, and to study the convergence properties of these algorithms for dose-volume and biological objective functions. With these aims in mind, we implemented Newton's, conjugate gradient (CG), and the steepest decent (SD) algorithms for dose-volume- and EUD-based objective functions. Our implementation of Newton's algorithm approximates the second derivative matrix (Hessian) by its diagonal. The standard SD algorithm and the CG algorithm with "line minimization" were also implemented. In addition, we investigated the use of a variation of the CG algorithm, called the "scaled conjugate gradient" (SCG) algorithm. To accelerate the optimization process, we investigated the validity of the use of a "hybrid optimization" strategy, in which approximations to calculated dose distributions are used during most of the iterations. Published studies have indicated that getting trapped in local minima is not a significant problem. To investigate this issue further, we first obtained, by trial and error, and starting with uniform intensity distributions, the parameters of the dose-volume- or EUD-based objective functions which produced IMRT plans that satisfied the clinical requirements. Using the resulting optimized intensity distributions as the initial guess, we investigated the possibility of getting trapped in a local minimum. For most of the results presented, we used a lung cancer case. To illustrate the generality of our methods, the results for a prostate case are also presented. For both dose-volume and EUD based objective functions, Newton's method far outperforms other algorithms in terms of speed. The SCG algorithm, which avoids expensive "line minimization," can speed up the standard CG algorithm by at least a factor of 2. For the same initial conditions, all algorithms converge essentially to the same plan. However, we demonstrate that for any of the algorithms studied, starting with previously optimized intensity distributions as the initial guess but for different objective function parameters, the solution frequently gets trapped in local minima. We found that the initial intensity distribution obtained from IMRT optimization utilizing objective function parameters, which favor a specific anatomic structure, would lead to a local minimum corresponding to that structure. Our results indicate that from among the gradient algorithms tested, Newton's method appears to be the fastest by far. Different gradient algorithms have the same convergence properties for dose-volume- and EUD-based objective functions. The hybrid dose calculation strategy is valid and can significantly accelerate the optimization process. The degree of acceleration achieved depends on the type of optimization problem being addressed (e.g., IMRT optimization, intensity modulated beam configuration optimization, or objective function parameter optimization). Under special conditions, gradient algorithms will get trapped in local minima, and reoptimization, starting with the results of previous optimization, will lead to solutions that are generally not significantly different from the local minimum.

Sensitivity of treatment plan optimisation for prostate cancer using the equivalent uniform dose (EUD) with respect to the rectal wall volume parameter

BACKGROUND AND PURPOSE

To analyse the sensitivity of plan optimisation of prostate cancer treatments with respect to changes in the volume parameter (n), when the EUD is used to control the dose in the rectal wall.

PATIENTS AND METHODS

A series of plans was defined, by varying n over a range between 0.08 and 1, and testing different cost functions and beam arrangements. In all cases, the aim was to minimise the EUD in the rectal wall, while ensuring specific dose coverage of the PTV, and limiting the dose in the other OARs. The results were evaluated in terms of 3-D dose distribution and with respect to the current clinical knowledge about late rectal toxicity after irradiation.

RESULTS

Different values of n lead to very similar dose distributions over the PTV (differences in mean dose < 1 Gy, differences in dose given to 99% of the volume < 1%). For the rectal wall, the following observations were made: (a) all cumulative DVH curves crossed each other around 60 Gy; (b) the rectal wall volume receiving doses between 30 and 45 Gy could change by 45 and 30%, respectively, depending on the value of n; (c) for doses higher than 70Gy the differences were typically within 5%. Different values of n also affected the position of isodose surfaces. The distance between the 70 and the 30 Gy isodose curves changed in the AP direction by a factor of 3 when n decreased from 1 to 0.08. High values of n were associated with less dose conformity and a larger volume (at least 20%) of normal tissues receiving 50 Gy or more. All DVHs for the rectal wall were below published dose toxicity thresholds except when the prescribed dose was escalated up to 86 Gy.

CONCLUSIONS

In most cases, the solutions associated with n values up to 0.25 produced similar dose distribution in the rectal wall for doses above 45 Gy, complying with the dose-toxicity thresholds we analysed. The choice of a specific value of n in the optimisation requires an analysis of its effects on the dose distribution for the rectal wall, but also on other aspects, such as the value of the dose to the non-involved normal tissues.

Degeneracy, frequency response and filtering in IMRT optimization

This paper attempts to provide an answer to some questions that remain either poorly understood, or not well documented in the literature, on basic issues related to intensity modulated radiation therapy (IMRT). The questions examined are: the relationship between degeneracy and frequency response of optimizations, effects of initial beamlet fluence assignment and stopping point, what does filtering of an optimized beamlet map actually do and how could image analysis help to obtain better optimizations? Two target functions are studied, a quadratic cost function and the log likelihood function of the dynamically penalized likelihood (DPL) algorithm. The algorithms used are the conjugate gradient, the stochastic adaptive simulated annealing and the DPL. One simple phantom is used to show the development of the analysis tools used and two clinical cases of medium and large dose matrix size (a meningioma and a prostate) are studied in detail. The conclusions reached are that the high number of iterations that is needed to avoid degeneracy is not warranted in clinical practice, as the quality of the optimizations, as judged by the DVHs and dose distributions obtained, does not improve significantly after a certain point. It is also shown that the optimum initial beamlet fluence assignment for analytical iterative algorithms is a uniform distribution, but such an assignment does not help a stochastic method of optimization. Stopping points for the studied algorithms are discussed and the deterioration of DVH characteristics with filtering is shown to be partially recoverable by the use of space-variant filtering techniques.

Multiobjective inverse planning for intensity modulated radiotherapy with constraint-free gradient-based optimization algorithms

We consider the behaviour of the limited memory L-BFGS algorithm as a representative constraint-free gradient-based algorithm which is used for multiobjective (MO) dose optimization for intensity modulated radiotherapy (IMRT). Using a parameter transformation, the positivity constraint problem of negative beam fluences is entirely eliminated: a feature which to date has not been fully understood by all investigators. We analyse the global convergence properties of L-BFGS by searching for the existence and the influence of possible local minima. With a fast simulated annealing (FSA) algorithm we examine whether the L-BFGS solutions are globally Pareto optimal. The three examples used in our analysis are a brain tumour, a prostate tumour and a test case with a C-shaped PTV. In 1% of the optimizations global convergence is violated. A simple mechanism practically eliminates the influence of this failure and the obtained solutions are globally optimal. A single-objective dose optimization requires less than 4 s for 5400 parameters and 40000 sampling points. The elimination of the problem of negative beam fluences and the high computational speed permit constraint-free gradient-based optimization algorithms to be used for MO dose optimization. In this situation, a representative spectrum of possible solutions is obtained which contains information such as the trade-off between the objectives and range of dose values. Using simple decision making tools the best of all the possible solutions can be chosen. We perform an MO dose optimization for the three examples and compare the spectra of solutions, firstly using recommended critical dose values for the organs at risk and secondly, setting these dose values to zero.

IMRT optimization: Variability of solutions and its radiobiological impact

We aim at (1) defining and measuring a "complexity" index for the optimization process of an intensity modulated radiation therapy treatment plan (IMRT TP), (2) devising an efficient approximate optimization strategy, and (3) evaluating the impact of the complexity of the optimization process on the radiobiological quality of the treatment. In this work, for a prostate therapy case, the IMRT TP optimization problem has been formulated in terms of dose-volume constraints. The cost function has been minimized in order to achieve the optimal solution, by means of an iterative procedure, which is repeated for many initial modulation profiles, and for each of them the final optimal solution is recorded. To explore the complexity of the space of such solutions we have chosen to minimize the cost function with an algorithm that is unable to avoid local minima. The size of the (sub)optimal solutions distribution is taken as an indicator of the complexity of the optimization problem. The impact of the estimated complexity on the probability of success of the therapy is evaluated using radiobiological indicators (Poissonian TCP model [S. Webb and A. E. Nahum, Phys. Med. Biol. 38(6), 653-666 (1993)] and NTCP relative seriality model [Kallman et al., Int. J. Radiat. Biol. 62(2), 249-262 (1992)]). We find in the examined prostate case a nontrivial distribution of local minima, which has symmetry properties allowing a good estimate of near-optimal solutions with a moderate computational load. We finally demonstrate that reducing the a priori uncertainty in the optimal solution results in a significant improvement of the probability of success of the TP, based on TCP and NTCP estimates.

Optimizer convergence and local minima errors and their clinical importance

Two of the errors common in the inverse treatment planning optimization have been investigated. The first error is the optimizer convergence error, which appears because of non-perfect convergence to the global or local solution, usually caused by a non-zero stopping criterion. The second error is the local minima error, which occurs when the objective function is not convex and/or the feasible solution space is not convex. The magnitude of the errors, their relative importance in comparison to other errors as well as their clinical significance in terms of tumour control probability (TCP) and normal tissue complication probability (NTCP) were investigated. Two inherently different optimizers, a stochastic simulated annealing and deterministic gradient method were compared on a clinical example. It was found that for typical optimization the optimizer convergence errors are rather small, especially compared to other convergence errors, e.g., convergence errors due to inaccuracy of the current dose calculation algorithms. This indicates that stopping criteria could often be relaxed leading into optimization speed-ups. The local minima errors were also found to be relatively small and typically in the range of the dose calculation convergence errors. Even for the cases where significantly higher objective function scores were obtained the local minima errors were not significantly higher. Clinical evaluation of the optimizer convergence error showed good correlation between the convergence of the clinical TCP or NTCP measures and convergence of the physical dose distribution. On the other hand, the local minima errors resulted in significantly different TCP or NTCP values (up to a factor of 2) indicating clinical importance of the local minima produced by physical optimization.

The method of intercepts in parameter space for the analysis of local minima caused by dose-volume constraints

The local minima problem in radiotherapy optimization has been a concern for both researchers and physicians. In this work, local minima induced by dose-volume histogram (DVH) constraints are discussed. The non-convex property of the feasible set formed by DVH constraints is discussed in beam weight space. An intuitive explanation of the origin of this type of local minima is given by a two-beam model setup. Some interesting properties and insights about the DVH-induced local minima are found. Based on this, a heuristic non-random initial guess sampling method is proposed and applied to a clinical nasopharyngeal case, where some significantly different local minima are located.

Global convergence analysis of fast multiobjective gradient-based dose optimization algorithms for high-dose-rate brachytherapy

We consider the problem of the global convergence of gradient-based optimization algorithms for interstitial high-dose-rate (HDR) brachytherapy dose optimization using variance-based objectives. Possible local minima could lead to only sub-optimal solutions. We perform a configuration space analysis using a representative set of the entire non-dominated solution space. A set of three prostate implants is used in this study. We compare the results obtained by conjugate gradient algorithms, two variable metric algorithms and fast-simulated annealing. For the variable metric algorithm BFGS from numerical recipes, large fluctuations are observed. The limited memory L-BFGS algorithm and the conjugate gradient algorithm FRPR are globally convergent. Local minima or degenerate states are not observed. We study the possibility of obtaining a representative set of non-dominated solutions using optimal solution rearrangement and a warm start mechanism. For the surface and volume dose variance and their derivatives, a method is proposed which significantly reduces the number of required operations. The optimization time, ignoring a preprocessing step, is independent of the number of sampling points in the planning target volume. Multiobjective dose optimization in HDR brachytherapy using L-BFGS and a new modified computation method for the objectives and derivatives has been accelerated, depending on the number of sampling points, by a factor in the range 10-100.

Absence of multiple local minima effects in intensity modulated optimization with dose-volume constraints

This paper reports on the analysis of intensity modulated radiation treatment optimization problems in the presence of non-convex feasible parameter spaces caused by the specification of dose-volume constraints for the organs-at-risk (OARs). The main aim was to determine whether the presence of those non-convex spaces affects the optimization of clinical cases in any significant way. This was done in two phases: (1) Using a carefully designed two-dimensional mathematical phantom that exhibits two controllable minima and with randomly initialized beamlet weights, we developed a methodology for exploring the nature of the convergence characteristics of quadratic cost function optimizations (deterministic or stochastic). The methodology is based on observing the statistical behaviour of the residual cost at the end of optimizations in which the stopping criterion is progressively more demanding and carrying out those optimizations to very small error changes per iteration. (2) Seven clinical cases were then analysed with dose-volume constraints that are stronger than originally used in the clinic. The clinical cases are two prostate cases differently posed, a meningioma case, two head-and-neck cases, a spleen case and a spine case. Of the 14 different sets of optimizations (with and without the specification of maximum doses allowed for the OARs), 12 fail to show any effect due to the existence of non-convex feasible spaces. The remaining two sets of optimizations show evidence of multiple minima in the solutions, but those minima are very close to each other in cost and the resulting treatment plans are practically identical, as measured by the quality of the dose-volume histograms (DVHs). We discuss the differences between fluence maps resulting from those similar treatment plans. We provide a possible reason for the observed results and conclude that, although the study is necessarily limited, the annealing characteristics of a simulated annealing method may not be justified in clinical optimization in the presence of dose-volume constraints. The results of optimizations by the Newton gradient (NG) method with a quadratic cost function are reported in detail. An adaptive simulated annealing method, optimizing the same function, and the dynamically penalized likelihood method, optimizing a log likelihood function, have also been used in the study. The results of the latter two methods have only been discussed briefly, as they yielded the same conclusions as the NG method.