Mathematical optimization of high dose-rate brachytherapy—derivation of a linear penalty model from a dose-volume model

Robust stochastic optimization of needle configurations for robotic HDR prostate brachytherapy

BACKGROUND

Ideally, inverse planning for HDR brachytherapy (BT) should include the pose of the needles which define the trajectory of the source. This would be particularly interesting when considering the additional freedom and accuracy in needle pose which robotic needle placement enables. However, needle insertion typically leads to tissue deformation, resulting in uncertainty regarding the actual pose of the needles with respect to the tissue.

PURPOSE

To efficiently address uncertainty during inverse planning for HDR BT in order to robustly optimize the pose of the needles before insertion, that is, to facilitate path planning for robotic needle placement.

METHODS

We use a form of stochastic linear programming to model the inverse treatment planning problem. To account for uncertainty, we consider random tissue displacements at the needle tip to simulate tissue deformation. Conventionally for stochastic linear programming, each simulated deformation is reflected by an addition to the linear programming problem which increases problem size and computational complexity substantially and leads to impractical runtime. We propose two efficient approaches for stochastic linear programming. First, we consider averaging dose coefficients to reduce the problem size. Second, we study weighting of the slack variables of an adjusted linear problem to approximate the full stochastic linear program. We compare different approaches to optimize the needle configurations and evaluate their robustness with respect to different amounts of tissue deformation.

RESULTS

Our results illustrate that stochastic planning can improve the robustness of the treatment with respect to deformation. The proposed approaches approximating stochastic linear programming better conform to the tissue deformation compared to conventional linear programming. They show good correlation with the plans computed after deformation while reducing the runtime by two orders of magnitude compared to the complete stochastic linear program. Robust optimization of needle configurations takes on average 59.42 s. Skew needle configurations lead to mean coverage improvements compared to parallel needles from 0.39 to 2.94 percentage points, when 8 mm tissue deformation is considered. Considering tissue deformations from 4  to 10 mm during planning with weighted stochastic optimization and skew needles generally results in improved mean coverage from 1.77 to 4.21 percentage points.

CONCLUSIONS

We show that efficient stochastic optimization allows selecting needle configurations which are more robust with respect to potentially negative effects of target deformation and displacement on the achievable prescription dose coverage. The approach facilitates robust path planning for robotic needle placement.

Benchmark of the PenRed Monte Carlo framework for HDR brachytherapy

PURPOSE

The purpose of this study is to validate the PenRed Monte Carlo framework for clinical applications in brachytherapy. PenRed is a C++ version of Penelope Monte Carlo code with additional tallies and utilities.

METHODS AND MATERIALS

Six benchmarking scenarios are explored to validate the use of PenRed and its improved bachytherapy-oriented capabilities for HDR brachytherapy. A new tally allowing the evaluation of collisional kerma for any material using the track length kerma estimator and the possibility to obtain the seed positions, weights and directions processing directly the DICOM file are now implemented in the PenRed distribution. The four non-clinical test cases developed by the Joint AAPM-ESTRO-ABG-ABS WG-DCAB were evaluated by comparing local and global absorbed dose differences with respect to established reference datasets. A prostate and a palliative lung cases, were also studied. For them, absorbed dose ratios, global absorbed dose differences, and cumulative dose-volume histograms were obtained and discussed.

RESULTS

The air-kerma strength and the dose rate constant corresponding to the two sources agree with the reference datatests within 0.3% (Sk) and 0.1% (Λ). With respect to the first three WG-DCAB test cases, more than 99.8% of the voxels present local (global) differences within ±1%(±0.1%) of the reference datasets. For test Case 4 reference dataset, more than 94.9%(97.5%) of voxels show an agreement within ±1%(±0.1%), better than similar benchmarking calculations in the literature. The track length kerma estimator scorer implemented increases the numerical efficiency of brachytherapy calculations two orders of magnitude, while the specific brachytherapy source allows the user to avoid the use of error-prone intermediate steps to translate the DICOM information into the simulation. In both clinical cases, only minor absorbed dose differences arise in the low-dose isodoses. 99.8% and 100% of the voxels have a global absorbed dose difference ratio within ±0.2% for the prostate and lung cases, respectively. The role played by the different segmentation and composition material in the bone structures was discussed, obtaining negligible absorbed dose differences. Dose-volume histograms were in agreement with the reference data.

CONCLUSIONS

PenRed incorporates new tallies and utilities and has been validated for its use for detailed and precise high-dose-rate brachytherapy simulations.

Dosimetric impact of a robust optimization approach to mitigate effects from rotational uncertainty in prostate intensity-modulated brachytherapy

BACKGROUND

Intensity-modulated brachytherapy (IMBT) is an emerging technology for cancer treatment, in which radiation sources are shielded to shape the dose distribution. The rotatable shields provide an additional degree of freedom, but also introduce an additional, directional, type of uncertainty, compared to conventional high-dose-rate brachytherapy (HDR BT).

PURPOSE

We propose and evaluate a robust optimization approach to mitigate the effects of rotational uncertainty in the shields with respect to planning criteria.

METHODS

A previously suggested prototype for platinum-shielded prostate ¹⁶⁹ Yb-based dynamic IMBT is considered. We study a retrospective patient data set (anatomical contours and catheter placement) from two clinics, consisting of six patients that had previously undergone conventional ¹⁹² Ir HDR BT treatment. The Monte Carlo-based treatment planning software RapidBrachyMCTPS is used for dose calculations. In our computational experiments, we investigate systematic rotational shield errors of ±10° and ±20°, and the same systematic error is applied to all dwell positions in each scenario. This gives us three scenarios, one nominal and two with errors. The robust optimization approach finds a compromise between the average and worst-case scenario outcomes.

RESULTS

We compare dose plans obtained from standard models and their robust counterparts. With dwell times obtained from a linear penalty model (LPM), for 10° errors, the dose to urethra ( D 0.1 c c $D_{0.1cc}$ ) and rectum ( D 0.1 c c $D_{0.1cc}$ and D 1 c c $D_{1cc}$ ) increase with up to 5% and 7%, respectively, in the worst-case scenario, while with the robust counterpart, the corresponding increases were 3% and 3%. For all patients and all evaluated criteria, the worst-case scenario outcome with the robust approach had lower deviation compared to the standard model, without compromising target coverage. We also evaluated shield errors up to 20° and while the deviations increased to a large extent with the standard models, the robust models were capable of handling even such large errors.

CONCLUSIONS

We conclude that robust optimization can be used to mitigate the effects from rotational uncertainty and to ensure the treatment plan quality of IMBT.

Incorporating Treatment Time into Butterfly Optimization to Reduce Total Treatment Time for Vaginal Cylinder Brachytherapy

Purpose

For patient comfort and safety, irradiation times should be kept at a minimum while maintaining high treatment quality. In this study of high dose rate (HDR) therapy with a vaginal cylinder, we used the butterfly optimization algorithm (BOA) to simultaneously optimize individual dwell times for precise dose conformity and for the reduction of total dwell time.

Material and methods

BOA is a population-based, meta-heuristic algorithm that averts local minima by conducting intensive local and global searching based on switching probability. We constructed an objective function (a stimulus intensity function) that consisted of two components. The first one was the root-mean-squared dose error (RMSE) defined as the square root of the sum of squared differences between the prescribed and delivered dose at the constraint points. The second component was weighted total treatment time. Eight previously treated cases were retrospectively reviewed by re-optimizing the clinical treatment plans with BOA. 

Results

Compared to the eight original plans generated with the commercial adaptive volume optimization algorithm (AVOA), the BOA-optimized plans reduced treatment times by 5.4% to 8.9%, corresponding to a time-saving of 13.1 to 47.7 seconds with the activities on the treatment day and saving from 29.3 to 64.6 seconds if treated with an activity of 5 CI. Dose deviations from the prescription were smaller than in the original plans.

Conclusion 

Dose optimizations based on the BOA algorithm yield closer dose conformity in vaginal HDR treatment than AVOA. Incorporating total treatment time into the optimization algorithm reduces the delivery time while having only a small effect on dose conformity.

Emerging technologies in brachytherapy

Brachytherapy is a mature treatment modality. The literature is abundant in terms of review articles and comprehensive books on the latest established as well as evolving clinical practices. The intent of this article is to part ways and look beyond the current state-of-the-art and review emerging technologies that are noteworthy and perhaps may drive the future innovations in the field. There are plenty of candidate topics that deserve a deeper look, of course, but with practical limits in this communicative platform, we explore four topics that perhaps is worthwhile to review in detail at this time. First, intensity modulated brachytherapy (IMBT) is reviewed. The IMBT takes advantage ofanisotropicradiation profile generated through intelligent high-density shielding designs incorporated onto sources and applicators such to achieve high quality plans. Second, emerging applications of 3D printing (i.e. additive manufacturing) in brachytherapy are reviewed. With the advent of 3D printing, interest in this technology in brachytherapy has been immense and translation swift due to their potential to tailor applicators and treatments customizable to each individual patient. This is followed by, in third, innovations in treatment planning concerning catheter placement and dwell times where new modelling approaches, solution algorithms, and technological advances are reviewed. And, fourth and lastly, applications of a new machine learning technique, called deep learning, which has the potential to improve and automate all aspects of brachytherapy workflow, are reviewed. We do not expect that all ideas and innovations reviewed in this article will ultimately reach clinic but, nonetheless, this review provides a decent glimpse of what is to come. It would be exciting to monitor as IMBT, 3D printing, novel optimization algorithms, and deep learning technologies evolve over time and translate into pilot testing and sensibly phased clinical trials, and ultimately make a difference for cancer patients. Today's fancy is tomorrow's reality. The future is bright for brachytherapy.

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.

Feasibility Study of Robust Optimization to Reduce Dose Delivery Uncertainty by Potential Applicator Displacements for a Cervix Brachytherapy

Brachytherapy is an important technique to increase the overall survival of cervical cancer patients. However, a possible shift of the applicators in relation to the target and organs at risk may occur between imaging and treatment. Without daily adaptive brachytherapy planning, these applicator displacements can lead to a significant change in dose distribution. In order to resolve it, a robust optimization method had been developed using a genetic algorithm combined with a median absolute deviation as a robustness evaluation function. The resulting robustness plans from our strategy might be worth considering according to the GEC-ESTRO guidelines. From the point of view of dose delivery uncertainty from applicator displacement, the robust optimization may be considered with caution in a single-plan approach for High Dose Rate brachytherapy treatment planning and should be confirmed by a more thorough investigation.

Simultaneous needle catheter selection and dwell time optimization for preplanning of high-dose-rate brachytherapy of prostate cancer

PURPOSE

Needle catheter positions critically affect the quality of treatment plans in prostate cancer high-dose-rate (HDR) brachytherapy. The current standard needle positioning approach is based on human intuition, which cannot guarantee a high-quality plan. This study proposed a method to simultaneously select needle catheter positions and determine dwell time for preplanning of HDR brachytherapy of prostate cancer.

METHODS

We formulated the needle catheter selection problem and inverse dwell time optimization problem in a unified framework. In addition to the dose objectives of the planning target volume (PTV) and organs at risk (OARs), the objective function incorporated a group-sparsity term with a needle-specific adaptive weighting scheme to generate high-quality plans with the minimal number of needle catheters. The optimization problem was solved by a fast-iterative shrinkage-thresholding algorithm. For validation purposes, we tested the proposed algorithm on 10 patient cases previously treated at our institution and compared the resulting plans with plans generated using needle catheters selected manually.

RESULTS

Compared to the plan with manually selected needle catheters, when normalizing both plans to the same PTV coverage V _100% = 95%, the plans generated by the proposed algorithm reduced median V _125% from 65% to 64%, but increased median V _150% from 35% to 38%, and V _200% from 14% to 16%. All planning objectives were met. All clinically important dosimetric parameters of OARs were reduced. D _1cc of bladder and rectum were reduced from 8.57 Gy to 8.50 Gy and from 7.24 Gy to 6.80 Gy, respectively. D _max of urethra was reduced from 15.85 Gy to 15.77 Gy. The median number of selected needle catheters was reduced by two. The computational time for solving the proposed optimization problem was ∼90 s using MATLAB.

CONCLUSION

The proposed algorithm was able to generate plans for prostate cancer HDR brachytherapy preplanning with increased median conformity index (0.73-0.77) and slightly lower median homogeneity index (0.64-0.62) with the number of selected needles reduced by two compared to the manual needle selection approach.

A new optimization algorithm for HDR brachytherapy that improves DVH-based planning: Truncated Conditional Value-at-Risk (TCVaR)

Purpose:To introduce a new optimization algorithm that improves DVH results and is designed for the type of heterogeneous dose distributions that occur in brachytherapy.Methods:The new optimization algorithm is based on a prior mathematical approach that uses mean doses of the DVH metric tails. The prior mean dose approach is referred to as conditional value-at-risk (CVaR), and unfortunately produces noticeably worse DVH metric results than gradient-based approaches. We have improved upon the CVaR approach, using the so-called Truncated CVaR (TCVaR), by excluding the hottest or coldest voxels in the structure from the calculations of the mean dose of the tail. Our approach applies an iterative sequence of convex approximations to improve the selection of the excluded voxels. Data Envelopment Analysis was used to quantify the sensitivity of TCVaR results to parameter choice and to compare the quality of a library of 256 TCVaR plans created for each of prostate, breast, and cervix treatment sites with commercially-generated plans.Results:In terms of traditional DVH metrics, TCVaR outperformed CVaR and the improvements increased monotonically as more iterations were used to identify and exclude the hottest/coldest voxels from the optimization problem. TCVaR also outperformed the Eclipse-Brachyvision TPS, with an improvement in PTVD95% (for equivalent organ-at-risk doses) of up to 5% (prostate), 3% (breast), and 1% (cervix).Conclusions:A novel optimization algorithm for HDR treatment planning produced plans with superior DVH metrics compared with a prior convex optimization algorithm as well as Eclipse-Brachyvision. The algorithm is computationally efficient and has potential applications as a primary optimization algorithm or quality assurance for existing optimization approaches.

Investigating the role of constrained CVT and CVT in HIPO inverse planning for HDR brachytherapy of prostate cancer

PURPOSE

The purpose of this study is to investigate the role of the centroidal Voronoi tessellation (CVT) and constrained CVT (CCVT) in inverse planning in combination with the Hybrid Inverse Planning Optimization (HIPO) algorithm in HDR brachytherapy of prostate cancer. HIPO implemented in Oncentra© Prostate treatment planning system, is used for three-dimensional (3D)-ultrasound-based intraoperative treatment planning in high dose rate brachytherapy. HIPO utilizes a hybrid iterative process to determine the most appropriate placement of a given number of catheters to fulfil predefined dose-volume constraints. The main goals of the current investigation were to identify a way of improving the performance of HIPO inverse planning; accelerating the HIPO, and to evaluate the effect of the two CVT-based initialization methods on the dose distribution in the sub-region of prostate that is not accessible by catheters, when trying to avoid perforation of urethra.

METHODS

We implemented the CVT algorithm to generate initial catheter configurations before the initialization of the HIPO algorithm. We introduced the CCVT algorithm to improve the dose distribution to the sub-volume of prostate within the bounding box of the urethra contours including its upper vertical extension (U-P). For the evaluation, we considered a total of 15 3D ultrasound-based HDRBT prostate implants. Execution time and treatment plan quality were evaluated based on the dose-volume histograms of prostate (PTV), its sub-volume U-P, and organs at risk (OARs). Furthermore, the conformity index COIN, the homogeneity index HI and the complication-free tumor control probability (P₊ ) were used for our treatment plan comparisons. Finally, the plans with the recommended HIPO execution mode were compared to the clinically used intraoperative pre-plans.

RESULTS

The plan quality achieved with CCVT-based HIPO initialization was superior to the default HIPO initialization method. Focusing on the U-P sub-region of the prostate, the CCVT method resulted in a significant improvement of all dosimetric indices compared to the default HIPO, when both were executed in the adaptive mode. For that recommended HIPO execution mode, and for U-P, CCVT demonstrated in general higher dosimetric indices than CVT. Additionally, the execution time of CCVT initialized HIPO was lower compared to both alternative initialization methods. This is also valid for the values of the aggregate objective function with the differences to the default initialization method being highly significant. Paired non-parametric statistical tests (Wilcoxon signed-rank) showed a significant improvement of dose-volume indices, COIN and P₊ for the plans generated by the CCVT-based catheter configuration initialization in HIPO compared to the default HIPO initialization process. Furthermore, in ten out of 15 cases, the CCVT-based HIPO plans fulfilled all the clinical dose-volume constraints in a single trial without any need for further catheter position adaption.

CONCLUSION

HIPO with CCVT-based initialization demonstrates better performance regarding the aggregate objective function and convergence when compared to the CVT-based and default catheter configuration initialization methods. This improved performance of HIPO inverse planning is clearly not at the cost of the dosimetric and radiobiologically evaluated plan quality. We recommend the use of the CCVT method for HIPO initialization especially in the adaptive planning mode.