A pencil-beam photon dose algorithm for stereotactic radiosurgery using a miniature multileaf collimator

Dose Calculation Algorithms for External Radiation Therapy: An Overview for Practitioners

Radiation therapy (RT) is a constantly evolving therapeutic technique; improvements are continuously being introduced for both methodological and practical aspects. Among the features that have undergone a huge evolution in recent decades, dose calculation algorithms are still rapidly changing. This process is propelled by the awareness that the agreement between the delivered and calculated doses is of paramount relevance in RT, since it could largely affect clinical outcomes. The aim of this work is to provide an overall picture of the main dose calculation algorithms currently used in RT, summarizing their underlying physical models and mathematical bases, and highlighting their strengths and weaknesses, referring to the most recent studies on algorithm comparisons. This handy guide is meant to provide a clear and concise overview of the topic, which will prove useful in helping clinical medical physicists to perform their responsibilities more effectively and efficiently, increasing patient benefits and improving the overall quality of the management of radiation treatment.

Angular under-sampling effect on VMAT dose calculation: An analysis and a solution strategy

PURPOSE

Most VMAT algorithms compute the dose on discretized apertures with small angular separations for practical reasons. However, machines deliver the VMAT dose with a continuously moving MLC and gantry and a continuously changing dose rate. The computed dose can deviate from the delivered dose, especially if no, or loose, MLC movement constraints are applied for the VMAT optimization. The goal of this paper is to establish a simplified mathematical model to analyze the discrepancy between the VMAT plan calculation dose and the delivered dose and to provide a reasonable solution for clinical implementation.

METHODS

A simplified metric is first introduced to describe the discrepancy between doses computed with discretized apertures and a continuous delivery model. The delivery fluences were formed separately for six different leaf movement scenarios. The formula was then rewritten in a more general form. The correlation between discretized and continuous fluence is summarized using this general form. The Fourier analysis for the impacts from three separate factors - dose kernel width, aperture width, aperture distance - to the dose discrepancy is also presented in order to provide insight into the dose discrepancy caused by under-sampling in the frequency domain. Finally, a weighting-based interpolation (WBI) algorithm, which can improve the aperture interpolation efficiency, is proposed. The associated evaluation methods and criteria for the proposed algorithm are also given.

RESULTS

The comparisons between the WBI algorithm and the equal angular interpolation (EAI) method suggested that the proposed algorithm has a great advantage with regard to aperture number efficiency. To achieve a 90% gamma passing rate using the dose computed with apertures generated with 0.5° EAI, with the initial optimization apertures as the standard for the comparison, the WBI needs only 66% and 54% of the aperture numbers that the EAI method needs for a 2° and a 4° angular separation of the VMAT optimization, respectively. The results also suggested that the weighted dose error index value, Θ, can be used as a stopping criterion for an interpolation algorithm, e.g., WBI or EAI, or as an indicator for sampling level evaluations. The phantom results indicate that the gamma passing rate decreases with increasing depth, from the phantom surface to the iso center, for the plans computed with under-sampled apertures. No obvious variation trends were observed for the plans computed with well-sampled apertures.

CONCLUSIONS

The mathematical analysis suggests that the dose discrepancies due to under-sampling are strongly correlated with the aperture width, the distance between apertures, and the width of the dose kernel. The WBI algorithm proves to be an efficient aperture interpolation strategy and is useful for dose computation of VMAT plans.

Dosimetric characteristics of dual-layer multileaf collimation for small-field and intensity-modulated radiation therapy applications

The purpose of the present work was to measure the performance characteristics in the penumbra region and on the leaf‐end of an innovative dual‐layer micro multileaf collimator (DmMLC). The DmMLC consists of two orthogonal (upper and lower) layers of leaves; a standard MLC consists of one layer. The DmMLC provides unique performance characteristics in smoothing dose undulation, reducing leaf‐end transmission, and reducing MLC field dependence of the leaf stepping angle. Two standard MLCs (80‐leaf and 120‐leaf versions: Varian Medical Systems, Palo Alto, CA), a DmMLC (AccuKnife: Initia Medical Technology, Canton, MA), and a Cerrobend (Cerro Metal Products, Bellefonte, PA) block were used in performance studies involving a triangular field, a cross leaf‐end field, and a circular field. Measurements were made with 6‐MV X‐rays and extended dose range film at a depth of 5 cm in Solid Water (Gammex rmi, Middleton, WI) at a source–axis distance of 100 cm. The field penumbra width measured between the 20% and 80% isodose lines through the MLC‐80, MLC‐120, DmMLC, and Cerrobend block were 9.0, 5.0, 3.0, and 2.0 mm respectively. The dose undulation amplitude of the 50% isodose line was measured as 5.5, 2.0, and 0.5 mm for the MLC‐80, MLC‐120, and DmMLC respectively. The planar dose difference between the MLC‐80, MLC‐120, and DmMLC against Cerrobend block was measured as ranging at ±52.5%,±35.0%, and ±20.0% respectively. The leaf‐end transmission was measured at 22.4% in maximum and 15.4% in average when closing a single layer of the DmMLC, and at 2.4% in maximum and 2.1% in average when closing both layers. The MLC dependence of the leaf stepping angle with the DmMLC ranged from 45 degrees to 90 degrees. The standard MLC leaf stepping angle ranged from 0 degrees to 90 degrees. In conclusion, the dose undulation, leaf‐end transmission, and MLC field dependence of the leaf stepping angle with the DmMLC were remarkably reduced as compared with those of the standard MLCs. And as compared with Cerrobend block, the DmMLC provided very comparable performance in field‐edge smoothing and in the shaping of complex fields.

PACS numbers: 87.56.Jk, 87.56.Nk, 87.56.Nj, 87.57.Nt

Effects of static dosimetric leaf gap on MLC-based small-beam dose distribution for intensity-modulated radiosurgery

The aim of the present study was to evaluate the effect of various specific dosimetric leaf gaps on the multileaf collimator (MLC)–based small‐beam dose distribution. The dosimetric static leaf gap was determined by comparing the profiles of small MLC‐based beams with those of small collimated fields (square fields of 1, 2, 3, and 4cm). The results showed that an approximately 2‐mm gap was optimal with the Millennium 120‐leaf MLC (Varian Medical Systems, Palo Alto, CA) and a Varian 21EX 6‐MV photon beam. We also investigated how much the leaf gap affects the planning results and the actual dose distribution. A doughnut‐shaped planning target volume (PTV, 6.1 cm³) and inner organ at risk (OAR, 0.3 cm³) were delineated for delicate intensity‐modulated radiosurgery test planning. The applied leaf gaps were 0, 1, and 2 mm. The measured dose distributions were compared with the dose distribution in the treatment planning system. The maximum dose differences at inside PTV, outside PTV, and inner OAR were, respectively, 22.3%, 20.2%, and 35.2% for the 0‐mm leaf gap; 17.8%, 22.8%, and 30.8% for the 1‐mm leaf gap; and 5.5%, 8.5%, and 6.3% for the 2‐mm leaf gap. In a human head phantom (model 605: CIRS, Norfolk, VA) study, large dose differences of 1.3% – 12.7% were noted for the measurements made using the MLC files generated by the three different leaf gaps. The planned results were similar, and measurements showed a large dose difference associated with the various leaf gaps. These results strongly suggest that plans generated by a commercial inverse planning system commissioned using general collimated field data will probably demonstrate discrepancies between the planned treatments and the measured results.

PACS number: 87.53.Dq

Determination of depth and field size dependence of multileaf collimator transmission in intensity-modulated radiation therapy beams

Intensity‐modulated radiation therapy (IMRT) plans for the treatment of large and complex volumes may contain a relatively large contribution from multileaf collimator (MLC) transmission. In such cases, comprehensive characterization of direct and scatter MLC transmission is important. We designed a set of tests (open beam, closed static MLC, and dynamic MLC gap) to determine dosimetric MLC properties as a function of field size and depth at the central axis.

We developed a generalized model of MLC transmission to account for direct MLC transmission, MLC scatter, beam hardening, and leaf‐end transmission (dosimetric gap). The model is consistent with the beam model used in IMRT optimization. We tested the model for extreme asymmetric fields relevant for large targets and for split IMRT fields. We applied our MLC scatter estimation formula to clinically relevant cases and showed that MLC scatter is contributing an undesired background dose. This contribution is relatively large, especially in low‐dose regions. (For instance, a uniform extra dose may dramatically increase normal‐lung toxicity in thorax treatment.) For complex IMRT of large‐volume targets, we found direct MLC transmission dose to be as high as 30%, and MLC scatter, up to 10% within the target volume for the selected cases. We identified that the dose discrepancies between the IMRT planning system [Eclipse (Varian Medical Systems, Palo Alto, CA)] and ionization chamber measurements (inside and outside of the field) are attributable to an inadequate model of MLC transmission in the planning system (constant‐value model).

In the present study, we measured MLC transmission properties for Varian 6EX (6 MV) and 21EXs (6 and 10 MV) linear accelerators; however, the experimental method and theoretical model are more general.

PACS number: 87.53.‐j

An independent dose calculation algorithm for MLC-based stereotactic radiotherapy

We have developed an algorithm to calculate dose in a homogeneous phantom for radiotherapy fields defined by multi-leaf collimator (MLC) for both static and dynamic MLC delivery. The algorithm was developed to supplement the dose algorithms of the commercial treatment planning systems (TPS). The motivation for this work is to provide an independent dose calculation primarily for quality assurance (QA) and secondarily for the development of static MLC field based inverse planning. The dose calculation utilizes a pencil-beam kernel. However, an explicit analytical integration results in a closed form for rectangular-shaped beamlets, defined by single leaf pairs. This approach reduces spatial integration to summation, and leads to a simple method of determination of model parameters. The total dose for any static or dynamic MLC field is obtained by summing over all individual rectangles from each segment which offers faster speed to calculate two-dimensional dose distributions at any depth in the phantom. Standard beam data used in the commissioning of the TPS was used as input data for the algorithm. The calculated results were compared with the TPS and measurements for static and dynamic MLC. The agreement was very good (<2.5%) for all tested cases except for very small static MLC sizes of 0.6 cm x 0.6 cm (<6%) and some ion chamber measurements in a high gradient region (<4.4%). This finding enables us to use the algorithm for routine QA as well as for research developments.

The use of modified single pencil beam dose kernels to improve IMRT dose calculation accuracy

Intensity modulated radiation therapy (IMRT) is used to deliver highly conformal radiation doses to tumors while sparing nearby sensitive tissues. Discrepancies between calculated and measured dose distributions have been reported for regions of high dose gradients corresponding to complex radiation fluence patterns. For the single pencil beam convolution dose calculation algorithm, the ability to resolve areas of high dose structure is partly related to the shape of the pencil beam dose kernel (similar to how a photon detector's point spread function relates to imaging resolution). Improvements in dose calculation accuracy have been reported when the treatment planning system (TPS) is recommissioned using high-resolution measurement data as input. This study proposes to improve the dose calculation accuracy for IMRT planning by modifying clinical dose kernel shapes already present in the TPS, thus avoiding the need to reacquire higher resolution commissioning data. The in-house optimization program minimizes a cost-function based on a two-dimensional composite dose subtraction/distance-to-agreement (gamma) analysis. The final modified kernel shapes are reintroduced into the treatment planning system and improvements to the dose calcula tion accuracy for complex IMRT dose distributions evaluated. The central kernel value (radius =0 cm) has the largest effect on the dose calculation resolution and is the focus of this study.

Optimisation in stereotactic radiosurgery of AVMs: I. Mathematical considerations

One forward-planning method and five inverse-planning methods for optimisation of treatment in radiation therapy were compared in the particular case of radiosurgery with micro-multi-leaves collimator (MMLC) and arc therapy. The "manual" method, two matrix methods (singular value decomposition and non-negative least square fit), two gradient methods (quasi-Newton and conjugate gradient algorithms) and the "simulated annealing" stochastic method were investigated. The performance of these methods was assessed in terms of the speed of convergence to an optimum, the ability to account for the organs at risk, and probability of targeted success. The study employed an adapted version of the GRATIS treatment planning system. A group of 22 patients previously treated by arc therapy for arteriovenous malformations (AVMs) were studied to evaluate the performance of the various optimisation methods for MMLC and arc therapy. The conjugate gradient method proved to be the most appropriate for most cases.

Exploring the limits of spatial resolution in radiation dose delivery

Flexibility and complexity in patient treatment due to advances in radiotherapy techniques necessitates a simple method for evaluating spatial resolution capabilities of the dose delivery device. Our purpose in this investigation is to evaluate a model that describes the ability of a radiation therapy device to deliver a desired dose distribution. The model is based on linear systems theory and is analogous to methods used to describe resolution degradation in imaging systems. A qualitative analysis of spatial resolution degradation using the model is presented in the spatial and spatial frequency domains. The ability of the model to predict the effects of geometric dose conformity to treatment volumes is evaluated by varying multileaf collimator leaf width and magnitude of dose spreading. Dose distributions for three clinical treatment shapes, circular shapes of varying diameter and one intensity modulated shape are used in the evaluation. We show that the model accurately predicts the dependence of dose conformity on these parameters. The spatial resolution capabilities of different radiation therapy devices can be quantified using the model, providing a simple method for comparing different treatment machine characteristics. Also, as different treatment sites have different resolution requirements this model may be used to tailor machine characteristics to the specific site.

Effects of intra-fraction motion on IMRT dose delivery: statistical analysis and simulation

There has been some concern that organ motion, especially intra-fraction organ motion due to breathing, can negate the potential merit of intensity-modulated radiotherapy (IMRT). We wanted to find out whether this concern is justified. Specifically, we wanted to investigate whether IMRT delivery techniques with moving parts, e.g., with a multileaf collimator (MLC), are particularly sensitive to organ motion due to the interplay between organ motion and leaf motion. We also wanted to know if, and by how much, fractionation of the treatment can reduce the effects. We performed a statistical analysis and calculated the expected dose values and dose variances for volume elements of organs that move during the delivery of the IMRT. We looked at the overall influence of organ motion during the course of a fractionated treatment. A linear-quadratic model was used to consider fractionation effects. Furthermore, we developed software to simulate motion effects for IMRT delivery with an MLC, with compensators, and with a scanning beam. For the simulation we assumed a sinusoidal motion in an isocentric plane. We found that the expected dose value is independent of the treatment technique. It is just a weighted average over the path of motion of the dose distribution without motion. If the treatment is delivered in several fractions, the distribution of the dose around the expected value is close to a Gaussian. For a typical treatment with 30 fractions, the standard deviation is generally within 1% of the expected value for MLC delivery if one assumes a typical motion amplitude of 5 mm (1 cm peak to peak). The standard deviation is generally even smaller for the compensator but bigger for scanning beam delivery. For the latter it can be reduced through multiple deliveries ('paintings') of the same field. In conclusion, the main effect of organ motion in IMRT is an averaging of the dose distribution without motion over the path of the motion. This is the same as for treatments with conventional beams. Additional effects that are specific to the IMRT delivery technique appear to be relatively small, except for the scanning beam.

What is the optimum leaf width of a multileaf collimator?

The following question is investigated: How narrow do the leaves of a multileaf collimator have to be such that further reduction of the leaf width does not lead to physical improvements of the dose distribution. Because of the physical principles of interaction between radiation and matter, dose distributions in radiotherapy are generally relatively smooth. According to the theory of sampling, the dose distribution can therefore be represented by a set of evenly spaced samples. The distance between the samples is identified with the distance between the leaf centers of a multileaf collimator. The optimum sampling distance is derived from the 20% to 80% field edge penumbra through the concept of the dose deposition kernel, which is approximated by a Gaussian. The leaf width of the multileaf collimator is considered to be independent from the sampling distance. Two cases are studied in detail: (i) the leaf width equals the sampling distance, which is the regular case, and (ii) the leaf width is twice the sampling distance. The practical delivery of the latter treatment geometry requires a couch movement or a collimator rotation. The optimum sampling distance equals the 20%-80% penumbra divided by 1.7 and is on the order of 1.5-2 mm for a typical 6 MV beam in soft tissue. The optimum leaf width equals this sampling distance in the regular case. A relatively small deterioration results if the leaf width is doubled, while the sampling distance remains the same. The deterioration can be corrected for by deconvolving the fluence profile with an inverse filter.

CONCLUSIONS

With the help of the sampling theory and, more generally, the theory of linear systems, one can find a general answer to the question about the optimum leaf width of a multileaf collimator from a physical point of view. It is important to distinguish between the sampling distance and the leaf width. The sampling distance is more critical than the leaf width. The leaf width can be up to twice as large as the sampling width. Furthermore, the derived sampling distance can be used to select the optimum resolution of both the fluence and the dose grid in dose calculation and inverse planning algorithms.

A practical technique for verification of three-dimensional conformal dose distributions in stereotactic radiosurgery

The trend toward conformal techniques in stereotactic radiosurgery necessitates an accurate and practical method for verification of irregular three-dimensional dose distributions. This work presents the design and evaluation of a phantom system facilitating the measurement of conformal dose distributions using one or more arrays of up to 20 radiographic films separated by 3.2 mm-thick tissue-equivalent spacers. Using Electron Gamma Shower version 4 (EGS4) Monte Carlo simulation, we show that for 6 MV radiosurgical photon beams this arrangement preserves tissue-equivalence to within 1%. The phantom provides 0.25 mm in-plane spatial resolution and multiple sets of films may be used to resample the dose volume in orthogonal planes. Dedicated software has been developed to automate the process of ordering and orienting of scanned film images, conversion of scanned pixel value to dose, resampling of one or more sets of film images and subsequent export of images in DICOM format for coregistration of planned and measured dose volumes. Calculated and measured isodose surfaces for a simple, circular-beam treatment agree to within 1.5 mm throughout the dose volume. For conformal radiosurgical applications, the measured and planned dose distributions agree to within the uncertainty of the manufacture of irregularly shaped collimators. The sensitivity of this technique to minor spatial inaccuracies in beam shaping is also demonstrated.