L-curve analysis of radiotherapy optimization problems

Robust optimization in IMPT using quadratic objective functions to account for the minimum MU constraint

PURPOSE

Currently, in clinical practice of intensity-modulated proton therapy (IMPT), the influence of the minimum monitor unit (MU) constraint is taken into account through postprocessing after the optimization is completed. This may degrade the plan quality and plan robustness. This study aims to mitigate the impact of the minimum MU constraint directly during the plan robust optimization.

METHODS AND MATERIALS

Cao et al. have demonstrated a two-stage method to account for the minimum MU constraint using linear programming without the impact of uncertainties considered. In this study, we took the minimum MU constraint into consideration using quadratic optimization and simultaneously had the impact of uncertainties considered using robust optimization. We evaluated our method using seven cancer patients with different machine settings.

RESULT

The new method achieved better plan quality than the conventional method. The D_95% of the clinical target volume (CTV) normalized to the prescription dose was (mean [min-max]): (99.4% [99.2%-99.6%]) vs. (99.2% [98.6%-99.6%]). Plan robustness derived from these two methods was comparable. For all seven patients, the CTV dose-volume histogram band gap (narrower band gap means more robust plans) at D_95% normalized to the prescription dose was (mean [min-max]): (1.5% [0.5%-4.3%]) vs. (1.2% [0.6%-3.8%]).

CONCLUSION

Our new method of incorporating the minimum MU constraint directly into the plan robust optimization can produce machine-deliverable plans with better tumor coverage while maintaining high-plan robustness.

Multicriteria optimization for volumetric-modulated arc therapy by decomposition into a fluence-based relaxation and a segment weight-based restriction

PURPOSE

To develop a method for inverse volumetric-modulated arc therapy (VMAT) planning that combines multicriteria optimization (MCO) with direct machine parameter optimization. The ultimate goal is to provide an efficient and intuitive method for generating high quality VMAT plans.

METHODS

Multicriteria radiation therapy treatment planning amounts to approximating the relevant treatment options by a discrete set of plans, and selecting the combination thereof that strikes the best possible balance between conflicting objectives. This approach is applied to two decompositions of the inverse VMAT planning problem: a fluence-based relaxation considered at a coarsened gantry angle spacing and under a regularizing penalty on fluence modulation, and a segment weight-based restriction in a neighborhood of the solution to the relaxed problem. The two considered variable domains are interconnected by direct machine parameter optimization toward reproducing the dose-volume histogram of the fluence-based solution.

RESULTS

The dose distribution quality of plans generated by the proposed MCO method was assessed by direct comparison with benchmark plans generated by a conventional VMAT planning method. The results for four patient cases (prostate, pancreas, lung, and head and neck) are highly comparable between the MCO plans and the benchmark plans: Discrepancies between studied dose-volume statistics for organs at risk were-with the exception of the kidneys of the pancreas case-within 1 Gy or 1 percentage point. Target coverage of the MCO plans was comparable with that of the benchmark plans, but with a small tendency toward a shift from conformity to homogeneity.

CONCLUSIONS

MCO allows tradeoffs between conflicting objectives encountered in VMAT planning to be explored in an interactive manner through search over a continuous representation of the relevant treatment options. Treatment plans selected from such a representation are of comparable dose distribution quality to conventionally optimized VMAT plans.

Inverse planning for IMRT with nonuniform beam profiles using total-variation regularization (TVR)

PURPOSE

Radiation therapy with high dose rate and flattening filter-free (FFF) beams has the potential advantage of greatly reduced treatment time and out-of-field dose. Current inverse planning algorithms are, however, not customized for beams with nonuniform incident profiles and the resultant IMRT plans are often inefficient in delivery. The authors propose a total-variation regularization (TVR)-based formalism by taking the inherent shapes of incident beam profiles into account.

METHODS

A novel TVR-based inverse planning formalism is established for IMRT with nonuniform beam profiles. The authors introduce a TVR term into the objective function, which encourages piecewise constant fluence in the nonuniform FFF fluence domain. The proposed algorithm is applied to lung and prostate and head and neck cases and its performance is evaluated by comparing the resulting plans to those obtained using a conventional beamlet-based optimization (BBO).

RESULTS

For the prostate case, the authors' algorithm produces acceptable dose distributions with only 21 segments, while the conventional BBO requires 114 segments. For the lung case and the head and neck case, the proposed method generates similar coverage of target volume and sparing of the organs-at-risk as compared to BBO, but with a markedly reduced segment number.

CONCLUSIONS

TVR-based optimization in nonflat beam domain provides an effective way to maximally leverage the technical capacity of radiation therapy with FFF fields. The technique can generate effective IMRT plans with improved dose delivery efficiency without significant deterioration of the dose distribution.

Search for IMRT inverse plans with piecewise constant fluence maps using compressed sensing techniques

An intensity-modulated radiation therapy (IMRT) field is composed of a series of segmented beams. It is practically important to reduce the number of segments while maintaining the conformality of the final dose distribution. In this article, the authors quantify the complexity of an IMRT fluence map by introducing the concept of sparsity of fluence maps and formulate the inverse planning problem into a framework of compressing sensing. In this approach, the treatment planning is modeled as a multiobjective optimization problem, with one objective on the dose performance and the other on the sparsity of the resultant fluence maps. A Pareto frontier is calculated, and the achieved dose distributions associated with the Pareto efficient points are evaluated using clinical acceptance criteria. The clinically acceptable dose distribution with the smallest number of segments is chosen as the final solution. The method is demonstrated in the application of fixed-gantry IMRT on a prostate patient. The result shows that the total number of segments is greatly reduced while a satisfactory dose distribution is still achieved. With the focus on the sparsity of the optimal solution, the proposed method is distinct from the existing beamlet- or segment-based optimization algorithms.

Interpretation of the dosimetric results of three uniformity regularization methods in terms of expected treatment outcome

In IMRT treatment plan optimization there are various methods that try to regularize the variation of dose nonuniformity using purely dosimetric measures. However, although these methods can help in finding a good dose distribution, they do not provide any information regarding the expected treatment outcome. When a treatment plan optimization is performed using biological measures, the final goal should be some indication about the expected tumor control or normal tissue complications, which is the primary goal of treatment planning (the association of treatment configurations and dose prescription with the treatment outcome). In this study, this issue is analyzed distinguishing the dose-oriented treatment plan optimization from the response-oriented optimization. Three different dose distributions were obtained by using a dose-based optimization technique, an EUD-based optimization without applying any technique for regularizing the nonuniformity of the dose distribution, and an EUD-based optimization using a variational regularization technique, which controls dose nonuniformity. The clinical effectiveness of the three dose distributions was investigated by calculating the response probabilities of the tumors and organs-at-risk (OARs) involved in two head and neck and prostate cancer cases. The radiobiological models used are the linear-quadratic-Poisson and the Relative Seriality models. Furthermore, the complication-free tumor control probability and the biologically effective uniform dose (D) were used for treatment plan evaluation and comparison. The radiobiological comparison shows that the EUD-based optimization using L-curve regularization gives better results than the EUD-based optimization without regularization and dose-based optimization in both clinical cases. Concluding, it appears that the applied dose nonuniformity regularization technique is expected to improve the effectiveness of the optimized IMRT dose distributions. However, more patient cases are needed to validate the statistical significance of the results and conclusions presented in this paper.

Optimization of equivalent uniform dose using the L-curve criterion

Optimization of equivalent uniform dose (EUD) in inverse planning for intensity-modulated radiation therapy (IMRT) prevents variation in radiobiological effect between different radiotherapy treatment plans, which is due to variation in the pattern of dose nonuniformity. For instance, the survival fraction of clonogens would be consistent with the prescription when the optimized EUD is equal to the prescribed EUD. One of the problems in the practical implementation of this approach is that the spatial dose distribution in EUD-based inverse planning would be underdetermined because an unlimited number of nonuniform dose distributions can be computed for a prescribed value of EUD. Together with ill-posedness of the underlying integral equation, this may significantly increase the dose nonuniformity. To optimize EUD and keep dose nonuniformity within reasonable limits, we implemented into an EUD-based objective function an additional criterion which ensures the smoothness of beam intensity functions. This approach is similar to the variational regularization technique which was previously studied for the dose-based least-squares optimization. We show that the variational regularization together with the L-curve criterion for the regularization parameter can significantly reduce dose nonuniformity in EUD-based inverse planning.

Dynamic-MLC leaf control utilizing on-flight intensity calculations: A robust method for real-time IMRT delivery over moving rigid targets

An algorithm is presented that allows for the control of multileaf collimation (MLC) leaves based entirely on real-time calculations of the intensity delivered over the target. The algorithm is capable of efficiently correcting generalized delivery errors without requiring the interruption of delivery (self-correcting trajectories), where a generalized delivery error represents anything that causes a discrepancy between the delivered and intended intensity profiles. The intensity actually delivered over the target is continually compared to its intended value. For each pair of leaves, these comparisons are used to guide the control of the following leaf and keep this discrepancy below a user-specified value. To demonstrate the basic principles of the algorithm, results of corrected delivery are shown for a leading leaf positional error during dynamic-MLC (DMLC) IMRT delivery over a rigid moving target. It is then shown that, with slight modifications, the algorithm can be used to track moving targets in real time. The primary results of this article indicate that the algorithm is capable of accurately delivering DMLC IMRT over a rigid moving target whose motion is (1) completely unknown prior to delivery and (2) not faster than the maximum MLC leaf velocity over extended periods of time. These capabilities are demonstrated for clinically derived intensity profiles and actual tumor motion data, including situations when the target moves in some instances faster than the maximum admissible MLC leaf velocity. The results show that using the algorithm while calculating the delivered intensity every 50 ms will provide a good level of accuracy when delivering IMRT over a rigid moving target translating along the direction of MLC leaf travel. When the maximum velocities of the MLC leaves and target were 4 and 4.2 cm/s, respectively, the resulting error in the two intensity profiles used was 0.1 +/- 3.1% and -0.5 +/- 2.8% relative to the maximum of the intensity profiles. For the same target motion, the error was shown to increase rapidly as (1) the maximum MLC leaf velocity was reduced below 75% of the maximum target velocity and (2) the system response time was increased.

Iterative regularization in intensity-modulated radiation therapy optimization

A common way to solve intensity-modulated radiation therapy (IMRT) optimization problems is to use a beamlet-based approach. The approach is usually employed in a three-step manner: first a beamlet-weight optimization problem is solved, then the fluence profiles are converted into step-and-shoot segments, and finally postoptimization of the segment weights is performed. A drawback of beamlet-based approaches is that beamlet-weight optimization problems are ill-conditioned and have to be regularized in order to produce smooth fluence profiles that are suitable for conversion. The purpose of this paper is twofold: first, to explain the suitability of solving beamlet-based IMRT problems by a BFGS quasi-Newton sequential quadratic programming method with diagonal initial Hessian estimate, and second, to empirically show that beamlet-weight optimization problems should be solved in relatively few iterations when using this optimization method. The explanation of the suitability is based on viewing the optimization method as an iterative regularization method. In iterative regularization, the optimization problem is solved approximately by iterating long enough to obtain a solution close to the optimal one, but terminating before too much noise occurs. Iterative regularization requires an optimization method that initially proceeds in smooth directions and makes rapid initial progress. Solving ten beamlet-based IMRT problems with dose-volume objectives and bounds on the beamlet-weights, we find that the considered optimization method fulfills the requirements for performing iterative regularization. After segment-weight optimization, the treatments obtained using 35 beamlet-weight iterations outperform the treatments obtained using 100 beamlet-weight iterations, both in terms of objective value and of target uniformity. We conclude that iterating too long may in fact deteriorate the quality of the deliverable plan.