Reconstruction of electron spectra using singular component decomposition

Beam modeling and beam model commissioning for Monte Carlo dose calculation-based radiation therapy treatment planning: Report of AAPM Task Group 157

Dose calculation plays an important role in the accuracy of radiotherapy treatment planning and beam delivery. The Monte Carlo (MC) method is capable of achieving the highest accuracy in radiotherapy dose calculation and has been implemented in many commercial systems for radiotherapy treatment planning. The objective of this task group was to assist clinical physicists with the potentially complex task of acceptance testing and commissioning MC-based treatment planning systems (TPS) for photon and electron beam dose calculations. This report provides an overview on the general approach of clinical implementation and testing of MC-based TPS with a specific focus on models of clinical photon and electron beams. Different types of beam models are described including those that utilize MC simulation of the treatment head and those that rely on analytical methods and measurements. The trade-off between accuracy and efficiency in the various source-modeling approaches is discussed together with guidelines for acceptance testing of MC-based TPS from the clinical standpoint. Specific recommendations are given on methods and practical procedures to commission clinical beam models for MC-based TPS.

Monte Carlo calculations of an Elekta Precise SL-25 photon beam model

Background

Monte Carlo (MC) simulations have been used extensively for benchmarking photon dose calculations in modern radiotherapy using linear accelerators (linacs). Moreover, a major barrier to widespread clinical implementation of MC dose calculation is the difficulty in characterising the radiation source using data reported from manufacturers.

Purpose

This work aims to develop a generalised full MC histogram source model of an Elekta Precise SL-25 linac (electron exit window, target, flattening filter, monitor chambers and collimators) for 6 MV photon beams used in standard therapies. The inclusion of many different probability processes such as scatter, nuclear reactions, decay, capture cross-sections and more led to more realistic dose calculations in treatment planning and quality assurance.

Materials and methods

Two different codes, MCNPX 2·6 and EGSr-BEAM, were used for the calculation of particle transport, first in the geometry of the internal/external accelerator source, and then followed by tracking the transport and energy deposition in phantom-equivalent tissues. A full phase space file was scored directly above the upper multilayer collimator’s jaws to derive the beam characteristics such as planar fluence, angular distribution and energy spectrum. To check the quality of the generated photon beam, its depth dose curves and cross-beam profiles were calculated and compared with measured data.

Results

In-field dose distributions calculated using the accelerator models were tuned to match measurement data with preliminary calculations performed using the accelerator information provided by the manufacturer. Field sizes of 3×3, 5×5, 10×10, 15×15 and 20×20 cm2were analysed. Local differences between calculated and measured curve doses beneath 2% were obtained for all the studied field sizes. Higher discrepancies were obtained in the air–water interface, where measurements of dose distributions with the ionisation chamber need to be shifted for the effective point of measurement.

Conclusion

The agreements between MC-calculated and measured dose distributions were excellent for both codes, showing the strength and stability of the proposed model. Beam reconstruction methods as direct input to dose-calculation codes using the recorded histograms can be implemented for more accurate patient dose estimation.

Phase space determination from measured dose data for intraoperative electron radiation therapy

A procedure to characterize beams of a medical linear accelerator for their use in Monte Carlo (MC) dose calculations for intraoperative electron radiation therapy (IOERT) is presented. The procedure relies on dose measurements in homogeneous media as input, avoiding the need for detailed simulations of the accelerator head. An iterative algorithm (EM-ML) has been employed to extract the relevant details of the phase space (PHSP) of the particles coming from the accelerator, such as energy spectra, spatial distribution and angle of emission of particles. The algorithm can use pre-computed dose volumes in water and/or air, so that the machine-specific tuning with actual data can be performed in a few minutes. To test the procedure, MC simulations of a linear accelerator with typical IOERT applicators and energies, have been performed and taken as reference. A solution PHSP derived from the dose produced by the simulated accelerator has been compared to the reference PHSP. Further, dose delivered by the simulated accelerator for setups not included in the fit of the PHSP were compared to the ones derived from the solution PHSP. The results show that it is possible to derive from dose measurements PHSP accurate for IOERT MC dose estimations.

Comparison of a finite-element multigroup discrete-ordinates code with Monte Carlo for radiotherapy calculations

Radiotherapy calculations often involve complex geometries such as interfaces between materials of vastly differing atomic number, such as lung, bone and/or air interfaces. Monte Carlo methods have been used to calculate accurately the perturbation effects of the interfaces. However, these methods can be computationally expensive for routine clinical calculations. An alternative approach is to solve the Boltzmann equation deterministically. We present one such deterministic code, Attila. Further, we computed a brachytherapy example and an external beam benchmark to compare the results with data previously calculated by MCNPX and EGS4. Our data suggest that the presented deterministic code is as accurate as EGS4 and MCNPX for the transport geometries examined in this study.

Reconstruction of electron spectra from depth doses with adaptive regularization

Electron spectral reconstruction of medical accelerators from measured depth doses is a practical method for providing the input initial phase space distribution at the patient surface that is required by Monte Carlo treatment planning systems. The posed inverse problem of spectral reconstruction is ill conditioned and this may lead to nonphysical oscillations in the reconstructed spectra. Use of a variational method of solution with a regularization technique removes the oscillations but tends to smooth the sharp (deltalike) energy peak that is a common feature in electron spectra generated by medical accelerators. Because the sharp peak contains a large percentage of the electrons in the spectrum, an accurate estimate of the peak width, height and position is critical to the success of the technique for spectrum reconstruction with regularization. We propose use of an adaptive regularization term as a special form of the general Tichonov regularization function. The variational method with the adaptive regularization term is applied to reconstruct electron spectra for the 6, 9, and 18 MeV electron beams of a Varian Clinac 2100C accelerator and proves to be a very simple, effective and accurate approach. Results using this variational method with adaptive regularization almost perfectly reconstruct electron spectra from depth dose distributions.

L-curve analysis of radiotherapy optimization problems

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

Regularization of inverse planning for intensity-modulated radiotherapy

The performance of a variational regularization technique to improve robustness of inverse treatment planning for intensity modulated radiotherapy is analyzed and tested. Inverse treatment planning is based on the numerical solutions to the Fredholm integral equation of the first kind which is ill-posed. Therefore, a fundamental problem with inverse treatment planning is that it may exhibit instabilities manifested in nonphysical oscillations in the beam intensity functions. To control the instabilities, we consider a variational regularization technique which can be applied for the methods which minimize a quadratic objective function. In this technique, the quadratic objective function is modified by adding of a stabilizing functional that allows for arbitrary order regularization. An optimal form of stabilizing functional is selected which allows for both regularization and good approximation of beam intensity functions. The regularized optimization algorithm is shown, by comparison for a typical case of a head-and-neck cancer treatment, to be significantly more accurate and robust than the standard approach, particularly for the smaller beamlet sizes.

Angular correction in reconstruction of electron spectra from depth dose distributions

Techniques for reconstruction of electron spectra from the depth-dose curves used to date have ignored the angular distribution of incident electrons scattered at large angles. The approximation adopted is to employ a database of monoenergetic depth-dose curves generated for normal incidence of electrons at the surface. This approximation is acceptable for direct electrons with small angular spread. However, electrons scattered from the treatment head and collimating system may have large average angles of incidence which affects the depth-dose distribution significantly at shallow depths by increasing energy deposition close to the surface. We show that ignoring the electron incident angular distribution leads to systematic errors in the low energy region of reconstructed electron spectra. We propose a simple 1-D model to correct for these systematic errors using only electron angular distribution at the central beam axis. This model provides reconstructed spectra in excellent agreement with Monte Carlo simulation in the low energy region.

Influence of initial electron beam characteristics on monte carlo calculated absorbed dose distributions for linear accelerator electron beams

The least known parameters in a Monte Carlo simulation of a linear accelerator treatment head are often the properties of the initial electron beam directed onto the exit vacuum window. Several initial beams with different spatial fluence distributions, angular divergences and energy spectra have been transported through the geometry of a scattering foil accelerator. The electron beam characteristics (energy spectrum and angular distribution) at the phantom surface and the subsequent relative absorbed dose distribution in a water phantom were calculated. The dose distribution was found to be insensitive to the geometrical properties of the initial beam. Furthermore, the lateral dose profiles are unaffected by the energy spectrum of the initial beam. The effect on the depth-dose curve is negligible if the initial energy spectrum is symmetric (e.g., Gaussian shaped) and its full width at half maximum (FWHM) is less than approximately 10% of the most probable energy. A larger FWHM will decrease the normalized dose gradient, but will not affect the dose in the build-up region. An asymmetric wedge shaped spectrum with a low-energy extension simultaneously increases the dose in the build-up region and decreases the dose gradient. The relationship between the energy spectral width and the normalized dose gradient is, however, smaller than published analytical expressions indicate. Some well-established energy-range relationships were shown to be accurate for most of the initial beams studied. The energy spectrum at the phantom surface was also derived from a measured depth-dose curve through different methods. The extracted spectrum depends on the beam model and the spectral reconstruction algorithm. Even though the depth-dose curve is fairly independent of initial beam characteristics, a correct description of the low-energy tail of the energy spectrum is important to obtain good agreement between measured and Monte Carlo calculated doses in the build-up region.