Regularization of inverse planning for intensity-modulated radiotherapy

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.

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.

The IMRT information process—mastering the degrees of freedom in external beam therapy

The techniques and procedures for intensity-modulated radiation therapy (IMRT) are reviewed in the context of the information process central to treatment planning and delivery of IMRT. A presentation is given of the evolution of the information based radiotherapy workflow and dose delivery techniques, as well as the volume and planning concepts for relating the dose information to image based patient representations. The formulation of the dose shaping process as an optimization problem is described. The different steps in the calculation flow for determination of machine parameters for dose delivery are described starting from the formulation of optimization objectives over dose calculation to optimization procedures. Finally, the main elements of the quality assurance procedure necessary for implementing IMRT clinically are reviewed.

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.

L-curve analysis of radiotherapy optimization problems

We attempt to select an optimal value of regularization parameter in the optimization problems for intensity-modulated radiotherapy which are solved using a variational regularization technique. We apply to inverse treatment planning the L-curve method which was developed to determine the regularization parameter in the discrete ill-posed problems. The L-curve method is based on finding the regularization parameter which minimizes the residual norm which is a measure of accuracy of fit and the solution norm which is a measure of smoothness of solution. The main idea of the L-curve method is to plot the smoothing norm as a function of the residual norm for all values of the regularization parameter. This characteristic curve has an L-shaped dependence and the optimal value of regularization parameter can be found at the "corner" of the L-curve. We plot the L-curves for the optimization problems which simulate prostate radiotherapy cancer treatment with intensity-modulated beams. Different numerical methods are applied to calculate the point of maximum curvature of the L-curves which is a criterion to locate the corner. We show that the point of maximum curvature can be located in a most robust way using a formula derived from the singular value decomposition analysis.

A particle swarm optimization algorithm for beam angle selection in intensity-modulated radiotherapy planning

Automatic beam angle selection is an important but challenging problem for intensity-modulated radiation therapy (IMRT) planning. Though many efforts have been made, it is still not very satisfactory in clinical IMRT practice because of overextensive computation of the inverse problem. In this paper, a new technique named BASPSO (Beam Angle Selection with a Particle Swarm Optimization algorithm) is presented to improve the efficiency of the beam angle optimization problem. Originally developed as a tool for simulating social behaviour, the particle swarm optimization (PSO) algorithm is a relatively new population-based evolutionary optimization technique first introduced by Kennedy and Eberhart in 1995. In the proposed BASPSO, the beam angles are optimized using PSO by treating each beam configuration as a particle (individual), and the beam intensity maps for each beam configuration are optimized using the conjugate gradient (CG) algorithm. These two optimization processes are implemented iteratively. The performance of each individual is evaluated by a fitness value calculated with a physical objective function. A population of these individuals is evolved by cooperation and competition among the individuals themselves through generations. The optimization results of a simulated case with known optimal beam angles and two clinical cases (a prostate case and a head-and-neck case) show that PSO is valid and efficient and can speed up the beam angle optimization process. Furthermore, the performance comparisons based on the preliminary results indicate that, as a whole, the PSO-based algorithm seems to outperform, or at least compete with, the GA-based algorithm in computation time and robustness. In conclusion, the reported work suggested that the introduced PSO algorithm could act as a new promising solution to the beam angle optimization problem and potentially other optimization problems in IMRT, though further studies need to be investigated.