Treatment planning for brachytherapy: an integer programming model, two computational approaches and experiments with permanent prostate implant planning

CT-guided Radioactive 125I Seed Implantation for Abdominal Incision Metastases of Colorectal Cancer: Safety and Efficacy in 17 Patients

INTRODUCTION

To retrospectively evaluate the safety and efficacy of computed tomography (CT)-guided iodine-125 (¹²⁵I) seed implantation for patients with abdominal incision metastases from colorectal cancer.

MATERIALS AND METHODS

Data of patients with abdominal incision metastases of colorectal cancer from November 2010 to October 2020 were retrospectively reviewed. Each incisional metastasis was percutaneously treated with ¹²⁵I seed implantation under CT guidance. Follow-up contrast-enhanced CT was reviewed, and the outcomes were evaluated in terms of objective response rate, complications, and overall survival.

RESULTS

A total of 17 patients were enrolled in this study. The median follow-up was 18 months (range, 2.7-22.1 months). At 3, 6, 12, and 18 months after the treatment, objective response rate was 52.9%, 63.6%, 33.3%, and 0%, respectively. A small amount of local hematoma occurred in two patients and resolved spontaneously without any treatment. Two patients experienced a minor displacement of radioactive seeds with no related symptoms. Severe complications, such as massive bleeding and radiation injury, were not observed. No ≥ grade 3 adverse events were identified. By the end of follow-up, 14 patients died of multiple hematogenous metastases. The one-year overall survival rate was 41.6%, and the median overall survival was 8.6 months.

CONCLUSION

CT-guided ¹²⁵I seed implantation brachytherapy is safe and feasible for patients with abdominal incision metastases from colorectal cancer.

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.

Feasibility and Clinical Value of CT-guided (125)I Brachytherapy for Bilateral Lung Recurrences from Colorectal Carcinoma

PURPOSE

To prospectively evaluate the feasibility and clinical value of computed tomography (CT)-guided iodine 125 ((125)I) brachytherapy to treat bilateral lung recurrences from colorectal carcinoma.

MATERIALS AND METHODS

This study was approved by Sun Yat-sen University Cancer Center Institutional Review Board and all patients provided informed written consent. Seventy-two patients with bilateral lung recurrences from colorectal carcinoma were enrolled and randomly divided into two groups. Thirty-three were percutaneously treated with CT-guided (125)I brachytherapy (group A) and the other 39 were only given symptomatic and supportive treatments (group B). Follow-up contrast agent-enhanced CT scans were reviewed and efficacy of treatment was evaluated. (125)I brachytherapy was considered a success if it achieved the computerized treatment planning system criteria 1 month after procedure. Analyses included Kaplan-Meier, Mantel-Cox log-rank test, and Cox proportional hazards regression.

RESULTS

In group A, 37 (125)I brachytherapy procedures were performed in 33 patients with 126 lung metastatic lesions and the success rate was 87.9% (29 of 33 patients). The local control rate of 3, 6, 12, 24, and 36 months was 75.8%, 51.5%, 33.3%, 24.2%, and 9.1%, respectively. A small amount of pulmonary hematoma occurred in five patients, and six patients presented with pneumothorax with pulmonary compression of 30%-40%. No massive bleeding or radiation pneumonitis occurred. The mean overall survival (OS) of group A was significantly longer than that of group B, and (125)I brachytherapy was an independent factor that affected the OS (group A, 18.8 months; group B, 8.6 months; hazard ratio, 0.391 [95% confidence interval: 0.196, 0.779]; P = .008).

CONCLUSION

CT-guided (125)I brachytherapy is feasible and safe for the treatment of bilateral lung recurrences from colorectal carcinoma.

An open-source genetic algorithm for determining optimal seed distributions for low-dose-rate prostate brachytherapy

PURPOSE

An open source optimizer that generates seed distributions for low-dose-rate prostate brachytherapy was designed, tested, and validated.

METHODS

The optimizer was a simple genetic algorithm (SGA) that, given a set of prostate and urethra contours, determines the optimal seed distribution in terms of coverage of the prostate with the prescribed dose while avoiding hotspots within the urethra. The algorithm was validated in a retrospective study on 45 previously contoured low-dose-rate prostate brachytherapy patients. Dosimetric indices were evaluated to ensure solutions adhered to clinical standards. The SGA performance was further benchmarked by comparing solutions obtained from a commercial optimizer (inverse planning simulated annealing [IPSA]) with the same cohort of 45 patients.

RESULTS

Clinically acceptable target coverage by the prescribed dose (V100) was obtained for both SGA and IPSA, with a mean ± standard deviation of 98 ± 2% and 99.5 ± 0.5%, respectively. For the prostate D90, SGA and IPSA yielded 177 ± 8 Gy and 186 ± 7 Gy, respectively, which were both clinically acceptable. Both algorithms yielded reasonable dose to the rectum, with V100 < 0.3 cc. A reduction in dose to the urethra was seen using SGA. SGA solutions showed a slight prostate volume dependence, with smaller prostates (<25 cc) yielding less desirable, although still clinically viable, dosimetric outcomes. SGA plans used, on average, fewer needles than IPSA (21 vs. 24, respectively), which may lead to a reduction in urinary toxicity and edema that alters post-implant dosimetry.

CONCLUSIONS

An open source SGA was validated that provides a research tool for the brachytherapy community.

A novel greedy heuristic-based approach to intraoperative planning for permanent prostate brachytherapy

This paper presents a greedy heuristic‐based double iteration and rectification (DIR) approach to intraoperative planning for permanent prostate brachytherapy. The DIR approach adopts a greedy seed selection (GSS) procedure to obtain a preliminary plan. In this process, the potential seeds are evaluated according to their ability to irradiate target while spare organs at risk (OARs), and their impact on dosimetric homogeneity within target volume. A flexible termination condition is developed for the GSS procedure, which guarantees sufficient dose within target volume while avoids overdosing the OARs. The preliminary treatment plan generated by the GSS procedure is further refined by the double iteration (DI) and rectification procedure. The DI procedure removes the needles containing only one seed (single seed) and implements the GSS procedure again to get a temporary plan. The DI procedure terminates until the needles number of the temporary plan does not decrease. This process is guided by constantly removing undesired part rather than imposing extra constrains. Following the DI procedure, the rectification procedure attempts to replace the remaining single seeds with the acceptable ones within the existing needles. The change of dosimetric distribution (DD) after the replacement is evaluated to determine whether to accept or to withdraw the replacement. Experimental results demonstrate that the treatment plans obtained by the DIR approach caters to all clinical considerations. Compared with currently available methods, DIR approach is faster, more reliable, and more suitable for intraoperative treatment planning in the operation room.

PACS number: 87

Treatment planning and image guidance for radiofrequency ablation of large tumors

This article addresses the two key challenges in computer-assisted percutaneous tumor ablation: planning multiple overlapping ablations for large tumors while avoiding critical structures, and executing the prescribed plan. Toward semiautomatic treatment planning for image-guided surgical interventions, we develop a systematic approach to the needle-based ablation placement task, ranging from preoperative planning algorithms to an intraoperative execution platform. The planning system incorporates clinical constraints on ablations and trajectories using a multiple objective optimization formulation, which consists of optimal path selection and ablation coverage optimization based on integer programming. The system implementation is presented and validated in both phantom and animal studies. The presented system can potentially be further extended for other ablation techniques such as cryotherapy.

Fast patient-specific Monte Carlo brachytherapy dose calculations via the correlated sampling variance reduction technique

PURPOSE

To demonstrate potential of correlated sampling Monte Carlo (CMC) simulation to improve the calculation efficiency for permanent seed brachytherapy (PSB) implants without loss of accuracy.

METHODS

CMC was implemented within an in-house MC code family (PTRAN) and used to compute 3D dose distributions for two patient cases: a clinical PSB postimplant prostate CT imaging study and a simulated post lumpectomy breast PSB implant planned on a screening dedicated breast cone-beam CT patient exam. CMC tallies the dose difference, ΔD, between highly correlated histories in homogeneous and heterogeneous geometries. The heterogeneous geometry histories were derived from photon collisions sampled in a geometrically identical but purely homogeneous medium geometry, by altering their particle weights to correct for bias. The prostate case consisted of 78 Model-6711 (125)I seeds. The breast case consisted of 87 Model-200 (103)Pd seeds embedded around a simulated lumpectomy cavity. Systematic and random errors in CMC were unfolded using low-uncertainty uncorrelated MC (UMC) as the benchmark. CMC efficiency gains, relative to UMC, were computed for all voxels, and the mean was classified in regions that received minimum doses greater than 20%, 50%, and 90% of D(90), as well as for various anatomical regions.

RESULTS

Systematic errors in CMC relative to UMC were less than 0.6% for 99% of the voxels and 0.04% for 100% of the voxels for the prostate and breast cases, respectively. For a 1 × 1 × 1 mm(3) dose grid, efficiency gains were realized in all structures with 38.1- and 59.8-fold average gains within the prostate and breast clinical target volumes (CTVs), respectively. Greater than 99% of the voxels within the prostate and breast CTVs experienced an efficiency gain. Additionally, it was shown that efficiency losses were confined to low dose regions while the largest gains were located where little difference exists between the homogeneous and heterogeneous doses. On an AMD 1090T processor, computing times of 38 and 21 sec were required to achieve an average statistical uncertainty of 2% within the prostate (1 × 1 × 1 mm(3)) and breast (0.67 × 0.67 × 0.8 mm(3)) CTVs, respectively.

CONCLUSIONS

CMC supports an additional average 38-60 fold improvement in average efficiency relative to conventional uncorrelated MC techniques, although some voxels experience no gain or even efficiency losses. However, for the two investigated case studies, the maximum variance within clinically significant structures was always reduced (on average by a factor of 6) in the therapeutic dose range generally. CMC takes only seconds to produce an accurate, high-resolution, low-uncertainly dose distribution for the low-energy PSB implants investigated in this study.

IPIP: A new approach to inverse planning for HDR brachytherapy by directly optimizing dosimetric indices

PURPOSE

Many planning methods for high dose rate (HDR) brachytherapy require an iterative approach. A set of computational parameters are hypothesized that will give a dose plan that meets dosimetric criteria. A dose plan is computed using these parameters, and if any dosimetric criteria are not met, the process is iterated until a suitable dose plan is found. In this way, the dose distribution is controlled by abstract parameters. The purpose of this study is to develop a new approach for HDR brachytherapy by directly optimizing the dose distribution based on dosimetric criteria.

METHODS

The authors developed inverse planning by integer program (IPIP), an optimization model for computing HDR brachytherapy dose plans and a fast heuristic for it. They used their heuristic to compute dose plans for 20 anonymized prostate cancer image data sets from patients previously treated at their clinic database. Dosimetry was evaluated and compared to dosimetric criteria.

RESULTS

Dose plans computed from IPIP satisfied all given dosimetric criteria for the target and healthy tissue after a single iteration. The average target coverage was 95%. The average computation time for IPIP was 30.1 s on an Intel(R) Core 2 Duo CPU 1.67 GHz processor with 3 Gib RAM.

CONCLUSIONS

IPIP is an HDR brachytherapy planning system that directly incorporates dosimetric criteria. The authors have demonstrated that IPIP has clinically acceptable performance for the prostate cases and dosimetric criteria used in this study, in both dosimetry and runtime. Further study is required to determine if IPIP performs well for a more general group of patients and dosimetric criteria, including other cancer sites such as GYN.

A greedy heuristic using adjoint functions for the optimization of seed and needle configurations in prostate seed implant

We continue our work on the development of an efficient treatment-planning algorithm for prostate seed implants by incorporation of an automated seed and needle configuration routine. The treatment-planning algorithm is based on region of interest (ROI) adjoint functions and a greedy heuristic. As defined in this work, the adjoint function of an ROI is the sensitivity of the average dose in the ROI to a unit-strength brachytherapy source at any seed position. The greedy heuristic uses a ratio of target and critical structure adjoint functions to rank seed positions according to their ability to irradiate the target ROI while sparing critical structure ROIs. Because seed positions are ranked in advance and because the greedy heuristic does not modify previously selected seed positions, the greedy heuristic constructs a complete seed configuration quickly. Isodose surface constraints determine the search space and the needle constraint limits the number of needles. This study additionally includes a methodology that scans possible combinations of these constraint values automatically. This automated selection scheme saves the user the effort of manually searching constraint values. With this method, clinically acceptable treatment plans are obtained in less than 2 min. For comparison, the branch-and-bound method used to solve a mixed integer-programming model took close to 2.5 h to arrive at a feasible solution. Both methods achieved good treatment plans, but the speedup provided by the greedy heuristic was a factor of approximately 100. This attribute makes this algorithm suitable for intra-operative real-time treatment planning.

Overview of image-guided radiation therapy

Radiation therapy has gone through a series of revolutions in the last few decades and it is now possible to produce highly conformal radiation dose distribution by using techniques such as intensity-modulated radiation therapy (IMRT). The improved dose conformity and steep dose gradients have necessitated enhanced patient localization and beam targeting techniques for radiotherapy treatments. Components affecting the reproducibility of target position during and between subsequent fractions of radiation therapy include the displacement of internal organs between fractions and internal organ motion within a fraction. Image-guided radiation therapy (IGRT) uses advanced imaging technology to better define the tumor target and is the key to reducing and ultimately eliminating the uncertainties. The purpose of this article is to summarize recent advancements in IGRT and discussed various practical issues related to the implementation of the new imaging techniques available to radiation oncology community. We introduce various new IGRT concepts and approaches, and hope to provide the reader with a comprehensive understanding of the emerging clinical IGRT technologies. Some important research topics will also be addressed.

Beam orientation optimization for intensity-modulated radiation therapy using mixed integer programming

The purpose of this study is to extend an algorithm proposed for beam orientation optimization in classical conformal radiotherapy to intensity-modulated radiation therapy (IMRT) and to evaluate the algorithm's performance in IMRT scenarios. In addition, the effect of the candidate pool of beam orientations, in terms of beam orientation resolution and starting orientation, on the optimized beam configuration, plan quality and optimization time is also explored. The algorithm is based on the technique of mixed integer linear programming in which binary and positive float variables are employed to represent candidates for beam orientation and beamlet weights in beam intensity maps. Both beam orientations and beam intensity maps are simultaneously optimized in the algorithm with a deterministic method. Several different clinical cases were used to test the algorithm and the results show that both target coverage and critical structures sparing were significantly improved for the plans with optimized beam orientations compared to those with equi-spaced beam orientations. The calculation time was less than an hour for the cases with 36 binary variables on a PC with a Pentium IV 2.66 GHz processor. It is also found that decreasing beam orientation resolution to 10 degrees greatly reduced the size of the candidate pool of beam orientations without significant influence on the optimized beam configuration and plan quality, while selecting different starting orientations had large influence. Our study demonstrates that the algorithm can be applied to IMRT scenarios, and better beam orientation configurations can be obtained using this algorithm. Furthermore, the optimization efficiency can be greatly increased through proper selection of beam orientation resolution and starting beam orientation while guaranteeing the optimized beam configurations and plan quality.

Potential dose-conformity advantages with multi-source intensity-modulated brachytherapy (IMBT)

The possibilities for optimizing brachytherapy by including additional degrees of freedom in source design were investigated. This included examining optimised dose delivery with a brachytherapy source that can provide intensity-modulated dose delivery in angle about the source travel direction (to achieve intensity-modulated brachytherapy-IMBT). A prostate HDR case was selected as an example. An inverse planning algorithm was used to define how an asymmetric radiation source can be controlled in multiple source catheters to maximize tumour dose coverage and minimize urethral and rectal doses. Substantial improvements in conformity in terms of tumour coverage and urethral dose reduction could be achieved when conventional HDR source positioning was used with IMBT. With the objective definition used in the example however, rectal doses could not be improved over those delivered via conventional HDR. When source position was included as a variable in IMBT, significant conformity improvements result for all structures. IMBT would be a technically challenging form of therapy that would be strongly influenced by the type of sources that could be created for it. This study has shown however that there is a potential for improving dose conformity with such a therapy. Introduction of IMBT techniques would require conventional brachytherapy concepts to be radically modified.

Simultaneous beam geometry and intensity map optimization in intensity-modulated radiation therapy

PURPOSE

In current intensity-modulated radiation therapy (IMRT) plan optimization, the focus is on either finding optimal beam angles (or other beam delivery parameters such as field segments, couch angles, gantry angles) or optimal beam intensities. In this article we offer a mixed integer programming (MIP) approach for simultaneously determining an optimal intensity map and optimal beam angles for IMRT delivery. Using this approach, we pursue an experimental study designed to (a) gauge differences in plan quality metrics with respect to different tumor sites and different MIP treatment planning models, and (b) test the concept of critical-normal-tissue-ring--a tissue ring of 5 mm thickness drawn around the planning target volume (PTV)--and its use for designing conformal plans.

METHODS AND MATERIALS

Our treatment planning models use two classes of decision variables to capture the beam configuration and intensities simultaneously. Binary (0/1) variables are used to capture "on" or "off" or "yes" or "no" decisions for each field, and nonnegative continuous variables are used to represent intensities of beamlets. Binary and continuous variables are also used for each voxel to capture dose level and dose deviation from target bounds. Treatment planning models were designed to explicitly incorporate the following planning constraints: (a) upper/lower/mean dose-based constraints, (b) dose-volume and equivalent-uniform-dose (EUD) constraints for critical structures, (c) homogeneity constraints (underdose/overdose) for PTV, (d) coverage constraints for PTV, and (e) maximum number of beams allowed. Within this constrained solution space, five optimization strategies involving clinical objectives were analyzed: optimize total intensity to PTV, optimize total intensity and then optimize conformity, optimize total intensity and then optimize homogeneity, minimize total dose to critical structures, minimize total dose to critical structures and optimize conformity simultaneously. We emphasize that the objectives that include optimizing conformity make use of the critical-normal-tissue-ring. Three tumor sites: head-and-neck, pediatric brain, and prostate are used for comparison.

RESULTS

The critical-normal-tissue-ring acts as a good device for enforcing conformity. Trends in the characteristics and quality of plans resulting from each model were observed. Attempts to reduce dose to critical structures tend to worsen PTV conformity (1.542 to 3.092) and homogeneity (1.223 to 1.984), depending on the relative size and spatial distance of the critical structures to the PTV. When the critical structures are relatively small compared with the PTV (as in the case for the pediatric brain tumor, where each is less than 15% in volume), dose reduction to critical structures is accompanied by much worse scores in conformity (2.482) and homogeneity (1.984). When the critical structures are larger, as in the case of head-and-neck (approximately 50%), the conformity and homogeneity deterioration is less significant (1.542 and 1.239, respectively). There is a clear tradeoff between homogeneity, conformity, and minimum dose to organs at risk (OARs). For head-and-neck and pediatric brain tumor, the model that minimizes total dose to critical structures and optimizes conformity simultaneously offers a compromise among these factors, resulting in reduced critical structure dose with conformal and homogeneous plans. In the prostate case, the tumor is smaller than the two large nearby critical structures, and all models provide very homogeneous PTV dose distribution. However, minimizing dose to critical structures worsens conformity, as it spreads the radiation to the area surrounding the PTV. The maximum dose to the critical structures also increases slightly. Compared with plans used in the clinic which generally have uniformly spaced beam angles, the optimal clinically acceptable plans obtained via the methods herein do not have equispaced beams. The optimal beam angles returned appear to be nonintuitive, and depend on PTV size and geometry and the spatial relationship between the tumor and critical structures.

CONCLUSIONS

The MIP model described allows simultaneous optimization over the space of beamlet fluence weights and beam and couch angles. Based on experiments with tumor data, this approach can return good plans that are clinically acceptable and practical. This work distinguishes itself from recent IMRT research in several ways. First, in previous methods beam angles are selected before intensity map optimization. Herein, we employ 0/1 variables to model the set of candidate beams, and thereby allow the optimization process itself to select optimal beams. Second, instead of incorporating dose-volume criteria within the objective function as in previous work, herein, a combination of discrete and continuous variables associated with each voxel provides a mechanism to strictly enforce dose-volume criteria within the constraints. Third, using the construct of critical-normal-tissue-ring within the objective function can enhance the achievement of conformal plans. Based on the three tumor sites considered, it appears that volume and spatial geometry with respect to the PTV are important factors to consider when selecting objectives to optimize, and in estimating how well suited a particular model is for achieving a specified goal.

Selection of beam orientations in intensity-modulated radiation therapy using single-beam indices and integer programming

While the process of IMRT planning involves optimization of the dose distribution, the procedure for selecting the beam inputs for this process continues to be largely trial-and-error. We have developed an integer programming (IP) optimization method to optimize beam orientation using mean organ-at-risk (MOD) data from single-beam plans. Two test cases were selected in which one organ-at-risk (OAR) and four OARs were simulated, respectively, along with a PTV. Beam orientation space was discretized in 10 degrees increments. For each beam orientation, a single-beam plan without intensity modulation and without constraints on OAR dose was generated and normalized to yield a mean PTV dose of 2 Gy and the corresponding MOD was calculated. The degree of OAR sparing was related to the average OAR MODs resulting from the beam orientations utilized with improvements of up to 10% at some dose levels. On the other hand, OAR DVHs in the IMRT plans were insensitive to beam numbers (in the 6-9 range) for similar average single-beam MODs. These MOD data were input to an IP optimization process, which then selected specified numbers of beam angles as inputs to a treatment planning system. Our results show that sets of beam angles with lower average single-beam MODs produce IMRT plans with better OAR sparing than manually selected beam angles. To optimize beam orientations, weights were assigned to each OAR following MOD input to the IP which was subsequently solved using the branch-and-cut algorithm. Seven-beam orientations obtained from solving the IP were applied to the test case with four OARs and the resulting plan with a dose prescription of 63 Gy was compared with an equi-spaced beam plan. The IP selected beams produced dose-volume improvements of up to 40% for OARs proximal to the PTV. Further improvement in the DVH can be obtained by increasing the weights assigned to these OARs but at the expense of the remaining OARs.

Treatment planning for prostate brachytherapy using region of interest adjoint functions and a greedy heuristic

We have developed an efficient treatment-planning algorithm for prostate implants that is based on region of interest (ROI) adjoint functions and a greedy heuristic. For this work, we define the adjoint function for an ROI as the sensitivity of the average dose in the ROI to a unit-strength brachytherapy source at any seed position. The greedy heuristic uses a ratio of target and critical structure adjoint functions to rank seed positions according to their ability to irradiate the target ROI while sparing critical structure ROIs. This ratio is computed once for each seed position prior to the optimization process. Optimization is performed by a greedy heuristic that selects seed positions according to their ratio values. With this method, clinically acceptable treatment plans are obtained in less than 2 s. For comparison, a branch-and-bound method to solve a mixed integer-programming model took more than 50 min to arrive at a feasible solution. Both methods achieved good treatment plans, but the speedup provided by the greedy heuristic was a factor of approximately 1500. This attribute makes this algorithm suitable for intra-operative real-time treatment planning.

Intraoperative dynamic dose optimization in permanent prostate implants

PURPOSE

With the advent of intraoperative optimized planning, the treatment of prostate cancer with permanent implants has reached an unprecedented level of dose conformity. However, because of well-documented (and unavoidable) inaccuracies in seed placement into the gland, carrying out a plan results in a large degree of variability relative to the intended dose distribution. This brings forth the need to periodically readjust the plan to allow for the real positions of seeds already implanted. In this paper, an algorithm for performing this task, hereby described as intraoperative dynamic dose optimization (IDDO), is presented and assessed.

METHODS AND MATERIALS

The general scheme for performing IDDO consists of three steps: (1) at some point during the implant, coordinates of implanted seeds are identified; (2) seed images are projected onto the reference frame of the ultrasound images for planning; and (3) the plan is reoptimized. Work on the first two steps is reported elsewhere. Here, we focus on the strategy for implementing the reoptimization step. An optimal treatment plan is first obtained based on initial operating room-acquired ultrasound images. We analyze the sensitivity and effect of the IDDO procedure with respect to the total number of reoptimizations performed. Specifically, we consider reoptimizing 2, 3, and 4 times. When two reoptimizations are used, half of the seeds from the initial optimal plan are implanted. The first reoptimization is performed on the remaining possible seed positions, and all the seeds designated in this reoptimized plan are implanted. The second (final) reoptimization is done on the remaining unused seed positions to ensure 100% coverage of the gland and to eliminate possible cold spots in the gland. Similarly, when three reoptimization steps are used, one-third of the seeds from the initial optimized plan, one-half of the seeds from the first reoptimization, and all seeds from the second reoptimization are implanted. The third (final) reoptimization is performed to assist in eliminating possible cold spots. Reoptimizing four times proceeds in a like manner. Fifteen patient cases are used for comparison. Strict dose bounds of 100% and 120% of the prescription dose are imposed on the urethra, and 100% coverage is imposed on the prostate volume. To assist in achieving good conformity, prostate contour points are assigned a target upper dose bound of 150% of the prescription dose.

RESULTS

A two-way comparison is performed: (a) initial optimized plan, (b) IDDO plan. Postimplant dose analysis, coverage and conformity measures, as well as actual dose received by urethra and rectum are used to gauge the results. The initial optimized plan consistently provides 93% prescription dose coverage to the gland with average conformity index of 1.32. The urethra dose ranges within 100% to 150%, and the maximum dose delivered to the rectum reaches 91% of the prescription dose. On average, about 50% of the urethra receives more than 120% of the prescription dose, and 19% of the rectum volume receives more than the 78% upper dose limit. For the IDDO plan, 100% postimplant coverage with 1.16 conformity is achieved. Urethra and rectum dose is maintained within the prescribed 100% to 120% range and 78% upper bound, respectively.

CONCLUSIONS

With real-time treatment planning, it is possible to dynamically reoptimize treatment plans to account for actual seed positions (as opposed to planned positions) and needle-induced swelling to the gland during implantation. Postimplant analysis shows that the final seed configuration resulting from the IDDO method yields improved dosimetry. The algorithmic design ensures that one can achieve complete coverage while maintaining good conformity, thus sparing excess radiation to external tissue. The study also provides evidence of the possibility of morbidity reduction to urethra and rectum (because of reduced dose delivered to these structures) via the use of IDDO planning. Clinical studies are needed to validate the importance of our approach.

Intraoperative dynamic dosimetry for prostate implants

This paper describes analytic tools in support of a paradigm shift in brachytherapy treatment planning for prostate cancer--a shift from standard pre-planning to intraoperative planning using dosimetric feedback based on the actual deposited seed positions within the prostate. The method proposed is guided by several desiderata: (a) bringing both planning and evaluation in the operating room (i.e. make post-implant evaluation superfluous) therefore making rectifications--if necessary--still achievable; (b) making planning and implant evaluation consistent by using the same imaging system (ultrasound); and (c) using only equipment commonly found in a hospital operating room. The intraoperative dosimetric evaluation is based on the fusion between ultrasound images and 3D seed coordinates reconstructed from fluoroscopic projections. Automatic seed detection and registration of the fluoroscopic and ultrasound information, two of the three key ingredients needed for the intraoperative dynamic dosimetry optimization (IDDO), are explained in detail. The third one, the reconstruction of 3D coordinates from projections, was reported in a previous article. The algorithms were validated using a custom-designed phantom with non-radioactive (dummy) seeds. Also, fluoroscopic images were taken at the conclusion of an actual permanent prostate implant and compared with data on the same patient obtained from radiographic-based post-implant evaluation. To offset the effect of organ motion the comparison was performed in terms of the proximity function of the two seed distributions. The agreement between the intra- and post-operative seed distributions was excellent.

Possibilities for intensity-modulated brachytherapy: technical limitations on the use of non-isotropic sources

An investigation was undertaken into possible dose conformity advantages and technical limitations of utilizing radially asymmetric internally applied radiation sources for intensity-modulated brachytherapy (IMBT). A feasible form of a source for IMBT would be a linear source with a high-intensity angular region, with some fractional transmission through the remainder of the source, which inhibits the resolution achievable in intensity modulation. Indexed rotation of the source about its axis would provide radial intensity modulation, which could compensate for variations in the spatial relationship between the source position and location of the target edge. Two treatment situations were simulated--one two-dimensional and one three-dimensional--both utilizing a single source (single catheter). The optimal intensity distribution of the source was determined by simulated annealing optimization using a conformality-based objective. The parameters in the optimization included the angular size of the source high-intensity region, and the fractional transmission through the low-intensity part of the source. Results indicate that limitations in source design suggest an optimal high-intensity resolution of approximately pi/4 to pi/8. The advantages of IMBT are rapidly reduced when fractional transmission through the low-intensity side of the source is increased.

Automated planning volume definition in soft-tissue sarcoma adjuvant brachytherapy

In current practice, the planning volume for adjuvant brachytherapy treatment for soft-tissue sarcoma is either not determined a priori (in this case, seed locations are selected based on isodose curves conforming to a visual estimate of the planning volume), or it is derived via a tedious manual process. In either case, the process is subjective and time consuming, and is highly dependent on the human planner. The focus of the work described herein involves the development of an automated contouring algorithm to outline the planning volume. Such an automatic procedure will save time and provide a consistent and objective method for determining planning volumes. In addition, a definitive representation of the planning volume will allow for sophisticated brachytherapy treatment planning approaches to be applied when designing treatment plans, so as to maximize local tumour control and minimize normal tissue complications. An automated tumour volume contouring algorithm is developed utilizing computational geometry and numerical interpolation techniques in conjunction with an artificial intelligence method. The target volume is defined to be the slab of tissue r cm perpendicularly away from the curvilinear plane defined by the mesh of catheters. We assume that if adjacent catheters are over 2r cm apart, the tissue between the two catheters is part of the tumour bed. Input data consist of the digitized coordinates of the catheter positions in each of several cross-sectional slices of the tumour bed, and the estimated distance r from the catheters to the tumour surface. Mathematically, one can view the planning volume as the volume enclosed within a minimal smoothly-connected surface which contains a set of circles, each circle centred at a given catheter position in a given cross-sectional slice. The algorithm performs local interpolation on consecutive triplets of circles. The effectiveness of the algorithm is evaluated based on its performance on a collection of soft-tissue sarcoma tumour beds within various anatomical structures. For each of 15 patient cases considered, the algorithm takes approximately 2 min to generate the planning volume. Although the tumour shapes are rather different, the algorithm consistently generates planning volumes that visually demonstrate smooth curves compactly encapsulating the circles. This general-purpose contouring algorithm works well whether the catheters are all close together, spread far apart in the plane or arranged in a convoluted way. The automatic contouring algorithm significantly reduces labour time and provides a consistent and objective method for determining planning volumes for soft-tissue sarcoma. Further studies are needed to validate the significance of the resulting planning volumes in designing treatment plans and the role that sophisticated brachytherapy treatment planning optimization may have in producing good plans.

Intraoperative planning and evaluation of permanent prostate brachytherapy: report of the American Brachytherapy Society

PURPOSE

The preplanned technique used for permanent prostate brachytherapy has limitations that may be overcome by intraoperative planning. The goal of the American Brachytherapy Society (ABS) project was to assess the current intraoperative planning process and explore the potential for improvement in intraoperative treatment planning (ITP).

METHODS AND MATERIALS

Members of the ABS with expertise in ITP performed a literature review, reviewed their clinical experience with ITP, and explored the potential for improving the technique.

RESULTS

The ABS proposes the following terminology in regard to prostate planning process: *Preplanning--Creation of a plan a few days or weeks before the implant procedure. *Intraoperative planning--Treatment planning in the operating room (OR): the patient and transrectal ultrasound probe are not moved between the volume study and the seed insertion procedure. * Intraoperative preplanning--Creation of a plan in the OR just before the implant procedure, with immediate execution of the plan. *Interactive planning--Stepwise refinement of the treatment plan using computerized dose calculations derived from image-based needle position feedback. *Dynamic dose calculation--Constant updating of dose distribution calculations using continuous deposited seed position feedback. Both intraoperative preplanning and interactive planning are currently feasible and commercially available and may help to overcome many of the limitations of the preplanning technique. Dosimetric feedback based on imaged needle positions can be used to modify the ITP. However, the dynamic changes in prostate size and shape and in seed position that occur during the implant are not yet quantifiable with current technology, and ITP does not obviate the need for postimplant dosimetric analysis. The major current limitation of ITP is the inability to localize the seeds in relation to the prostate. Dynamic dose calculation can become a reality once these issues are solved. Future advances can be expected in methods of enhancing seed identification, in imaging techniques, and in the development of better source delivery systems. Additionally, ITP should be correlated with outcome studies, using dosimetric, toxicity, and efficacy endpoints.

CONCLUSION

ITP addresses many of the limitations of current permanent prostate brachytherapy and has some advantages over the preplanned technique. Further technologic advancement will be needed to achieve dynamic real-time calculation of dose distribution from implanted sources, with constant updating to allow modification of subsequent seed placement and consistent, ideal dose distribution within the target volume.

A theoretical derivation of the nomograms for permanent prostate brachytherapy

This study calculates the required minimum radioactivity to deliver a prescribed dose of radiation to a target using radioisotopes in permanent prostate brachytherapy. Assuming the radioactivity to be in a continuous form, an integral equation--Fredholm equation of the first kind, can be formulated with the radioactivity density used as the variable. The density distribution to produce a uniform volume dose rate is determined using a quadrature method and the radial profile behaves smoothly from the zero radius, and peaks sharply approaching the volume boundary. The density for Pd-103 is about 1.5 times that of I-125 due to its higher spatial attenuation. A nomogram is the relationship between the total activity per unit dose (A) and the dimension of the volume (d). Expressing the nomogram as A=c X dn U/Gy, then (c,n)= [(0.0098, 2.09) I-125] and [(0.031, 2.25) Pd-103]. Compared with the Memorial nomogram, (c,n)=[(0.011,2.2) I-125] and [(0.036,2.56) Pd-103], or that quoted by AAPM TG64, (c,n)=[(0.014,2.05) I-125] and [(0.056,2.22) Pd-103], our calculation determined an average 33% and 35% decrease for I-125, and 89% and 77% decrease for Pd-103, respectively. Two reasons for the extra total activity found in the Memorial and AAPM nomograms are: (a) An imperfect clinical situation limited by the restraints of implant techniques (e.g., use of templates) associated with the presence of adjacent normal organs, and (b) source discretization into seeds. When radioactivity is clumped as discrete seeds, higher activity is needed because of "wastage" in two aspects: (a) Dose cold-spots at intersource spaces, (b) hot-spots around the sources. Thus in theory, use of lower activity seeds will require less total activity to deliver a prescribed dose. Based on our study, Pd-103 delivers a higher therapeutic ratio and a lower integral dose to the patient compared to I-125.

On the determination of an effective planning volume for permanent prostate implants

PURPOSE

In current practice, planning for prostate brachytherapy is based on the state of the prostate at a particular instant in time. Because treatment occurs over an extended period, changes in the prostate volume (gland shrinkage) and seed displacement lead to disagreement between planned dosimetry to the prostate and the dose actually received by the prostate. Discrepancies between planned and actual dose to the rectum and urethra also occur. The purpose of this study is to investigate the possibility of defining an "effective planning volume" that compensates for changes in prostate volume and seed displacement.

METHODS AND MATERIALS

Waterman's formula is used to estimate prostate shrinkage and seed displacement. The prostate volume and potential seed positions at days 0, 6, 12, 18, 24, and 30 are used in formulating time-dependent dosimetric treatment planning models. Both single-period and multi-period models are proposed and analyzed. A state-of-the-art computational engine generates unbiased, high-quality treatment plans in a matter of minutes. Plans are evaluated using coverage and conformity indices computed at specific times over a period of 30 days. The models allow dose to urethra and rectum to be strictly controlled at specific instants in time, or throughout the 30-day horizon.

RESULTS

For plans generated from the single-period models-based on projected prostate volumes and potential seed positions on days t = 0, 6, 12, 18, 24, 30, respectively-as t increases, the conformity index improves while the coverage worsens. In particular, the best coverage and worst conformity are achieved for the plan generated using t = 0 (day 0) information. This plan provides over 99% coverage over the entire 30-day period, and while it has initial conformity index 1.24, the conformity index climbs to 1.58 by day 30. Conversely, the worst coverage and best conformity are achieved when the plan is generated using projected information from t = 30 (day 30). Plans based on projected data at day 30 yield an initial coverage of only 84%, with conformity scores less than 1.34 over the entire 30-day period. Among the multi-period plans, with the exception of the two-period plan obtained using day 0 and projected day 6 data, the average coverage is 98% while conformity indices below 1.46 are maintained throughout the 30-day horizon. Excessive dose to the urethra and rectum is observed when only day 0 dosimetric and volumetric data are imposed in the planning procedure. In this case, by day 30, 89% of urethra volume receives dose in excess of 120% of the remaining prescription dose. Similarly, 40% of rectum volume receives dose in excess of the prescribed upper dose bound of 78% of the remaining prescription dose. When multi-period dosimetric constraints for urethra and rectum are imposed, dose to these structures is controlled throughout the 30-day period.

CONCLUSIONS

A planning method that takes into account prostate shrinkage and seed displacement over time can be used to adjust the balance between coverage and conformity. Incorporating projected future volumetric information is useful in providing more conformal plans, in some cases improving conformity by as much as 21% while sacrificing roughly 7% of initial coverage. Evidence of possible morbidity reduction to urethra and rectum via the use of multi-period dosimetric constraints on these structures is demonstrated. Among all plans considered, the plan obtained via the six-period model provides the best coverage and conformity over the 30-day horizon.

An iterative sequential mixed-integer approach to automated prostate brachytherapy treatment plan optimization

Conventional treatment planning for interstitial prostate brachytherapy is generally a 'trial and error' process in which improved treatment plans are generated by iteratively changing, via expert judgement, the configuration of sources within the target volume in order to achieve a satisfactory dose distribution. We have utilized linear mixed-integer programming (MIP) and the branch-and-bound method, a deterministic search algorithm, to generate treatment plans. The rapidity of dose falloff from an interstitial radioactive source requires fine sampling of the space in which dose is calculated. This leads to a large and complex model that is difficult to solve as a single 3D problem. We have therefore implemented an iterative sequential approach that optimizes pseudo-independent 2D slices to achieve a fine-grid 3D solution. Using our approach, treatment plans can be generated in 20-45 min on a 200 MHz processor. A comparison of our approach with the manual 'trial and error' approach shows that the optimized plans are generally superior. The dose to the urethra and rectum is usually maintained below harmful levels without sacrificing target coverage. In the event that the dose to the urethra is undesirably high, we present a refined optimization approach that lowers urethra dose without significant loss in target coverage. An analysis of the sensitivity of the optimized plans to seed misplacement during the implantation process is also presented that indicates remarkable stability of the dose distribution in comparison with manual treatment plans.

The combined use of the natural and the cumulative dose-volume histograms in planning and evaluation of permanent prostatic seed implants

BACKGROUND AND PURPOSE

To investigate prostate dose coverage and overdosage in planned and realized permanent iodine seed prostate implants and to explore the use of the natural dose-volume histogram (NDVH) and the cumulative dose-volume histogram (CDVH) as tools to optimize prostate implants.

MATERIALS AND METHODS

The optimal prescription dose (PD) or natural prescription dose (NPD) was derived from the NDVH. The mismatch between the NPD and the given PD was called the natural dose ratio (NDR). For an ideal implant the NDR should be 1. The target is overdosed if NDR >1 and underdosed if NDR <1. The NDR and prostate coverage were evaluated in implants of nine patients. Prostate coverage was determined from the CDVH based on pre-implant ultrasound or post-implant MRI for the planned and realized implants, respectively. The use of the NDVH to further optimize the planned prostate implants was also explored.

RESULTS

The mean values of the NDRs were 1.30+/-0.34 (range 0.76-1.79), 1.22+/-0.31 (0.76-1.74) and 1.22+/-0.12 (0.98-1.33) for the planned, realized and optimized seed distributions, respectively. The realized prostatic implants showed smaller prostate coverage than the planned implants. The prostate volume fractions receiving 100% of the prescription dose were V(100)=79+/-6% and V(100)=97+/-3% for the realized and the planned implants, respectively.

CONCLUSIONS

The NDVH and the CDVH proved to be valuable tools in plan evaluation. The NDVH and its derived parameter NDR quantify the risk of under or overdosage for a given PD. The CDVH is valuable in evaluation of prostate coverage realized prostate. Our strategy to implant just the prostate and not the prostate plus a margin led to NDR values between 1.1 and 1.3 and a prostate coverage of V(100)=79+/-6% in the nine patients. The planned coverage of V(100)=95% was not realized, mainly due to inadequate coverage of the base of the prostate.

A mixed-encoding genetic algorithm with beam constraint for conformal radiotherapy treatment planning

In this paper we propose a new hierarchical evolutionary algorithm that combines binary encoding and floating-point encoding to automatically select the beam directions and determine the weights of the selected beams. With traditional optimization methods the beam directions are fixed a priori by the operator in recognition of the fact that computer selection of beam directions is a difficult problem. In this investigation, we used a hybrid-encoding scheme. The binary encoding part of each chromosome was used to select the beam directions, and its corresponding floating-point encoding part of the same chromosome was used to determine the weights of those selected beams. Before beginning the optimization process, we set a constraint on the number of the beam directions we wanted in the final solution. We present three examples to verify this method. These examples differ with each other in tumor sites, problem sizes, and optimization parameters. Three-dimensional optimization results and statistical data showed that this method is feasible. We think this method can be easily extended to solve more complex target problems (such as nonconvex target problems).

Treatment planning for prostate implants using magnetic-resonance spectroscopy imaging

PURPOSE

Recent studies have demonstrated that magnetic-resonance spectroscopic imaging (MRSI) of the prostate may effectively distinguish between regions of cancer and normal prostatic epithelium. This diagnostic imaging tool takes advantage of the increased choline plus creatine versus citrate ratio found in malignant compared to normal prostate tissue. The purpose of this study is to describe a novel brachytherapy treatment-planning optimization module using an integer programming technique that will utilize biologic-based optimization. A method is described that registers MRSI to intraoperative-obtained ultrasound images and incorporates this information into a treatment-planning system to achieve dose escalation to intraprostatic tumor deposits.

METHODS

MRSI was obtained for a patient with Gleason 7 clinically localized prostate cancer. The ratios of choline plus creatine to citrate for the prostate were analyzed, and regions of high risk for malignant cells were identified. The ratios representing peaks on the MR spectrum were calculated on a spatial grid covering the prostate tissue. A procedure for mapping points of interest from the MRSI to the ultrasound images is described. An integer-programming technique is described as an optimization module to determine optimal seed distribution for permanent interstitial implantation. MRSI data are incorporated into the treatment-planning system to test the feasibility of dose escalation to positive voxels with relative sparing of surrounding normal tissues. The resultant tumor control probability (TCP) is estimated and compared to TCP for standard brachytherapy-planned implantation.

RESULTS

The proposed brachytherapy treatment-planning system is able to achieve a minimum dose of 120% of the 144 Gy prescription to the MRS positive voxels using (125)I seeds. The preset dose bounds of 100-150% to the prostate and 100-120% to the urethra were maintained. When compared to a standard plan without MRS-guided optimization, the estimated TCP for the MRS-optimized plan is superior. The enhanced TCP was more pronounced for smaller volumes of intraprostatic tumor deposits compared to estimated TCP values for larger lesions.

CONCLUSIONS

Using this brachytherapy-optimization system, we could demonstrate the feasibility of MRS-optimized dose distributions for (125)I permanent prostate implants. Based on probability estimates of anticipated improved TCP, this approach may have an impact on the ability to safely escalate dose and potentially improve outcome for patients with organ-confined but aggressive prostatic cancers. The magnitude of the TCP enhancement, and therefore the risks of ignoring the MR data, appear to be more substantial when the tumor is well localized; however, the gain achievable in TCP may depend quite considerably on the MRS tumor-detection efficiency.

Dosimetric and volumetric criteria for selecting a source activity and a source type ((125)I or (103)Pd) in the presence of irregular seed placement in permanent prostate implants

PURPOSE

The dosimetric merit of a permanent prostate implant relies on two factors: the quality of the plan itself, and the fidelity of its implementation. The former factor depends on source type and on source strength, while the latter is a combination of skill and experience. The purpose of this study is to offer criteria by which to select a source type ((125)I or (103)Pd) and activity.

METHODS AND MATERIALS

Given a prescription dose and potential seed positions along needles, treatment plans were designed for a number of seed types and activities, specifically for (125)I with activities ranging from 0.3 to 0.7 mCi, and for (103)Pd with activities in the range of 0.8 to 1.6 mCi. To avoid human planner bias, an automated computerized planning system based on integer programming was used to obtain optimal seed configurations for each seed type and activity. To simulate the effect of seed-placement inaccuracies, random seed-displacement "errors" were generated for all plans. The displacement errors were assumed to be uniformly distributed within a cube with side equal to 2sigma. The resulting treatment plans were assessed using two volumetric and two dosimetric indices.

RESULTS

For (125)I implants a coverage index (CI) of 98.5% or higher can be achieved for all activities (CI is the fraction of the target volume receiving the prescribed or larger dose). The external volume index (EI) (i.e., the amount of healthy tissue, as percentage of the target volume, receiving the prescribed or larger dose) increases from 13.9% to 20% as the activity increases from 0.3 to 0.7 mCi. For implants using (103)Pd, the external volume index increases from 10. 2% to 13.9% whenever CI exceeds 98.5%. Volumetric and dosimetric indices (coverage index, external volume index, D90, and D80) are all sensitive to seed displacement, although the activity dependence of these indices is more pronounced for (125)I than for (103)Pd implants.

CONCLUSIONS

For both isotopes, the lower activities studied systematically result in lower EIs. If seeds can be placed within approximately 0.5 cm of their intended position (103)Pd should be preferred because its EI is lower than that of (125)I. For all activities the coverage indices and D90 are within the required range. If seed placement uncertainties are larger than 0.5 cm, (125)I provides slightly better target coverage; however, in terms of external volume (healthy tissue) covered, (103)Pd is superior to (125)I.

Generation of uniformly distributed dose points for anatomy-based three-dimensional dose optimization methods in brachytherapy

We have studied the accuracy of statistical parameters of dose distributions in brachytherapy using actual clinical implants. These include the mean, minimum and maximum dose values and the variance of the dose distribution inside the PTV (planning target volume), and on the surface of the PTV. These properties have been studied as a function of the number of uniformly distributed sampling points. These parameters, or the variants of these parameters, are used directly or indirectly in optimization procedures or for a description of the dose distribution. The accurate determination of these parameters depends on the sampling point distribution from which they have been obtained. Some optimization methods ignore catheters and critical structures surrounded by the PTV or alternatively consider as surface dose points only those on the contour lines of the PTV. D(min) and D(max) are extreme dose values which are either on the PTV surface or within the PTV. They must be avoided for specification and optimization purposes in brachytherapy. Using D(mean) and the variance of D which we have shown to be stable parameters, achieves a more reliable description of the dose distribution on the PTV surface and within the PTV volume than does D(min) and D(max). Generation of dose points on the real surface of the PTV is obligatory and the consideration of catheter volumes results in a realistic description of anatomical dose distributions.

Optimization of radiosurgery treatment planning via mixed integer programming

An automated optimization algorithm based on mixed integer programming techniques is presented for generating high-quality treatment plans for LINAC radiosurgery treatment. The physical planning in radiosurgery treatment involves selecting among a large collection of beams with different physical parameters an optimal beam configuration (geometries and intensities) to deliver the clinically prescribed radiation dose to the tumor volume while sparing the nearby critical structure and normal tissue. The proposed mixed integer programming models incorporate strict dose restrictions on tumor volume, and constraints on the desired number of beams, isocenters, couch angles, and gantry angles. The model seeks to deliver full prescription dose coverage and uniform radiation dose to the tumor volume while minimizing the excess radiation to the periphery normal tissue. In particular, it ensures that proximal normal tissues receive minimal dose via rapid dose fall-off. Preliminary numerical tests on a single patient case indicate that this approach can produce exceptionally high-quality plans in a fraction of the time required using the procedure currently employed by clinicians. The resulting plans provide highly uniform prescription dose to the tumor volume while drastically reducing the irradiation received by the proximal critical normal tissue.