Reconstruction of megavoltage photon spectra by attenuation analysis

Hybrid Monte Carlo source model: Advantages and deficiencies

Monte Carlo (MC) simulation is the gold standard for dose calculation. An accurate mathematical source model can be used for the radiation beams. Source models can consist of sub-sources or fewer sources with data that need to be measured. This can speed up treatment plan verification without the need for a full simulation of the radiation treatment machine.

Aims: This study aimed to construct a novel hybrid source model for 6 MV photon beams for an Elekta Synergy accelerator and to commission it against measured beam data and treatments plans.

Methods and Material: The model comprised of a circular photon and planar electron contamination source. The modified Schiff formula provided off-axis variable bremsstrahlung spectra. Collimation and scatter were modelled with error functions. An exponential function modelled the transmitted fluence through the collimators. The source model was commissioned by comparing simulated and measured MC data. Dose data included profiles, depth dose and film measurements in a Rando phantom. Field sizes ranged from 1 × 1 cm2 to 40 × 40 cm2.

Results: Regular, wedged and asymmetrical fields could be modelled within 1.5% or 1.5 mm. More than 95% of all points lie within 3% or 3 mm for the multi-leaf collimators contours data. A gamma criterion of 3% or 3 mm was met for a complex treatment case.

Conclusions: The two sub-source model replicated clinical 6 MV Elekta Synergy photons beams and could calculate the dose accurately for conformal treatments in complex geometries such as a head-and-neck case.

An improved physics-based approach for unfolding megavoltage bremsstrahlung spectra using transmission analysis

PURPOSE

To develop a physics-based approach to improve the accuracy and robustness of the ill-conditioned problem of unfolding megavoltage bremsstrahlung spectra from transmission data.

METHODS

Spectra are specified using a rigorously-benchmarked functional form. Since ion chambers are the typical detector used in transmission measurements, the energy response of a Farmer chamber is calculated using the EGSnrc Monte Carlo code, and the effect of approximating the energy response on the accuracy of the unfolded spectra is studied. A proposal is introduced to enhance spectral sensitivity by combining transmission data measured with multiple detectors of different energy response and by combining data from multiple attenuating materials. Monte Carlo methods are developed to correct for nonideal exponential attenuation (e.g., scatter effects and secondary attenuation). The performance of the proposed methods is evaluated for a diverse set of validated clinical spectra (3.5-25 MV) using analytical transmission data with simulated experimental noise.

RESULTS

The approximations commonly used in previous studies for the ion-chamber energy response lead to significant errors in the unfolded spectra. Of the configurations studied, the one with best spectral sensitivity is to measure four full transmission curves using separate low-Z and high-Z attenuators in conjunction with two detectors of different energy response (the authors propose a Farmer-type ion chamber, once with a low-Z, and once with a high-Z buildup cap material), then to feed the data simultaneously to the unfolding algorithm. Deviations from ideal exponential attenuation are as much as 1.5% for the smallest transmission signals, and the proposed methods properly correct for those deviations. The transmission data with enhanced spectral sensitivity, combined with the accurate and flexible spectral functional form, lead to robust unfolding without requiring a priori knowledge of the spectrum. Compared with the commonly-used methods, the accuracy is improved for the unfolded spectra and for the unfolded mean incident electron kinetic energy by at least factors of three and four, respectively. With simulated experimental noise and a lowest transmission of 1%, the unfolded energy fluence spectra agree with the original spectra with a normalized root-mean-square deviation, %Δ(ψ), of 2.3%. The unfolded mean incident electron kinetic energies agree, on average, with the original values within 1.4%. A lowest transmission of only 10% still allows unfolding with %Δ(ψ) of 3.3%.

CONCLUSIONS

In the presence of realistic experimental noise, the proposed approach significantly improves the accuracy and robustness of the spectral unfolding problem for all therapy and MV imaging beams of clinical interest.

Dose profile measurement using an imaging plate: Evaluation of filters using Monte Carlo simulation of 4 MV x-rays

Computed radiography (CR) is gradually replacing film. The application of CR for two-dimensional profiles and off-axis ratio (OAR) measurement using an imaging plate (IP) in a CR system is currently under discussion. However, a well known problem for IPs in dosimetry is that they use high atomic number (Z) materials, such as Ba, which have an energy dependency in a photon interaction. Although there are some reports that it is possible to compensate for the energy dependency with metal filters, the appropriate thicknesses of these filters and where they should be located have not been investigated. The purpose of this study is to find the most suitable filter for use with an IP as a dosimetric tool. Monte Carlo simulation (Geant4 8.1) was used to determine the filter to minimize the measurement error in OAR measurements of 4 MV x-rays. In this simulation, the material and thickness of the filter and distance between the IP and the filter were varied to determine most suitable filter conditions that gave the best fit to the MC calculated OAR in water. With regard to changing the filter material, we found that using higher Z and higher density material increased the effectiveness of the filter. Also, increasing the distance between the filter and the IP reduced the effectiveness, whereas increasing the thickness of the filter increased the effectiveness. The result of this study showed that the most appropriate filter conditions consistent with the calculated OAR in water were the ones with the IP sandwiched between two 2 mm thick lead filters at a distance of 5 mm from the IP or the IP sandwiched directly between two 1 mm lead filters. Using these filters, we measured the OAR at 10 cm depth with 100 cm source-to-surface distance and surface 10x10 cm(2) field size. The results of this measurement represented that it is possible to achieve measurements with less than within 2.0% and 2.0% in the field and with less than 1.1% and 0.6% out of the field by using 2 and 1 mm lead filters, respectively.

Photon spectral characteristics of dissimilar 6 MV linear accelerators

This work measures and compares the energy spectra of four dosimetrically matched 6 MV beams, generated from four physically different linear accelerators. The goal of this work is twofold. First, this study determines whether the spectra of dosimetrically matched beams are measurably different. This study also demonstrates that the spectra of clinical photon beams can be measured as a part of the beam data collection process for input to a three-dimensional (3D) treatment planning system. The spectra of 6 MV beams that are dosimetrically matched for clinical use were studied to determine if the beam spectra are similarly matched. Each of the four accelerators examined had a standing waveguide, but with different physical designs. The four accelerators were two Varian 2100C/Ds (one 6 MV/18 MV waveguide and one 6 MV/10 MV waveguide), one Varian 600 C with a vertically mounted waveguide and no bending magnet, and one Siemens MD 6740 with a 6 MV/10 MV waveguide. All four accelerators had percent depth dose curves for the 6 MV beam that were matched within 1.3%. Beam spectra were determined from narrow beam transmission measurements through successive thicknesses of pure aluminum along the central axis of the accelerator, made with a graphite Farmer ion chamber with a Lucite buildup cap. An iterative nonlinear fit using a Marquardt algorithm was used to find each spectrum. Reconstructed spectra show that all four beams have similar energy distributions with only subtle differences, despite the differences in accelerator design. The measured spectra of different 6 MV beams are similar regardless of accelerator design. The measured spectra show excellent agreement with those found by the auto-modeling algorithm in a commercial 3D treatment planning system that uses a convolution dose calculation algorithm. Thus, beam spectra can be acquired in a clinical setting at the time of commissioning as a part of the routine beam data collection.

Photon spectra calculation for an Elekta linac beam using experimental scatter measurements and Monte Carlo techniques

The present work is centered in reconstructing by means of a scatter analysis method the primary beam photon spectrum of a linear accelerator. This technique is based on irradiating the isocenter of a rectangular block made of methacrylate placed at 100 cm distance from surface and measuring scattered particles around the plastic at several specific positions with different scatter angles. The MCNP5 Monte Carlo code has been used to simulate the particles transport of mono-energetic beams to register the scatter measurement after contact the attenuator. Measured ionization values allow calculating the spectrum as the sum of mono-energetic individual energy bins using the Schiff Bremsstrahlung model. The measurements have been made in an Elekta Precise linac using a 6 MeV photon beam. Relative depth and profile dose curves calculated in a water phantom using the reconstructed spectrum agree with experimentally measured dose data to within 3%.

Spectral reconstruction by scatter analysis for a linear accelerator photon beam

Pre-existing methods for photon beam spectral reconstruction are briefly reviewed. An alternative reconstruction method by scatter analysis for linear accelerators is introduced. The method consists in irradiating a small plastic phantom at standard 100 cm SSD and inferring primary beam energy spectral information based on the measurement with a standard Farmer chamber of scatter around the phantom at several specific scatter angles: a scatter curve is measured which is indicative of the primary spectrum at hand. A Monte Carlo code is used to simulate the scatter measurement set-up and predict the relative magnitude of scatter measurements for mono-energetic primary beams. Based on mono-energetic primary scatter data, measured scatter curves are analysed and the spectrum unfolded as the sum of mono-energetic individual energy bins using the Schiff bremsstrahlung model. The method is applied to an Elekta/SL18 6 MV photon beam. The reconstructed spectrum matches the Monte Carlo calculated spectrum for the same beam within 6.2% (average error when spectra are compared bin by bin). Depth dose values calculated for the reconstructed spectrum agree with physically measured depth dose data to within 1%. Scatter analysis is preliminarily shown to have potential as a practical spectral reconstruction method requiring few measurements under standard 100 cm SSD and feasible in any radiotherapy department using a phantom and a Farmer chamber.

Spectrum reconstruction from dose measurements as a linear inverse problem

There are three ways to determine the spectrum of a clinical photon beam: direct measurement, modelling the source and reconstruction from ion-chamber measurements. We focus on reconstruction because the necessary equipment is readily available and it provides independent confirmation of source models for a given machine. Reconstruction methods involve measuring the dose in an ion chamber after the beam passes through an attenuator. We gain information about the spectrum from measurements using attenuators of differing compositions and thicknesses since materials have energy dependent attenuation. Unlike the procedures used in other papers, we do not discretize or parametrize the spectrum. With either of these two approximations, reconstruction is a least squares problem. The forward problem of going from a spectrum to a series of dose measurements is a linear operator, with the composition and thickness of the attenuators as parameters. Hence the singular value decomposition (SVD) characterizes this operator. The right singular vectors form a basis for the spectrum, and, at first approximation, only those corresponding to singular values above a threshold are measurable. A more rigorous error analysis shows with what confidence different components of the spectrum can be measured. We illustrate this theory with simulations and an example utilizing six sets of dose measurements with water and lead as attenuators.

[Improvement of the accuracy of the Laplace transform method for the determination of radiotherapy spectra of clinical linear accelerators]

The present study focused on the reconstruction of the bremsstrahlung spectrum of a clinical linear accelerator from the measured transmission curve, with the aim of improving the accuracy of this method. The essence of the method was the analytic inverse Laplace transform of a parameter function fitted to the measured transmission curve. We tested known fitting functions, however they resulted in considerable fitting inaccuracy, leading to inaccuracies of the bremsstrahlung spectrum. In order to minimise the fitting errors, we employed a linear combination of n equations with 2n-1 parameters. The fitting errors are now considerably smaller. The measurement of the transmission function requires that the energy-dependent detector response is taken into account. We analysed the underlying physical context and developed a function that corrects for the energy-dependent detector response. The factors of this function were experimentally determined or calculated from tabulated values.

A simple method for bremsstrahlung spectra reconstruction from transmission measurements

A new method for evaluation of bremsstrahlung spectra from transmission measurements has been developed. In this method some very well known facts relating to thick target bremsstrahlung spectra are a priori included in the calculation procedure. Some characteristics of the method are preliminarily illustrated on a 6 MV therapy linear accelerator.

Reconstruction of electron spectra using singular component decomposition

Reconstruction of electron spectra of medical accelerators from measured depth dose distributions is an attractive tool for commissioning of a Monte Carlo treatment planning system. However, the reconstruction method is an inverse radiation transport problem which is poorly conditioned, in the sense it may become unstable due to small perturbations in the input data. Predicting the sharp (delta-like) peak in the electron spectrum provides an additional challenge for the numerical reconstruction technique. To improve efficiency and robustness of the reconstruction technique, we developed an algorithm based on a separation of the electron spectrum into singular and regular components. We approximate the singular peak of the spectrum by a narrow weighted Gaussian function. The parameters of this Gaussian function are sought using only the fall-off and toe regions of the depth-dose curve. Analytical representation of the spectral peak by a Gaussian has benefit since only one weight and the mean and variance must be derived from the depth-dose curve instead of multiple spectra weights. The regular part of the spectrum is reconstructed from the residual depth-dose distribution using a variational method combined with a regularization technique to avoid the nonphysical oscillations. The effectiveness of the method is demonstrated by comparing predictions to "benchmark" spectra and depth-dose distributions from Monte Carlo simulation of medical accelerators.

Monte Carlo calculation of nine megavoltage photon beam spectra using the BEAM code

A recent paper analyzed the sensitivity to various simulation parameters of the Monte Carlo simulations of nine beams from three major manufacturers of commercial medical linear accelerators, ranging in energy from 4-25 MV. In this work the nine models are used: to calculate photon energy spectra and average energy distributions and compare them to those published by Mohan et al. [Med. Phys. 12, 592-597 (1985)]; to separate the spectra into primary and scatter components from the primary collimator, the flattening filter and the adjustable collimators; and to calculate the contaminant-electron fluence spectra and the electron contribution to the depth-dose curves. Notwithstanding the better precision of the calculated spectra, they are similar to those calculated by Mohan et al. The three photon spectra at 6 MV from the machines of three different manufacturers show differences in their shapes as well as in the efficiency of bremsstrahlung production in the corresponding target and filter combinations. The contribution of direct photons to the photon energy fluence in a 10 x 10 field varies between 92% and 97%, where the primary collimator contributes between 0.6% and 3.4% and the flattening filter contributes between 0.6% and 4.5% to the head-scatter energy fluence. The fluence of the contaminant electrons at 100 cm varies between 5 x 10(-9) and 2.4 x 10(-7) cm(-2) per incident electron on target, and the corresponding spectrum for each beam is relatively invariant inside a 10 x 10 cm2 field. On the surface the dose from electron contamination varies between 5.7% and 11% of maximum dose and, at the depth of maximum dose, between 0.16% and 2.5% of maximum dose. The photon component of the percentage depth-dose at 10 cm depth is compared with the general formula provided by AAPM's task group 51 and confirms the claimed accuracy of 2%.

Computer algebra for x-ray spectral reconstruction between 6 and 25 MV

A previously presented algorithm for the reconstruction of bremsstrahlung spectra from transmission data has been implemented into MATHEMATICA. Spectra vectorial algebra has been used to solve the matrix system A * F = T. The new implementation has been tested by reconstructing photon spectra from transmission data acquired in narrow beam conditions, for nominal energies of 6, 15, and 25 MV. The results were in excellent agreement with the original calculations. Our implementation has the advantage to be based on a well-tested mathematical kernel. Furthermore it offers a comfortable user interface.

Reconstruction of megavoltage photon spectra from electronic portal imager derived transmission measurements

Three variants of the Schiff equation are investigated to model the spectra produced by megavoltage linear accelerators. These models are tested against well-validated Monte Carlo (MC) generated spectra on the central axis of large-area fields, and show excellent agreement. Numerical reconstructions of 6 and 10 MV spectra using the same models are then presented, using experimental attenuation data derived from an electronic portal imager. The process of deriving spectra from experimental attenuation data is shown to be inherently badly constrained mathematically, with the derived spectrum being highly sensitive to noise in the source data, and non-unique. By placing a priori constraints on the Schiff model from both physical knowledge of the construction of the accelerator and MC data, physically useful results are gained and presented for both the energy dependence and off-axis behaviour of photon spectra.

Determining clinical photon beam spectra from measured depth dose with the Cimmino algorithm

A method to determine the spectrum of a clinical photon beam from measured depth-dose data is described. At shallow depths, where the range of Compton-generated electrons increases rapidly with photon energy, the depth dose provides the information to discriminate the spectral contributions. To minimize the influence of contaminating electrons, small (6 x 6 cm2) fields were used. The measured depth dose is represented as a linear combination of basis functions, namely the depth doses of monoenergetic photon beams derived by Monte Carlo simulations. The weights of the basis functions were obtained with the Cimmino feasibility algorithm, which examines in each iteration the discrepancy between predicted and measured depth dose. For 6 and 15 MV photon beams of a clinical accelerator, the depth dose obtained from the derived spectral weights was within about 1% of the measured depth dose at all depths. Because the problem is ill conditioned, solutions for the spectrum can fluctuate with energy. Physically realistic smooth spectra for these photon beams appeared when a small margin (about +/- 1%) was attributed to the measured depth dose. The maximum energy of both derived spectra agreed with the measured energy of the electrons striking the target to within 1 MeV. The use of a feasibility method on minimally relaxed constraints provides realistic spectra quickly and interactively.

Dose calculations for external photon beams in radiotherapy

Dose calculation methods for photon beams are reviewed in the context of radiation therapy treatment planning. Following introductory summaries on photon beam characteristics and clinical requirements on dose calculations, calculation methods are described in order of increasing explicitness of particle transport. The simplest are dose ratio factorizations limited to point dose estimates useful for checking other more general, but also more complex, approaches. Some methods incorporate detailed modelling of scatter dose through differentiation of measured data combined with various integration techniques. State-of-the-art methods based on point or pencil kernels, which are derived through Monte Carlo simulations, to characterize secondary particle transport are presented in some detail. Explicit particle transport methods, such as Monte Carlo, are briefly summarized. The extensive literature on beam characterization and handling of treatment head scatter is reviewed in the context of providing phase space data for kernel based and/or direct Monte Carlo dose calculations. Finally, a brief overview of inverse methods for optimization and dose reconstruction is provided.

Analytical considerations of beam hardening in medical accelerator photon spectra

Beam hardening is a well-known phenomenon for therapeutic accelerator beams passing through matter in narrow beam geometry. This study assesses quantitatively the magnitude of beam hardening of therapeutic beams in water. A formal concept of beam hardening is proposed which is based on the decrease of the mean attenuation coefficient with depth. On the basis of this concept calculations of beam hardening effects are easily performed by means of a commercial spreadsheet program. Published accelerator spectra and the tabulated values of attenuation coefficients serve as input for these calculations. It is shown that the mean attenuation coefficient starts at depth zero with an almost linear decrease and then slowly levels off to a limit value. A similar behavior is found for the beam hardening coefficient. A physically reasonable, semianalytical model is given which fits the data better than previously published functions. The energy dependence of the initial attenuation coefficient is evaluated and shown. It fits well to published experimental data. The initial beam hardening coefficient, however, shows no energy dependence. Its mean value (eta0) approximately 0.006 cm(-1)) is also in close agreement to the measured data.

X-ray spectra estimation using attenuation measurements from 25 kVp to 18 MV

Attenuation measurements for primary x-ray spectra from 25 kVp to 18 MV were made using aluminum filters for all energies except for orthovoltage where copper filters were used. An iterative perturbation method, which utilized these measurements, was employed to derive the apparent x-ray spectrum. An initial spectrum or pre-spectrum was used to start the process. Each energy value of the pre-spectrum was perturbed positively and negatively, and an attenuation curve was calculated using the perturbed values. The value of x-rays in the given energy bin was chosen to minimize the difference between the measured and calculated transmission curves. The goal was to derive the minimum difference between the measured transmission curve and the calculated transmission curve using the derived x-ray spectrum. The method was found to yield useful information concerning the lower photon energy and the actual operating potential versus the nominal potential. Mammographic, diagnostic, orthovoltage, and megavoltage x-ray spectra up to 18 MV nominal were derived using this method. The method was validated using attenuation curves from published literature. The method was also validated using attenuation curves calculated from published spectra. The attenuation curves were then used to derive the x-ray spectra.

Spectral reconstruction of clinical megavoltage photon beams and the implications of spectral determination on the dosimetry of such beams

An analysis technique, based on simulated annealing, is described which is employed to derive megavoltage photon beam spectral information from narrow beam attenuation measurements. Megavoltage photon beam spectra have been determined using this technique for linear accelerators from different manufacturers, and different models from individual manufacturers at a range of energies from nominal 6 MV to nominal 25 MV. All of the photon beams included in the study are in routine clinical use. The subsequent effects on dosimetry of employing derived primary spectra to specify beam quality are examined. The results suggest that the quality index TPR(20)10 may be insensitive to beam quality changes for high-energy beams in the range of 15 MV to 25 MV. Although the quality index may be insensitive as a beam quality specifier at these higher qualities, the actual difference in the calculated dose delivered using derived spectra as the quality specifier rather than TPR(20)10 is likely to be small, the results obtained indicating a difference of between 0.2% and 0.7% in the calculated dose delivered.

Extraction of the photon spectra from measured beam parameters

Knowledge of the photon spectrum of a radiotherapy beam is often needed for three-dimensional (3-D) dose calculations using Monte Carlo methods and/or algorithms employing energy deposition kernels. Direct measurement of the x-ray energy fluence spectrum is not feasible for the high-energy photon beams used clinically. In this paper, the spectrum is extracted from basic beam data that are readily obtained for a clinical beam. We describe the photon spectrum using just two parameters. One parameter, which determines the high-energy part of the spectrum, is obtained using the measured dose in the buildup region for a small field, where electron contamination of the beam can be neglected. The other parameter is extracted from the photon beam attenuation in water. The results compare favorably to spectra generated from Monte Carlo simulations.

Reconstruction of 6 MV photon spectra from measured transmission including maximum energy estimation

Photon spectra from a nominally 6 MV beam under standard clinical conditions and at higher and lower beam qualities have been derived from narrow-beam transmission measurements using a previously published three-parameter reconstruction model. Estimates of the maximum photon energy present in each spectrum were derived using a reduced number of model parameters. An estimate of the maximum contribution of background, or room, scatter to transmission measurements has been made for this study and is shown to be negligible in terms of the quality index and percentage depth-dose of the derived spectra. Percentage depth-dose data for standard beam conditions derived from the reconstructed spectrum were found to agree with direct measurements to within approximately 1% for depths of up to 25 cm in water. Quality indices expressed in terms of TPR10(20) for all spectra were found to agree with directly measured values to within 1%. The experimental procedure and reconstruction model are therefore shown to produce photon spectra whose derived quality indices and percentage depth-dose values agree with direct measurement to within expected experimental uncertainty.